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

Idea

A Kan extension provides a method for extending a functor along another functor. It generalizes the concept of extending a function from a subspace to a larger space by finding the best possible approximation that factors through a given morphism.

Definition

Let $F:\mathcal{C}\Rightarrow \mathcal{E}$ and $K:\mathcal{C}\Rightarrow \mathcal{D}$ be functors.

Left Kan Extension

The left Kan extension of $F$ along $K$ is a functor $La{n}_{K}F:\mathcal{D}\Rightarrow \mathcal{E}$ equipped with a natural transformation $\eta :F\Rrightarrow (La{n}_{K}F)\circ K$ such that for any functor $G:\mathcal{D}\Rightarrow \mathcal{E}$ and natural transformation $\alpha :F\Rrightarrow G\circ K$, there exists a unique natural transformation $\sigma :La{n}_{K}F\Rrightarrow G$ such that $\alpha =(\sigma K)\circ \eta$.

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
\mathcal{C} \arrow[rr, "K"] \arrow[rd, "F"'] & & \mathcal{D} \arrow[ld, "Lan_K F"] \\
& \mathcal{E}
\end{tikzcd}
\end{document}

Right Kan Extension

The right Kan extension of $F$ along $K$ is a functor $Ra{n}_{K}F:\mathcal{D}\Rightarrow \mathcal{E}$ equipped with a natural transformation $ϵ:(Ra{n}_{K}F)\circ K\Rrightarrow F$ such that for any functor $G:\mathcal{D}\Rightarrow \mathcal{E}$ and natural transformation $\alpha :G\circ K\Rrightarrow F$, there exists a unique natural transformation $\sigma :G\Rrightarrow Ra{n}_{K}F$ such that $\alpha =ϵ\circ (\sigma K)$.

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
\mathcal{C} \arrow[rr, "K"] \arrow[rd, "F"'] & & \mathcal{D} \arrow[ld, "Ran_K F"] \\
& \mathcal{E}
\end{tikzcd}
\end{document}

Adjunction Perspective

Kan extensions are characterized as adjoints to the precomposition functor $K^{\ast }:[\mathcal{D},\mathcal{E}]\Rightarrow [\mathcal{C},\mathcal{E}]$ defined by $K^{\ast }(G)=G\circ K$. When they exist for all $F$, we have a sequence of adjoints: $La{n}_K⊣K^{\ast }⊣Ra{n}_K$ This yields the natural isomorphisms: $Nat(La{n}_{K}F,G)\cong Nat(F,G\circ K)$ $Nat(G\circ K,F)\cong Nat(G,Ra{n}_{K}F)$

Formulae via (Co)ends

If $\mathcal{C}$ is small and $\mathcal{E}$ is cocomplete, the left Kan extension exists and can be computed pointwise using an end:

$$(La{n}_{K}F)(d)={\int }^{c\in \mathcal{C}}\mathcal{D}(Kc,d)\cdot Fc$$

Dually, if $\mathcal{E}$ is complete, the right Kan extension exists and is computed using a coend:

$$(Ra{n}_{K}F)(d)={\int }_{c\in \mathcal{C}}F{c}^{\mathcal{D}(d,Kc)}$$

References

• riehl2016-category-theory
• maclane1978-categories
• riehl2014-categorical-homotopy-theory
• kan58-kan-extensions

Link to original

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