**Time and place:** Tuesday (first session: February 14), 13.30-15.30, room: HG 03.085 (aka. Hilbert space)

The notion of a simplicial set is a powerful (combinatorial) tool for studying topological spaces up to weak homotopy equivalence. Simplicial sets have fundamental applications throughout mathematics, whenever homotopy theory plays a role. In this seminar we discuss some aspects of simplicial homotopy theory. Along the way, we also develop some basics of the theory of model categories. As an upshot of the first eight talk we can give a precise theorem showing that simplicial sets and topological spaces model the same homotopy theory.

The seminar starts with looking at the basics of simplicial sets and their geometric realization to topological spaces. From this we motivate fundamental notions like Kan fibration of simplicial sets, simplicial homotopy, and simplicial homotopy groups. After a short detour in model category theory we establish the Serre model structure on topological spaces. We then see that the above notions from simplicial homotopy theory are central ingredients for the model structure on simplicial sets. The theory of model categories permits us to derive certain well-behaved functors, the so-called Quillen functors, in not necessarily additive contexts. There is the important result establishing a Quillen equivalence between simplicial sets and topological spaces.

In the last block of talks we turn to applications in logic. Topics to be discussed include aspects from type theory, homotopy type theory, and univalence.

The seminar starts with looking at the basics of simplicial sets and their geometric realization to topological spaces. From this we motivate fundamental notions like Kan fibration of simplicial sets, simplicial homotopy, and simplicial homotopy groups. After a short detour in model category theory we establish the Serre model structure on topological spaces. We then see that the above notions from simplicial homotopy theory are central ingredients for the model structure on simplicial sets. The theory of model categories permits us to derive certain well-behaved functors, the so-called Quillen functors, in not necessarily additive contexts. There is the important result establishing a Quillen equivalence between simplicial sets and topological spaces.

In the last block of talks we turn to applications in logic. Topics to be discussed include aspects from type theory, homotopy type theory, and univalence.

**Schedule of the seminar:**

Talk 01: Basic definitions and examples (Matija Bašić)

Talk 02: Geometric realization (Giovanni Caviglia)

Talk 03: Serre fibrations, Kan fibrations, and anodyne extensions (Frank Roumen)

Talk 04: Simplicial homotopy (Matija Bašić)

Talk 05: Model categories and Quillen functors (Giovanni Caviglia)

Talk 06: Model structures on groupoids, categories, and chain complexes (Urs Schreiber)

Talk 07: Kan model structure on simplicial sets (Moritz Groth)

Talk 08: Serre model structure on topological spaces and the homotopy theorem (Moritz Groth)

Talk 09: Basics of type theory (Andrew Polonsky)

Talk 10: Type theory and category theory (Bas Spitters)

Talk 11: Homotopy theory in type theory (Bas Spitters)

Talk 12: Categorical models of homotopy type theory (Bas Spitters)

**References:**There are many references for the respective parts of such a seminar including:

Curtis: Simplicial Homotopy Theory

Friedman: An elementary illustrated introduction to simplicial sets

Gabriel, Zisman: Calculus of Fractions and Homotopy Theory

Goerss, Jardine: Simplicial Homotopy Theory

Dwyer, Spalinski: Homotopy theories and model categories

Goerss, Jardine: Simplicial Homotopy Theory

Hovey: Model Categories

Awodey: Type theory and homotopy

Coquand: Various texts on type theory

Shulman: Minicourse on homotopy type theory