Spahn HTT, A.2 model categories (Rev #4, changes)

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This is a subentry of a reading guide to HTT.

Contents

A.2.1 The model category axioms

Definition A.2.1.1

A model category is a category

A model category is a category CC equipped with three distinguished classes of morphisms in CC: The classes (C)(C), (F)(F), (W)(W) of cofibrations, fibrations, and weak equivalences, respectively, satisfying the following axioms:

  • CC admits (small) limits and colimits.

  • The class of weak equivalences satisfies 2-out-of-3.

  • (C)(C),(F)(F)and (W)(W) are closed under retracts.

  • (CW,F)(C\cup W,F) and (C,FW)(C,F\cup W) are weak factorization systems.

A.2.2 The homotopy category of a model category

Definition A.2.2.1

A.2.3 A lifting criterion

A.2.4 Left properness and homotopy push out squares

A.2.5 Quillen adjunctions and Quillen equivalences

A.2.6 Combinatorial model categories

Definition A.2.6.1
Proposition A.2.6.13

A.2.7 Simplicial sets

A.2.8 Diagram categories and homotopy colimits

Definition A.2.8.1
Proposition A.2.8.2
Remark A.2.8.6
Proposition A.2.8.7
Remark A.2.8.8
Proposition A.2.8.9
Remark A.2.8.11

Revision on June 23, 2012 at 17:31:32 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.