Thu 5, Feb — 09.30h: Categories (Luis).

Thu 12, Feb — 09.30h: Compact Hausdorff spaces I (O'Bryan).

The aim of this talk is to provide a general overview of some categorical aspects, motivated by classical constructions and concrete examples. The emphasis is on compact Hausdorff spaces and on the use of categorical language to clarify and unify familiar topological notions.

The main reference for the talk is Le Stum's notes, in particular Chapter 1 (Sections 1.4.3 and 1.5.4) and Chapter 2 (Sections 2.1 and 2.2).

The outline of the talk is as follows:

1. Projective and injective objects in a category.
2. Reflective subcategories and adjoint functors arising from topological constructions.
3. A brief review of basic topology, with emphasis on compact Hausdorff spaces.
4. The Stone—Čech compactification as motivating example. The book Counterexamples in topology, by Seebach and Steen contains some useful information.

These topics provide a natural entry point to the categorical framework underlying the theory of condensed sets.

Fri 20, Feb — 09.30h: Compact Hausdorff spaces II (Sergio).

In this talk we are going to introduce the definition of free compact Hausdorff space, Stone space and Stonean space, together with some key properties of these spaces and the categories they belong to.

Furthermore, throughout the talk we'll show some examples. Finally, our main goal is to provide characterizations of Stone spaces and Stonean spaces, as well as to introduce the Stone representation theorem.

We will follow section 2.2 of Le Stum's notes.

Thu 26, Feb — 09.30h: Compact Hausdorff spaces III (Mario).

In this talk we will study the concept of closed cartesian category, along with associated concepts as the one of internal hom.

In particular, in the category of topological spaces, we consider the compact-open topology on spaces of maps. In order to have a cartesian closed category, we have to restrict to a subcategory, namely, the one of compactly generated Hausdorff spaces, consisting of those spaces that are colimits of compact Hausdorff spaces.

We will follow section 2.3 of Le Stum's notes.

Fri 6, Mar — 09.30h: Condensed sets I (Álvaro).

This talk gives a brief introduction to condensed sets and some of their basic properties.

We discuss how topological spaces naturally embed into the category of condensed sets, present examples that do not arise from topological spaces, and provide some intuition for the richer nature of this framework.

Thu 12, Mar — 09.30h: Condensed sets II (Javier).

In this talk we make essentially two constructions: we extend condensed sets to act on compact Hausdorff spaces by means of profinite resolutions and asign a topological space to every condensed set.

Those constructions are instances of the notion of Kan extension, which we also introduce. Moreover, we study properties of some of the functors involved.

Thu 19 and Thu 26, Mar — 09.30h: Grothendieck topologies and sheaves on sites (Aarón).

In these two talks we will introduce the definition of topology on a category and study the associated theory of sheaves.

We discuss the notion of covering sieves on an object and the various interpretations of the concept, both in general, and in concrete examples relevant to our study.

We explain the sheafification process of a presheaf and give an overview of the properties underlying a category of sheaves. Furthermore, we give some insight on how condensed sets can be interpreted as sheaves on a given site, and explain the relation between previously studied properties of condensed sets and those arising from sheaves.

The first talk follows sections 3.1 and 3.2 of Le Stum's notes.

The second one follows sections 3.2 and 3.3 of Le Stum's notes.

Thu 9 and Thu 16, Apr — 09.30h: Solid modules wout/ condensed formalism (Luis).

In this talk we study solid modules over a discrete commutative ring following [Ked24, Section 1].

Solid modules over a commutative ring (in particular, solid abelian groups) are a generalization of the 'usual' discrete modules. In this category of solid modules, morphisms between finitely and infinitely generated free modules modules behave, in a certain sense, in the same way.

The first goal is to establish the basic definitions necessary to define solid modules and their operations.

Thu 30, Apr and Thu 7, May — 09.30h: More on solid modules (Fran).

We continue reading [Ked24, Section 1].

We give detailed proofs of some facts about solid modules, making clear how do we work with them.

Some highlights are: extension of scalars from discrete to solids is fully faithful but restriction isn't, and $\operatorname{Mod}_{R_\square}$ is a closed symmetric monoidal category where duality exchanges direct products and direct sums.