riehl2014-categorical-homotopy-theory

Abstract

The text develops the foundations of abstract homotopy theory using Quillen model categories and weak factorization systems, progressing to homotopy limits and colimits, Reedy categories, and Bousfield localization.

Introduction

Homotopy Theory Categorical Homotopy Theory

All concepts are Kan extensions

Kan Extension

Definition

Let $F:\mathcal{C}\Rightarrow \mathcal{E};K:\mathcal{C}\to \mathcal{D}$ be functors. A left Kan extension of $F$ along $K$ is a functor ${\mathrm{Lan}}_{K}F:\mathcal{D}\Rightarrow \mathcal{E}$ Together with a natural transformation $\eta :F\Rrightarrow {\mathrm{Lan}}_{K}F$ such that any other pair $(G:\mathcal{D}\Rightarrow \mathcal{E},\epsilon :F\Rrightarrow G)$

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Derived functors via deformations

Basic concepts of enriched category theory

The unreasonably effective (co)bar construction

Homotopy limits and colimits: the theory

Homotopy limits and colimits: the practice

Weighted limits and colimits

Derived enrichment

Weak factorization systems in model categories

Algebraic perspectives on the small object argument

Enriched factorizations and enriched lifting properties

A brief tour of Reed category theory

Preliminaries on quasi-categories

Simplicial categories and homotopy coherence

Isomorphisms in quasi-category

A sampling of 2-categorical aspects of quasi-categories