The New York City

Category Theory Seminar

Department of Computer Science
Department of Mathematics
The Graduate Center of The City University of New York

THIS SEMESTER, SOME TALKS WILL BE IN-PERSON AND SOME WILL BE ON ZOOM.
Time: Wednesdays 07:00 PM Eastern Time (US and Canada)

IN-PERSON INFORMATION:
365 Fifth Avenue (at 34th Street) map
(Diagonally across from the Empire State Building)
New York, NY 10016-4309
Room 6496
The videos of the lectures will be put up on YouTube a few hours after the lecture.


ZOOM INFORMATION:
https://brooklyn-cuny-edu.zoom.us/j/87212617091?pwd=IuHqYMUumYIJgdGVLTZiPVeKPxPAMP.1
Meeting ID: 872 1261 7091
Passcode: NYCCTS

Seminar web page.
Videoed talks.
Previous semesters.
List of previous speakers.
Researchseminars.org page.

Contact N. Yanofsky to schedule a speaker
or to add a name to the seminar mailing list.

Fall 2024



  • Speaker:     Jake Araujo-Simon, Cornell Tech.

  • Date and Time:     Wednesday September 18, 2024, 7:00 - 8:30 PM. IN-PERSON TALK

  • Title:     Categorifying the Volterra series: towards a compositional theory of nonlinear signal processing.

  • Abstract:The Volterra series is a model of nonlinear behavior that extends the convolutional representation of linear and time-invariant systems to the nonlinear regime. Though well-known and applied in electrical, mechanical, biomedical, and audio engineering, its abstract and especially compositional properties have been less studied. In this talk, we present an approach to categorifying the Volterra series, in which a Volterra series 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 category, which we call Volt. We study three monoidal structures on Volt, and outline connections of our work to the field of time-frequency analysis. We also include an audio demo.

    Paper link: https://arxiv.org/abs/2308.07229.


  • Speaker:     Noah Chrein, University of Maryland.

  • Date and Time:     Wednesday September 25, 2024, 7:00 - 8:30 PM. IN-PERSON TALK

  • Title:     A formal category theory for oo-T-multicategories.

  • Abstract: We will explore a framework for 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 yoneda 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-multicategories.

    We 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] has other useful structure. One such structure is a join operation. This join 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-theories and type theories.


  • Speaker:     Sam McCrosson, Montana State University.

  • Date and Time:     Wednesday October 9, 2024, 7:00 - 8:30 PM. ZOOM TALK.

  • Title:     Exodromy.

  • Abstract: A favorite result of first semester algebraic topology is the “monodromy theorem,” which states that for a suitable topological space X, there is a triple equivalence between the categories of covering spaces of X, sets with an action from the fundamental group of X, and locally constant sheaves on X. This result has recently been upgraded by MacPherson and others to a stratified setting, where the underlying space may be carved into a poset of subspaces. In this talk, we’ll look at the main ingredients of the so-called “exodromy theorem,” reviewing stratified spaces and developing “constructible sheaves” and the “exit-path category” along the way.


  • Speaker:     Bruno Gavranović, Symbolica AI.

  • Date and Time:     Wednesday October 30, 2024, 2:00PM NYC Time. NOTE SPECIAL TIME. ZOOM TALK.

  • Title:     Categorical Deep Learning: An Algebraic Theory of Architectures.

  • Abstract: We present our position on the elusive quest for a general-purpose framework for specifying and studying deep learning architectures. Our opinion is that the key attempts made so far lack a coherent bridge between specifying constraints which models must satisfy and specifying their implementations. Focusing on building such a bridge, we propose to apply category theory— precisely, the universal algebra of monads valued in a 2-category of parametric maps—as a single theory elegantly subsuming both of these flavours of neural network design. To defend our position, we show how this theory recovers constraints induced by geometric deep learning, as well as implementations of many architectures drawn from the diverse landscape of neural networks, such as RNNs. We also illustrate how the theory naturally encodes many standard constructs in computer science and automata theory.


  • Speaker:     David Jaz Myers, Topos Research UK.

  • Date and Time:     Wednesday November 6, 2024, ZOOM TALK. SPECIAL TIME: 2:00 PM NYC TIME

  • Title:     Contextads: Para and Kleisli constructions as wreath products.

  • Abstract: Given a comonad D on a category C, we can produce a double category whose tight maps are those of C and whose loose maps are Kleisli maps for D --- this is the Kleisli double category kl(D). Given a monoidal right action & : C x M --> C, we can produce a double category Para(&) whose tight maps are those of C and whose loose maps A -|-> B are pairs (P, f : A & P --> B) of a parameter space P in M and a parameterised map f.

    In this talk, we'll see both these as special cases of a general construction: the Ctx construction which takes a *contextad* on a (double) category and produces a new double category. We'll see that this construction is "just" the wreath product of pseudo-monads in Span(Cat). We'll then exploit this observation to find 2-algebraic structure on the Ctx constructions of suitably structured contextads; vastly generalizing the old observation that a colax monoidal comonad has a monoidal Kleisli category.

    This is joint work with Matteo Capucci.


  • Speaker:    Emilio Minichiello, CUNY CityTech.

  • Date and Time:     Wednesday November 13, 2024, 7:00 - 8:30 PM.IN-PERSON TALK.

  • Title:     Decision Problems on Graphs with Sheaves.

  • Abstract: This semester I don’t feel like talking about my research. Instead I’ll talk about what I’ve learned from reading the paper Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves by Althaus, Bumpus, Fairbanks and Rosiak. This paper is about how we can use sheaf theory to break apart a computational problem, 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 monomorphisms as a main example to showcase their results.


  • Speaker:     Arnon Avron, Tel-Aviv University.

  • Date and Time:     Wednesday November 20, 2024, 7:00 - 8:30 PM. IN-PERSON TALK

  • Title:     What is the Structure of the Natural numbers?

  • Abstract: We present some theorems that show that the notion of a structure, which is central for both Structuralism and category theory, has the very serious defect of having no satisfactory notion of identity which can be associated with it. We use those theorems to show that in particular, there are at least two completely different structures that are entitled to be taken as `the structure of the natural numbers', and any choice between them would arbitrarily favor one of them over the equally legitimate other. This fact refutes (so we believe) the structuralist thesis that the natural numbers are just positions (or places) in "the structure of the natural numbers". Finally, we argue for the high plausibility of the identification of the natural numbers with the finite von Neumann ordinals.


  • Speaker:     Tim Hosgood, Topos Institute.

  • Date and Time:     Wednesday November 27, 2024, 2PM NEW YORK TIME. NOTE SPECIAL TIME. ZOOM TALK

  • Title:     Loose simplicial objects.

  • Abstract: There are two stories that are historically reasonably unrelated, but that both lead to the same definition of a "loose simplicial object", namely (i) the proof that totalisation of a Reedy fibrant cosimplicial simplicial set computes the homotopy limit (via the Bousfield–Kan map), and (ii) the construction of global simplicial resolutions of coherent analytic sheaves (via Toledo–Tong twisting cochains). In this talk, we will look at both of these stories and see what common definition they suggest, and then examine how this definition might be useful.

    This talk is on (incomplete) work in progress, joint with Cheyne Glass.


  • Speaker:     Charlotte Aten, University of Colorado, Boulder.

  • Date and Time:     Wednesday December 4, 2024, 7:00 - 8:30 PM. ZOOM TALK

  • Title:     Invariants of structures.

  • Abstract: I will discuss one part of my PhD thesis, in which I provide a categorification of the notion of a mathematical structure originally given by Bourbaki in their set theory textbook. The main result is that any isomorphism-invariant property of a finite structure can be checked by computing the number of isomorphic copies of small substructures it contains. A special case of this theorem is the classical result of Hilbert about elementary symmetric polynomials generating the algebra of all symmetric polynomials. I will also discuss how the logical complexity of a positive formula controls the size of the small substructures one must count.


  • Speaker:     Matthew Cushman, CUNY.

  • Date and Time:     Wednesday December 11, 2024, 7:00 - 8:30 PM. IN PERSON TALK

  • Title:     Recollements: gluing and fracture for categories.

  • Abstract: Recollements provide a way of gluing two categories together along a left-exact functor, or conversely of obtaining a semi-orthogonal decomposition of a category by two full subcategories. Every recollement comes with a fracture square, which in some circumstances can be extended to a hexagon-shaped diagram of fiber sequences. In this talk we will discuss concrete examples from topological spaces and graphs before moving to smooth manifolds and the recollement that gives rise to differential cohomology theories.

    Spring 2025





  • Speaker:     Raymond Puzio.

  • Date and Time:     Wednesday February 5, 2025, 7:00 - 8:30 PM. IN PERSON TALK!!!

  • Title:     Gentle Introduction to Synthetic Differential Geometry - Part 1.

  • Abstract: Calculations and constructions with infinitesimals make for a handy, intuitive way of doing calculus and differential 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 justifications for manipulations involving infinitesimals.

    One such justification emerged from scheme theory and category theory and goes by the name "synthetic differential geometry". This talk will be an elementary pedagogical introduction to the subject. We will begin by showing how one can re-interpret computing with square zero infinitesimals in terms of homomorphisms from an algebra of smooth functions to the algebra of dual numbers. Using concepts from scheme theory, we will correctly interpret the of corresponding picture of infinitesimally near points. Moving on, we will introduce the axiomatic approach to synthetic differential geometry and describe how passing 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.



  • Speaker:     Jacob S. Zelko, Northeastern University.

  • Date and Time:     Wednesday February 19, 2025, 7:00 - 8:30 PM. IN PERSON TALK!

  • Title:     An Introduction to Compositional Public Health.

  • Abstract: Compositional public health is an emerging research field that exists to address the complexity in public health responses. The field lies at the intersection of category theory, epidemiology, and engineering and utilizes tools from applied category theory for public health applications. This talk will present the motivation of this field, an overview of the mathematics involved in its approaches, current state of the art, live demonstrations, and future research directions within this developing field.



  • Speaker:     Thiago Alexandre

  • Date and Time:     Wednesday February 26, 2025, 7:00 - 8:30 PM IN-PERSON TALK!!!.

  • Title:     Topological Derivators.

  • Abstract: The theory of derivators was originally developed by Grothendieck with high inspiration in topos cohomology. In a letter sent to Thomason, where he explains the main ideas and motivations guiding the formal reasoning of derivators, Grothendieck also remarks that those are Morita-invariant. This means that, if two small categories A and B have equivalent topoi of presheaves, 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 any derivator D to the entire 2-category of topoi and geometric morphisms between them. Grothendieck conjectures that such an extension is always possible and essentially unique. In this case, every derivator D defined over small categories would be coming from a derivator D′ defined over topoi via natural equivalences of categories of the form D(A) = D′(A^), where A varies through small categories and A^ denotes the category of presheaves over A. However, despite these considerations, a theory of derivators over topoi has not yet been developed. To address this gap, I am currently developing a theory of topological derivators. With this theory, I aim 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 between the later and homotopical algebra. In my talk, I would like to present the theory of topological derivators and some of its main results.



  • Speaker:     Grigorios Giotopoulos, NYU Abu Dhabi.

  • Date and Time:     Wednesday March 5, 2025---- 10:00AM, ZOOM TALK!!!

  • Title:     Thickened smooth sets as a natural setting for Lagrangian field theory.

  • Abstract: I will describe how a particularly convenient model for synthetic differential geometry -- the sheaf topos of infinitesimally thickened smooth sets -- serves as a powerful context to host classical Lagrangian field theory. As motivation, I will recall the textbook description of variational Lagrangian field theory, and list desiderata for an ambient category in 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 field theoretic concepts. Time permitting, I will indicate how the setting naturally generalizes to include the description of fermionic fields, and (gauge) fields with internal symmetries. This is based on joint work with Hisham Sati and Urs Schreiber.



  • Speaker:     Jonathon Funk, Queensborough, CUNY.

  • Date and Time:     Wednesday March 12, 2025, 7:00 - 8:30 PM. IN PERSON TALK

  • Title:     Toposes and rings.

    • Slides of the talk.

  • Abstract: I shall attempt to explain a part of a broader program of how topos theory and operator algebra theory match. Following the example of what I call a supported C*-algebra [1], such as a von Neumann algebra, we extend to an arbitrary ring the notions and constructions introduced there. (Familiarity with [1] is not necessary for the purposes of this talk.) I have included an explanation of the Zariski spectrum of a commutative ring in terms 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 supported ones.

    This is joint work with Simon Henry.

    [1] J. Funk, Toposes and C*-algebras, preprint, March 2024.



  • Speaker:     Sophie d'Espalungue.

  • Date and Time:     Wednesday March 19, 2025, 2:00 - 3:00 PM. Zoom talk. NOTE SPECIAL TIME!!

  • Title:     Building All of Mathematics Without Axioms: An n-Categorical Manifesto.

  • Abstract: The formalization of mathematical language traditionally relies on undefined terms - such as Set, Type, universes - whose properties are specified by axioms and inference rules. In this talk, I present an alternative approach in which mathematical language is entirely built from definitions. At its core are n-category constructors - an internal alternative to typing judgments - denoted as (X : Cat_n) for a variable X, which are inductively assigned a truth value - a meaning. Defining an n-category here consists of constructing an element (a proof) of the corresponding truth value. To give meaning to these constructors, (n-1)-categories and (n-1)-functors are inductively organised as an n-category, resulting in a graded structure of nested n-categories (Cat_{n-1} : Cat_n). By treating each mathematical object 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 as sameness, size, and ∞-categories, as well as its connexions to homotopy type theory.



  • Speaker:     Hannah Aizenman, The Graduate Center, CUNY.

  • Date and Time:     Wednesday March 26, 2025, 7:00 - 8:30 PM. IN-PERSON TALK!

  • Title:     Topologically Equivalent Artist Model.

  • Abstract: The contract data visualization tools make with their users 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 propose 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 underlying data - tables, images, networks, etc. - in a manner that independently 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 to work regardless of how the dataset is stored - whether the data is on disk, distributed, or on demand. Specifying the visualization library components as natural transforms of sheaves means that the constraints that the component must satisfy to be structure preserving can be specified as the set of morphisms on the data and graphic sheaves, including the structure on the topology and fields of the data. Using category theory to formally express how visual elements are constructed means we can translate those expectations into code, which can then be used to enforce the expectation that a visualization tool is faithfully translating between numbers and charts.



  • Speaker:     Emilio Minichiello, CUNY CityTech.

  • Date and Time:     Wednesday April 9, 2025, 7:00 - 8:30 PM.IN PERSON TALK.

  • Title:     Structured Decomposition Categories.

  • Abstract: In this talk I’ll report on some new work, joint with Ben Bumpus, Zoltan Kocsis and Jade Master. The idea here is to come up with a categorical framework to talk about decompositions. In graph theory, there are all kinds of ways of decomposing graphs, the most 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 decompositions, one obtains treewidth, the most important graph invariant in algorithmics. In this paper, we abstract this notion, coming up with the definition of structured decomposition categories. To each such category, we can assign to each of its objects a width number. We prove that this number is monotone under monomorphisms, and come up with an appropriate definition of structured decomposition functor such that we get a relationship between widths. We construct several examples of structured decomposition categories, whose widths coincide with several important examples from the literature.


  • Speaker:     Andrei Rodin, University of Lorraine.

  • Date and Time:     Wednesday April 23, 2025, 7:00 - 8:30 PM. IN PERSON TALK!!!

  • Title:     The concept of mathematical structure according to Voevodsky.

  • Abstract: In our email exchange dating back to 2016 Vladimir Voevodsky suggested an original conception 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 reading of Proclus’ commentary on Euclid’s definition of plane angle (Def. 1.8. of the Elements). In my talk I present Vladimir’s conception of mathematical structure, compare it with standard conceptions, and discuss some questions asked by Vladimir during the same exchange. The talk is based on this paper: arXiv:2409.02935



  • Speaker:     Sergei Artemov, Graduate Center CUNY.

  • Date and Time:     Wednesday May 7, 2025, 7:00 - 8:30 PM. IN PERSON TALK!!!

  • Title:     Consistency of PA is a serial property, and it is provable in PA.

  • Abstract: We show that PA consistency is mathematically equivalent to the serial property, which we call the consistency scheme ConS(PA):

    "n is not a proof of 0=1", for n=0,1,2,... .

    The proof of this equivalence is formalizable in PA. Since the standard consistency formula Con(PA)

    "for all x, x is not a code of a proof of 0=1"

    is strictly stronger than the scheme ConS(PA) in PA, Goedel's Second Incompleteness theorem, stating that PA |-\- Con(PA) does not yield the unprovability of PA consistency. Hence, the widespread belief that a consistent theory cannot establish its consistency has never been justified.

    Moreover, 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 build on Hilbert's ideas and prove ConS(PA) in PA.

    This talk is a "dress rehearsal" for the speaker's plenary talk at the ASL meeting on May 13, 2025.

    Reference:
    S.Artemov "Serial Properties, Selector Proofs, and the Provability of Consistency," Journal of Logic and Computation, Volume 35, Issue 3, April 2025.
    https://doi.org/10.1093/logcom/exae034
    Published: 26 July 2024.

    Slides of the talk.



  • Speaker:     Raymond Puzio.

  • Date and Time:     Wednesday May 14, 2025, 7:00 - 8:30 PM. IN PERSON TALK

  • Title:     Gentle Introduction to Synthetic Differential Geometry --- Part two.

  • Abstract: This is part II of "Gentle introduction to synthetic differential geometry". This talk will be self contained 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.

    In part one, we introduce the topic in a "bottom-up" manner starting with the simplest instance and building 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 properties and proceeding from these postulates.

    After introducing the topic, we will turn to Lie groups as an illustrative application. Intuitively, 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 employ various work-arounds. However, in synthetic differential geometry, infinitessimal generators are well defined and we can build up Lie theory in a way which accords with naive intuition. In this talk, we shall go through the first few steps of this development. Then we shall note how the synthetic approach is not only more intuitive but more powerful because it allows us to extend the notion of Lie group beyond finite-dimensional manifolds to which the classical approach is limited. We will also say a few words about how the some of these infinite-dimensional generalizations are of use in in practical applications.



  • Speaker:     Thiago Alexandre.

  • Date and Time:     Wednesday May 28, 2025, 4:00 - 5:30 PM. SPECIAL TIME!!! ZOOM TALK

  • Title:     Topological Derivators --- Part two.

  • Abstract: In this second part, I begin by recalling the axioms of topological derivators and presenting some elementary consequences of these axioms. Following this, I explain how topological derivators can be constructed by sheafifying homotopy theories. I conclude with the deepest theorem I have obtained in the theory of topological derivators, which provides strong evidence for Grothendieck’s conjecture: if a derivator can be extended to a topological derivator, then this extension is essentially unique.