**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 6417

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/82359069037?pwd=wpvpZqXvQHcXUWSPCeQpya33a4a18q.1

Meeting ID: 823 5906 9037

Passcode: NYCCTS

Seminar web page.

Videoed talks.

Previous semesters.

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, TBA.**

Date and Time: ** Wednesday November 27, 2024, 7:00 - 8:30 PM.**

Title:** TBA. **

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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.

Speaker: ** TBA, TBA.**

Date and Time: ** Wednesday December 18, 2024, 7:00 - 8:30 PM.**

Title:** TBA. **

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**Spring 2025 **

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Date and Time: ** Wednesday February 5, 2025, 7:00 - 8:30 PM. . **

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Date and Time: ** Wednesday February 19, 2025, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday February 26, 2025, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday March 5, 2025, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday March 12, 2025, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday March 19, 2025, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday March 26, 2025, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday April 2, 2025, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday May 7, 2025, 7:00 - 8:30 PM. .**

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Date and Time: ** Wednesday May 14, 2025, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday May 21, 2025, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday May 28, 2025, 7:00 - 8:30 PM. **

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