**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/83243451066?pwd=V3BkMCtxTnM3WTQ0QlN3K3NRRHNSQT09

Meeting ID: 832 4345 1066

Passcode: NYCCTS1

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 2023 **

Speaker: ** Tomáš Gonda, University of Innsbruck.**

Date and Time: ** Wednesday September 27, 2023, 5:00 - 6:00 PM. ZOOM TALK. NOTE SPECIAL TIME!**

Title:** A Framework for Universality in Physics, Computer Science, and Beyond. **

Abstract: Turing machines and spin models share a notion of universality according to which some simulate all others. We set up a categorical framework for universality which includes as instances universal Turing machines, universal spin models, NP completeness, top of a preorder, denseness of a subset, and others. By identifying necessary conditions for universality, we show that universal spin models cannot be finite. We also characterize when universality can be distinguished from a trivial one and use it to show that universal Turing machines are non-trivial in this sense. We leverage a Fixed Point Theorem inspired by a result of Lawvere to establish that universality and negation give rise to unreachability (such as uncomputability). As such, this work sets the basis for a unified approach to universality and invites the study of further examples within the framework.

Speaker: ** Thiago Alexandre, University of São Paulo (Brazil).**

Date and Time: ** Wednesday October 11, 2023, 7:00 - 8:30 PM.**

Title:** Internal homotopy theories. **

Abstract: The idea of 'Homotopy theories' was introduced by Heller in his seminal paper from 1988. Two years later, Grothendieck discovered the theory of derivators (1990), exposed in his late manuscript Les Dérivateurs, and developed further by several authors. Essentially, there are no significant differences between Heller's homotopy theories and Grothendieck's derivators. They are tautologically the same 2-categorical yoga. However, they come from distinct motivations. For Heller, derivators should be a definitive answer to the question "What is a homotopy theory?", while for Grothendieck, who was strongly inspired by topos cohomology, the first main motivation for derivators was to surpass some technical deficiencies that appeared in the theory of triangulated categories. Indeed, Grothendieck designed the axioms of derivators in light of a certain 2-functorial construction, which associates the corresponding (abelian) derived category to each topos, and more importantly, inverse and direct cohomological images to each geometric morphism. It was from this 2-functorial construction, from where topos cohomology arises, that Grothendieck discovered the axioms of derivators, which are surprisingly the same as Heller's homotopy theories. Nowadays, it is commonly accepted that a homotopy theory is a quasi-category, and they can all be presented by a localizer (M,W), i.e., a couple composed by a category M and a class of arrows in W. This point of view is not so far from Heller, since pre-derivators, quasi-categories, and localizers, are essentially equivalent as an answer to the question "What is a homotopy theory?". In my talk, I will expose these subjects in more detail, and I am also going to explore how to internalize a homotopy theory in an arbitrary (Grothendieck) topos, a problem which strongly relates formal logic and homotopical algebra.

Speaker: ** Michael Shulman, University of San Diego.**

Date and Time: ** Wednesday October 18, 2023, 7:00 - 8:30 PM.**

Title:** The derivator of setoids. **

Abstract:

Speaker: ** Emilio Minichiello, CUNY Graduate Center.**

Date and Time: ** Wednesday October 25, 2023, 7:00 - 8:30 PM. IN PERSON TALK.**

Title:** A Mathematical Model of Package Management Systems. **

Abstract: In this talk, I will review some recent joint work with Gershom Bazerman and Raymond Puzio. The motivation is simple: provide a mathematical model of package management systems, such as the Hackage package respository for Haskell, or Homebrew for Mac users. We introduce Dependency Structures with Choice (DSC) which are sets equipped with a collection of possible dependency sets for every element and satisfying some simple conditions motivated from real life use cases. We define a notion of morphism of DSCs, and prove that the resulting category of DSCs is equivalent to the category of antimatroids, which are mathematical structures found in combinatorics and computer science. We analyze this category, proving that it is finitely complete, has coproducts and an initial object, but does not have all coequalizers. Further, we construct a functor from a category of DSCs equipped with a certain subclass of morphisms to the opposite of the category of finite distributive lattices, making use of a simple finite characterization of the Bruns-Lakser completion.

Speaker: ** Larry Moss, Indiana University, Bloomington
.**

Date and Time: ** Wednesday November 8, 2023, 7:00 - 8:30 PM. ZOOM TALK**

Title:** On Kripke, Vietoris, and Hausdorff Polynomial Functors. **

Abstract: The Vietoris space of compact subsets of a given Hausdorff space yields an endofunctor V on the category of Hausdorff spaces. Vietoris polynomial endofunctors on that category are built from V, the identity and constant functors by forming products, coproducts and compositions. These functors are known to have terminal coalgebras and we deduce that they also have initial algebras. We present an analogous class of endofunctors on the category of extended metric spaces, using in lieu of V the Hausdorff functor H. We prove that the ensuing Hausdorff polynomial functors have terminal coalgebras and initial algebras. Whereas the canonical constructions of terminal coalgebras for Vietoris polynomial functors takes omega steps, one needs \omega + \omega steps in general for Hausdorff ones. We also give a new proof that the closed set functor on metric spaces has no fixed points.

Speaker: ** Pedro Sota, TBA.**

Date and Time: ** Wednesday November 22, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

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Speaker: ** Charlotte Aten, University of Denver.**

Date and Time: ** Wednesday November 29, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

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Speaker: ** TBA, TBA.**

Date and Time: ** Wednesday December 6, 2023, 7:00 - 8:30 PM.**

Title:** TBA. **

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**Spring 2024 **

Speaker: ** TBA, TBA.**

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

Title:** TBA. **

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Date and Time: ** Wednesday February 14, 2024, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday February 28, 2024, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday March 6, 2024, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday March 20, 2024, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday April 10, 2024, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday April 17, 2024, 7:00 - 8:30 PM. .**

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

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Date and Time: ** Wednesday May 22, 2024, 7:00 - 8:30 PM. **

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Date and Time: ** Wednesday May ??, 2024, 7:00 - 8:30 PM. **

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