The course will give an introduction to algebraic topology with a focus on homotopy theory. We will consider the classifying space of a small category and develop the homotopy theoretical methods used to study its homotopy type. Later in the course, we will define and study algebraic K-theory. Here is a more detailed syllabus:

**Time and place:** Thursday 1:00-2:30 in 555.

**Lecture notes:**

- Lecture 1: The classifying space of a category.

- Lecture 2: The geometric realization of a simplicial set.

- Lecture 3: Limits and colimit; filtered colimits and finite limits commute.

- Lecture 4: Adjoint functors, limits and colimits.

- Lecture 5: Characterizing the geometric realization by maps from it.

- Lecture 6: Geometric realization preserves finite products; k-spaces.

- Lecture 7: Homotopy groups; mapping fiber; long-exact sequence.

- Lecture 8: The classifying space of a finite group. (I recommend to skip this lecture.)

- Lecture 9: Weak equivalences, Serre fibrations, Serre cofibrations; Quillen model categories.

- Lecture 10: The homotopy category; Quillen functors.

- Lecture 11: Reedy model structure on the category of simplicial spaces; geometric realization is a left Quillen functor.

- Lecture 12: Bi-simplicial sets and their geometric realization; geometric realization and weak equivalences; Quillen's Theorem A and B.

- Lecture 13: Algebraic K-theory.

- Lecture 14: The additivity theorem.

- Lecture 15: Hochschild homology of a ringoid: An introduction.