# nLab chain homotopy

## Theorems

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Idea

A chain homotopy is a homotopy in a category of chain complexes with respect to the standard interval object in chain complexes.

## Definition

Let $𝒜=$ Ab be the abelian category of abelian groups. Write ${\mathrm{Ch}}_{•}\left(𝒜\right)$ for the category of chain complexes in $𝒜$.

A chain homotopy is a homotopy in ${\mathrm{Ch}}_{•}\left(𝒜\right)$. We first give the explicit definition, the more abstract characterization is below in prop. 1.

###### Definition

A chain homotopy $\psi :f⇒g$ between two chain maps $f,g:{C}_{•}\to {D}_{•}$ in ${\mathrm{Ch}}_{•}\left(𝒜\right)$ is a sequence of morphisms

$\left\{\left({\psi }_{n}:{C}_{n}\to {D}_{n+1}\right)\in 𝒜\mid n\in ℕ\right\}$\{ (\psi_n : C_n \to D_{n+1}) \in \mathcal{A} | n \in \mathbb{N} \}

in $𝒜$ such that

${f}_{n}-{g}_{n}={\partial }^{D}\circ {\psi }_{n}+{\psi }_{n-1}\circ {\partial }^{C}\phantom{\rule{thinmathspace}{0ex}}.$f_n - g_n = \partial^D \circ \psi_n + \psi_{n-1} \circ \partial^C \,.
###### Remark

It may be useful to illustrate this with the following graphics, which however is not a commuting diagram:

$\begin{array}{ccc}⋮& & ⋮\\ ↓& & ↓\\ {C}_{n+1}& \stackrel{{f}_{n+1}-{g}_{n+1}}{\to }& {D}_{n+1}\\ {↓}^{{\partial }_{n}^{C}}& {↗}_{{\psi }_{n}}& {↓}^{{\partial }_{n}^{D}}\\ {C}_{n}& \stackrel{{f}_{n}-{g}_{n}}{\to }& {D}_{n}\\ {↓}^{{\partial }_{n-1}^{C}}& {↗}_{{\psi }_{n-1}}& {↓}^{{\partial }_{n-1}^{D}}\\ {C}_{n-1}& \stackrel{{f}_{n-1}-{g}_{n-1}}{\to }& {D}_{n-1}\\ ↓& & ↓\\ ⋮& & ⋮\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \vdots && \vdots \\ \downarrow && \downarrow \\ C_{n+1} &\stackrel{f_{n+1} - g_{n+1}}{\to}& D_{n+1} \\ \downarrow^{\mathrlap{\partial^C_{n}}} &\nearrow_{\mathrlap{\psi_{n}}}& \downarrow^{\mathrlap{\partial^D_{n}}} \\ C_n &\stackrel{f_n - g_n}{\to}& D_n \\ \downarrow^{\mathrlap{\partial^C_{n-1}}} &\nearrow_{\mathrlap{\psi_{n-1}}}& \downarrow^{\mathrlap{\partial^D_{n-1}}} \\ C_{n-1} &\stackrel{f_{n-1} - g_{n-1}}{\to}& D_{n-1} \\ \downarrow && \downarrow \\ \vdots && \vdots } \,.

Instead, a way to encode chain homotopies by genuine diagrammatics is below in prop. 1.

## Properties

### In terms of general homotopy

###### Definition

Let

${I}_{•}≔{N}_{•}\left(C\left(\Delta \left[1\right]\right)\right)$I_\bullet \coloneqq N_\bullet(C(\Delta[1]))

be the normalized chain complex in $𝒜$ of the simplicial chains on the simplicial 1-simplex:

${I}_{•}=\left[\cdots \to 0\to 0\to 𝟙\stackrel{\left(\mathrm{id},-\mathrm{id}\right)}{\to }𝟙\oplus 𝟙\right]\phantom{\rule{thinmathspace}{0ex}}.$I_\bullet = [ \cdots \to 0 \to 0 \to \mathbb{1} \stackrel{(id,-id)}{\to} \mathbb{1} \oplus \mathbb{1} ] \,.

This is the standard interval in chain complexes. Indeed it is manifestly the “abelianization” of the standard interval object in sSet/Top.

###### Proposition

A chain homotopy $\psi :f⇒g$ is equivalently a commuting diagram

$\begin{array}{c}{C}_{•}\\ ↓& {↘}^{f}\\ {C}_{•}\otimes {I}_{•}& \stackrel{\left(f,g,\psi \right)}{\to }& {D}_{•}\\ ↑& {↗}_{g}\\ {C}_{•}\end{array}$\array{ C_\bullet \\ \downarrow & \searrow^{\mathrlap{f}} \\ C_\bullet \otimes I_\bullet &\stackrel{(f,g,\psi)}{\to}& D_\bullet \\ \uparrow & \nearrow_{\mathrlap{g}} \\ C_\bullet }

in ${\mathrm{Ch}}_{•}\left(𝒜\right)$, hence a genuine left homotopy with respect to the interval object in chain complexes.

###### Proof

For notational simplicity we discuss this in $𝒜=$ Ab.

Observe that ${N}_{•}\left(ℤ\left(\Delta \left[1\right]\right)\right)$ is the chain complex

$\left(\cdots \to 0\to 0\to ℤ\stackrel{\left(\mathrm{id},-\mathrm{id}\right)}{\to }ℤ\oplus ℤ\to 0\to 0\to \cdots \right)$( \cdots \to 0 \to 0 \to \mathbb{Z} \stackrel{(id,-id)}{\to} \mathbb{Z} \oplus \mathbb{Z} \to 0 \to 0 \to \cdots)

where the term $ℤ\oplus ℤ$ is in degree 0: this is the free abelian group on the set $\left\{0,1\right\}$ of 0-simplices in $\Delta \left[1\right]$. The other copy of $ℤ$ is the free abelian group on the single non-degenerate edge in $\Delta \left[1\right]$. All other cells of $\Delta \left[1\right]$ are degenerate and hence do not contribute to the normalized chain complex. The single nontrivial differential sends $1\in ℤ$ to $\left(1,-1\right)\mathrm{in}ℤ\oplus ℤ$, reflecting the fact that one of the vertices is the 0-boundary the other the 1-boundary of the single nontrivial edge.

It follows that the tensor product of chain complexes ${C}_{•}\otimes {I}_{•}$ is

$\cdots \to {C}_{1}\oplus {C}_{2}\oplus {C}_{2}\to {C}_{0}\oplus {C}_{1}\oplus {C}_{1}\to {C}_{-1}\oplus {C}_{0}\oplus {C}_{0}\to \cdots \phantom{\rule{thinmathspace}{0ex}}.$\cdots \to C_1 \oplus C_{2} \oplus C_2 \to C_0 \oplus C_{1} \oplus C_{1} \to C_{-1} \oplus C_0 \oplus C_0 \to \cdots \,.

Therefore a chain map $\left(f,g,\psi \right):{C}_{•}\otimes {I}_{\right\}}\mathrm{bullet}\to {D}_{•}$ that restricted to the two copies of ${C}_{•}$ is $f$ and $g$, respectively, is characterized by a collection of commuting diagrams

$\begin{array}{ccc}{C}_{n+1}\oplus {C}_{n+1}\oplus {C}_{n}& \stackrel{\left({f}_{n+1},{g}_{n+1},{\psi }_{n}\right)}{\to }& {D}_{n}\\ {}^{}↓& & {↓}^{{\partial }^{D}}\\ {C}_{n}\oplus {C}_{n}\oplus {C}_{n-1}& \stackrel{\left({f}_{n},{g}_{n},{\psi }_{n-1}\right)}{\to }& {D}_{n-1}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ C_{n+1}\oplus C_{n+1} \oplus C_{n} &\stackrel{(f_{n+1},g_{n+1}, \psi_n)}{\to}& D_n \\ {}^{\mathllap{}}\downarrow && \downarrow^{\mathrlap{\partial^D}} \\ C_{n} \oplus C_{n} \oplus C_{n-1} &\stackrel{(f_n,g_n,\psi_{n-1})}{\to} & D_{n-1} } \,.

On the elements $\left(1,0,0\right)$ and $\left(0,1,0\right)$ in the top left this reduces to the chain map condition for $f$ and $g$, respectively. On the element $\left(0,0,1\right)$ this is the equation for the chain homotopy

${f}_{n}-{g}_{n}-{\psi }_{n-1}{d}_{C}={d}_{D}{\psi }_{n}\phantom{\rule{thinmathspace}{0ex}}.$f_n - g_n - \psi_{n-1} d_C = d_D \psi_{n} \,.

### Homotopy equivalence

Let ${C}_{•},{D}_{•}\in {\mathrm{Ch}}_{•}\left(𝒜\right)$ be two chain complexes.

###### Definition

Define the relation chain homotopic on $\mathrm{Hom}\left({C}_{•},{D}_{•}\right)$ by

$\left(f\sim g\right)⇔\exists \left(\psi :f⇒g\right)\phantom{\rule{thinmathspace}{0ex}}.$(f \sim g) \Leftrightarrow \exists (\psi : f \Rightarrow g) \,.
###### Proposition

Chain homotopy is an equivalence relation on $\mathrm{Hom}\left({C}_{•},{D}_{•}\right)$.

###### Proposition

Write $\mathrm{Hom}\left({C}_{•},{D}_{•}{\right)}_{\sim }$ for the quotient of the hom set $\mathrm{Hom}\left({C}_{•},{D}_{•}\right)$ by chain homotopy.

###### Proposition

This quotient is compatible with composition of chain maps.

Accordingly the following category exists:

###### Definition

Write $𝒦\left(𝒜\right)$ for the category whose objects are those of ${\mathrm{Ch}}_{•}\left(𝒜\right)$, and whose morphisms are chain homotopy classes of chain maps:

${\mathrm{Hom}}_{𝒦\left(𝒜\right)}\left({C}_{•},{D}_{•}\right)≔{\mathrm{Hom}}_{{\mathrm{Ch}}_{•}\left(𝒜\right)}\left({C}_{•},{D}_{•}{\right)}_{\sim }\phantom{\rule{thinmathspace}{0ex}}.$Hom_{\mathcal{K}(\mathcal{A})}(C_\bullet, D_\bullet) \coloneqq Hom_{Ch_\bullet(\mathcal{A})}(C_\bullet, D_\bullet)_\sim \,.

This is usually called the homotopy category of chain complexes in $𝒜$.

###### Remark

Beware, as discussed there, that another category that would deserve to carry this name instead is called the derived category of $𝒜$. In the derived category one also quotients out chain homotopy, but one allows that first the domain of the two chain maps $f$ and $g$ is refined along a quasi-isomorphism.

## References

Section 1.4 of

Revised on September 25, 2012 14:47:17 by Anonymous Coward (130.230.176.186)