Department of Mathematics

The Graduate Center of The City University of New York

THE TALKS WILL ALL BE DONE ON ZOOM THIS SEMESTER.

Time: Wednesdays 07:00 PM Eastern Time (US and Canada)

https://us02web.zoom.us/j/84134331639?pwd=TVRzVjlaZW5CNVh5ampxOGJ0RE5QQT09

Meeting ID: 841 3433 1639

Passcode: NYCCTS

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

Room 6417

Wednesdays 7:00 - 8:30 PM

Videoed talks.

Previous semesters.

Research seminars page.

Contact N. Yanofsky to schedule a speaker

or to add a name to the seminar mailing list.

visiblein Lawvere’s Theorem.

use this function defined over this small set to generate predictions over that larger set.Extrapolation, interpolation, statistical inference and forecasting all reduce to this problem. The Kan extension is a powerful tool in category theory that generalizes this notion. In this work we explore several applications of Kan extensions to data science.

Related paper.

Slides

Motivated by these considerations, we develop a general axiomatic framework for studying enriched structure-semantics adjunctions and monad-theory equivalences for subcategories of arities, which generalizes all of the aforementioned results and also provides substantial new examples of relevance for topology and differential geometry. For a subcategory of arities J in a V-category C over a symmetric monoidal closed category V, Linton’s notion of clone generalizes to provide enriched notions of J-theory and J-pretheory, which were also employed by Bourke and Garner (2019). We say that J is amenable if every J-theory admits free algebras, and is strongly amenable if every J-pretheory admits free algebras. If J is amenable, then we obtain an idempotent structure-semantics adjunction between certain J-pretheories and J-tractable V-categories over C, which yields an equivalence between J-theories and J-nervous V-monads on C. If J is strongly amenable, then we also obtain a rich theory of presentations for J-theories and J-nervous V-monads. We show that many previously studied subcategories of arities are (strongly) amenable, from which we recover the aforementioned structure-semantics adjunctions and monad-theory equivalences. We conclude with the result that any small subcategory of arities in a locally bounded closed category is strongly amenable, from which we obtain structure-semantics adjunctions and monad-theory equivalences in (e.g.) many convenient categories of spaces.

Joint work with Rory Lucyshyn-Wright.

In this expository talk I will first give a streamlined introduction to the original notion of defect of a finitely presented functor defined on a module category and then show how to generalize it to arbitrary additive functors. Along the way I will give a dozen or so examples illustrating various use cases for the defect. The ultimate goal of this lecture is to provide a background for the upcoming talk of Alex Sorokin, who will report on his vast generalization of the defect to arbitrary profunctors enriched in a cosmos.

This presentation is based on joint work in progress with Jeremy Russell.