We will continue on Thursday, September 26, in M5 at 1pm by the talk

R. Stenzel

From Univalence to descent via "split indexed quasi-categories"

Univalence is a type theoretical notion at the heart of Voevodsky’s Univalent Foundations Program, and Descent is a property of presentable (infinity,1)-categories introduced by Rezk as a slick way to define higher toposes. In recent years it has been understood that univalence (i.e. the existence of certain univalent maps) and descent are two sides of the same coin when the (infinity,1)-category is presentable.

In the talk I will explain this correspondence in the world of model categories, but instead via local classes in the sense of Gepner and Kock, we will take an excursion to complete Segal objects and their associated "indexed
quasi-categories". We will see that in this way a third property arises naturally - call it P for short - which relates univalence and descent directly to two properties crucial for the model theory of HoTT: the fibration extension property and the weak equivalence extension property.

In the first half of the talk I will briefly introduce all notions referred to above and draw a big diagram relating them.
In the second half I will introduce the property P and elaborate on its relation to descent both in the presentable and non-presentable case, with a view towards the space between logoi in the sense of Anel and elementary higher toposes in the sense of Rasekh.

Module Theory in Sup and its applications

Ulrich Höhle

The lecture series will take place at the Department of Mathematics and Statics, Faculty of Science, Masaryk University, Kotlarska 2, 611 37 Brno, Czech Republic. Each talk will be approx. 90 minutes.

1. Talk: Module Theory in Sup and Enriched Order Theory. (Thursday, October 3, 10.00, Seminar room)

2. Talk: Topological Representation of Semi-Unital Quantales with Applications to C*-algebras. (Thursday, October 3, 14.30, Seminar room)

3. Talk: Part I: Three-Valuedness --- The First Step Towards Many-Valuedness (Tutorial).
Part II: Applications of Module Theory in Sup to Linear Stochastic Programming. (Friday, October 4, 13.00, Seminar room)

4. Talk: Preservation of Projective Right Modules in Sup under Duality. (Monday, October 7, 14.00, lecture room M3)

Abstract

1. Since many people are educated in module theory over abelian groups and  NOT over complete lattices, as a first step I will give an introduction into the general principles of module theory in Sup. Among other things I will refer to A. Joyal and M. Tierney 1984.  One interesting thing is of course how we derive the related Q-preorder from a given right action over Q. Since the tensor product in Sup plays here an important role, I will also say something about its construction.
   
   
2. Applications of module theory in Sup. I proceed now from pure to applied mathematics.
   
   (a) Topological  representation of semi-unital quantales with applications to C*-algebras.
   (b) Three-valuedness ---- the first step towards many-valuedness.
   (c) Applications of module theory in Sup to linear stochastic programming.
   (d) Projective right modules  and their dual right modules over  involutive and unital quantales: The problem of the preservation of their projectivity.

We will continue on Thursday, September 19, in M5 at 1pm by the talk

J. Adamek

Ultrafilters in Locally Presentable Categories

Abstract:
For a number of locally finitely presentable categories K we describe the codensity monad of the full embedding of all finitely presentable objects into K. We introduce the concept of D-ultrafilter on an object, where D is a “nice” cogenerating object of K.
Example: in Pos we choose the 2-chain as D. A D-ultrafilter on a poset X is a prime up-set, closed under finite intersecitions, in the poset of all up-sets of X.

We prove that the above codensity monad assigns to every object an object representing all D-ultrafilters on it. Our result covers e.g. the categories of sets, vector spaces, posets, semilattices, graphs and M-sets for finite commutative monoids M.

Dear Friends of Hikes and Mathematics,

you are invited to our 25th mathematical hike planned on September 28th. Start at 09:44 at the train station "Níhov". There is a train from Brno that leaves the station "Dolní nádraží" at 08:50.

We have planned a 16 km hike from Níhov to Tišnov regardless of the weather. We follow yellow and green paths, also we can visit the abbey "Porta coeli".

We would love to see you there, please join us. It's a great opportunity for a meeting of students, graduates and employees of our department of all age or mathematical categories.

We are planning our next hikes on 2nd of November, 14th of December.

All information and photos can be found at http://conference.math.muni.cz/vylety/. (in CZ)

Have a nice start of the semester. Looking forward to see you soon,

Jonatan Kolegar and Jana Bartoňová, organizers,
Jan Slovák, Director of the Department of Mathematics and Statistics