BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Rick Jardine (University of Western Ontario.) DTSTART:20200916T230000Z DTEND:20200917T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/1 DESCRIPTION:Title: Posets\, metric spaces\, and topological data analysis.\nby Ri ck Jardine (University of Western Ontario.) as part of New York City Categ ory Theory Seminar\n\n\nAbstract\nTraditional TDA is the analysis of homot opy invariants of systems of spaces V(X) that arise from finite metric spa ces X\, via distance measures. These spaces can be expressed in terms of p osets\, which are barycentric subdivisions of the usual Vietoris-Rips comp lexes V(X). The proofs of stability theorems in TDA are sharpened consider ably by direct use of poset techniques.\n\nExpanding the domain of definit ion to extended pseudo metric spaces enables the construction of a realiza tion functor on diagrams of spaces\, which has a right adjoint Y |--> S(Y) \, called the singular functor. The realization of the Vietoris-Rips syste m V(X) for an ep-metric space X is the space itself. The counit of the adj unction defines a map \\eta: V(X) --> S(X)\, which is a sectionwise weak e quivalence - the proof uses simplicial approximation techniques.\n\nThis i s the context for the Healy-McInnes UMAP construction\, which will be disc ussed if time permits. UMAP is non-traditional: clusters for UMAP are defi ned by paths through sequences of neighbour pairs\, which can be a highly efficient process in practice.\n LOCATION:https://researchseminars.org/talk/Category_Theory/1/ END:VEVENT BEGIN:VEVENT SUMMARY:David Ellerman (University of Ljubljana) DTSTART:20200930T230000Z DTEND:20201001T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/2 DESCRIPTION:Title: The Logical Theory of Canonical Maps: The Elements & Distinctions Analysis of the Morphisms\, Duality\, Canonicity\, and Universal Construct ions in Sets.\nby David Ellerman (University of Ljubljana) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbstract: Category th eory gives a mathematical characterization of naturality but not of canoni city. The purpose of this paper is to develop the logical theory of canoni cal maps based on the broader demonstration that the dual notions of eleme nts & distinctions are the basic analytical concepts needed to unpack and analyze morphisms\, duality\, canonicity\, and universal constructions in Sets\, the category of sets and functions. The analysis extends directly t o other Sets-based concrete categories (groups\, rings\, vector spaces\, e tc.). Elements and distinctions are the building blocks of the two dual lo gics\, the Boolean logic of subsets and the logic of partitions. The parti al orders (inclusion and refinement) in the lattices for the dual logics d efine morphisms. The thesis is that the maps that are canonical in Sets ar e the ones that are defined (given the data of the situation) by these two logical partial orders and by the compositions of those maps.\n LOCATION:https://researchseminars.org/talk/Category_Theory/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Jonathon Funk (Queensborough CUNY) DTSTART:20201014T220000Z DTEND:20201014T233000Z DTSTAMP:20260422T214840Z UID:Category_Theory/3 DESCRIPTION:Title: Pseudogroup Torsors.\nby Jonathon Funk (Queensborough CUNY) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbstract: We use sheaf theory to analyze the topos of etale actions on the germ groupo id of a pseudogroup in the sense that we present a site for this topos\, w hich we call the classifying topos of the pseudogroup. Our analysis carrie s us further into how pseudogroup morphisms and geometric morphisms are re lated. Ultimately\, we shall see that the classifying topos classifies wha t we call a pseudogroup torsor. In hindsight\, we see that pseudogroups fo rm a bicategory of `flat' bimodules.\n LOCATION:https://researchseminars.org/talk/Category_Theory/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrei V. Rodin (Saint Petersburg State University.) DTSTART:20201021T230000Z DTEND:20201022T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/4 DESCRIPTION:Title: Vladimir Voevodsky’s Unachieved Project\nby Andrei V. Rodin (Saint Petersburg State University.) as part of New York City Category The ory Seminar\n\n\nAbstract\nSoon after receiving the Fields Medal for his p roof of Milnor Conjecture and the related work in the Motivic Theory\, Vla dimir delivered a series of two public lectures in the Wuhan University (C hina) titled “What is most important for mathematics in the near future? ” where he described the most urgent tasks as follows: 1) to build a com puterised version of Bourbaki’s ‘Elements’ and 2) to bridge pure and applied mathematics. The first project resulted into the Univalent founda tions of mathematics. The second project remained unachieved in spite of s ignificant time and efforts that Vladimir spent for its realisation. More specifically\, during 2007-2009 Vladimir worked on a mathematical theory o f Population Dynamics but then abandoned this project and focused on the U nivalent Foundations until the sudden end of his life in 2017. \n

\ nUsing extensive unpublished materials available via Vladimir Voevodsky’ s memorial webpage (https:// www.math.ias.edu/Voevodsky/)\n I reconstruct Vladimir’s vision of ma thematics and its role in science incuding his original strategy of bridgi ng the gap between the pure and applied mathematics. Finally\, I show a re levance of Univalent Foundations to Vladimir’s unachieved project and sp eculate about a possible role of Univalent Foundations in science.\n LOCATION:https://researchseminars.org/talk/Category_Theory/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Larry Moss (Indiana University) DTSTART:20201028T230000Z DTEND:20201029T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/5 DESCRIPTION:Title: Coalgebra in Continuous Mathematics.\nby Larry Moss (Indiana U niversity) as part of New York City Category Theory Seminar\n\n\nAbstract\ nAbstract: A slogan from coalgebra in the 1990's holds that\n\n'discrete m athematics : algebra :: continuous mathematics : coalgebra'\n\nThe idea is that objects in continuous math\, like real numbers\, are often understoo d via their approximations\, and coalgebra gives tools for understanding a nd working with those objects. Some examples of this are Pavlovic and Esca rdo's relation of ordinary differential equations with coinduction\, and a lso Freyd's formulation of the unit interval as a final coalgebra. My talk will be an organized survey of several results in this area\, including ( 1) a new proof of Freyd's Theorem\, with extensions to fractal sets\; (2) other presentations of sets of reals as corecursive algebras and final coa lgebras\; (3) a coinductive proof of the correctness of policy iteration f rom Markov decision processes\; and (4) final coalgebra presentations of u niversal Harsanyi type spaces from economics.\n\nThis talk reports on join t work with several groups in the past 5-10 years\, and also some ongoing work.\n LOCATION:https://researchseminars.org/talk/Category_Theory/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Luis Scoccola (Michigan State University) DTSTART:20201105T000000Z DTEND:20201105T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/6 DESCRIPTION:Title: Locally persistent categories and approximate homotopy theory. \nby Luis Scoccola (Michigan State University) as part of New York City Ca tegory Theory Seminar\n\n\nAbstract\nAbstract: In applied homotopy theory and topological data analysis\, procedures use homotopy invariants of spac es to study and classify discrete data\, such as finite metric spaces. To show that such a procedure is robust to noise\, one endows the collection of possible inputs and the collection of outputs with metrics\, and shows that the procedure is continuous with respect to these metrics\, so one is interested in doing some kind of approximate homotopy theory. I will show that a certain type of enriched categories\, which I call locally persist ent categories\, provide a natural framework for the study of approximate categorical structures\, and in particular\, for the study of metrics rele vant to applied homotopy theory and metric geometry.\n LOCATION:https://researchseminars.org/talk/Category_Theory/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Noah Chrein (University of Maryland) DTSTART:20201112T000000Z DTEND:20201112T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/7 DESCRIPTION:Title: Yoneda ontologies.\nby Noah Chrein (University of Maryland) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbstract: We will discuss a 2-categorical model of ontology\, and how to view certain higher categories as ontologies in this language. We can translate the var ious Yoneda lemmas associated to higher categories into the language of on tology\, and in turn\, discuss what it means for a generic ontology to hav e a yoneda lemma. These will be the "Yoneda Ontologies".\n LOCATION:https://researchseminars.org/talk/Category_Theory/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Enrico Ghiorzi (Appalachian State University) DTSTART:20201119T000000Z DTEND:20201119T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/8 DESCRIPTION:Title: Internal enriched categories.\nby Enrico Ghiorzi (Appalachian State University) as part of New York City Category Theory Seminar\n\n\nAb stract\nAbstract: Internal categories feature a notion of completeness whi ch is remarkably well behaved. For example\, the internal adjoint functor theorem requires no solution set condition. Indeed\, internal categories a re intrinsically small\, and thus immune from the size issues commonly aff licting standard category theory. Unfortuntely\, they are not quite as exp ressive as we would like: for example\, there is no internal Yoneda lemma. To increase the expressivity of internal category theory\, we define a no tion of internal enrichment over an internal monoidal category and develop its theory of completeness. The resulting theory unites the good properti es of internal categories with the expressivity of enriched category theor y\, thus providing a powerful framework to work with.\n LOCATION:https://researchseminars.org/talk/Category_Theory/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Dan Shiebler (Oxford University) DTSTART:20201210T000000Z DTEND:20201210T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/9 DESCRIPTION:Title: Functorial Manifold Learning and Overlapping Clustering.\nby D an Shiebler (Oxford University) as part of New York City Category Theory S eminar\n\n\nAbstract\nAbstract: We adapt previous research on functorial c lustering and topological unsupervised learning to develop a functorial pe rspective on manifold learning algorithms. Our framework characterizes a m anifold learning algorithm in terms of the loss function that it optimizes \, which allows us to focus on the algorithm's objective rather than the d etails of the learning process. We demonstrate that we can express several state of the art manifold learning algorithms\, including Laplacian Eigen maps\, Metric Multidimensional Scaling\, and UMAP\, as functors in this fr amework. This functorial perspective allows us to reason about the invaria nces that these algorithms preserve and prove refinement bounds on the kin ds of loss functions that any such functor can produce. Finally\, we exper imentally demonstrate how this perspective enables us to derive and analyz e novel manifold learning algorithms.\n LOCATION:https://researchseminars.org/talk/Category_Theory/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrew Winkler DTSTART:20201203T000000Z DTEND:20201203T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/10 DESCRIPTION:Title: Functors as homomorphisms of quivered algebras.\nby Andrew Wi nkler as part of New York City Category Theory Seminar\n\n\nAbstract\nAbst ract: A quiver induces a minimalist algebraic structure which is\, nonethe less\, balanced\, associative\, elementwise strongly irreducible\, and bot h left and right quivered\, in a functorial way\; a homomorphism of quiver s induces a homomorphism of algebras. Q balanced\, quivered algebra posses ses a quiver structure\, but it is not true in general that a homomorphism for the algebra is also a homomorphism for the quiver. It will be precise ly when it is also a homomorphism for the algebra structure induced by the quiver structure it induces. Such a bihomorphism\, in the special case of categories\, (where the associativity property and a composition-inducing property hold)\, is precisely a functor. This facet of categories\, as po ssessing two compatible composition structures\, explains in some sense a bifurcation in the structure of monads.\n LOCATION:https://researchseminars.org/talk/Category_Theory/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Arthur Parzygnat (IHES) DTSTART:20201216T180000Z DTEND:20201216T193000Z DTSTAMP:20260422T214840Z UID:Category_Theory/11 DESCRIPTION:Title: A functorial characterization of classical and quantum entropies. \nby Arthur Parzygnat (IHES) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbstract: Entropy appears as a useful concept in a wide variety of academic disciplines. As such\, one would suspect that cat egory theory would provide a suitable language to encompass all or most of these definitions. The Shannon entropy has recently been given a characte rization as a certain affine functor by Baez\, Fritz\, and Leinster. This characterization is the only characterization I know of that uses linear a ssumptions (as opposed to additive\, exponential\, logarithmic\, etc). Her e\, we extend that characterization to include the von Neumann entropy as well as highlight the new categorical structures that arise when trying to do so. In particular\, we introduce Grothendieck fibrations of convex cat egories\, and we review the notion of a disintegration\, which is a key pa rt of conditional probability and Bayesian statistics and plays a crucial role in our characterization theorem. The characterization of Baez\, Fritz \, and Leinster interprets Shannon entropy in terms of the information los s associated to a deterministic process\, which is possible since the entr opy difference associated to such a process is always non-negative. This f ails for quantum entropy\, and has important physical consequences. \n

\nReferences:
\n Paper (and references therein)
\n Paper (original paper of Baez\, Fritz\, and Leinster)\n LOCATION:https://researchseminars.org/talk/Category_Theory/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Jason Parker (Brandon University) DTSTART:20210204T000000Z DTEND:20210204T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/12 DESCRIPTION:Title: Isotropy Groups of Quasi-Equational Theories.\nby Jason Parke r (Brandon University) as part of New York City Category Theory Seminar\n\ n\nAbstract\nAbstract: In [2]\, my PhD supervisors (Pieter Hofstra and Phi lip Scott) and I studied the new topos-theoretic phenomenon of isotropy (a s introduced in [1]) in the context of single-sorted algebraic theories\, and we gave a logical/syntactic characterization of the isotropy group of any such theory\, thereby showing that it encodes a notion of inner automo rphism or conjugation for the theory. In the present talk\, I will summari ze the results of my recent PhD thesis\, in which I build on this earlier work by studying the isotropy groups of (multi-sorted) quasi-equational th eories (also known as essentially algebraic\, cartesian\, or finite limit theories). In particular\, I will show how to give a logical/syntactic cha racterization of the isotropy group of any such theory\, and that it encod es a notion of inner automorphism or conjugation for the theory. I will al so describe how I have used this characterization to exactly characterize the ‘inner automorphisms’ for several different examples of quasi-equa tional theories\, most notably the theory of strict monoidal categories an d the theory of presheaves valued in a category of models. In particular\, the latter example provides a characterization of the (covariant) isotrop y group of a category of set-valued presheaves\, which had been an open qu estion in the theory of categorical isotropy.\n\n[1] J. Funk\, P. Hofstra\ , B. Steinberg. Isotropy and crossed toposes. Theory and Applications of C ategories 26\, 660-709\, 2012.\n\n[2] P. Hofstra\, J. Parker\, P.J. Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Computer Science 341\, 201-217\, 2018.\n LOCATION:https://researchseminars.org/talk/Category_Theory/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Peter Hines (University of York) DTSTART:20210211T000000Z DTEND:20210211T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/13 DESCRIPTION:Title: Shuffling cards as an operad.\nby Peter Hines (University of York) as part of New York City Category Theory Seminar\n\n\nAbstract\nThe theory of how two packs of cards may be shuffled together to form a single pack has been remarkably well-studied in combinatorics\, group theory\, s tatistics\, and other areas of mathematics. This talk aims to study natura l extensions where 1/ We may have infinitely many cards in a deck\, 2/ We may take the result of a previous shuffle as one of our decks of cards (i. e. shuffles are hierarchical)\, and 3/ There may even be an infinite numbe r of decks of cards.\n\nFar from being 'generalisation for generalisation' s sake'\, the original motivation came from theoretical & practical comput er science. The mathematics of card shuffles is commonly used to describe processing in multi-threaded computations. Moving to the infinite case giv es a language in which one may talk about potentially non-terminating proc esses\, or servers with an unbounded number of clients\, etc.\n\nHowever\, this talk is entirely about algebra & category theory -- just as in the f inite case\, the mathematics is of interest in its own right\, and should be studied as such.\n\nWe model shuffles using operads. The intuition behi nd them of allowing for arbitrary n-ary operations that compose in a hiera rchical manner makes them a natural\, inevitable choice for describing suc h processes such as merging multiple packs of cards.\n\nWe use very concre te examples\, based on endomorphism operads in groupoids of arithmetic ope rations. The resulting structures are at the same time both simple (i.e. e lementary arithmetic operations)\, and related to deep structures in mathe matics and category theory (topologies\, tensors\, coherence\, associahedr a\, etc.)\n\nWe treat this as a feature\, not a bug\, and use it to descri be complex structures in elementary terms. We also aim to give previously unobserved connections between distinct areas of mathematics.\n LOCATION:https://researchseminars.org/talk/Category_Theory/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Richard Blute (University of Ottawa) DTSTART:20210218T000000Z DTEND:20210218T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/14 DESCRIPTION:Title: Finiteness Spaces\, Generalized Polynomial Rings and Topological Groupoids.\nby Richard Blute (University of Ottawa) as part of New Yor k City Category Theory Seminar\n\n\nAbstract\nAbstract: The category of fi niteness spaces was introduced by Thomas Ehrhard as a model of classical l inear logic\, where a set is equipped with a class of subsets to be though t of as finitary. Morphisms are relations preserving the finitary structur e. The notion of finitary subset allows for a sharp analysis of computatio nal structure.\n\nWorking with finiteness spaces forces the number of summ ands in certain calculations to be finite and thus avoid convergence quest ions. An excellent example of this is how Ribenboim’s theory of generali zed power series rings can be naturally interpreted by assigning finitenes s monoid structure to his partially ordered monoids. After Ehrhard’s lin earization construction is applied\, the resulting structures are the ring s of Ribenboim’s construction.\n\nThere are several possible choices of morphism between finiteness spaces. If one takes structure-preserving part ial functions\, the resulting category is complete\, cocomplete and symmet ric monoidal closed. Using partial functions\, we are able to model topolo gical groupoids\, when we consider composition as a partial function. We c an associate to any hemicompact etale Hausdorff groupoid a complete convol ution ring. This is in particular the case for the infinite paths groupoid associated to any countable row-finite directed graph.\n LOCATION:https://researchseminars.org/talk/Category_Theory/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Joshua Sussan (Medgar Evers) DTSTART:20210304T000000Z DTEND:20210304T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/15 DESCRIPTION:Title: Categorification and quantum topology.\nby Joshua Sussan (Med gar Evers) as part of New York City Category Theory Seminar\n\n\nAbstract\ nAbstract: The Jones polynomial of a link could be defined through the rep resentation theory of quantum sl(2). It leads to a 3-manifold invariant an d 2+1 dimensional TQFT. In the mid 1990s\, Crane and Frenkel outlined the categorification program with the aim of constructing a 3+1 dimensional TQ FT by upgrading the representation theory of quantum sl(2) to some categor ical structures. We will review these ideas and give examples of various c ategorifications of quantum sl(2) constructions.\n LOCATION:https://researchseminars.org/talk/Category_Theory/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Tobias Fritz (University of Innsbruck) DTSTART:20210317T230000Z DTEND:20210318T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/16 DESCRIPTION:Title: Categorical Probability and the de Finetti Theorem\nby Tobias Fritz (University of Innsbruck) as part of New York City Category Theory Seminar\n\n\nAbstract\nI will give an introduction to categorical probabil ity in terms of Markov categories\, followed by a discussion of the classi cal de Finetti theorem within that framework. Depending on whether current ideas work out or not\, I may (or may not) also present a sketch of a pur ely categorical proof of the de Finetti theorem based on the law of large numbers. Joint work with Tomáš Gonda\, Paolo Perrone and Eigil Fjeldgren Rischel.\n LOCATION:https://researchseminars.org/talk/Category_Theory/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Ross Street (Macquarie University) DTSTART:20210414T230000Z DTEND:20210415T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/17 DESCRIPTION:Title: Absolute colimits for differential graded categories.\nby Ros s Street (Macquarie University) as part of New York City Category Theory S eminar\n\n\nAbstract\nA little enriched category theory will be reviewed\, in particular\, absolute colimits and Cauchy completion. Then the focus w ill be on the monoidal category DGAb of chain complexes of abelian groups which is at the heart of homological and homotopical algebra. Categories e nriched in DGAb are called differential graded categories (DG-categories). Recent joint work with Branko Nikolic and Giacomo Tendas on the absolute colimit completion of a DG-category will be described. The talk is dedicat ed to the memory of two great New Yorkers\, Sammy Eilenberg and Alex Helle r.\n LOCATION:https://researchseminars.org/talk/Category_Theory/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Juan Orendain (University of Mexico\, UNAM.) DTSTART:20210505T230000Z DTEND:20210506T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/18 DESCRIPTION:Title: How long does it take to frame a bicategory?\nby Juan Orendai n (University of Mexico\, UNAM.) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbstract: Framed bicategories are double categories having all companions and conjoints. Many structures naturally organize i nto framed bicategories\, e.g. open Petri nets\, polynomials functors\, po lynomial comonoids\, structured cospans\, algebras\, etc. Symmetric monoid al structures on framed bicategories descend to symmetric monoidal structu res on the corresponding horizontal bicategories. The axioms defining symm etric monoidal double categories are much more tractible than those defini ng symmetric monoidal bicategories. It is thus convenient to study ways of lifting a given bicategory into a framed bicategory along an appropriate category of vertical morphisms. Solutions to the problem of lifting bicate gories to double categories have classically being useful in expressing Ke lly and Street's mates correspondence and in proving the higher dimensiona l Seifert-van Kampen theorem of Brown et. al.\, amongst many other applica tions. We consider lifting problems in their full generality.\n\nGlobularl y generated double categories are minimal solutions to lifting problems of bicategories into double categories along given categories of vertical ar rows. Globularly generated double categories form a coreflective sub-2-cat egory of general double categories. This\, together with an analysis of th e internal structure of globularly generated double categories yields a nu merical invariant on general double categories. We call this invariant the vertical length. The vertical length of a double category C measures the complexity of mixed compositions of globular and horizontal identity squar es of C and thus provides a measure of complexity for lifting problems on the horizontal bicategory HC of C. I will explain recent results on the th eory of globularly generated double categories and the vertical length inv ariant. The ultimate goal of the talk is to present conjectures on the ver tical length of framed bicategories and possible applications.\n LOCATION:https://researchseminars.org/talk/Category_Theory/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Tobias Fritz (University of Innsbruck) DTSTART:20210324T230000Z DTEND:20210325T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/19 DESCRIPTION:Title: Categorical Probability and the de Finetti Theorem\nby Tobias Fritz (University of Innsbruck) as part of New York City Category Theory Seminar\n\n\nAbstract\nI will give an introduction to categorical probabil ity in terms of Markov categories\, followed by a discussion of the classi cal de Finetti theorem within that framework. Depending on whether current ideas work out or not\, I may (or may not) also present a sketch of a pur ely categorical proof of the de Finetti theorem based on the law of large numbers. Joint work with Tomáš Gonda\, Paolo Perrone and Eigil Fjeldgren Rischel.\n LOCATION:https://researchseminars.org/talk/Category_Theory/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Gemma De las Cuevas (University of Innsbruck) DTSTART:20211006T230000Z DTEND:20211007T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/20 DESCRIPTION:Title: From simplicity to universality and undecidability\nby Gemma De las Cuevas (University of Innsbruck) as part of New York City Category Theory Seminar\n\n\nAbstract\nWhy is it so easy to generate complexity? I will suggest that this is due to the phenomenon of universality — essent ially every non-trivial system is universal\, and thus able to explore all complexity in its domain. We understand universality in spin models\, aut omata and neural networks. I will present the first step toward rigorously linking the first two\, where we cast classical spin Hamiltonians as form al languages and classify the latter in the Chomsky hierarchy. We prove th at the language of (effectively) zero-dimensional spin Hamiltonians is reg ular\, one-dimensional spin Hamiltonians is deterministic context-free\, a nd higher-dimensional and all-to-all spin Hamiltonians is context-sensitiv e. I will also talk about the other side of the coin of universality\, nam ely undecidability\, and will raise the question of whether universality i s "visible" in Lawvere’s Theorem.\n LOCATION:https://researchseminars.org/talk/Category_Theory/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Dan Shiebler (Oxford University) DTSTART:20211020T230000Z DTEND:20211021T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/21 DESCRIPTION:Title: Out of Sample Generalization with Kan Extensions\nby Dan Shie bler (Oxford University) as part of New York City Category Theory Seminar\ n\n\nAbstract\nA common problem in data science is "use this function defi ned over this small set to generate predictions over that larger set." Ext rapolation\, interpolation\, statistical inference and forecasting all red uce to this problem. The Kan extension is a powerful tool in category theo ry that generalizes this notion. In this work we explore several applicati ons of Kan extensions to data science.\n LOCATION:https://researchseminars.org/talk/Category_Theory/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Dusko Pavlovic (University of Hawai‘i at Mānoa) DTSTART:20211103T230000Z DTEND:20211104T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/22 DESCRIPTION:Title: Geometry of computation and string-diagram programming in monoida l computer\nby Dusko Pavlovic (University of Hawai‘i at Mānoa) as p art of New York City Category Theory Seminar\n\n\nAbstract\nA monoidal com puter is a monoidal category with a distinguished type carrying the struct ure of a single-instruction programming language. The instruction would be written as "run"\, but it is usually drawn as a string diagram. Equivalen tly\, the monoidal computer structure can be viewed as a typed lambda-calc ulus without lambda abstraction\, even implicit. Any Turing complete progr amming language\, including Turing machines and partial recursive function s\, gives rise to a monoidal computer. We have thus added yet another item to the Church-Turing list of models of computation. It differs from other models by its categoricity. While the other Church-Turing models can be p rogrammed to simulate each other in many different ways\, and each interpr ets even itself in infinitely many non-isomorphic ways\, the structure of a monoidal computer is unique up to isomorphism. A monoidal category can b e a monoidal computer in at most one way\, just like it can be closed in a t most one way\, up to isomorphism. In other words\, being a monoidal comp uter is a property\, not structure. Computability is thus a categorical pr operty\, like completeness. This opens an alley towards an abstract treatm ent of parametrized complexity\, one-way and trapdoor functions on one han d\, and of algorithmic learning in the other.\n LOCATION:https://researchseminars.org/talk/Category_Theory/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Marco Schorlemmer (Spanish National Research Council) DTSTART:20211118T000000Z DTEND:20211118T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/23 DESCRIPTION:Title: A Uniform Model of Computational Conceptual Blending\nby Marc o Schorlemmer (Spanish National Research Council) as part of New York City Category Theory Seminar\n\n\nAbstract\nWe present a mathematical model fo r the cognitive operation of conceptual blending that aims at being unifor m across different representation formalisms\, while capturing the relevan t structure of this operation. The model takes its inspiration from amalga ms as applied in case-based reasoning\, but lifts them into category theor y so as to follow Joseph Goguen’s intuition for a mathematically precise characterisation of conceptual blending at a representation-independent l evel of abstraction. We prove that our amalgam-based category-theoretical model of conceptual blending is essentially equivalent to the pushout mode l in the ordered category of partial maps as put forward by Goguen. But un like Goguen’s approach\, our model is more suitable to capture computati onal realisations of conceptual blending\, and we exemplify this by concre tising our model to computational conceptual blends for various representa tion formalisms and application domains.\n LOCATION:https://researchseminars.org/talk/Category_Theory/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Geroch (University of Chicago) DTSTART:20211202T000000Z DTEND:20211202T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/24 DESCRIPTION:Title: An Alien's Perspective on Mathematics (and Physics).\nby Robe rt Geroch (University of Chicago) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbstract: We describe what might be called a "poin t of view" toward mathematics. This view touches on such issues as how God el's theorem might be interpreted\, the relevance to physics of mathematic al axioms such as the axiom of choice\, and the possibility of using physi cs to "solve" unsolvable mathematical problems.\n LOCATION:https://researchseminars.org/talk/Category_Theory/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Samantha Jarvis (The CUNY Graduate Center) DTSTART:20211216T000000Z DTEND:20211216T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/25 DESCRIPTION:Title: Language as an Enriched Category.\nby Samantha Jarvis (The CU NY Graduate Center) as part of New York City Category Theory Seminar\n\n\n Abstract\nWe review enriched category theory\, with particular focus on en riching over posets such as [0\,1]. We then apply this to natural language \, making a language category into an enriched language category as in Bra dley-Vlassopoulos-Terilla (our advisor!) [2106.07890.pdf (arxiv.org)]. The statements of enriched category theory have concrete (and interesting!) i nterpretations when applied to this enriched language category.\n LOCATION:https://researchseminars.org/talk/Category_Theory/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Todd Trimble (Western Connecticut State University) DTSTART:20211223T000000Z DTEND:20211223T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/26 DESCRIPTION:Title: Categorifying negatives: roadblocks and detours.\nby Todd Tri mble (Western Connecticut State University) as part of New York City Categ ory Theory Seminar\n\n\nAbstract\nThe challenge of finding meaningful cate gorified interpretations of "reciprocals" of objects and "negatives" of ob jects poses some intriguing problems. In this talk\, we consider a few res ponses to this challenge\, with particular attention to extending the subs titution product on species to "negative species" and "virtual species".\n LOCATION:https://researchseminars.org/talk/Category_Theory/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Ralph Wojtowicz (Shenandoah University) DTSTART:20220203T000000Z DTEND:20220203T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/27 DESCRIPTION:Title: On Logic-Based Artificial Intelligence and Categorical Logic. \nby Ralph Wojtowicz (Shenandoah University) as part of New York City Cate gory Theory Seminar\n\n\nAbstract\nThe objective of this talk is to reform ulate the logic-based artificial intelligence algorithms and examples from the text of Russell and Norvig using the syntax and categorical semantics of Johnstone’s Sketches of an Elephant in order to: (1) identify the fr agments of first-order logic required\; (2) enable symbolic reasoning abou t richly-structured semantic objects (e.g.\, graphs\, dynamic systems and objects in categories other than Set)\; (3) clarify the separation between syntax and semantics\; and (4) support use of other category-theoretic in frastructure such as Morita equivalence and transformations between theori es and sketches.\n LOCATION:https://researchseminars.org/talk/Category_Theory/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Emilio Minichiello (CUNY Graduate Center) DTSTART:20220217T000000Z DTEND:20220217T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/28 DESCRIPTION:Title: Category Theory ∩ Differential Geometry.\nby Emilio Minichi ello (CUNY Graduate Center) as part of New York City Category Theory Semin ar\n\n\nAbstract\nIn this talk we will take a tour through some areas of m ath at the intersection of category theory and differential geometry. We w ill talk about how the use of category theory works towards solving 2 prob lems: 1) to give rigorous definitions and techniques to study increasingly complicated objects in differential geometry that are coming from physics \, like orbifolds and bundle gerbes\, and 2) to find good categories in wh ich to embed the category of finite dimensional smooth manifolds\, without losing too much geometric intuition. This involves the study of Lie group oids\, sheaves\, diffeological spaces\, stacks\, and infinity stacks. I wi ll try to motivate the use of these mathematical objects and how they help mathematicians understand differential geometry and expand its scope.\n LOCATION:https://researchseminars.org/talk/Category_Theory/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Jens Hemelaer (University of Antwerp) DTSTART:20211209T000000Z DTEND:20211209T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/29 DESCRIPTION:by Jens Hemelaer (University of Antwerp) as part of New York C ity Category Theory Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/Category_Theory/29/ END:VEVENT BEGIN:VEVENT SUMMARY:David Roberts DTSTART:20220224T000000Z DTEND:20220224T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/30 DESCRIPTION:Title: Do you have what it takes to use the diagonal argument?\nby D avid Roberts as part of New York City Category Theory Seminar\n\n\nAbstrac t\nLawvere's reformulation of the diagonal argument captured many instance s from the literature in an elegant and abstract category-theoretic treatm ent. The original version used cartesian closed categories\, but gave a no d to how the statement of the argument could be adjusted so as to make few er demands on the category. In fact the argument\, and the fixed-point the orem that Lawvere provided as the positive version of the argument\, both require much less than Lawvere stated. This talk will give an outline of L awvere's version of the diagonal argument\, his corresponding fixed-point theorem\, and then cover a few versions obtained recently that drop assump tions on the properties/structure of the category at hand.\n LOCATION:https://researchseminars.org/talk/Category_Theory/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Morgan Rogers (Universit`a degli Studi dell’Insubria.) DTSTART:20220330T230000Z DTEND:20220331T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/32 DESCRIPTION:Title: Toposes of Topological Monoid Actions.\nby Morgan Rogers (Uni versit`a degli Studi dell’Insubria.) as part of New York City Category T heory Seminar\n\n\nAbstract\nAnyone encountering topos theory for the firs t time will be familiar with the fact that the category of actions of a mo noid on sets is a special case of a presheaf topos. It turns out that if w e equip the monoid with a topology and consider the subcategory of continu ous actions\, the result is still a Grothendieck topos. It is possible to characterize such toposes in terms of their points\, and along the way ext ract canonical representing topological monoids\, the complete monoids. I' ll sketch the trajectory of this story\, present some positive and negativ e results about Morita-equivalence of topological monoids\, and explain ho w one can extract a semi-Galois theory from this set-up.\n LOCATION:https://researchseminars.org/talk/Category_Theory/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Jin-Cheng Guu (Stony Brook University) DTSTART:20220316T230000Z DTEND:20220317T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/33 DESCRIPTION:Title: Topological Quantum Field Theories from Monoidal Categories\n by Jin-Cheng Guu (Stony Brook University) as part of New York City Categor y Theory Seminar\n\n\nAbstract\nAbstract: We will introduce the notion of a topological quantum field theory (tqft) and a monoidal category. We will then construct a few (extended) tqfts from monoidal categories\, and show how quantum invariants of knots and 3-manifolds were obtained. If time pe rmits\, I will discuss (higher) values in (higher) codimensions based on m y recent work on categorical center of higher genera (joint with A. Kirill ov and Y. H. Tham).\n LOCATION:https://researchseminars.org/talk/Category_Theory/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Joseph Dimos DTSTART:20220323T230000Z DTEND:20220324T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/34 DESCRIPTION:Title: Introduction to Fusion Categories and Some Applications.\nby Joseph Dimos as part of New York City Category Theory Seminar\n\n\nAbstrac t\nAbstract: Tensor categories and multi-tensor categories have strong ali gnment with module categories. We can use the multi-tensor categories C in conjunction with classifying tensor algebras wrt C. From here\, we can il lustrate some examples of tensor categories: the category Vec of k-vector spaces that gives us a fusion category. This is defined as a category Rep( G) of some finite dimensional k-representations of a group G. From here\, I will walk through the correspondence of tensor categories (Etingof) and fusion categories. Throughout\, I will indicate a few unitary and non-unit ary cases of fusion categories. Those unitary fusion categories are those that admit a uniquely monoidal structure. For example\, this draws upon [J ones 1983] for finite index and finite depth that bridges a subfactor A-bi module B to provide a full subcategory of some category A by its module st ructure. I will discuss some of these components throughout and explain th e Morita equivalence of fusion categories.\n LOCATION:https://researchseminars.org/talk/Category_Theory/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Jason Parker (Brandon University in Manitoba.) DTSTART:20220406T230000Z DTEND:20220407T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/35 DESCRIPTION:Title: Enriched structure-semantics adjunctions and monad-theory equival ences for subcategories of arities.\nby Jason Parker (Brandon Universi ty in Manitoba.) as part of New York City Category Theory Seminar\n\n\nAbs tract\nSeveral structure-semantics adjunctions and monad-theory equivalenc es have been established in category theory. Lawvere (1963) developed a st ructure-semantics adjunction between Lawvere theories and tractable Set-va lued functors\, which was subsequently generalized by Linton (1969)\, whil e Dubuc (1970) established a structure-semantics adjunction between V-theo ries and tractable V-valued V-functors for a symmetric monoidal closed cat egory V. It is also well known (and due to Linton) that there is an equiva lence between Lawvere theories and finitary monads on Set. Generalizing th is result\, Lucyshyn-Wright (2016) established a monad-theory equivalence for eleutheric systems of arities in arbitrary closed categories. Building on earlier work by Nishizawa and Power\, Bourke and Garner (2019) subsequ ently proved a general monad-theory equivalence for arbitrary small subcat egories of arities in locally presentable enriched categories. However\, n either of these equivalences generalizes the other\, and there has not yet been a general treatment of enriched structure-semantics adjunctions that specializes to those established by Lawvere\, Linton\, and Dubuc.\n\nMoti vated by these considerations\, we develop a general axiomatic framework f or studying enriched structure-semantics adjunctions and monad-theory equi valences for subcategories of arities\, which generalizes all of the afore mentioned results and also provides substantial new examples of relevance for topology and differential geometry. For a subcategory of arities J in a V-category C over a symmetric monoidal closed category V\, Linton’s no tion of clone generalizes to provide enriched notions of J-theory and J-pr etheory\, which were also employed by Bourke and Garner (2019). We say tha t J is amenable if every J-theory admits free algebras\, and is strongly a menable if every J-pretheory admits free algebras. If J is amenable\, then we obtain an idempotent structure-semantics adjunction between certain J- pretheories and J-tractable V-categories over C\, which yields an equivale nce between J-theories and J-nervous V-monads on C. If J is strongly amena ble\, then we also obtain a rich theory of presentations for J-theories an d J-nervous V-monads. We show that many previously studied subcategories o f arities are (strongly) amenable\, from which we recover the aforemention ed structure-semantics adjunctions and monad-theory equivalences. We concl ude with the result that any small subcategory of arities in a locally bou nded closed category is strongly amenable\, from which we obtain structure -semantics adjunctions and monad-theory equivalences in (e.g.) many conven ient categories of spaces.\n\nJoint work with Rory Lucyshyn-Wright.\n LOCATION:https://researchseminars.org/talk/Category_Theory/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Alex Martsinkovsky (Northeastern University) DTSTART:20220413T230000Z DTEND:20220414T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/36 DESCRIPTION:Title: A Reflector in Search of a Category.\nby Alex Martsinkovsky ( Northeastern University) as part of New York City Category Theory Seminar\ n\n\nAbstract\nThe last several months have seen an explosive growth of ac tivities centered around the defect of a finitely presented functor. This notion made its first appearance in M. Auslander's fundamental work on coh erent functors in the mid-1960s\, although at that time it was mostly used just as a technical tool. A phenomenological study of that concept was in itiated by Jeremy Russell in 2016. What transpired in the recent months is the ubiquitous nature of the defect\, explained in part by the fact that it is adjoint to the Yoneda embedding. Thus any branch of mathematics\, co mputer science\, physics\, or any applied science that references the Yone da embedding automatically becomes a candidate for applications of the def ect.\n\nIn this expository talk I will first give a streamlined introducti on to the original notion of defect of a finitely presented functor define d on a module category and then show how to generalize it to arbitrary add itive functors. Along the way I will give a dozen or so examples illustrat ing various use cases for the defect. The ultimate goal of this lecture is to provide a background for the upcoming talk of Alex Sorokin\, who will report on his vast generalization of the defect to arbitrary profunctors e nriched in a cosmos.\n\nThis presentation is based on joint work in progre ss with Jeremy Russell.\n LOCATION:https://researchseminars.org/talk/Category_Theory/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Alex Sorokin (Northeastern University) DTSTART:20220427T230000Z DTEND:20220428T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/37 DESCRIPTION:Title: The defect of a profunctor.\nby Alex Sorokin (Northeastern Un iversity) as part of New York City Category Theory Seminar\n\n\nAbstract\n In the mid 1960s Auslander introduced a notion of the defect of a finitely presented functor on a module category. In 2021 Martsinkovsky generalized it to arbitrary additive functors. In this talk I will show how to define a defect of any enriched functor with a codomain a cosmos. Under mild ass umptions\, the covariant (contravariant) defect functor turns out to be a left covariant (right contravariant) adjoint to the covariant (contravaria nt) Yoneda embedding. Both defects can be defined for any profunctor enric hed in a cosmos V. They happen to be adjoints to the embeddings of V-Cat i n V-Prof. Moreover\, the Isbell duals of a profunctor are completely deter mined by the profunctor's covariant and contravariant defects. These resul ts are based on applications of the Tensor-Hom-Cotensor adjunctions and th e (co)end calculus.\n LOCATION:https://researchseminars.org/talk/Category_Theory/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Gershom Bazerman (Arista Networks.) DTSTART:20220504T230000Z DTEND:20220505T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/38 DESCRIPTION:Title: Classes of Closed Monoidal Functors which Admit Infinite Traversa ls.\nby Gershom Bazerman (Arista Networks.) as part of New York City C ategory Theory Seminar\n\n\nAbstract\nIn functional programming\, functors that are equipped with a traverse\noperation can be thought of as data st ructures which permit an\nin-order traversal of their elements. This has b een made precise by\nthe correspondence between traversable functors and f initary\ncontainers (aka polynomial functors). This correspondence was\nes tablished in the context of total\, necessarily terminating\,\nfunctions. However\, the Haskell language is non-strict and permits\nfunctions that d o not terminate. It has long been observed that\ntraversals can at times\, in practice\, operate over infinite lists\, for\nexample in distributing the Reader applicative. We present work in\nprogress that characterizes wh en this situation occurs\, making use of\nthe toolkit of guarded recursion .\n LOCATION:https://researchseminars.org/talk/Category_Theory/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Sergei Burkin (University of Tokyo) DTSTART:20220907T230000Z DTEND:20220908T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/39 DESCRIPTION:Title: Segal conditions and twisted arrow categories of operads\nby Sergei Burkin (University of Tokyo) as part of New York City Category Theo ry Seminar\n\n\nAbstract\nSeveral categories\, including the simplex categ ory Delta and Moerdijk-Weiss dendroidal category Omega\, allow to encode s tructures (in this case categories and operads reprectively) as Segal pres heaves. There are other examples of such categories\, which were defined i ntuitively\, by analogy with Delta. We will describe a general constructio n of categories from operads that produces categories that admit Segal pre sheaves. This construction explains why these categories appear in homotop y theory\, why these allow to encode homotopy coherent structures as simpl icial presheaves that satisfy weak Segal condition. Further generalization of this construction to clones shows that these categories are not as can onical as one might have hoped.\n LOCATION:https://researchseminars.org/talk/Category_Theory/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Prakash Panangaden (McGill University) DTSTART:20220914T230000Z DTEND:20220915T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/40 DESCRIPTION:Title: Quantitative Equational Logic\nby Prakash Panangaden (McGill University) as part of New York City Category Theory Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/Category_Theory/40/ END:VEVENT BEGIN:VEVENT SUMMARY:David Ellerman (University of Ljubljana) DTSTART:20221019T230000Z DTEND:20221020T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/41 DESCRIPTION:Title: To Interpret Quantum Mechanics:``Follow the Math'': The math of Q M as the linearization of the math of partitions\nby David Ellerman (U niversity of Ljubljana) as part of New York City Category Theory Seminar\n \n\nAbstract\nAbstract: Set partitions are dual to subsets\, so there is a logic of partitions dual to the Boolean logic of subsets. Partitions are the mathematical tool to describe definiteness and indefiniteness\, distin ctions and distinctions\, as well as distinguishability and indistinguisha bility. There is a semi-algorithmic process or ``Yoga'' of linearization t o transform the concepts of partition math into the corresponding vector s pace concepts. Then it is seen that those vector space concepts\, particul arly in Hilbert spaces\, are the mathematical framework of quantum mechani cs. (QM). This shows that those concepts\, e.g.\, distinguishability versu s indistinguishability\, are the central organizing concepts in QM to desc ribe an underlying reality of objective indefiniteness--as opposed to the classical physics and common sense view of reality as ``definite all the w ay down'' This approach thus supports what Abner Shimony called the ``Lite ral Interpretation'' of QM which interprets the formalism literally as des cribing objective indefiniteness and objective probabilities--as well as b eing complete in contrast to the other realistic interpretations such as t he Bohmian\, spontaneous localization\, and many world interpretations whi ch embody other variables\, other equations\, or other worldly ideas.\n LOCATION:https://researchseminars.org/talk/Category_Theory/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrei Rodin (University of Lorraine (Nancy\, France)) DTSTART:20221110T000000Z DTEND:20221110T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/42 DESCRIPTION:Title: Kolmogorov's Calculus of Problems and Homotopy Type theory\nb y Andrei Rodin (University of Lorraine (Nancy\, France)) as part of New Yo rk City Category Theory Seminar\n\n\nAbstract\nA. N. Kolmogorov in 1932 pr oposed an original version of mathematical intuitionism where the concept of problem plays a central role\, and which differs in its content from th e versions of intuitionism developed by A. Heyting and other followers of L. Brouwer. The popular BHK-semantics of Intuitionistic logic follows Heyt ing's line and conceals the original features of Kolmogorov's logical idea s. Homotopy Type theory (HoTT) implies a formal distinction between senten ces and higher-order constructions and thus provides a mathematical argume nt in favour of Kolmogorov's approach and against Heyting's approach. At t he same time HoTT does not support the constructive notion of negation app licable to general problems\, which is informally discussed by Kolmogorov in the same context. Formalisation of Kolmogorov-style constructive negati on remains an interesting open problem.\n LOCATION:https://researchseminars.org/talk/Category_Theory/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Saeed Salehi (University of Tabriz) DTSTART:20221124T000000Z DTEND:20221124T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/43 DESCRIPTION:Title: Self-Reference and Diagonalization: their difference and a short history.\nby Saeed Salehi (University of Tabriz) as part of New York C ity Category Theory Seminar\n\n\nAbstract\nWhat is now called the Diagonal (or the Self-Reference) Lemma\, is the statement that for every formula F(x)\, with the only free variable x\, there exists a sentence &sigma\; such that &sigma\; is equivalent to the F of the Gö\;del code of &sigma\;\, i.e.\, &sigma\; &equiv\; F(#&sigma\;)\; and this equivalence is provable in certain weak arithmetics. This lemma is credited to Gö\;del (1931)\, in the special case when F is the < i>unprovability predicate\, and to Carnap (1934) in the more general case.\n

\nIn this talk \, we will argue that Gö\;del-Carnap's original Diag onal Lemma is not the modern formulation and was more similar to\, but not exactly identical with\, the Strong Diagonal (or Direct Self-Reference) L emma. This lemma\, so-called recently\, says that for every formula F(x)\, in a suffi ciently expressive language\, there exists a sentence &sigma\; such that &sigma\; is equal to the F of the Gö \;del code of &sigma\;\, i.e.\, &sigma\; = F(#&sigma\; )\; and this equality is provable in sufficiently strong theories. We will attempt at tracking down the first appearance of the modern formulati on of the Diagonal Lemma in the equivalent form\, also in the strong direc t form of equality.\n LOCATION:https://researchseminars.org/talk/Category_Theory/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Pare (Dalhousie University) DTSTART:20221208T000000Z DTEND:20221208T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/45 DESCRIPTION:Title: The horizontal/vertical synergy of double categories\nby Robe rt Pare (Dalhousie University) as part of New York City Category Theory Se minar\n\n\nAbstract\nA double category is a category with two types of arr ows\, horizontal and vertical\, related by double cells. Think of sets wit h functions and relations as arrows and implications as double cells. The theory is 2-dimensional just like for 2-categories. In fact 2-categories w ere originally defined as double categories in which all vertical arrows w ere identities. Most of the theory of 2-categories extends to double categ ories resulting in a deeper understanding. This is one aspect of double ca tegories: they’re “new and improved” 2-categories.\n\nFrom a purely formal point of view\, a double category is a category object in CAT. Once a familiarity with double categories has developed\, it is amusing and in structive to see how the various constructs of formal category theory play out in this setting.\n\nBut these two aspects of double categories\, fanc y 2-categories or internal categories\, are only part of the picture. Perh aps the most important thing is the interplay between the horizontal and t he vertical.\n\nI will start with some examples of double categories to gi ve a feeling for the objects I will be discussing\, and then look at sever al concepts indicative of the rich interplay between the horizontal and th e vertical.\n LOCATION:https://researchseminars.org/talk/Category_Theory/45/ END:VEVENT BEGIN:VEVENT SUMMARY:James Torre DTSTART:20220928T230000Z DTEND:20220929T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/46 DESCRIPTION:Title: Diagonalization\, and the Limits of Limitative Theorems\nby J ames Torre as part of New York City Category Theory Seminar\n\nAbstract: T BA\n LOCATION:https://researchseminars.org/talk/Category_Theory/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Astra Kolomatskaia (Stony Brook) DTSTART:20221102T230000Z DTEND:20221103T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/47 DESCRIPTION:Title: The Objective Metatheory of Simply Typed Lambda Calculus\nby Astra Kolomatskaia (Stony Brook) as part of New York City Category Theory Seminar\n\n\nAbstract\nLambda calculus is the language of functions. One r educes the application of a function to an argument by substituting the ar gument for the function's formal parameter inside of the function's body. The result of such a reduction may have further instances of function appl ication. We can write down expressions\, such as ((λ f. f f) (λ f. f f)) \, in which this process does not terminate. In the presence of types\, ho wever\, one has a normalisation theorem\, which effectively states that "p rograms can be run". One proof of this theorem\, which only works for the most elementary of type theories\, is to assign some monotone well-founded invariant to a given reduction algorithm. A much more surprising proof pr oceeds by constructing the normal form of a term by structural recursion o n the term's syntactic representation\, without ever performing reduction. Such normalisation algorithms fall under the class of Normalisation by Ev aluation. Since the accidental discovery of the first such algorithm\, it was clear that NbE had some underlying categorical content\, and\, in 1995 \, Altenkirch\, Hofmann\, and Streicher published the first categorical no rmalisation proof. Discovering this content requires first asking the ques tion “What is STLC?”\, perhaps preceded by the question “What is a t ype theory?”. In this talk we will lay out the details of Altenkirch's s eminal paper and explore conceptual refinements discovered in the process of its formalisation in Cubical Agda.\n LOCATION:https://researchseminars.org/talk/Category_Theory/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Ross Street (Macquarie University) DTSTART:20221026T230000Z DTEND:20221027T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/48 DESCRIPTION:Title: The core groupoid can suffice\nby Ross Street (Macquarie Univ ersity) as part of New York City Category Theory Seminar\n\n\nAbstract\nAb stract: Let V be the monoidal category of modules over a commuative ring R . I am interested in categories A for which there is a groupoid G such tha t the functor categories [A\,V] and [G\,V] are equivalent. In particular\, G could be the core groupoid of A\; that is\, the subcategory with the sa me objects and with only the invertible morphisms. Every category A can be regarded as a V-category (that is\, an R-linear category)\, denoted RA\, with the same objects and with hom R-module RA(a\,b) free on the homset A( a\,b). Indeed\, RA is the free V-category on A so that the V-functor categ ory [RA\,V] is the ordinary functor category [A\,V] with the pointwise R-l inear structure. In these terms\, we are interested in when RA and RG are Morita equivalent V-categories. In my joint work with Steve Lack on Dold-K an-type equivalences\, we had many examples of this phenomenon. However\, the example of Nick Kuhn\, where A is the category of finite vector spaces over a fixed finite field F with all F-linear functions and G is the gene ral linear groupoid over F\, does not fit our theory. Yet the ``kernel'' o f the equivalence is of the same type. The present work shows that the cat egory theory behind the Kuhn result also covers our Dold-Kan-type setting. I plan to start with a baby example which highlights the ideas.\n\nI am g rateful to Nick Kuhn and Ben Steinberg for their patient email corresponde nce with me on this topic.\n LOCATION:https://researchseminars.org/talk/Category_Theory/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Igor Baković (University of Osijek\, Croatia.) DTSTART:20230202T000000Z DTEND:20230202T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/49 DESCRIPTION:Title: Enhanced 2-adjunctions.\nby Igor Baković (University of Osij ek\, Croatia.) as part of New York City Category Theory Seminar\n\n\nAbstr act\nWhenever one has a class of objects possessing certain structure and a hierarchy of morphisms that preserve structure more or less tightly\, we are in an enhanced context. Enhanced 2-categories were introduced by Lack and Shulman in 2012 with a paradigmatic example of an enhanced 2-category T-alg of strict algebras for a 2-monad and whose tight and loose 1-cells are pseudo- and lax morphisms of algebras\, respectively. They can be defi ned in two equivalent ways: either as 2-functors\, which are the identity on objects\, faithful\, and locally fully faithful\, or as categories enri ched over the cartesian closed category F\, whose objects are functors tha t are fully faithful and injective on objects. Lack and Shulman called obj ects of F full embeddings\, but we will call them "enhanced categories" be cause they are nothing else but categories with a distinguished class of o bjects\, which we call tight.The 2-category F has a much richer structure besides being cartesian closed\; there are additional closed (but not mono idal) structures\, and we show how 2-categories with a right ideal of 1-ce lls as in 2-categories with Yoneda structure on them can be presented as c ategories enriched in F in the sense of Eilenberg and Kelly. Since Lack an d Shulman were mainly motivated by limits in enhanced 2-categories\, they didn't further develop the theory of enhanced (co)lax functors and their e nhanced lax adjunctions. The purpose of this talk is to lay the foundation s of the theory of enhanced 2-adjunctions and give their examples througho ut mathematics and theoretical computer science.\n LOCATION:https://researchseminars.org/talk/Category_Theory/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikhail Khovanov (Columbia University.) DTSTART:20230209T000000Z DTEND:20230209T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/50 DESCRIPTION:Title: Universal construction and its applications.\nby Mikhail Khov anov (Columbia University.) as part of New York City Category Theory Semin ar\n\n\nAbstract\nUniversal construction starts with an evaluation of clos ed n-manifolds and builds a topological theory (a lax TQFT) for n-cobordis ms. A version of it has been used for years as an intermediate step in con structing link homology theories\, by evaluating foams embedded in 3-space . More recently\, universal construction in low dimensions has been used t o find interesting structures related to Deligne categories\, formal langu ages and automata. In the talk we will describe the universal construction and review these developments.\n LOCATION:https://researchseminars.org/talk/Category_Theory/50/ END:VEVENT BEGIN:VEVENT SUMMARY:Mee Seong Im (United States Naval Academy\, Annapolis) DTSTART:20230216T000000Z DTEND:20230216T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/51 DESCRIPTION:Title: Automata and topological theories.\nby Mee Seong Im (United S tates Naval Academy\, Annapolis) as part of New York City Category Theory Seminar\n\n\nAbstract\nTheory of regular languages and finite state automa ta is part of the foundations of computer science. Topological quantum fie ld theories (TQFT) are a key structure in modern mathematical physics. We will interpret a nondeterministic automaton as a Boolean-valued one-dimens ional TQFT with defects labelled by letters of the alphabet for the automa ton. We will also describe how a pair of a regular language and a circular regular language gives rise to a lax one-dimensional TQFT.\n LOCATION:https://researchseminars.org/talk/Category_Theory/51/ END:VEVENT BEGIN:VEVENT SUMMARY:Joshua Sussan (CUNY) DTSTART:20230223T000000Z DTEND:20230223T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/52 DESCRIPTION:Title: Non-semisimple Hermitian TQFTs.\nby Joshua Sussan (CUNY) as p art of New York City Category Theory Seminar\n\n\nAbstract\nTopological qu antum field theories coming from semisimple categories build upon interest ing structures in representation theory and have important applications in low dimensional topology and physics. The construction of non-semisimple TQFTs is more recent and they shed new light on questions that seem to be inaccessible using their semisimple relatives. In order to have potential applications to physics\, these non-semisimple categories and TQFTs should possess Hermitian structures. We will define these structures and give so me applications.\n LOCATION:https://researchseminars.org/talk/Category_Theory/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Jens Hemelaer (University of Antwerp.) DTSTART:20230315T230000Z DTEND:20230316T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/53 DESCRIPTION:Title: EILC toposes.\nby Jens Hemelaer (University of Antwerp.) as p art of New York City Category Theory Seminar\n\n\nAbstract\nIn topos theor y\, local connectedness of a geometric morphism is a very geometric proper ty\, in the sense that it is stable under base change\, can be checked loc ally\, and so on. In some situations however\, the weaker property of bein g essential is easier to verify. In this talk\, we will discuss EILC topos es: toposes E such that any essential geometric morphism with codomain E i s automatically locally connected. It turns out that many toposes of inter est are EILC\, including toposes of sheaves on Hausdorff spaces and classi fying toposes of compact groups.\n LOCATION:https://researchseminars.org/talk/Category_Theory/53/ END:VEVENT BEGIN:VEVENT SUMMARY:Jim Otto DTSTART:20230329T230000Z DTEND:20230330T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/54 DESCRIPTION:Title: P Time\, A Bounded Numeric Arrow Category\, and Entailments.\ nby Jim Otto as part of New York City Category Theory Seminar\n\n\nAbstrac t\nWe revisit the characterization of the P Time functions from our McGill thesis.\n\n1. We build on work of L. Roman (89) on primitive recursion an d of A. Cobham (65) and Bellantoni-Cook(92) on P Time.\n\n2. We use base 2 numbers with the digits 1 & 2. Let N be the set of these numbers. We spli t the tapes of a multi-tape Turing machine each into 2 stacks of digits 1 & 2. These are (modulo allowing an odd numberof stacks) the multi-stack ma chines we use to study P Time.\n\n3. Let Num be the category with objects the finite products of N and arrows the functions between these. From its arrow category Num^2 we abstract the doctrine (here a category of small ca tegories with chosen structure) PTime of categories with with finite produ cts\, base 2 numbers\, 2-comprehensions\, flat recursion\, & safe recursio n. Since PTime is a locally finitely presentable category\, it has an init ial category I. Our characterization is that the bottom of the image of I in Num^2 consists of the P Time functions.\n\n4. We can use I (thinking of its arrows as programs) to run multi-stack machines long enough to get P Time.This is the completeness of the characterization.\n\n5. We cut down t he numeric arrow category Num^2\, using Bellantoni-Cook growth & time boun ds on the functions\, to get a bounded numeric arrow category B. B is in t he doctrine PTime. This yields the soundness of the characterization.\n\n6 . For example\, the doctrine of toposes with base 1 numbers\, choice\, & p recisely 2 truth values (which captures much of ZC set theory) likely lack s an initial category\, much as there is an initial ring\, but no initial field.\n\n7. On the other hand\, the L. Roman doctrine PR of categories wi th finite products\, base 1 numbers\, & recursion (that is\, product stabl e natural numbers objects) does have an initial category as it consists of the strong models of a finite set of entailments. And is thus locally fin itely presentable. We sketch the signature graph for these entailments. An d some of these entailments. Similarly (but with more complexity) there ar e entaiments for the doctrine PTime.\n LOCATION:https://researchseminars.org/talk/Category_Theory/54/ END:VEVENT BEGIN:VEVENT SUMMARY:Walter Tholen (York University) DTSTART:20230419T230000Z DTEND:20230420T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/55 DESCRIPTION:Title: What does “smallness” mean in categories of topological space s?\nby Walter Tholen (York University) as part of New York City Catego ry Theory Seminar\n\n\nAbstract\nQuillen’s notion of small object and th e Gabriel-Ulmer notion of finitely presentable or generated object are fun damental in homotopy theory and categorical algebra. Do these notions alwa ys lead to rather uninteresting classes of objects in categories of topolo gical spaces\, such as the class of finite discrete spaces\, or just the e mpty space \, as the examples and remarks in the existing literature may s uggest?\n\nIn this talk we will demonstrate that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces\, such as those of T_i-spaces (i= 0\ , 1\, 2)\, can be quite challenging and may lead to unexpected surprises. In fact\, we will show that there are significant differences in this rega rd even amongst the categories defined by the standard separation conditio ns\, with the T1-separation condition standing out. The findings about the se specific categories lead us to insights also when considering rather ar bitrary full reflective subcategories of Top.\n\n(Based on joint work with J. Adamek\, M. Husek\, and J. Rosicky.)\n LOCATION:https://researchseminars.org/talk/Category_Theory/55/ END:VEVENT BEGIN:VEVENT SUMMARY:Dusko Pavlovic (University of Hawai‘i at Mānoa) DTSTART:20230426T230000Z DTEND:20230427T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/56 DESCRIPTION:Title: Program-closed categories.\nby Dusko Pavlovic (University of Hawai‘i at Mānoa) as part of New York City Category Theory Seminar\n\n\ nAbstract\nLet CC be a symmetric monoidal category with a comonoid on ever y object. Let CC* be the cartesian subcategory with the same objects and j ust the comonoid homomorphisms. A *programming language* is a well-ordered object P with a *program closure*: a family of X-natural surjections\n\nC C(XA\,B) <<--run_X-- CC*(X\,P)\n\none for every pair A\,B. In this talk\, I will sketch a proof that program closure is a property: Any two programm ing languages are isomorphic along run-preserving morphisms. The result co unters Kleene's interpretation of the Church-Turing Thesis\, which has bee n formalized categorically as the suggestion that computability is a struc ture\, like a group presentation\, and not a property\, like completeness. We prove that it is like completeness. The draft of a book on categorical computability is available from the web site dusko.org.\n LOCATION:https://researchseminars.org/talk/Category_Theory/56/ END:VEVENT BEGIN:VEVENT SUMMARY:Arthur Parzygnat (Nagoya University.) DTSTART:20230517T230000Z DTEND:20230518T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/58 DESCRIPTION:Title: Inferring the past and using category theory to define retrodicti on.\nby Arthur Parzygnat (Nagoya University.) as part of New York City Category Theory Seminar\n\n\nAbstract\nClassical retrodiction is the act of inferring the past based on knowledge of the present. The primary examp le is given by Bayes' rule P(y|x) P(x) = P(x|y) P(y)\, where we use prior information\, conditional probabilities\, and new evidence to update our b elief of the state of some system. The question of how to extend this idea to quantum systems has been debated for many years. In this talk\, I will lay down precise axioms for (classical and quantum) retrodiction using ca tegory theory. Among a variety of proposals for quantum retrodiction used in settings such as thermodynamics and the black hole information paradox\ , only one satisfies these categorical axioms. Towards the end of my talk\ , I will state what I believe is the main open question for retrodiction\, formalized precisely for the first time. This work is based on the prepri nt https://arxiv.org/abs/2210.13531 and is joint work with Francesco Busce mi.\n LOCATION:https://researchseminars.org/talk/Category_Theory/58/ END:VEVENT BEGIN:VEVENT SUMMARY:Tomáš Gonda (University of Innsbruck) DTSTART:20230927T230000Z DTEND:20230928T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/59 DESCRIPTION:Title: A Framework for Universality in Physics\, Computer Science\, and Beyond.\nby Tomáš Gonda (University of Innsbruck) as part of New Yor k City Category Theory Seminar\n\n\nAbstract\nTuring machines and spin mod els share a notion of universality according to which some simulate all ot hers. We set up a categorical framework for universality which includes as instances universal Turing machines\, universal spin models\, NP complete ness\, top of a preorder\, denseness of a subset\, and others. By identify ing necessary conditions for universality\, we show that universal spin mo dels cannot be finite. We also characterize when universality can be disti nguished from a trivial one and use it to show that universal Turing machi nes are non-trivial in this sense. We leverage a Fixed Point Theorem inspi red by a result of Lawvere to establish that universality and negation giv e rise to unreachability (such as uncomputability). As such\, this work se ts the basis for a unified approach to universality and invites the study of further examples within the framework.\n\nTALK AT 5PM. Not 7PM. NOTE SP ECIAL TIME!!!\n LOCATION:https://researchseminars.org/talk/Category_Theory/59/ END:VEVENT BEGIN:VEVENT SUMMARY:Thiago Alexandre (University of São Paulo (Brazil)) DTSTART:20231011T230000Z DTEND:20231012T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/60 DESCRIPTION:Title: Internal homotopy theories\nby Thiago Alexandre (University o f São Paulo (Brazil)) as part of New York City Category Theory Seminar\n\ n\nAbstract\nThe idea of 'Homotopy theories' was introduced by Heller in h is seminal paper from 1988. Two years later\, Grothendieck discovered the theory of derivators (1990)\, exposed in his late manuscript Les Dérivate urs\, and developed further by several authors. Essentially\, there are no significant differences between Heller's homotopy theories and Grothendie ck's derivators. They are tautologically the same 2-categorical yoga. Howe ver\, they come from distinct motivations. For Heller\, derivators should be a definitive answer to the question "What is a homotopy theory?"\, whil e for Grothendieck\, who was strongly inspired by topos cohomology\, the f irst main motivation for derivators was to surpass some technical deficien cies that appeared in the theory of triangulated categories. Indeed\, Grot hendieck designed the axioms of derivators in light of a certain 2-functor ial construction\, which associates the corresponding (abelian) derived ca tegory to each topos\, and more importantly\, inverse and direct cohomolog ical images to each geometric morphism. It was from this 2-functorial cons truction\, from where topos cohomology arises\, that Grothendieck discover ed the axioms of derivators\, which are surprisingly the same as Heller's homotopy theories. Nowadays\, it is commonly accepted that a homotopy theo ry is a quasi-category\, and they can all be presented by a localizer (M\, W)\, i.e.\, a couple composed by a category M and a class of arrows in W. This point of view is not so far from Heller\, since pre-derivators\, quas i-categories\, and localizers\, are essentially equivalent as an answer to the question "What is a homotopy theory?". In my talk\, I will expose the se subjects in more detail\, and I am also going to explore how to interna lize a homotopy theory in an arbitrary (Grothendieck) topos\, a problem wh ich strongly relates formal logic and homotopical algebra.\n LOCATION:https://researchseminars.org/talk/Category_Theory/60/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Shulman (University of San Diego) DTSTART:20231018T230000Z DTEND:20231019T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/61 DESCRIPTION:Title: The derivator of setoids\nby Michael Shulman (University of S an Diego) as part of New York City Category Theory Seminar\n\n\nAbstract\n The question of "what is a homotopy theory" or "what is a higher category" is already interesting in classical mathematics\, but in constructive mat hematics (such as the internal logic of a topos) it becomes even more subt le. In particular\, existing constructive attempts to formulate a homotop y theory of spaces (infinity-groupoids) have the curious property that the ir "0-truncated objects" are more general than ordinary sets\, being inste ad some kind of "free exact completion" of the category of sets (a.k.a. "s etoids"). It is at present unclear whether this is a necessary feature of a constructive homotopy theory or whether it can be avoided somehow. One way to find some evidence about this question is to use the "derivators" of Heller\, Franke\, and Grothendieck\, as they give us access to higher h omotopical structure without depending on a preconcieved notion of what su ch a thing should be. It turns out that constructively\, the free exact c ompletion of the category of sets naturally forms a derivator that has a u niversal property analogous to the classical category of sets and to the c lassical homotopy theory of spaces: it is the "free cocompletion of a poin t" in a certain universe. This suggests that either setoids are an unavoi dable aspect of constructive homotopy theory\, or more radical modificatio ns to the notion of homotopy theory are needed.\n LOCATION:https://researchseminars.org/talk/Category_Theory/61/ END:VEVENT BEGIN:VEVENT SUMMARY:Emilio Minichiello (CUNY Graduate Center) DTSTART:20231025T230000Z DTEND:20231026T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/62 DESCRIPTION:Title: A Mathematical Model of Package Management Systems.\nby Emili o Minichiello (CUNY Graduate Center) as part of New York City Category The ory Seminar\n\n\nAbstract\nAbstract: In this talk\, I will review some rec ent joint work with Gershom Bazerman and Raymond Puzio. The motivation is simple: provide a mathematical model of package management systems\, such as the Hackage package respository for Haskell\, or Homebrew for Mac users . We introduce Dependency Structures with Choice (DSC) which are sets equi pped with a collection of possible dependency sets for every element and s atisfying some simple conditions motivated from real life use cases. We de fine a notion of morphism of DSCs\, and prove that the resulting category of DSCs is equivalent to the category of antimatroids\, which are mathemat ical structures found in combinatorics and computer science. We analyze th is category\, proving that it is finitely complete\, has coproducts and an initial object\, but does not have all coequalizers. Further\, we constru ct a functor from a category of DSCs equipped with a certain subclass of m orphisms to the opposite of the category of finite distributive lattices\, making use of a simple finite characterization of the Bruns-Lakser comple tion.\n LOCATION:https://researchseminars.org/talk/Category_Theory/62/ END:VEVENT BEGIN:VEVENT SUMMARY:Larry Moss (Indiana University\, Bloomington) DTSTART:20231109T000000Z DTEND:20231109T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/63 DESCRIPTION:Title: On Kripke\, Vietoris\, and Hausdorff Polynomial Functors.\nby Larry Moss (Indiana University\, Bloomington) as part of New York City Ca tegory Theory Seminar\n\n\nAbstract\nThe Vietoris space of compact subsets of a given Hausdorff space yields an endofunctor V on the category of Hau sdorff spaces. Vietoris polynomial endofunctors on that category are built from V\, the identity and constant functors by forming products\, coprodu cts and compositions. These functors are known to have terminal coalgebras and we deduce that they also have initial algebras. We present an analogo us class of endofunctors on the category of extended metric spaces\, using in lieu of V the Hausdorff functor H. We prove that the ensuing Hausdorff polynomial functors have terminal coalgebras and initial algebras. Wherea s the canonical constructions of terminal coalgebras for Vietoris polynomi al functors takes omega steps\, one needs \\omega + \\omega steps in gener al for Hausdorff ones. We also give a new proof that the closed set functo r on metric spaces has no fixed points.\n LOCATION:https://researchseminars.org/talk/Category_Theory/63/ END:VEVENT BEGIN:VEVENT SUMMARY:Pedro Sota (CANCELLED) DTSTART:20231123T000000Z DTEND:20231123T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/64 DESCRIPTION:Title: CANCELLED\nby Pedro Sota (CANCELLED) as part of New York City Category Theory Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/Category_Theory/64/ END:VEVENT BEGIN:VEVENT SUMMARY:Charlotte Aten (University of Denver) DTSTART:20231130T000000Z DTEND:20231130T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/65 DESCRIPTION:Title: A categorical semantics for neural networks\nby Charlotte Ate n (University of Denver) as part of New York City Category Theory Seminar\ n\n\nAbstract\nIn recent work on discrete neural networks\, I considered s uch networks whose activation functions are polymorphisms of finite\, disc rete relational structures. The general framework I provided was not entir ely categorical in nature but did provide a steppingstone to a categorical treatment of neural nets which are definitionally incapable of overfittin g. In this talk I will outline how to view neural nets as categories of fu nctors from certain multicategories to a target multicategory. Moreover\, I will show that the results of my PhD thesis allow one to systematically define polymorphic learning algorithms for such neural nets in a manner ap plicable to any reasonable (read: functorial) finite data structure.\n LOCATION:https://researchseminars.org/talk/Category_Theory/65/ END:VEVENT BEGIN:VEVENT SUMMARY:Juan Orendain (Case Western Univeristy) DTSTART:20240508T230000Z DTEND:20240509T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/66 DESCRIPTION:Title: Canonical squares in fully faithful and absolutely dense equipmen ts.\nby Juan Orendain (Case Western Univeristy) as part of New York Ci ty Category Theory Seminar\n\n\nAbstract\nZOOM TALK. \nAbstract: Equipment s are categorical structures of dimension 2 having two separate types of 1 -arrows -vertical and horizontal- and supporting restriction and extension of horizontal arrows along vertical ones. Equipments were defined by Wood in [W] as 2-functors satisfying certain conditions\, but can also be unde rstood as double categories satisfying a fibrancy condition as in [Sh]. In the zoo of 2-dimensional categorical structures\, equipments nicely fit i n between 2-categories and double categories\, and are generally considere d as the 2-dimensional categorical structures where synthetic category the ory is done\, and in some cases\, where monoidal bicategories are more nat urally defined.\n\nIn a previous talk in the seminar\, I discussed the pro blem of lifting a 2-category into a double category along a given category of vertical arrows\, and how this problem allows us to define a notion of length on double categories. The length of a double category is a number that roughly measures the amount of work one needs to do to reconstruct th e double category from a bicategory along its set of vertical arrows.\n\nI n this talk I will review the length of double categories\, and I will dis cuss two recent developments in the theory: In the paper [OM] a method for constructing different double categories from a given bicategory is prese nted. I will explain how this construction works. One of the main ingredie nts of the construction are so-called canonical squares. In the preprint [ O] it is proven that in certain classes of equipments -fully faithful and absolutely dense- every square that can be canonical is indeed canonical. I will explain how from this\, it can be concluded that fully faithful and absolutely dense equipments are of length 1\, and so they can be 'easily' reconstructed from their horizontal bicategories.\n\nReferences:\n[O] Len gth of fully faithful framed bicategories. arXiv:2402.16296.\n\n[OM] J. Or endain\, R. Maldonado-Herrera\, Internalizations of decorated bicategories via π-indexings. To appear in Applied Categorical Structures. arXiv:2310 .18673.\n\n[W] R. K. Wood\, Abstract Proarrows I\, Cahiers de topologie et géométrie différentielle 23 3 (1982) 279-290.\n\n[Sh] M. Shulman\, Fra med bicategories and monoidal fibrations. Theory and Applications of Categ ories\, Vol. 20\, No. 18\, 2008\, pp. 650–738.\n\nZoom Talk.\n LOCATION:https://researchseminars.org/talk/Category_Theory/66/ END:VEVENT BEGIN:VEVENT SUMMARY:Raymond Puzio DTSTART:20240515T230000Z DTEND:20240516T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/67 DESCRIPTION:Title: Uniqueness of Classical Retrodiction\nby Raymond Puzio as par t of New York City Category Theory Seminar\n\n\nAbstract\nIN PERSON TALK.\ nAbstract: In previous talks at this Category seminar and at the Topology\ , Geometry and Physics seminar\, Arthur Parzygnat showed how Bayesian inve rsion and its generalization to quantum mechanics may be interpreted as a functor on a suitable category of states which satisfies certain axioms. S uch a functor is called a retrodiction and Parzygnat and collaborators con jectured that retrodiction is unique. In this talk\, I will present a proo f of this conjecture for the special case of classical probability theory on finite state spaces.\n\nIn this special case\, the category in question has non-degenerate probability distributions on finite sets as its object s and stochastic matrices as its morphisms. After preliminary definitions and lemmas\, the proof proceeds in three main steps.\n\nIn the first step\ , we focus on certain groups of automorphisms of certain objects. As a con sequence of the axioms\, it follows that these groups are preserved under any retrodiction functor and that the restriction of the functor to such a group is a certain kind of group automorphism. Since this group is isomor phic to a Lie group\, it is easy to prove that the restriction of a retrod iction to such a group must equal Bayesian inversion if we assume continui ty. If we do not make that assumption\, we need to work harder and derive continuity "from scratch" starting from the positivity condition in the de finition of stochastic matrix.\n\nIn the second step\, we broaden our atte ntion to the full automorphism groups of objects of our category correspon ding to uniform distributions. We show that these groups are generated by the union of the subgroup consisting of permutation matrices and the subgr oup considered in the first step. From this fact\, it follows that the res triction of a retrodiction to this larger group must equal Bayesian invers ion.\n\nIn the third step\, we finally consider all the objects and morphi sms of our category. As a consequence of what we have shown in the first t wo steps and some preliminary lemmas\, it follows that retrodiction is giv en by matrix conjugation. Furthermore\, Bayesian inversion is the special case where the conjugating matrices are diagonal matrices. Because the hom sets of our category are convex polytopes and a retrodiction functor is a continuous bijection of such sets\, a retodiction must map polytope faces to faces. By an algebraic argument\, this fact implies that the conjugati ng matrices are diagonal\, answering the conjecture in the affirmative.\n\ nIN PERSON TALK.\n LOCATION:https://researchseminars.org/talk/Category_Theory/67/ END:VEVENT BEGIN:VEVENT SUMMARY:Emilio Minichiello (The CUNY Graduate Center) DTSTART:20240522T230000Z DTEND:20240523T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/68 DESCRIPTION:Title: Presenting Profunctors\nby Emilio Minichiello (The CUNY Gradu ate Center) as part of New York City Category Theory Seminar\n\n\nAbstract \nIN PERSON TALK.\nAbstract: In categorical database theory\, profunctors are ubiquitous. For example\, they are used to define schemas in the algeb raic data model. However\, they can also be used to query and migrate data . In this talk\, we will discuss an interesting phenomenon that arises whe n trying to model profunctors in a computer. We will introduce two notions of profunctor presentations: the UnCurried and Curried presentations. The y are modeled on thinking of profunctors as functors P: C^op x D -> Set an d as functors P: C^op -> Set^D\, respectively. Semantically of course\, th ese are equivalent\, but their syntactic properties are quite different. T he UnCurried presentations are more intuitive and easier to work with\, bu t they carry a fatal flaw: there does not exist a semantics-preserving com position operation of UnCurried presentations that also preserves finitene ss. Therefore we introduce the Curried presentations and show that they re medy this flaw. In the process\, we characterize which UnCurried Presentat ions can be made Curried\, and discuss some applications. This talk will b e based off of this recent preprint which is joint work with Gabriel Goren Roig and Joshua Meyers.\n\nIN PERSON TALK.\n LOCATION:https://researchseminars.org/talk/Category_Theory/68/ END:VEVENT BEGIN:VEVENT SUMMARY:Samuel Mimram (École Polytechnique) DTSTART:20240529T230000Z DTEND:20240530T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/69 DESCRIPTION:by Samuel Mimram (École Polytechnique) as part of New York Ci ty Category Theory Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/Category_Theory/69/ END:VEVENT BEGIN:VEVENT SUMMARY:Jake Araujo-Simon (Cornell Tech---In-Person) DTSTART:20240918T230000Z DTEND:20240919T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/70 DESCRIPTION:Title: Categorifying the Volterra series: towards a compositional theory of nonlinear signal processing.\nby Jake Araujo-Simon (Cornell Tech-- -In-Person) as part of New York City Category Theory Seminar\n\n\nAbstract \nAbstract: The Volterra series is a model of nonlinear behavior that exte nds the convolutional representation of linear and time-invariant systems to the nonlinear regime. Though well-known and applied in electrical\, mec hanical\, biomedical\, and audio engineering\, its abstract and especially compositional properties have been less studied. In this talk\, we presen t an approach to categorifying the Volterra series\, in which a Volterra s eries is defined as a functor on a category of signals and linear maps\, a morphism between Volterra series is a lens map and natural transformation \, and together\, Volterra series and their morphisms assemble into a cate gory\, which we call Volt. We study three monoidal structures on Volt\, an d outline connections of our work to the field of time-frequency analysis. We also include an audio demo.\n\nPaper link: https://arxiv.org/abs/2308. 07229\n LOCATION:https://researchseminars.org/talk/Category_Theory/70/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam McCrosson (Montana State University---Zoom Talk) DTSTART:20241009T230000Z DTEND:20241010T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/71 DESCRIPTION:Title: Exodromy.\nby Sam McCrosson (Montana State University---Zoom Talk) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbst ract: A favorite result of first semester algebraic topology is the “mon odromy theorem\,” which states that for a suitable topological space X\, there is a triple equivalence between the categories of covering spaces o f X\, sets with an action from the fundamental group of X\, and locally co nstant sheaves on X. This result has recently been upgraded by MacPherson and others to a stratified setting\, where the underlying space may be car ved into a poset of subspaces. In this talk\, we’ll look at the main ing redients of the so-called “exodromy theorem\,” reviewing stratified sp aces and developing “constructible sheaves” and the “exit-path categ ory” along the way.\n LOCATION:https://researchseminars.org/talk/Category_Theory/71/ END:VEVENT BEGIN:VEVENT SUMMARY:Bruno Gavranović (Symbolica AI--- Zoom Talk - Special Time) DTSTART:20241030T180000Z DTEND:20241030T200000Z DTSTAMP:20260422T214840Z UID:Category_Theory/72 DESCRIPTION:Title: Categorical Deep Learning: An Algebraic Theory of Architectures-- -NOTE SPECIAL TIME.\nby Bruno Gavranović (Symbolica AI--- Zoom Talk - Special Time) as part of New York City Category Theory Seminar\n\n\nAbstr act\nWe present our position on the elusive quest for a general-purpose fr amework for specifying and studying deep learning architectures. Our opini on is that the key attempts made so far lack a coherent bridge between spe cifying constraints which models must satisfy and specifying their impleme ntations. Focusing on building such a bridge\, we propose to apply categor y theory— precisely\, the universal algebra of monads valued in a 2-cate gory of parametric maps—as a single theory elegantly subsuming both of t hese flavours of neural network design. To defend our position\, we show h ow this theory recovers constraints induced by geometric deep learning\, a s well as implementations of many architectures drawn from the diverse lan dscape of neural networks\, such as RNNs. We also illustrate how the theor y naturally encodes many standard constructs in computer science and autom ata theory.\n LOCATION:https://researchseminars.org/talk/Category_Theory/72/ END:VEVENT BEGIN:VEVENT SUMMARY:David Jaz Myers (Symbolica AI... 2PM TALK) DTSTART:20241107T000000Z DTEND:20241107T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/73 DESCRIPTION:Title: The Para and Kleisli constructions as wreath products.\nby Da vid Jaz Myers (Symbolica AI... 2PM TALK) as part of New York City Category Theory Seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/Category_Theory/73/ END:VEVENT BEGIN:VEVENT SUMMARY:Emilio Minichiello (CUNY CityTech.--- In-Person) DTSTART:20241114T000000Z DTEND:20241114T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/74 DESCRIPTION:Title: Decision Problems on Graphs with Sheaves. IN PERSON\nby Emili o Minichiello (CUNY CityTech.--- In-Person) as part of New York City Categ ory Theory Seminar\n\n\nAbstract\nThis semester I don’t feel like talkin g about my research. Instead I’ll talk about what I’ve learned from re ading the paper "Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves" by Althaus\, Bumpus\, Fairbanks and Rosiak. This pa per is about how we can use sheaf theory to break apart a computational pr oblem\, solve it on small pieces\, and then glue the solutions together to get a global solution to the computational problem. I’ll go through the main ideas of this paper\, using the category of simple graphs with monom orphisms as a main example to showcase their results.\n LOCATION:https://researchseminars.org/talk/Category_Theory/74/ END:VEVENT BEGIN:VEVENT SUMMARY:Noah Chrein (University of Maryland --- In-Person) DTSTART:20240925T230000Z DTEND:20240926T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/75 DESCRIPTION:Title: A formal category theory for oo-T-multicategories.\nby Noah C hrein (University of Maryland --- In-Person) as part of New York City Cate gory Theory Seminar\n\n\nAbstract\nAbstract: We will explore a framework f or oo-T-multicategories. To begin\, we build a schema for multicategories out of the simplex schema and the monoid schema. The multicategory schema\ , D_m\, inherits the structure of a monad from the +1 monad on the monoid schema. Simplicial T-multicategories are monad preserving functors out of the multicategory schema\, [D_m\, T]\, into another monad T. The framework is larger than just [D_m\,T]. A larger structure describes notions of yon eda lemma and fibration. Inner fibrant\, simplicial T-multicategories are oo-T-multicategories. oo-T-multicategories generalize oo-categories and oo -operads: oo-operads are fm-multicategories\, oo-categories are Id-multica tegories.\n\nWe use this framework to study oo-fc-multicategories\, or "oo - virtual double categories". In general\, under various assumptions on T (which hold for fc)\, the collection of oo-T-multicategories [D_m\, T] ha s other useful structure. One such structure is a join operation. This joi n operation points towards a synthetic definition of op/cartesian cells\, which we hope will model oo-virtual equipments. If there is time\, I will explain the motivation for this study as it relates to ontologies\, meta-t heories and type theories.\n LOCATION:https://researchseminars.org/talk/Category_Theory/75/ END:VEVENT BEGIN:VEVENT SUMMARY:Tim Hosgood (Topos Institute. ZOOM TALK) DTSTART:20241127T190000Z DTEND:20241127T200000Z DTSTAMP:20260422T214840Z UID:Category_Theory/76 DESCRIPTION:Title: Loose simplicial objects.\nby Tim Hosgood (Topos Institute. Z OOM TALK) as part of New York City Category Theory Seminar\n\n\nAbstract\n There are two stories that are historically reasonably unrelated\, but tha t both lead to the same definition of a "loose simplicial object"\, namely (i) the proof that totalisation of a Reedy fibrant cosimplicial simplicia l set computes the homotopy limit (via the Bousfield–Kan map)\, and (ii) the construction of global simplicial resolutions of coherent analytic sh eaves (via Toledo–Tong twisting cochains). In this talk\, we will look a t both of these stories and see what common definition they suggest\, and then examine how this definition might be useful.\n\nThis talk is on (inco mplete) work in progress\, joint with Cheyne Glass.\n LOCATION:https://researchseminars.org/talk/Category_Theory/76/ END:VEVENT BEGIN:VEVENT SUMMARY:Matthew Cushman (CUNY -- IN PERSON TALK) DTSTART:20241212T000000Z DTEND:20241212T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/77 DESCRIPTION:Title: Recollements: gluing and fracture for categories.\nby Matthew Cushman (CUNY -- IN PERSON TALK) as part of New York City Category Theory Seminar\n\nAbstract: TBA\n\nRecollements provide a way of gluing two cate gories together along a left-exact functor\, or conversely of obtaining a semi-orthogonal decomposition of a category by two full subcategories. Eve ry recollement comes with a fracture square\, which in some circumstances can be extended to a hexagon-shaped diagram of fiber sequences. In this ta lk we will discuss concrete examples from topological spaces and graphs be fore moving to smooth manifolds and the recollement that gives rise to dif ferential cohomology theories.\n LOCATION:https://researchseminars.org/talk/Category_Theory/77/ END:VEVENT BEGIN:VEVENT SUMMARY:Charlotte Aten (University of Colorado Boulder--- ZOOM TALK) DTSTART:20241205T000000Z DTEND:20241205T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/78 DESCRIPTION:Title: Invariants of structures.\nby Charlotte Aten (University of C olorado Boulder--- ZOOM TALK) as part of New York City Category Theory Sem inar\n\n\nAbstract\nI will discuss one part of my PhD thesis\, in which I provide a categorification of the notion of a mathematical structure origi nally given by Bourbaki in their set theory textbook. The main result is t hat any isomorphism-invariant property of a finite structure can be checke d by computing the number of isomorphic copies of small substructures it c ontains. A special case of this theorem is the classical result of Hilbert about elementary symmetric polynomials generating the algebra of all symm etric polynomials. I will also discuss how the logical complexity of a pos itive formula controls the size of the small substructures one must count. \n LOCATION:https://researchseminars.org/talk/Category_Theory/78/ END:VEVENT BEGIN:VEVENT SUMMARY:Arnon Avron (Tel Aviv U -- In Person talk) DTSTART:20241121T000000Z DTEND:20241121T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/79 DESCRIPTION:Title: What is the Structure of the Natural numbers?\nby Arnon Avron (Tel Aviv U -- In Person talk) as part of New York City Category Theory S eminar\n\n\nAbstract\nWe present some theorems that show that the notion o f a structure\,\nwhich is central for both Structuralism and category theo ry\, has the very serious\ndefect of having no satisfactory notion of iden tity which can be associated with it.\nWe use those theorems to show that in particular\, there are at least two completely \ndifferent structures t hat are entitled to be taken as `the structure of the natural \nnumbers'\, and any choice between them would arbitrarily favor one of them over\nthe equally legitimate other. This fact refutes (so we believe) the structura list thesis\nthat the natural numbers are just positions (or places) in "t he structure of the natural numbers". Finally\, we argue for the high plau sibility of the identification of the natural numbers with the finite von Neumann ordinals.\n LOCATION:https://researchseminars.org/talk/Category_Theory/79/ END:VEVENT BEGIN:VEVENT SUMMARY:Raymond Puzio (IN PERSON TALK) DTSTART:20250206T000000Z DTEND:20250206T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/80 DESCRIPTION:Title: Gentle Introduction to Synthetic Differential Geometry Part 1 \nby Raymond Puzio (IN PERSON TALK) as part of New York City Category Theo ry Seminar\n\n\nAbstract\nAbstract: Calculations and constructions with in finitesimals make for a handy\, intuitive way of doing calculus and differ ential geometry. They went out of favor in the nineteenth century when the real number system was defined precisely but were rehabilitated a century later when various people such as Robinson\, Lawvere\, and Kock realized that it is nonetheless possible to produce logically rigorous justificatio ns for manipulations involving infinitesimals.\n\nOne such justification e merged from scheme theory and category theory and goes by the name "synthe tic differential geometry". This talk will be an elementary pedagogical in troduction to the subject. We will begin by showing how one can re-interpr et computing with square zero infinitesimals in terms of homomorphisms fro m an algebra of smooth functions to the algebra of dual numbers. Using con cepts from scheme theory\, we will correctly interpret the of correspondin g picture of infinitesimally near points. Moving on\, we will introduce th e axiomatic approach to synthetic differential geometry and describe how p assing to the presheaf topos allows one to treat such infinite-dimensional entities as the totality of mappings between two given manifolds as well- defined spaces. We round off this introduction with a few words about Lie groups\, making precise the idea that the Lie algebra with its commutation relations forms a group presentation in terms of infinitesimal generators and their relations.\n LOCATION:https://researchseminars.org/talk/Category_Theory/80/ END:VEVENT BEGIN:VEVENT SUMMARY:Jacob Szelko (Northeastern University-IN PERSON TALK) DTSTART:20250220T000000Z DTEND:20250220T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/81 DESCRIPTION:Title: An Introduction to Compositional Public Health.\nby Jacob Sze lko (Northeastern University-IN PERSON TALK) as part of New York City Cate gory Theory Seminar\n\n\nAbstract\nCompositional public health is an emerg ing research field that exists to address the complexity in public health responses. The field lies at the intersection of category theory\, epidemi ology\, and engineering and utilizes tools from applied category theory fo r public health applications. This talk will present the motivation of thi s field\, an overview of the mathematics involved in its approaches\, curr ent state of the art\, live demonstrations\, and future research direction s within this developing field.\n LOCATION:https://researchseminars.org/talk/Category_Theory/81/ END:VEVENT BEGIN:VEVENT SUMMARY:Thiago Alexandre (IN PERSON TALK) DTSTART:20250227T000000Z DTEND:20250227T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/82 DESCRIPTION:Title: Topological Derivators\nby Thiago Alexandre (IN PERSON TALK) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbstract: The theory of derivators was originally developed by Grothendieck with hig h inspiration in topos cohomology. In a letter sent to Thomason\, where he explains the main ideas and motivations guiding the formal reasoning of d erivators\, Grothendieck also remarks that those are Morita-invariant. Thi s means that\, if two small categories A and B have equivalent topoi of pr esheaves\, then the categories D(A) and D(B ) are also equivalent for any derivator D. This observation suggests that it may be possible to extend a ny derivator D to the entire 2-category of topoi and geometric morphisms b etween them. Grothendieck conjectures that such an extension is always pos sible and essentially unique. In this case\, every derivator D defined ove r small categories would be coming from a derivator D′ defined over topo i via natural equivalences of categories of the form D(A) = D′(A^)\, whe re A varies through small categories and A^ denotes the category of preshe aves over A. However\, despite these considerations\, a theory of derivato rs over topoi has not yet been developed. To address this gap\, I am curre ntly developing a theory of topological derivators. With this theory\, I a im to provide answers to Grothendieck’s conjecture. Beyond applications in geometry\, the theory of topological derivators has strong connections to first-order categorical logic. In fact\, it lies in the intersection be tween the later and homotopical algebra. In my talk\, I would like to pres ent the theory of topological derivators and some of its main results.\n LOCATION:https://researchseminars.org/talk/Category_Theory/82/ END:VEVENT BEGIN:VEVENT SUMMARY:Grigorios Giotopoulos (NYU Abu Dhabi-ZOOM TALK 10AM-NYCity Time) DTSTART:20250306T000000Z DTEND:20250306T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/83 DESCRIPTION:Title: Thickened smooth sets as a natural setting for Lagrangian field t heory.\nby Grigorios Giotopoulos (NYU Abu Dhabi-ZOOM TALK 10AM-NYCity Time) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbst ract: I will describe how a particularly convenient model for synthetic di fferential geometry -- the sheaf topos of infinitesimally thickened smooth sets -- serves as a powerful context to host classical Lagrangian field t heory. As motivation\, I will recall the textbook description of variation al Lagrangian field theory\, and list desiderata for an ambient category i n which this can rigorously be formalized. I will then explain how sheaves over infinitesimally thickened Cartesian spaces naturally satisfy all the desiderata\, and furthermore allow to rigorously formalize several more f ield theoretic concepts. Time permitting\, I will indicate how the setting naturally generalizes to include the description of fermionic fields\, an d (gauge) fields with internal symmetries. This is based on joint work wit h Hisham Sati and Urs Schreiber.\n LOCATION:https://researchseminars.org/talk/Category_Theory/83/ END:VEVENT BEGIN:VEVENT SUMMARY:Jonathon Funk (Queensborough\, CUNY- IN PERSON TALK) DTSTART:20250312T230000Z DTEND:20250313T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/84 DESCRIPTION:Title: Toposes and Rings\nby Jonathon Funk (Queensborough\, CUNY- IN PERSON TALK) as part of New York City Category Theory Seminar\n\n\nAbstra ct\nI shall attempt to explain a part of a broader program of how topos th eory and operator algebra theory match. Following the example of what I ca ll a supported C*-algebra [1]\, such as a von Neumann algebra\, we extend to an arbitrary ring the notions and constructions introduced there. (Fami liarity with [1] is not necessary for the purposes of this talk.) I have i ncluded an explanation of the Zariski spectrum of a commutative ring in te rms of the constructions I explain. Ultimately\, our goal is to return to C*-algebras in order to generalize [1] to all C*-algebras\, not just the s upported ones.\n\nThis is joint work with Simon Henry.\n\n[1] J. Funk\, To poses and C*-algebras\, preprint\, March 2024.\n LOCATION:https://researchseminars.org/talk/Category_Theory/84/ END:VEVENT BEGIN:VEVENT SUMMARY:Emilio Minichiello (CUNY CityTech - IN PERSON TALK) DTSTART:20250409T230000Z DTEND:20250410T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/85 DESCRIPTION:Title: Structured Decomposition Categories.\nby Emilio Minichiello ( CUNY CityTech - IN PERSON TALK) as part of New York City Category Theory S eminar\n\n\nAbstract\nAbstract: In this talk I’ll report on some new wor k\, joint with Ben Bumpus\, Zoltan Kocsis and Jade Master. The idea here i s to come up with a categorical framework to talk about decompositions. In graph theory\, there are all kinds of ways of decomposing graphs\, the mo st important being tree decompositions. This is a way to decompose a graph into pieces in such a way that if you squint at it\, it looks like a tree . By looking at the biggest piece and minimizing over all tree decompositi ons\, one obtains treewidth\, the most important graph invariant in algori thmics. In this paper\, we abstract this notion\, coming up with the defin ition of structured decomposition categories. To each such category\, we c an assign to each of its objects a width number. We prove that this number is monotone under monomorphisms\, and come up with an appropriate definit ion of structured decomposition functor such that we get a relationship be tween widths. We construct several examples of structured decomposition ca tegories\, whose widths coincide with several important examples from the literature.\n LOCATION:https://researchseminars.org/talk/Category_Theory/85/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrei Rodin (University of Lorraine - IN PERSON TALK) DTSTART:20250423T230000Z DTEND:20250424T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/86 DESCRIPTION:Title: The concept of mathematical structure according to Voevodsky. \nby Andrei Rodin (University of Lorraine - IN PERSON TALK) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbstract: In our email e xchange dating back to 2016 Vladimir Voevodsky suggested an original conce ption of mathematical structure\, which was motivated\, on the one hand\, by his work in the Homotopy Type theory and\, on the other hand\, by his r eading of Proclus’ commentary on Euclid’s definition of plane angle (D ef. 1.8. of the Elements). In my talk I present Vladimir’s conception of mathematical structure\, compare it with standard conceptions\, and discu ss some questions asked by Vladimir during the same exchange. The talk is based on this paper: arXiv:2409.02935\n LOCATION:https://researchseminars.org/talk/Category_Theory/86/ END:VEVENT BEGIN:VEVENT SUMMARY:Sophie d'Espalungue. (ZOOM TALK Note Special Time) DTSTART:20250319T230000Z DTEND:20250320T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/88 DESCRIPTION:Title: Building All of Mathematics Without Axioms: An n-Categorical Mani festo.\nby Sophie d'Espalungue. (ZOOM TALK Note Special Time) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbstract: The form alization of mathematical language traditionally relies on undefined terms - such as Set\, Type\, universes - whose properties are specified by axio ms and inference rules. In this talk\, I present an alternative approach i n which mathematical language is entirely built from definitions. At its c ore are n-category constructors - an internal alternative to typing judgme nts - denoted as (X : Cat_n) for a variable X\, which are inductively assi gned a truth value - a meaning. Defining an n-category here consists of co nstructing an element (a proof) of the corresponding truth value. To give meaning to these constructors\, (n-1)-categories and (n-1)-functors are in ductively organised as an n-category\, resulting in a graded structure of nested n-categories (Cat_{n-1} : Cat_n). By treating each mathematical obj ect as an element of another object\, this framework offers a natural and expressive language for higher category theory\, set theory\, and logic\, all with vast generalisation potential. I will discuss key consequences of this approach\, including its implications for fundamental notions such a s sameness\, size\, and ∞-categories\, as well as its connexions to homo topy type theory.\n LOCATION:https://researchseminars.org/talk/Category_Theory/88/ END:VEVENT BEGIN:VEVENT SUMMARY:Hannah Aizenman (The Graduate Center\, CUNY) DTSTART:20250326T230000Z DTEND:20250327T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/89 DESCRIPTION:Title: Topologically Equivalent Artist Model.\nby Hannah Aizenman (T he Graduate Center\, CUNY) as part of New York City Category Theory Semina r\n\n\nAbstract\nThe contract data visualization tools make with their use rs is that a chart is a faithful and accurate visual representation of the numbers it is made from. Motivated by wanting to make better tools\, we p ropose a methodology for fully specifying arbitrary data to visualization mappings in a manner that easily translates to code. We propose that fiber bundles provide a uniform interface for describing a variety of underlyin g data - tables\, images\, networks\, etc. - in a manner that independentl y encodes the mathematical structure of the topology and the fields of the dataset. Modeling the data structures that store the datasets as sheaves provides a method for specifying visualization methods that are designed t o work regardless of how the dataset is stored - whether the data is on di sk\, distributed\, or on demand. Specifying the visualization library comp onents as natural transforms of sheaves means that the constraints that th e component must satisfy to be structure preserving can be specified as th e set of morphisms on the data and graphic sheaves\, including the structu re on the topology and fields of the data. Using category theory to formal ly express how visual elements are constructed means we can translate thos e expectations into code\, which can then be used to enforce the expectati on that a visualization tool is faithfully translating between numbers and charts.\n LOCATION:https://researchseminars.org/talk/Category_Theory/89/ END:VEVENT BEGIN:VEVENT SUMMARY:Raymond Puzio. (IN PERSON TALK) DTSTART:20250514T230000Z DTEND:20250515T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/90 DESCRIPTION:Title: Gentle Introduction to Synthetic Differential Geometry - Part two .\nby Raymond Puzio. (IN PERSON TALK) as part of New York City Categor y Theory Seminar\n\n\nAbstract\nAbstract: This is part II of "Gentle intro duction to synthetic differential geometry". This talk will be self contai ned and not assume familiarity with part one. Moreover\, the approach and topics covered this time will be sufficiently different that it will be of interest to people who attended part one.\n\nIn part one\, we introduce t he topic in a "bottom-up" manner starting with the simplest instance and b uilding up in complexity. In part two\, we will introduce the subject in a "top-down" manner where we begin by postulating a category with certain p roperties and proceeding from these postulates.\n\nAfter introducing the t opic\, we will turn to Lie groups as an illustrative application. Intuitiv ely\, to make a presentation of a Lie group by generators and relations\, we would want to pick infinitessimal transformations for generators. This is not possible in classical differential geometry so one must instead emp loy various work-arounds. However\, in synthetic differential geometry\, i nfinitessimal generators are well defined and we can build up Lie theory i n a way which accords with naive intuition. In this talk\, we shall go thr ough the first few steps of this development. Then we shall note how the s ynthetic approach is not only more intuitive but more powerful because it allows us to extend the notion of Lie group beyond finite-dimensional mani folds to which the classical approach is limited. We will also say a few w ords about how the some of these infinite-dimensional generalizations are of use in in practical applications.\n LOCATION:https://researchseminars.org/talk/Category_Theory/90/ END:VEVENT BEGIN:VEVENT SUMMARY:Thiago Alexandre. (ZOOM TALK) DTSTART:20250528T200000Z DTEND:20250528T213000Z DTSTAMP:20260422T214840Z UID:Category_Theory/91 DESCRIPTION:Title: Topological Derivators --- Part two.\nby Thiago Alexandre. (Z OOM TALK) as part of New York City Category Theory Seminar\n\n\nAbstract\n Abstract: In this second part\, I begin by recalling the axioms of topolog ical derivators and presenting some elementary consequences of these axiom s. Following this\, I explain how topological derivators can be constructe d by sheafifying homotopy theories. I conclude with the deepest theorem I have obtained in the theory of topological derivators\, which provides str ong evidence for Grothendieck’s conjecture: if a derivator can be extend ed to a topological derivator\, then this extension is essentially unique. \n LOCATION:https://researchseminars.org/talk/Category_Theory/91/ END:VEVENT BEGIN:VEVENT SUMMARY:Sergei Artemov (IN PERSON TALK) DTSTART:20250507T230000Z DTEND:20250508T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/92 DESCRIPTION:Title: Consistency of PA is a serial property\, and it is provable in PA .\nby Sergei Artemov (IN PERSON TALK) as part of New York City Categor y Theory Seminar\n\n\nAbstract\nAbstract: We show that PA consistency is m athematically equivalent to the serial property\, which we call the consis tency scheme ConS(PA):\n\n"n is not a proof of 0=1"\, for n=0\,1\,2\,... . \nThe proof of this equivalence is formalizable in PA. Since the standard consistency formula Con(PA)\n\n"for all x\, x is not a code of a proof of 0=1"\nis strictly stronger than the scheme ConS(PA) in PA\, Goedel's Secon d Incompleteness theorem\, stating that PA |-\\- Con(PA) does not yield th e unprovability of PA consistency. Hence\, the widespread belief that a co nsistent theory cannot establish its consistency has never been justified. \n\nMoreover\, we show that this belief is false. The question of proving PA consistency in PA reduces to proving the scheme ConS(PA) in PA. We buil d on Hilbert's ideas and prove ConS(PA) in PA.\n\nThis talk is a "dress re hearsal" for the speaker's plenary talk at the ASL meeting on May 13\, 202 5.\n\nReference:\nS.Artemov "Serial Properties\, Selector Proofs\, and the Provability of Consistency\," Journal of Logic and Computation\, Volume 3 5\, Issue 3\, April 2025.\nhttps://doi.org/10.1093/logcom/exae034\nPublish ed: 26 July 2024.\n LOCATION:https://researchseminars.org/talk/Category_Theory/92/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam McCrosson (Montana State University...Zoom Talk) DTSTART:20250917T230000Z DTEND:20250918T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/93 DESCRIPTION:Title: TALK CANCELED.\nby Sam McCrosson (Montana State University... Zoom Talk) as part of New York City Category Theory Seminar\n\n\nAbstract\ nMicrolocal sheaf theory has been gaining popularity recently for its appl ications to symplectic geometry. In this talk\, we’ll explore a more top ological application of this subject: how the notion of the microsupport o f a sheaf can be used to tell if a sheaf is “constructible\,” i.e. loc ally constant on strata\, and if so\, what the coarsest stratification is with this property.\n\nVersions of this result can be found as far back as Kashiwara and Schapira’s 1990 book “Sheaves on Manifolds” (which pi oneered the subject of microlocal sheaf theory). Today\, all sorts of gene ralizations are possible using schemes\, \\infty-categories\, and other fa ncy machinery. This talk will focus on a particularly simple case: using 1 -category theory and sheaves of sets on topological spaces to illustrate t he key ideas with concrete examples.\n LOCATION:https://researchseminars.org/talk/Category_Theory/93/ END:VEVENT BEGIN:VEVENT SUMMARY:Amartya Shekhar Dubey (National Institute of Science Education and Research... Zoom Talk 2PM NYC time) DTSTART:20251022T180000Z DTEND:20251022T190000Z DTSTAMP:20260422T214840Z UID:Category_Theory/94 DESCRIPTION:Title: Unital k-restricted Infinity Operads.\nby Amartya Shekhar Dub ey (National Institute of Science Education and Research... Zoom Talk 2PM NYC time) as part of New York City Category Theory Seminar\n\n\nAbstract\n The goal is to understand unital \\infty-operads by their arity restrictio ns. Given k \\geq 1\, we develop a model for unital k-restricted \\infinty -operads\, which are variants of \\infinity-operads with (\\leq k)-arity m orphisms\, as complete Segal presheaves on closed k-dendroidal trees built from corollas with valence \\leq k. Furthermore\, we prove that the restr iction functors from unital \\infty-operads to unital k-restricted \\infty -operads admit fully faithful left and right adjoints by showing that the left and right Kan extensions preserve complete Segal objects. Varying k\, the left and right adjoints give a filtration and a co-filtration for any unital \\infty-operads by k-restricted \\infty-operads. This is joint wor k with Yu Leon Liu.\n LOCATION:https://researchseminars.org/talk/Category_Theory/94/ END:VEVENT BEGIN:VEVENT SUMMARY:Jonathon Funk (Queensborough\, CUNY.. In Person Talk) DTSTART:20251029T230000Z DTEND:20251030T003000Z DTSTAMP:20260422T214840Z UID:Category_Theory/95 DESCRIPTION:Title: More Toposes and C*-algebras.\nby Jonathon Funk (Queensboroug h\, CUNY.. In Person Talk) as part of New York City Category Theory Semina r\n\n\nAbstract\nLet \\( R \\) be a PID and let \\( \\vec{s} = (n_1\, \\do ts\, n_k) \\) be a "shape" vector with \\( k\, n_i \\ge 2 \\).\nWrite \\( X_\\bullet(\\vec{s}) \\) for the simplicial \\( R \\)-module generated by tensors of shape \\( \\vec{s} \\)\,\nwhere the simplicial structure is ind uced by diagonal application of the usual coface and codegeneracy operator s along all axes\;\nset \\( n = \\min(\\vec{s}) - 1 \\).\n
\nFor a nond egenerate \\( n \\)-simplex \\( T \\) (with nondegenerate boundary\, i.e.\ , no face \\( d_i T \\) lies in the degenerate submodule)\nand \\( 0 \\le j \\le n \\)\, let \\( \\Lambda^n_j(T) \\) be the corresponding horn and l et \\( L_\\bullet \\) denote the simplicial submodule\ngenerated by the fa ces of the horn.\n
\n\n\nUsing the standard fact that the set of filler s of \\( \\Lambda^n_j(T) \\) is a torsor under\n
\n\n\\[\nR_{n\,j} = \\ bigcap_{i \\ne j} \\ker\\!\\left(d_i : X_n(\\vec{s}) \\to X_{n-1}(\\vec{s} )\\right)\,\n\\]\n
\nwe show that \\( \\operatorname{rank}_R R_{n\,j} \ \) admits an explicit inclusion–exclusion formula counting the “missin g indices.”\nIn particular\, non-unique fillers occur iff \\( k \\ge n \ \).\n\n


\nUnder the Dold–Kan correspondence\, the relative ho mology\n\\[\nH_n\\left(N(X_\\bullet(\\vec{s}))\, N(L_\\bullet)\\right)\n\\ ]\nis the quotient of \\( R_{n\,j} \\) by the image of the normalized boun dary \\( \\partial_{n+1} \\)\,\nand is nontrivial iff \\( k \\ge n+1 \\)\; when nontrivial its rank equals the missing–index count.\n


\ nI will present the combinatorial formula\, and accompanying code that ver ifies the counts and the two dichotomies.\n LOCATION:https://researchseminars.org/talk/Category_Theory/95/ END:VEVENT BEGIN:VEVENT SUMMARY:Florian Lengyel (CUNY. Zoom Talk) DTSTART:20251106T000000Z DTEND:20251106T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/96 DESCRIPTION:Title: Horn-filling thresholds in simplicial tensor modules.\nby Flo rian Lengyel (CUNY. Zoom Talk) as part of New York City Category Theory Se minar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/Category_Theory/96/ END:VEVENT BEGIN:VEVENT SUMMARY:Emilio Minichiello (CUNY CityTech. In Person Talk) DTSTART:20251113T000000Z DTEND:20251113T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/97 DESCRIPTION:Title: Model Structures for Simplicial Complexes and Graphs.\nby Emi lio Minichiello (CUNY CityTech. In Person Talk) as part of New York City C ategory Theory Seminar\n\n\nAbstract\nI’ll talk about my new paper which constructs model structure s on the category of simplicial complexes and on reflexive graphs which ar e reminiscent of the Thomason model structure on categories. I’ll give s ome background and motivation for studying this and the surrounding questi ons of graph homotopy theory.\n
\n LOCATION:https://researchseminars.org/talk/Category_Theory/97/ END:VEVENT BEGIN:VEVENT SUMMARY:Evan Misshula (CUNY. In-Person Talk) DTSTART:20251204T000000Z DTEND:20251204T013000Z DTSTAMP:20260422T214840Z UID:Category_Theory/98 DESCRIPTION:Title: From Outer Measures to Adjunctions: A Category-Theoretic Recastin g of Caratheodory’s Extension Theorem.\nby Evan Misshula (CUNY. In-P erson Talk) as part of New York City Category Theory Seminar\n\n\nAbstract \nAbstract: The Caratheodory Extension Theorem underpins modern measure th eory by extending a pre-measure on an algebra to a complete measure on the generated -algebra. Traditionally\, the proof proceeds through outer mea sures\, Caratheodory measurability\, and a series of delicate technical ve rifications – a process that reveals the "monsters" lurking in the power set but often obscures the structural simplicity of the result. In this t alk\, I will first outline the classical construction to highlight these s ubtleties\, and then present a category-theoretic reformulation: the exten sion theorem emerges naturally from an adjunction. The categorical perspec tive streamlines the argument\, but seeing both sides illuminates why rigo r is indispensable and how category theory captures the essence of the con struction.\n LOCATION:https://researchseminars.org/talk/Category_Theory/98/ END:VEVENT BEGIN:VEVENT SUMMARY:Sergei Artemov (The CUNY Graduate Center.) DTSTART:20260204T190000Z DTEND:20260204T203000Z DTSTAMP:20260422T214840Z UID:Category_Theory/99 DESCRIPTION:Title: Non-compact proofs\nby Sergei Artemov (The CUNY Graduate Cent er.) as part of New York City Category Theory Seminar\n\nLecture held in R oom 4214.03 in The Graduate Center\, CUNY.\n\nAbstract\nNon-compact proofs are used in mathematics but overlooked in the analysis of (un)provability of consistency. We focus on arithmetical proofs of universal statements ( *) "for any natural number n\, F(n)." A proof of (*) is compact if all pro ofs of F(n)'s for n=0\,1\,2\,... fit into some finitely axiomatized fragme nt of Peano Arithmetic PA. An example of non-compact reasoning is the stan dard proof of Mostowski's 1952 reflexivity theorem: PA proves the consiste ncy of its finite fragments.\n\nIt turns out that Gödel's Second Incomple teness Theorem\, G2\, prohibits compact proofs but does not rule out non-c ompact proofs of PA-consistency formalizable in PA. This explains why and how the recent proofs of PA-consistency in PA work: they essentially forma lize in PA the explicit version of Mostowski's non-compact proof and use G ödelian provable explicit reflection to rid redundant provability operato rs. \n\nThese findings yield a new foundational reading of G2: "the consis tency of PA is not provable within a finite fragment of PA\," complemented with the positive message: "the consistency of PA is provable within the whole PA." This perspective suggests that Gödel's theorem does not repres ent a failure of the system to "know" its own consistency\, but rather a s tructural limit on how that knowledge can be packaged into a single finite string.\n LOCATION:https://researchseminars.org/talk/Category_Theory/99/ END:VEVENT BEGIN:VEVENT SUMMARY:David Ellerman (University of Ljubljana) DTSTART:20260211T190000Z DTEND:20260211T203000Z DTSTAMP:20260422T214840Z UID:Category_Theory/100 DESCRIPTION:Title: A Fundamental Duality in the Exact Sciences: An Introduction to Mathematical Metaphysics.\nby David Ellerman (University of Ljubljana) as part of New York City Category Theory Seminar\n\nLecture held in Room 4214.03 in The Graduate Center\, CUNY.\n\nAbstract\nThere is a fundamental duality that runs through the exact sciences. At the logical level\, it i s the duality between (Boolean) logic of subsets and the logic of partitio ns. The quantitative versions of the dual logics are logical probability t heory and logical information theory. The duality accounts for the duality in the category of Sets and its opposite Sets^{op}. The partial order in the two dual logics gives the two fundamental canonical functions and the claim is that all canonical morphisms in Sets arise from those two morphis ms. In physics\, there is the notion of "definiteness all the way down" wh ich arises in classical physics (Boolean logic of subsets) and dually ther e is the notion of definiteness only down to a certain level and then obje ctive indefiniteness that arises in quantum physics (logic of partitions). \n LOCATION:https://researchseminars.org/talk/Category_Theory/100/ END:VEVENT BEGIN:VEVENT SUMMARY:Ellis Cooper DTSTART:20260218T190000Z DTEND:20260218T203000Z DTSTAMP:20260422T214840Z UID:Category_Theory/101 DESCRIPTION:Title: Algebraic String Diagrams and a Manifest Covariance Theorem\ nby Ellis Cooper as part of New York City Category Theory Seminar\n\nLectu re held in Room 4214.03 in The Graduate Center\, CUNY.\n\nAbstract\nBook t itles such as "Covariant Physics" (Moataz H. Emam) and "Covariant Loop Qua ntum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam T heory" (Carlo Rovelli and Francesca Vidotto) shout the important of covari ance in modern mathematical physics. In categorical terms\, covariance is a family of natural isomorphisms of pairs of functors defined on the group oid of diffeomorphisms in a category of ``domains." "Manifest covariance" is a syntactic concept arising from preservation of covariance of basic co variant tensor calculations combined by composition and product maps. Diff erential geometry and general relativity theory calculations are expressed by algebraic string diagrams\, including the Einstein Curvature Tensor. P hysical nature is categorically natural.\n LOCATION:https://researchseminars.org/talk/Category_Theory/101/ END:VEVENT BEGIN:VEVENT SUMMARY:James Austin Myer (CUNY) DTSTART:20260304T190000Z DTEND:20260304T203000Z DTSTAMP:20260422T214840Z UID:Category_Theory/102 DESCRIPTION:Title: The Bloch Material: a Simplicial Set Whose Homology is the Highe r Chow Groups of Spencer Bloch.\nby James Austin Myer (CUNY) as part o f New York City Category Theory Seminar\n\nLecture held in Room 4214.03 in The Graduate Center\, CUNY.\n\nAbstract\nTo this day\, it is still unknow n whether every variety enjoys a resolution of its singularities. Dennis S ullivan suggests in a 2004 memorial article for René Thom that the obstru ctions constructed to attack Steenrod’s problem could be adapted to hand le the outstanding scenario in positive characteristic. We will discuss a construction en route to the realization of this dream: to each variety\, we prescribe a simplicial set whose homology is the higher Chow groups of (Spencer) Bloch. In particular\, this simplicial set recovers the topology of the analytic space associated to a variety over the complex numbers\n LOCATION:https://researchseminars.org/talk/Category_Theory/102/ END:VEVENT BEGIN:VEVENT SUMMARY:Kristaps John Balodis (University of Calgary) DTSTART:20260311T180000Z DTEND:20260311T193000Z DTSTAMP:20260422T214840Z UID:Category_Theory/103 DESCRIPTION:Title: A geometric introduction to the local Langlands correspondence.< /a>\nby Kristaps John Balodis (University of Calgary) as part of New York City Category Theory Seminar\n\nLecture held in Room 4214.03 in The Gradua te Center\, CUNY.\n\nAbstract\nIn this talk I will provide a non-tradition al introduction to the local Langlands program for p-adic groups by framin g the so-called "Galois side" geometrically. For simplicity\, the primary focus will be on the "unramified" version for GL(n). The ultimate goal wil l be to articulate the p-adic Kazhdan-Lusztig hypothesis with accompanying examples. Along the way\, I will stop to discuss some categorical aspects of the theory.\n LOCATION:https://researchseminars.org/talk/Category_Theory/103/ END:VEVENT BEGIN:VEVENT SUMMARY:Morgan Rogers (University of Sorbonne) DTSTART:20260325T180000Z DTEND:20260325T193000Z DTSTAMP:20260422T214840Z UID:Category_Theory/104 DESCRIPTION:Title: Ultrarings. A categorical approach to unifying boolean and algeb raic descriptive complexity\nby Morgan Rogers (University of Sorbonne) as part of New York City Category Theory Seminar\n\nLecture held in Room 4214.03 in The Graduate Center\, CUNY.\n\nAbstract\nThe presentation of a commutative ring by generators and relations is (at least superficially) s imilar to the presentation of a first order theory in terms of sorts\, fun ction/relation symbols and axioms. More concretely\, we can associate cate gories to rings and to theories:\n\n In commutative algebra we can asso ciate to a ring its category of finitely generated projective modules\, wh ich is a monoidal category with coproducts (over which the monoidal produc t distributes) having several further special properties.\n Classical f irst-order theories are classified by Boolean lextensive categories. That is\, if we take a first order theory 𝕋\, we can associate to it a categ ory ℬ𝕋 (its "syntactic category") with finite limits and finite (pull back-stable\, disjoint) coproducts in which every subobject has a compleme nt\, such that models of 𝕋 in the category of sets correspond to functo rs ℬ𝕋 → Set preserving all that structure.\n\nOf course\, there are many other categories we could have chosen on each side\, but these parti cular constructions admit a mutual generalization which we call ultrarings . In this talk we explain what ultrarings are\, how their presentations ge neralize those of commutative rings and first-order theories\, and how the y connect to the logical approach to complexity theory (descriptive comple xity). We will also sketch how we hope to exploit this connection in the f uture to transport tools from algebraic geometry.\n\nThis talk is based on work with Baptiste Chanus and Damiano Mazza\, https://doi.org/10.4230/LIP Ics.FSCD.2025.13\, with slightly updated definitions.\n LOCATION:https://researchseminars.org/talk/Category_Theory/104/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Lesnick (University at Albany -- SUNY) DTSTART:20260415T180000Z DTEND:20260415T193000Z DTSTAMP:20260422T214840Z UID:Category_Theory/105 DESCRIPTION:Title: Limit Computation Over Posets via Minimal Initial Functors\n by Michael Lesnick (University at Albany -- SUNY) as part of New York City Category Theory Seminar\n\nLecture held in Room 4214.03 in The Graduate C enter\, CUNY.\n\nAbstract\nJoint work with Tamal Dey\, Department of CS\, Purdue University. \n\nIt is well known that limits can be computed by res tricting along an initial functor\, and that this often simplifies limit c omputation. We systematically study the algorithmic implications of this i dea for diagrams indexed by a finite poset. We say an initial functor $F\\ colon C\\to D$ with $C$ small is \\emph{minimal} if the sets of objects an d morphisms of $C$ each have minimum cardinality\, among the sources of al l initial functors with target $D$. For $Q$ a finite poset or $Q\\subseteq \\mathbb{N}^d$ an interval (i.e.\, a convex\, connected subposet)\, we de scribe all minimal initial functors $F\\colon P\\to Q$ and in particular\, show that $F$ is always a subposet inclusion. We give efficient algorithm s to compute a choice of minimal initial functor. In the case that $Q\\sub seteq \\mathbb{N}^d$ is an interval\, we give asymptotically optimal bound s on $|P|$\, the number of relations in $P$ (including identities)\, in te rms of the number $n$ of minima of $Q$: We show that $|P|=\\Theta(n)$ for $d\\leq 3$\, and $|P|=\\Theta(n^2)$ for $d>3$. We apply these results to g ive new bounds on the cost of computing $\\lim G$ for a functor $G \\colon Q\\to \\mathbf{Vec}$ valued in vector spaces. For $Q$ connected\, we also give new bounds on the cost of computing the \\emph{generalized rank} of $G$ (i.e.\, the rank of the induced map $\\lim G\\to \\operatorname{colim} G$)\, which is of interest in topological data analysis.\n LOCATION:https://researchseminars.org/talk/Category_Theory/105/ END:VEVENT BEGIN:VEVENT SUMMARY:Gabriel Goren-Roig (Universidad de Buenos Aires & CONICET.) DTSTART:20260422T180000Z DTEND:20260422T193000Z DTSTAMP:20260422T214840Z UID:Category_Theory/106 DESCRIPTION:Title: Arboreal Adjunctions from Shapes.\nby Gabriel Goren-Roig (Un iversidad de Buenos Aires & CONICET.) as part of New York City Category Th eory Seminar\n\nLecture held in Room 4214.03 in The Graduate Center\, CUNY .\n\nAbstract\nAbstract: Arboreal adjunctions are a categorical tool to st udy logical indistinguishability (i.e. when two models satisfy exactly the same formulas) for logics of relevance in theoretical computer science. I n this talk we present a general method for constructing arboreal adjuncti ons starting from a chosen class of "shapes" in a category of models\, e.g . relational structures. We first define a category of trees whose nodes a re labelled by shapes\, then we observe that these trees can be "realised" into a model by gluing the shapes together. Under suitable conditions\, t he category is arboreal and the realisation functor is comonadic\, thus ob taining the desired arboreal adjunction. The framework amounts to a vast g eneralisation of the arboreal adjunction for Basic Modal Logic and aims to capture a broad range of logical systems. Joint work with Tomáš Jakl (F aculty of Information Technology\, Czech Technical University) and Luca Re ggio (Department of Mathematics\, Università degli Studi di Milano).\n LOCATION:https://researchseminars.org/talk/Category_Theory/106/ END:VEVENT BEGIN:VEVENT SUMMARY:Emilio Minichiello (CUNY CityTech) DTSTART:20260513T180000Z DTEND:20260513T193000Z DTSTAMP:20260422T214840Z UID:Category_Theory/107 DESCRIPTION:Title: Introduction to locally presentable categories\nby Emilio Mi nichiello (CUNY CityTech) as part of New York City Category Theory Seminar \n\nLecture held in Room 4214.03 in The Graduate Center\, CUNY.\n\nAbstrac t\nThis will be an expository talk about locally presentable categories an d surrounding ideas\, corresponding to Appendix C of my notes https://arxi v.org/pdf/2503.20664.\n LOCATION:https://researchseminars.org/talk/Category_Theory/107/ END:VEVENT BEGIN:VEVENT SUMMARY:Cheyne Glass (The Graduate Center CUNY) DTSTART:20260506T180000Z DTEND:20260506T193000Z DTSTAMP:20260422T214840Z UID:Category_Theory/108 DESCRIPTION:Title: Homotopy theoretic least squares models.\nby Cheyne Glass (T he Graduate Center CUNY) as part of New York City Category Theory Seminar\ n\nLecture held in Room 4214.03 in The Graduate Center\, CUNY.\n\nAbstract \nThis talk will explore a potential application of "higher sheaf theory" in data analysis. Given a fixed ambient space\, a presheaf of complexes wi ll be constructed on the poset of finite subsets\, which encodes informati on about least squares solutions (LS) for a choice particular model (ie "y =mx+b"). Given a choice of cover of a data set\, the presheaf of complexes evaluated on the Čech nerve naturally assembles into a Čech-LS total co mplex where 0-cocycles represent choices of parameters on each covering su bset\, and homotopies between the discrepancies on overlaps. For the purpo ses of moving toward a predictive model\, additional evaluation and locali zation procedures (and data) are described which assemble instead into a t wisted complex. With time permitting we will try to work through a toy exa mple with 5 data points.\n LOCATION:https://researchseminars.org/talk/Category_Theory/108/ END:VEVENT BEGIN:VEVENT SUMMARY:Joseph Helfer CANCELED (Simons Center in Stony Brook) DTSTART:20260225T190000Z DTEND:20260225T203000Z DTSTAMP:20260422T214840Z UID:Category_Theory/109 DESCRIPTION:Title: Set-theoretic universes and paradoxes in elementary 2-topoi. \nby Joseph Helfer CANCELED (Simons Center in Stony Brook) as part of New York City Category Theory Seminar\n\nLecture held in Room 4214.03 in The G raduate Center\, CUNY.\n\nAbstract\nAbstract: In 1966\, Lawvere proposed a n axiomatization of the category of categories as a foundational theory of mathematics. Nowadays\, we recognize that the totality of categories shou ld rather be considered a *2-category*. Making this modification to Lawver e's idea results in the notion of elementary 2-topos\, introduced by M. We ber: an axiomatization of the 2-category of categories. This is part of th e more general\, ongoing program of "higher-categorical foundations" which includes M. Makkai's theory of First-Order Logic with Dependent Sorts\, a nd V. Voevodsky's Homotopy Type Theory.\n\nFollowing ideas of M. Makkai an d B. Boshuk\, I have been continuing to develop Weber's theory\, in partic ular adding crucial axioms which were missing from his definition. After g iving some general explanations of these notions\, I will explain two rela ted results about elementary 2-topoi which serve as "tests" of its adequac y as a foundational theory: that is subsumes usual ZF set theory\, and tha t it reproduces a version of the classical set-theoretic paradoxes\, provi ded one is not careful to impose some "size restrictions".\n LOCATION:https://researchseminars.org/talk/Category_Theory/109/ END:VEVENT BEGIN:VEVENT SUMMARY:Kristaps John Balodis (University of Calgary) DTSTART:20260429T180000Z DTEND:20260429T193000Z DTSTAMP:20260422T214840Z UID:Category_Theory/110 DESCRIPTION:Title: A geometric introduction to the local Langlands correspondence\, Part II.\nby Kristaps John Balodis (University of Calgary) as part of New York City Category Theory Seminar\n\nLecture held in Room 4214.03 in The Graduate Center\, CUNY.\n\nAbstract\nAbstract: After reviewing the mai n facts about the representation theory of GL(n) from part 1\, I will desc ribe the geometry of "Vogan varieties"\, with examples\, and introduce equ ivariant perverse sheaves on them. I will then articulate the p-adic Kazhd an-Lusztig hypothesis. From this point we will have several directions whi ch we can go in depending on time and interest. This could include\, discu ssing some categorical aspects of the theory\, talking about other manifes tations of representation theory in the geometry of Vogan varieties\, and the relationship with the theory of real groups.\n LOCATION:https://researchseminars.org/talk/Category_Theory/110/ END:VEVENT BEGIN:VEVENT SUMMARY:Joseph Helfer (Simons Center in Stony Brook) DTSTART:20260520T180000Z DTEND:20260520T193000Z DTSTAMP:20260422T214840Z UID:Category_Theory/111 DESCRIPTION:Title: Set-theoretic universes and paradoxes in elementary 2-topoi. \nby Joseph Helfer (Simons Center in Stony Brook) as part of New York City Category Theory Seminar\n\nLecture held in Room 4214.03 in The Graduate C enter\, CUNY.\n\nAbstract\nAbstract: In 1966\, Lawvere proposed an axiomat ization of the category of categories as a foundational theory of mathemat ics. Nowadays\, we recognize that the totality of categories should rather be considered a *2-category*. Making this modification to Lawvere's idea results in the notion of elementary 2-topos\, introduced by M. Weber: an a xiomatization of the 2-category of categories. This is part of the more ge neral\, ongoing program of "higher-categorical foundations" which includes M. Makkai's theory of First-Order Logic with Dependent Sorts\, and V. Voe vodsky's Homotopy Type Theory.\n\nFollowing ideas of M. Makkai and B. Bosh uk\, I have been continuing to develop Weber's theory\, in particular addi ng crucial axioms which were missing from his definition. After giving som e general explanations of these notions\, I will explain two related resul ts about elementary 2-topoi which serve as "tests" of its adequacy as a fo undational theory: that is subsumes usual ZF set theory\, and that it repr oduces a version of the classical set-theoretic paradoxes\, provided one i s not careful to impose some "size restrictions".\n LOCATION:https://researchseminars.org/talk/Category_Theory/111/ END:VEVENT BEGIN:VEVENT SUMMARY:Joseph Dorfer (Graz University of Technology) DTSTART:20260603T180000Z DTEND:20260603T193000Z DTSTAMP:20260422T214840Z UID:Category_Theory/112 DESCRIPTION:by Joseph Dorfer (Graz University of Technology) as part of Ne w York City Category Theory Seminar\n\nLecture held in Room 4214.03 in The Graduate Center\, CUNY.\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/Category_Theory/112/ END:VEVENT END:VCALENDAR