This course is a follow up of Lecture 19215101 "Topologie III" by Elmar Vogt and **STARTS** on 06.06.2017. The contents of both lectures are basic for (algebraic and geometric) topology, and we strongly recommend for students to take both as a single package.

Exercises are conducted by Filipp Levikov and run as scheduled to cover both courses.

**CONTENTS** of Categories and homotopy theory:

This will be an introduction to some of the basic techniques of homotopy theory that use category theory. We will cover basic categorical constructions: functors, adjunctions, monads, the yoneda lemma, (co) limits and we use these to describe the category of simplicial sets. We go back to homotopy theory with the concept of model categories, homotopy (co) limits. One of our main goals is to understand Quillen's Theorem A and why simplicial sets are combinatorial models for spaces. This course will complement the reading seminar in infinity categories 19233511.

For the content of "Topologie III" go to Lecture 19215101 in KVV or eVV.