We will continue online on Thursday, September 24, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Raffael Stenzel

Infinity-categorical comprehension schemes

Abstract:
Comprehension schemes arose as crucial notions in the early work on the foundations of set theory, and hence they found expression in a considerable variety of foundational settings for mathematics. Particularly, they have been introduced to the context of categorical logic first by Lawvere and then by Benabou in the 1970s.

In this talk we define and study a theory of comprehension schemes for fibered infinity-categories, generalizing Johnstone's respective notion for ordinary categories. This includes natural generalizations of all the fundamental instances originally defined by Benabou, and their application to Jacob's comprehension categories. Thereby, we can characterize

- numerous categorical structures arising in higher topos theory
- the notion of univalence
- internal infinity-categories

in terms of comprehension schemes, while some of the 1-categorical counterparts fail to hold in ordinary category theory. As an application, we can show that the universal cartesian fibration is represented via externalization by the "freely walking chain" in the infinity-category of small infinity-categories.

In the end, if my time management permits, we take a look at the externalization construction of internal infinity-categories from a model categorical perspective and review some examples from the literature in this light.

Institute of Mathematics and Statistics, No.8 building within the Faculty of Science, Kotlarska 2, Brno

September 16, 2020, at 2:00 p.m. in M1

Habilitation lecture: Mgr. Vojtěch Žádník, Ph.D.

"Geometric constructions old and new: from local coordinate manipulation to general extension functor and back again"

We will continue online on Thursday, September 10, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Edoardo Lanari (Czech Academy of Sciences)

Gray tensor products and lax functors of (∞,2)-categories

Abstract:
We give a definition of the Gray tensor product in the setting of scaled simplicial sets which is associative and forms a left Quillen bifunctor with respect to the bicategorical model structure of Lurie. We then introduce a notion of oplax functor in this setting, and use it in order to characterize the Gray tensor product by means of a universal property. A similar characterization was used by Gaitsgory and Rozenblyum in their definition of the Gray product, thus giving a promising lead for comparing the two settings.  This is a report on joint work with A. Gagna and Y. Harpaz.

Title: Annihilators of the ideal class group of imaginary cyclic fields

Author: Mgr. Pavel Francírek

Defence: Wednesday, September 9, 2020, 14.45, Meeting Room of the department

Supervisor: prof. RNDr. Radan Kučera, DSc.

The main objective of this thesis is to find for certain infinite family of imaginary cyclic fields annihilators of the ideal class group living outside the Sinnott's Stickelberger ideal. In this thesis we study a field L which is the compositum of a real cyclic field K whose degree over rationals is a power of an odd prime l and an imaginary cyclic field F whose degree over rationals is not divisible by l. In addition, we assume that the conductors of the fields K and F are relatively prime. The main idea of this thesis is to find in the field L a nontrivial root of a certain modified Gauss sum. The factorization of the principal ideal generated by this root gives rise to annihilators of the ideal class group of L. Then we show that these annihilators live outside the Sinnott's Stickelberger ideal if the number of primes ramified in K that split completely in the smallest imaginary subfield of F is sufficiently large. Assuming that l does not ramify in L, it is sufficient that the number of these primes is greater than or equal to two. In the case of F being a quadratic imaginary field our approach generally leads to a stronger annihilation result compared to a result of Greither and Kučera. At the end, we also obtain a result on the divisibility of the relative class number of L.