\nUsing extensive unpublished materials available via Vladimir Voevo dsky’s memorial webpage (h ttps://www.math.ias.edu/Voevodsky/)\n I reconstruct Vladimir’s visio n of mathematics and its role in science incuding his original strategy of bridging the gap between the pure and applied mathematics. Finally\, I sh ow a relevance of Univalent Foundations to Vladimir’s unachieved project and speculate about a possible role of Univalent Foundations in science.\ n END:VEVENT BEGIN:VEVENT SUMMARY:Larry Moss (Indiana University) DTSTART;VALUE=DATE-TIME:20201028T230000Z DTEND;VALUE=DATE-TIME:20201029T003000Z DTSTAMP;VALUE=DATE-TIME:20201029T111018Z UID:Category_Theory/5 DESCRIPTION:Title: Coalgebra in Continuous Mathematics.\nby Larry Moss (In diana University) as part of New York City Category Theory Seminar\n\n\nAb stract\nAbstract: A slogan from coalgebra in the 1990's holds that\n\n'dis crete mathematics : algebra :: continuous mathematics : coalgebra'\n\nThe idea is that objects in continuous math\, like real numbers\, are often un derstood via their approximations\, and coalgebra gives tools for understa nding and working with those objects. Some examples of this are Pavlovic a nd Escardo's relation of ordinary differential equations with coinduction\ , and also Freyd's formulation of the unit interval as a final coalgebra. My talk will be an organized survey of several results in this area\, incl uding (1) a new proof of Freyd's Theorem\, with extensions to fractal sets \; (2) other presentations of sets of reals as corecursive algebras and fi nal coalgebras\; (3) a coinductive proof of the correctness of policy iter ation from Markov decision processes\; and (4) final coalgebra presentatio ns of universal Harsanyi type spaces from economics.\n\nThis talk reports on joint work with several groups in the past 5-10 years\, and also some o ngoing work.\n END:VEVENT BEGIN:VEVENT SUMMARY:Luis Scoccola (Michigan State University) DTSTART;VALUE=DATE-TIME:20201105T000000Z DTEND;VALUE=DATE-TIME:20201105T013000Z DTSTAMP;VALUE=DATE-TIME:20201029T111018Z 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 Category Theory Seminar\n\n\nAbstract\nAbstract: In applied homotopy theory and topological data analysis\, procedures use homotopy invariants of spaces to study and classify discrete data\, such as finite metric spac es. To show that such a procedure is robust to noise\, one endows the coll ection 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 wi ll show that a certain type of enriched categories\, which I call locally persistent categories\, provide a natural framework for the study of appro ximate categorical structures\, and in particular\, for the study of metri cs relevant to applied homotopy theory and metric geometry.\n END:VEVENT BEGIN:VEVENT SUMMARY:Noah Chrein (University of Maryland) DTSTART;VALUE=DATE-TIME:20201112T000000Z DTEND;VALUE=DATE-TIME:20201112T013000Z DTSTAMP;VALUE=DATE-TIME:20201029T111018Z UID:Category_Theory/7 DESCRIPTION:Title: Yoneda ontologies.\nby Noah Chrein (University of Maryl and) as part of New York City Category Theory Seminar\n\n\nAbstract\nAbstr act: We will discuss a 2-categorical model of ontology\, and how to view c ertain higher categories as ontologies in this language. We can translate the various Yoneda lemmas associated to higher categories into the languag e of ontology\, and in turn\, discuss what it means for a generic ontology to have a yoneda lemma. These will be the "Yoneda Ontologies".\n END:VEVENT BEGIN:VEVENT SUMMARY:Enrico Ghiorzi (Appalachian State University) DTSTART;VALUE=DATE-TIME:20201119T000000Z DTEND;VALUE=DATE-TIME:20201119T013000Z DTSTAMP;VALUE=DATE-TIME:20201029T111018Z UID:Category_Theory/8 DESCRIPTION:Title: Internal enriched categories.\nby Enrico Ghiorzi (Appal achian State University) as part of New York City Category Theory Seminar\ n\n\nAbstract\nAbstract: Internal categories feature a notion of completen ess which is remarkably well behaved. For example\, the internal adjoint f unctor theorem requires no solution set condition. Indeed\, internal categ ories are intrinsically small\, and thus immune from the size issues commo nly afflicting standard category theory. Unfortuntely\, they are not quite as expressive as we would like: for example\, there is no internal Yoneda lemma. To increase the expressivity of internal category theory\, we defi ne a notion of internal enrichment over an internal monoidal category and develop its theory of completeness. The resulting theory unites the good p roperties of internal categories with the expressivity of enriched categor y theory\, thus providing a powerful framework to work with.\n END:VEVENT BEGIN:VEVENT SUMMARY:Dan Shiebler (Oxford University) DTSTART;VALUE=DATE-TIME:20201210T000000Z DTEND;VALUE=DATE-TIME:20201210T013000Z DTSTAMP;VALUE=DATE-TIME:20201029T111018Z UID:Category_Theory/9 DESCRIPTION:Title: Functorial Manifold Learning and Overlapping Clustering .\nby Dan Shiebler (Oxford University) as part of New York City Category T heory Seminar\n\n\nAbstract\nAbstract: We adapt previous research on funct orial clustering and topological unsupervised learning to develop a functo rial perspective on manifold learning algorithms. Our framework characteri zes a manifold learning algorithm in terms of the loss function that it op timizes\, which allows us to focus on the algorithm's objective rather tha n the details of the learning process. We demonstrate that we can express several state of the art manifold learning algorithms\, including Laplacia n Eigenmaps\, Metric Multidimensional Scaling\, and UMAP\, as functors in this framework. This functorial perspective allows us to reason about the invariances that these algorithms preserve and prove refinement bounds on the kinds of loss functions that any such functor can produce. Finally\, w e experimentally demonstrate how this perspective enables us to derive and analyze novel manifold learning algorithms.\n END:VEVENT BEGIN:VEVENT SUMMARY:Andrew Winkler DTSTART;VALUE=DATE-TIME:20201203T000000Z DTEND;VALUE=DATE-TIME:20201203T013000Z DTSTAMP;VALUE=DATE-TIME:20201029T111018Z UID:Category_Theory/10 DESCRIPTION:Title: Functors as homomorphisms of quivered algebras.\nby And rew Winkler as part of New York City Category Theory Seminar\n\n\nAbstract \nAbstract: A quiver induces a minimalist algebraic structure which is\, n onetheless\, balanced\, associative\, elementwise strongly irreducible\, a nd both left and right quivered\, in a functorial way\; a homomorphism of quivers induces a homomorphism of algebras. Q balanced\, quivered algebra possesses a quiver structure\, but it is not true in general that a homomo rphism for the algebra is also a homomorphism for the quiver. It will be p recisely when it is also a homomorphism for the algebra structure induced by the quiver structure it induces. Such a bihomorphism\, in the special c ase of categories\, (where the associativity property and a composition-in ducing property hold)\, is precisely a functor. This facet of categories\, as possessing two compatible composition structures\, explains in some se nse a bifurcation in the structure of monads.\n END:VEVENT BEGIN:VEVENT SUMMARY:Arthur Parzygnat (IHES) DTSTART;VALUE=DATE-TIME:20201216T180000Z DTEND;VALUE=DATE-TIME:20201216T193000Z DTSTAMP;VALUE=DATE-TIME:20201029T111018Z 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 T heory Seminar\n\n\nAbstract\nAbstract: Entropy appears as a useful concept in a wide variety of academic disciplines. As such\, one would suspect th at category theory would provide a suitable language to encompass all or m ost of these definitions. The Shannon entropy has recently been given a ch aracterization as a certain affine functor by Baez\, Fritz\, and Leinster. This characterization is the only characterization I know of that uses li near assumptions (as opposed to additive\, exponential\, logarithmic\, etc ). Here\, we extend that characterization to include the von Neumann entro py as well as highlight the new categorical structures that arise when try ing to do so. In particular\, we introduce Grothendieck fibrations of conv ex categories\, and we review the notion of a disintegration\, which is a key part of conditional probability and Bayesian statistics and plays a cr ucial role in our characterization theorem. The characterization of Baez\, Fritz\, and Leinster interprets Shannon entropy in terms of the informati on loss associated to a deterministic process\, which is possible since th e entropy difference associated to such a process is always non-negative. This fails 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 END:VEVENT BEGIN:VEVENT SUMMARY:Jason Parker (Brandon University) DTSTART;VALUE=DATE-TIME:20210204T000000Z DTEND;VALUE=DATE-TIME:20210204T013000Z DTSTAMP;VALUE=DATE-TIME:20201029T111018Z UID:Category_Theory/12 DESCRIPTION:Title: Isotropy Groups of Quasi-Equational Theories.\nby Jason Parker (Brandon University) as part of New York City Category Theory Semi nar\n\n\nAbstract\nAbstract: In [2]\, my PhD supervisors (Pieter Hofstra a nd Philip Scott) and I studied the new topos-theoretic phenomenon of isotr opy (as introduced in [1]) in the context of single-sorted algebraic theor ies\, and we gave a logical/syntactic characterization of the isotropy gro up of any such theory\, thereby showing that it encodes a notion of inner automorphism or conjugation for the theory. In the present talk\, I will s ummarize the results of my recent PhD thesis\, in which I build on this ea rlier work by studying the isotropy groups of (multi-sorted) quasi-equatio nal theories (also known as essentially algebraic\, cartesian\, or finite limit theories). In particular\, I will show how to give a logical/syntact ic characterization of the isotropy group of any such theory\, and that it encodes a notion of inner automorphism or conjugation for the theory. I w ill also describe how I have used this characterization to exactly charact erize the ‘inner automorphisms’ for several different examples of quas i-equational theories\, most notably the theory of strict monoidal categor ies and the theory of presheaves valued in a category of models. In partic ular\, the latter example provides a characterization of the (covariant) i sotropy group of a category of set-valued presheaves\, which had been an o pen question in the theory of categorical isotropy.\n\n[1] J. Funk\, P. Ho fstra\, B. Steinberg. Isotropy and crossed toposes. Theory and Application s of Categories 26\, 660-709\, 2012.\n\n[2] P. Hofstra\, J. Parker\, P.J. Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Com puter Science 341\, 201-217\, 2018.\n END:VEVENT END:VCALENDAR