The New York City
Category Theory Seminar
Department of Computer Science
Department of Mathematics
The Graduate Center of The City University of New York
THE TALKS WILL ALL BE DONE ON ZOOM THIS SEMESTER.
THIS WEEKS ZOOM INFORMATION:
Topic: New York City Category Theory Seminar
Time: Wednesdays 07:00 PM Eastern Time (US and Canada)
Join Zoom Meeting
Meeting ID: 811 3939 6554
Usually our talks take place at
365 Fifth Avenue (at 34th Street) map
(Diagonally across from the Empire State Building)
New York, NY 10016-4309
Wednesdays 7:00 - 8:30 PM.
Usually the talks are in
Room 6417 .
Some of the talks are videoed and available here.
Contact N. Yanofsky to
schedule a speaker
or to add a name to the
seminar mailing list.
Speaker: Rick Jardine, University of Western Ontario.
Date and Time:   Wednesday September 16, 2020, 7:00 - 8:30 PM., on Zoom.
Title: Posets, metric spaces, and topological data analysis.
Slides: Available here.
Abstract: Traditional TDA is the analysis of homotopy invariants of systems of spaces V(X) that arise from finite metric spaces X, via distance measures. These spaces can be expressed in terms of posets, which are barycentric subdivisions of the usual Vietoris-Rips complexes V(X). The proofs of stability theorems in TDA are sharpened considerably by direct use of poset techniques.
Expanding the domain of definition to extended pseudo metric spaces enables the construction of a realization functor on diagrams of spaces, which has a right adjoint Y |--> S(Y), called the singular functor. The realization of the Vietoris-Rips system V(X) for an ep-metric space X is the space itself. The counit of the adjunction defines a map \eta: V(X) --> S(X), which is a sectionwise weak equivalence - the proof uses simplicial approximation techniques.
This is the context for the Healy-McInnes UMAP construction, which will be discussed if time permits. UMAP is non-traditional: clusters for UMAP are defined by paths through sequences of neighbour pairs, which can be a highly efficient process in practice.
Speaker: David Ellerman, University of Ljubljana.
Date and Time:   Wednesday September 30, 2020, 7:00 - 8:30 PM., on Zoom.
Title: The Logical Theory of Canonical Maps: The Elements & Distinctions Analysis of the Morphisms, Duality, Canonicity, and Universal Constructions in Sets.
Paper: Available here.
Slides: Available here.
Abstract: Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements & 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 to other Sets-based concrete categories (groups, rings, vector spaces, etc.). Elements and distinctions are the building blocks of the two dual logics, the Boolean logic of subsets and the logic of partitions. The partial orders (inclusion and refinement) in the lattices for the dual logics define morphisms. The thesis is that the maps that are canonical in Sets are the ones that are defined (given the data of the situation) by these two logical partial orders and by the compositions of those maps.
Speaker: Jonathon Funk, Queensborough CUNY.
Date and Time:   Wednesday October 14, 2020, 6:00 - 7:30PM (NOTICE DIFFERENT TIME) on Zoom.
Title: Pseudogroup Torsors.
Abstract: We use sheaf theory to analyze the topos of etale actions on the germ groupoid of a pseudogroup in the sense that we present a site for this topos, which we call the classifying topos of the pseudogroup. Our analysis carries us further into how pseudogroup morphisms and geometric morphisms are related. Ultimately, we shall see that the classifying topos classifies what we call a pseudogroup torsor. In hindsight, we see that pseudogroups form a bicategory of `flat' bimodules.
Joint work with Pieter Hofstra.
Speaker: Andrei V. Rodin, Saint Petersburg State University.
Date and Time:   Wednesday October 21, 2020, 7:00 - 8:30 PM., on Zoom.
Title: Vladimir Voevodsky’s Unachieved Project.
Abstract: Soon after receiving the Fields Medal for his proof of Milnor Conjecture and the related work in the Motivic Theory, Vladimir delivered a series of two public lectures in the Wuhan University (China) titled “What is most important for mathematics in the near future?” where he described the most urgent tasks as follows: 1) to build a computerised version of Bourbaki’s ‘Elements’ and 2) to bridge pure and applied mathematics. The first project resulted into the Univalent foundations of mathematics. The second project remained unachieved in spite of significant time and efforts that Vladimir spent for its realisation. More specifically, during 2007-2009 Vladimir worked on a mathematical theory of Population Dynamics but then abandoned this project and focused on the Univalent Foundations until the sudden end of his life in 2017.
Using extensive unpublished materials available via Vladimir Voevodsky’s memorial webpage (https://www.math.ias.edu/Voevodsky/)
I reconstruct Vladimir’s vision of mathematics and its role in science incuding his original strategy of bridging the gap between the pure and applied mathematics. Finally, I show a relevance of Univalent Foundations to Vladimir’s unachieved project and speculate about a possible role of Univalent Foundations in science.
Speaker: Larry Moss, Indiana University.
Date and Time:   Wednesday October 28, 2020, 7:00 - 8:30 PM., on Zoom.
Title: Coalgebra in Continuous Mathematics.
Abstract: A slogan from coalgebra in the 1990's holds that
'discrete mathematics : algebra :: continuous mathematics : coalgebra'
The idea is that objects in continuous math, like real numbers, are often understood via their approximations,
and coalgebra gives tools for understanding and working with those objects. Some examples of this are
Pavlovic and 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, 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 coalgebras; (3) a coinductive proof of the correctness of policy iteration from Markov decision processes;
and (4) final coalgebra presentations of universal Harsanyi type spaces from economics.
This talk reports on joint work with several groups in the past 5-10 years, and also some ongoing work.
Speaker: Luis Scoccola, Michigan State University.
Date and Time:   Wednesday November 4, 2020, 7:00 - 8:30 PM., on Zoom.
Title: Locally persistent categories and approximate homotopy theory.
Abstract: In applied homotopy theory and topological data analysis, procedures use homotopy invariants of spaces 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 persistent categories, provide a natural framework for the study of approximate categorical structures, and in particular, for the study of metrics relevant to applied homotopy theory and metric geometry.
Speaker: Noah Chrein, University of Maryland.
Date and Time:   Wednesday November 11, 2020, 7:00 - 8:30 PM., on Zoom.
Title: Yoneda ontologies.
Abstract: 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 various Yoneda lemmas associated to higher categories into the language of ontology, and in turn, discuss what it means for a generic ontology to have a yoneda lemma. These will be the "Yoneda Ontologies".
Speaker: Enrico Ghiorzi, Appalachian State University.
Date and Time:   Wednesday November 18, 2020, 7:00 - 8:30 PM., on Zoom.
Title: Internal enriched categories.
Abstract: Internal categories feature a notion of completeness which is remarkably well behaved. For example, the internal adjoint functor theorem requires no solution set condition. Indeed, internal categories are intrinsically small, and thus immune from the size issues commonly 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 define a notion of internal enrichment over an internal monoidal category and develop its theory of completeness. The resulting theory unites the good properties of internal categories with the expressivity of enriched category theory, thus providing a powerful framework to work with.
Speaker: Andrew Winkler.
Date and Time:   Wednesday December 2, 2020, 7:00 - 8:30 PM., on Zoom.
Title: Functors as homomorphisms of quivered algebras.
Abstract: A quiver induces a minimalist algebraic structure which is, nonetheless,
balanced, associative, elementwise strongly irreducible, and 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
homomorphism for the algebra is also a homomorphism for the quiver. It will be precisely 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 possessing
two compatible composition structures, explains in some sense a bifurcation in the structure of monads.
Speaker: Dan Shiebler, Oxford University.
Date and Time:   Wednesday December 9, 2020, 7:00 - 8:30 PM., on Zoom.
Title: Functorial Manifold Learning and Overlapping Clustering.
Abstract: We adapt previous research on functorial clustering and topological unsupervised learning to develop a functorial perspective on manifold learning algorithms. Our framework characterizes a manifold learning algorithm in terms of the loss function that it optimizes, which allows us to focus on the algorithm's objective rather than the details of the learning process. We demonstrate that we can express several state of the art manifold learning algorithms, including Laplacian 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, we experimentally demonstrate how this perspective enables us to derive and analyze novel manifold learning algorithms.
Speaker: Arthur Parzygnat, IHES.
Date and Time:   Wednesday December 16, 2020, 1:00 - 2:30 PM. ***NOTICE THE SPECIAL TIME***, on Zoom.
Title: A functorial characterization of classical and quantum entropies.
Abstract: Entropy appears as a useful concept in a wide variety of academic disciplines. As such, one would suspect that category theory would provide a suitable language to encompass all or most of these definitions. The Shannon entropy has recently been given a characterization as a certain affine functor by Baez, Fritz, and Leinster. This characterization is the only characterization I know of that uses linear assumptions (as opposed to additive, exponential, logarithmic, etc). Here, 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 categories, and we review the notion of a disintegration, which is a key part 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 loss associated to a deterministic process, which is possible since the entropy difference associated to such a process is always non-negative. This fails for quantum entropy, and has important physical consequences.
Paper (and references therein)
Paper (original paper of Baez, Fritz, and Leinster)
Speaker: Jason Parker, Brandon University in Manitoba.
Date and Time:   Wednesday February 3, 2021, 7:00 - 8:30 PM., on Zoom.
Title: Isotropy Groups of Quasi-Equational Theories.
Abstract: In , my PhD supervisors (Pieter Hofstra and Philip Scott) and I studied the new topos-theoretic phenomenon of isotropy (as introduced in ) 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 automorphism or conjugation for the theory. In the present talk, I will summarize 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 theories (also known as essentially algebraic, cartesian, or finite limit theories). In particular, I will show how to give a logical/syntactic characterization of the isotropy group of any such theory, and that it encodes a notion of inner automorphism or conjugation for the theory. I will also describe how I have used this characterization to exactly characterize the ‘inner automorphisms’ for several different examples of quasi-equational theories, most notably the theory of strict monoidal categories and the theory of presheaves valued in a category of models. In particular, the latter example provides a characterization of the (covariant) isotropy group of a category of set-valued presheaves, which had been an open question in the theory of categorical isotropy.
 J. Funk, P. Hofstra, B. Steinberg. Isotropy and crossed toposes. Theory and Applications of Categories 26, 660-709, 2012.
 P. Hofstra, J. Parker, P.J. Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Computer Science 341, 201-217, 2018.
Speaker: Peter Hines, University of York.
Date and Time:   Wednesday February 10, 2021, 7:00 - 8:30 PM., on Zoom.
Title: Shuffling cards as an operad.
Abstract: The 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, statistics, and other areas of mathematics. This talk aims to study natural 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 number of decks of cards.
Far from being 'generalisation for generalisation's sake', the original motivation came from theoretical & practical computer science. The mathematics of card shuffles is commonly used to describe processing in multi-threaded computations. Moving to the infinite case gives a language in which one may talk about potentially non-terminating processes, or servers with an unbounded number of clients, etc.
However, this talk is entirely about algebra & category theory -- just as in the finite case, the mathematics is of interest in its own right, and should be studied as such.
We model shuffles using operads. The intuition behind them of allowing for arbitrary n-ary operations that compose in a hierarchical manner makes them a natural, inevitable choice for describing such processes such as merging multiple packs of cards.
We use very concrete examples, based on endomorphism operads in groupoids of arithmetic operations. The resulting structures are at the same time both simple (i.e. elementary arithmetic operations), and related to deep structures in mathematics and category theory (topologies, tensors, coherence, associahedra, etc.)
We treat this as a feature, not a bug, and use it to describe complex structures in elementary terms. We also aim to give previously unobserved connections between distinct areas of mathematics.
Speaker: Richard Blute, University of Ottawa.
Date and Time:   Wednesday February 17, 2021, 7:00 - 8:30 PM., on Zoom.
Title: Finiteness Spaces, Generalized Polynomial Rings and Topological Groupoids.
Abstract: The category of finiteness spaces was introduced by Thomas Ehrhard as a model of classical linear logic, where a set is equipped with a class of subsets to be thought of as finitary. Morphisms are relations preserving the finitary structure. The notion of finitary subset allows for a sharp analysis of computational structure.
Working with finiteness spaces forces the number of summands in certain calculations to be finite and thus avoid convergence questions. An excellent example of this is how Ribenboim’s theory of generalized power series rings can be naturally interpreted by assigning finiteness monoid structure to his partially ordered monoids. After Ehrhard’s linearization construction is applied, the resulting structures are the rings of Ribenboim’s construction.
There are several possible choices of morphism between finiteness spaces. If one takes structure-preserving partial functions, the resulting category is complete, cocomplete and symmetric monoidal closed. Using partial functions, we are able to model topological groupoids, when we consider composition as a partial function. We can associate to any hemicompact etale Hausdorff groupoid a complete convolution ring. This is in particular the case for the infinite paths groupoid associated to any countable row-finite directed graph.
Speaker: Joshua Sussan, Medgar Evers, CUNY.
Date and Time:   Wednesday March 3, 2021, 7:00 - 8:30 PM., on Zoom.
Title: Categorification and quantum topology.
Abstract: The Jones polynomial of a link could be defined through the representation theory of quantum sl(2). It leads to a 3-manifold invariant and 2+1 dimensional TQFT. In the mid 1990s, Crane and Frenkel outlined the categorification program with the aim of constructing a 3+1 dimensional TQFT by upgrading the representation theory of quantum sl(2) to some categorical structures. We will review these ideas and give examples of various categorifications of quantum sl(2) constructions.
Speaker: Paolo Perrone, University of Oxford.
Date and Time:   Wednesday March 17, 2021, 7:00 - 8:30 PM., on Zoom.
Title: Categorical probability, Markov categories, and the de Finetti theorem.
Abstract: An impromptu talk on the categorical probability and the probabilistic theorems that can be done categorically.
Speaker: Tobias Fritz, University of Innsbruck.
Date and Time:   Wednesday March 24, 2021, 7:00 - 8:30 PM., on Zoom.
Title: Categorical Probability and the de Finetti Theorem.
Abstract: I will give an introduction to categorical probability in terms of Markov categories, followed by a discussion of the classical 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 purely 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.
Speaker: Ross Street, Macquarie University.
Date and Time:   Wednesday April 14, 2021, 7:00 - 8:30 PM., on Zoom.
Title: Absolute colimits for differential graded categories.
Abstract: A little enriched category theory will be reviewed, in particular, absolute colimits and Cauchy completion. Then the focus will be on the monoidal category DGAb of chain complexes of abelian groups which is at the heart of homological and homotopical algebra. Categories enriched 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 dedicated to the memory of two great New Yorkers, Sammy Eilenberg and Alex Heller.
Speaker: Juan Orendain, University of Mexico, UNAM.
Date and Time:   Wednesday May 5, 2021, 7:00 - 8:30 PM., on Zoom.
Title: How long does it take to frame a bicategory?
Abstract: Framed bicategories are double categories having all companions and conjoints. Many structures naturally organize into framed bicategories, e.g. open Petri nets, polynomials functors, polynomial comonoids, structured cospans, algebras, etc. Symmetric monoidal structures on framed bicategories descend to symmetric monoidal structures on the corresponding horizontal bicategories. The axioms defining symmetric monoidal double categories are much more tractible than those defining 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 bicategories to double categories have classically being useful in expressing Kelly and Street's mates correspondence and in proving the higher dimensional Seifert-van Kampen theorem of Brown et. al., amongst many other applications. We consider lifting problems in their full generality.
Globularly generated double categories are minimal solutions to lifting problems of bicategories into double categories along given categories of vertical arrows. Globularly generated double categories form a coreflective sub-2-category of general double categories. This, together with an analysis of the internal structure of globularly generated double categories yields a numerical 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 squares 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 theory of globularly generated double categories and the vertical length invariant. The ultimate goal of the talk is to present conjectures on the vertical length of framed bicategories and possible applications.
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