This page host information on Anton Dochtermann's talk "What is..
Homotopical Algebra?" at the
WhatIsSeminar.
Video
A video recording with integrated slides is available at
http://www.scivee.tv/node/8089 .
Abstract
Homotopy theory of topological spaces represents a rich and interesting
interplay between relatively easy-to-define notions such as homotopy of
maps, homotopy groups of spaces, fibrations, etc.
In the sixties Quillen
realized that topological structures like these could be encoded in a set of
axioms which, if satisfied, allow one to talk about 'homotopy theory' in a
more abstract setting. Any category that satisfies these axioms is called a
'(closed) model category'. Although many instances arise in a geometric
context, a perhaps surprising application of model categories is in a more
algebraic setting: one of the early success of model categories was the
proof that the combinatorial notion of 'simplicial sets' sufficiently
'models' the homotopy category of topological spaces. It turns out that
chain complexes of modules also satisfy the axioms (this lead Quillen to the
notion of 'homotopical algebra') and hence we can talk about such things as
the 'suspension of a chain complex', etc. More recently model categories
have been introduced in algebraic geometry in the context of 'A^1 homotopy'
of schemes.
In this talk we will introduce the axioms for a model category
and discuss a couple of examples and applications (among those mentioned
above).