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/83479113097?pwd=Wis5OG80WUdScFQwcXdFRlVQdjZoZz09

Meeting ID: 834 7911 3097

Passcode: NYCCTS

Seminar web page.

YouTube videoed talks.

Previous semesters.

researchseminars.org page.

Contact N. Yanofsky to schedule a speaker

or to add a name to the seminar mailing list.

In this talk, we will construct semi-simplicial types in Displayed Type Theory [dTT], a fully semantically general homotopy type theory. Many of our main results are independent of type theory and will say something new and surprising about the homotopy theoretic notion of a classifier for semi-simplicial objects.

This talk is based on joint work with Michael Shulman. Reference: https://arxiv.org/abs/2311.18781

Lean formalization repository.

In a previous talk in the seminar, I discussed the problem of lifting a 2-category into a double category along a given category of vertical arrows, and how this problem allows us to define a notion of length on double categories. The length of a double category is a number that roughly measures the amount of work one needs to do to reconstruct the double category from a bicategory along its set of vertical arrows.

In this talk I will review the length of double categories, and I will discuss two recent developments in the theory: In the paper [OM] a method for constructing different double categories from a given bicategory is presented. I will explain how this construction works. One of the main ingredients of the construction are so-called canonical squares. In the preprint [O] it is proven that in certain classes of equipments -fully faithful and absolutely dense- every square that can be canonical is indeed canonical. I will explain how from this, it can be concluded that fully faithful and absolutely dense equipments are of length 1, and so they can be 'easily' reconstructed from their horizontal bicategories.

References:

[O] Length of fully faithful framed bicategories. arXiv:2402.16296.

[OM] J. Orendain, R. Maldonado-Herrera, Internalizations of decorated bicategories via π-indexings. To appear in Applied Categorical Structures. arXiv:2310.18673.

[W] R. K. Wood, Abstract Proarrows I, Cahiers de topologie et géométrie différentielle 23 3 (1982) 279-290.

[Sh] M. Shulman, Framed bicategories and monoidal fibrations. Theory and Applications of Categories, Vol. 20, No. 18, 2008, pp. 650–738.

In this special case, the category in question has non-degenerate probability distributions on finite sets as its objects and stochastic matrices as its morphisms. After preliminary definitions and lemmas, the proof proceeds in three main steps.

In the first step, we focus on certain groups of automorphisms of certain objects. As a consequence of the axioms, it follows that these groups are preserved under any retrodiction functor and that the restriction of the functor to such a group is a certain kind of group automorphism. Since this group is isomorphic to a Lie group, it is easy to prove that the restriction of a retrodiction to such a group must equal Bayesian inversion if we assume continuity. If we do not make that assumption, we need to work harder and derive continuity "from scratch" starting from the positivity condition in the definition of stochastic matrix.

In the second step, we broaden our attention to the full automorphism groups of objects of our category corresponding to uniform distributions. We show that these groups are generated by the union of the subgroup consisting of permutation matrices and the subgroup considered in the first step. From this fact, it follows that the restriction of a retrodiction to this larger group must equal Bayesian inversion.

In the third step, we finally consider all the objects and morphisms of our category. As a consequence of what we have shown in the first two steps and some preliminary lemmas, it follows that retrodiction is given by matrix conjugation. Furthermore, Bayesian inversion is the special case where the conjugating matrices are diagonal matrices. Because the hom sets of our category are convex polytopes and a retrodiction functor is a continuous bijection of such sets, a retodiction must map polytope faces to faces. By an algebraic argument, this fact implies that the conjugating matrices are diagonal, answering the conjecture in the affirmative.

Paper.