\ nUsing extensive unpublished materials available via Vladimir Voevodsky’ s memorial webpage (https:// www.math.ias.edu/Voevodsky/)\n I reconstruct Vladimir’s vision of ma thematics and its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;VALUE=DATE-TIME:20201028T230000Z DTEND;VALUE=DATE-TIME:20201029T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20201105T000000Z DTEND;VALUE=DATE-TIME:20201105T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20201112T000000Z DTEND;VALUE=DATE-TIME:20201112T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20201119T000000Z DTEND;VALUE=DATE-TIME:20201119T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20201210T000000Z DTEND;VALUE=DATE-TIME:20201210T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20201203T000000Z DTEND;VALUE=DATE-TIME:20201203T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20201216T180000Z DTEND;VALUE=DATE-TIME:20201216T193000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20210204T000000Z DTEND;VALUE=DATE-TIME:20210204T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20210211T000000Z DTEND;VALUE=DATE-TIME:20210211T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20210218T000000Z DTEND;VALUE=DATE-TIME:20210218T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20210304T000000Z DTEND;VALUE=DATE-TIME:20210304T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20210317T230000Z DTEND;VALUE=DATE-TIME:20210318T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20210414T230000Z DTEND;VALUE=DATE-TIME:20210415T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20210505T230000Z DTEND;VALUE=DATE-TIME:20210506T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20210324T230000Z DTEND;VALUE=DATE-TIME:20210325T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20211006T230000Z DTEND;VALUE=DATE-TIME:20211007T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20211020T230000Z DTEND;VALUE=DATE-TIME:20211021T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20211103T230000Z DTEND;VALUE=DATE-TIME:20211104T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20211118T000000Z DTEND;VALUE=DATE-TIME:20211118T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20211202T000000Z DTEND;VALUE=DATE-TIME:20211202T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20211216T000000Z DTEND;VALUE=DATE-TIME:20211216T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20211223T000000Z DTEND;VALUE=DATE-TIME:20211223T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20220203T000000Z DTEND;VALUE=DATE-TIME:20220203T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20220217T000000Z DTEND;VALUE=DATE-TIME:20220217T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20211209T000000Z DTEND;VALUE=DATE-TIME:20211209T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20220224T000000Z DTEND;VALUE=DATE-TIME:20220224T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20220330T230000Z DTEND;VALUE=DATE-TIME:20220331T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20220316T230000Z DTEND;VALUE=DATE-TIME:20220317T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20220323T230000Z DTEND;VALUE=DATE-TIME:20220324T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20220406T230000Z DTEND;VALUE=DATE-TIME:20220407T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20220413T230000Z DTEND;VALUE=DATE-TIME:20220414T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20220427T230000Z DTEND;VALUE=DATE-TIME:20220428T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20220504T230000Z DTEND;VALUE=DATE-TIME:20220505T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20220907T230000Z DTEND;VALUE=DATE-TIME:20220908T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20220914T230000Z DTEND;VALUE=DATE-TIME:20220915T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20221019T230000Z DTEND;VALUE=DATE-TIME:20221020T003000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20221110T000000Z DTEND;VALUE=DATE-TIME:20221110T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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;VALUE=DATE-TIME:20221124T000000Z DTEND;VALUE=DATE-TIME:20221124T013000Z DTSTAMP;VALUE=DATE-TIME:20240715T164924Z 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

\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