nLab Homotopical Algebra

This page is to record the reference:

• Daniel Quillen:

  
  

  Homotopical Algebra

  
  

  Lecture Notes in Mathematics 43, Springer (1967)

  doi:10.1007/BFb0097438

  TeX-typeset manuscript: pdf.

introducing the theory of model categories as a unifying tool in homotopy theory (“abstract homotopy theory”).

Chapter I. Axiomatic homotopy theory

1. The axioms

• fibration

• cofibration

• weak equivalence

• lifting property

2. The loop and suspension functors

• loop space

• suspension

3. Fibration and cofibration sequences

• homotopy fiber

• fiber sequence

• cofiber sequence

4. Equivalence of homotopy theories

• Quillen equivalence

5. Closed model categories

• model category

Chapter II. Examples of simplicial homotopy theories

1. Simplicial categories

• simplicially enriched category

2. Closed simplicial model categories

• simplicial model category

3. Topological spaces, sumplicial sets, and simplicial groups

• classical model structure on topological spaces

• on simplicial sets

• on simplicial groups

4. $s A$ as a model category

• simplicial object

• model structure on simplicial abelian groups

• model structure on chain complexes

• Dold-Kan correspondence

5. Homology and cohomology

• homological algebra

6. Modules over a simplicial ring

• simplicial ring

category: reference