Schreiber Seminar on simplicial methods


Spring 2012

Seminar on simplicial methods

The notion of simplicial set is a powerful tool for studying – by combinatorial means – topological spaces up to weak homotopy equivalence. It has fundamental applications throughout mathematics, whenever homotopy theory plays a role. One speaks of simplicial 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. These are the ingredients for the model structure on simplicial sets which allows to grasp their relation to topological spaces via the central theorem that establishes a Quillen equivalence between the homotopy theory of simplicial sets and that of topological spaces.

Standard references include

A pedagogical introduction to the basic notions is in

  • Greg Friedman, An elementary illustrated introduction to simplicial sets (arXiv:0809.4221)


Basics definitions and examples


Geometric realization



Simplicial homotopy

Model categories


Serre-Model structure on topological spaces

Kan-model structure on simplicial sets

Derived functors and the homotopy theorem

Homotopy type theory


Last revised on May 29, 2012 at 22:04:00. See the history of this page for a list of all contributions to it.