nLab chain homotopy

Idea

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

Sometimes a chain homotopy is called a homotopy operator. This is the terminology common for instance in the standard proof of the Poincaré lemma.

Definition

Let $\mathcal{A} =$ Ab be the abelian category of abelian groups. Write $Ch_\bullet(\mathcal{A})$ for the category of chain complexes in $\mathcal{A}$.

A chain homotopy is a homotopy in $Ch_\bullet(\mathcal{A})$. We first give the explicit definition, the more abstract characterization is below in prop. .

Definition

A chain homotopy $\psi : f \Rightarrow g$ between two chain maps $f,g : C_\bullet \to D_\bullet$ in $Ch_\bullet(\mathcal{A})$ is a sequence of morphisms

$$\{ (\psi_n : C_n \to D_{n+1}) \in \mathcal{A} | n \in \mathbb{N} \}$$

in $\mathcal{A}$ such that

$$f_n - g_n = \partial^D \circ \psi_n + \psi_{n-1} \circ \partial^C \,.$$

Remark

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

$$\array{ \vdots && \vdots \\ \downarrow && \downarrow \\ C_{n+1} &\stackrel{f_{n+1} - g_{n+1}}{\to}& D_{n+1} \\ \downarrow^{\mathrlap{\partial^C_{n}}} &\nearrow_{\mathrlap{\psi_{n}}}& \downarrow^{\mathrlap{\partial^D_{n}}} \\ C_n &\stackrel{f_n - g_n}{\to}& D_n \\ \downarrow^{\mathrlap{\partial^C_{n-1}}} &\nearrow_{\mathrlap{\psi_{n-1}}}& \downarrow^{\mathrlap{\partial^D_{n-1}}} \\ C_{n-1} &\stackrel{f_{n-1} - g_{n-1}}{\to}& D_{n-1} \\ \downarrow && \downarrow \\ \vdots && \vdots } \,.$$

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

Properties

In terms of general homotopy

Definition

Let

$$I_\bullet \coloneqq N_\bullet(C(\Delta[1]))$$

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

$$I_\bullet = [ \cdots \to 0 \to 0 \to \mathbb{1} \stackrel{(id,-id)}{\to} \mathbb{1} \oplus \mathbb{1} ] \,.$$

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 $\psi : f \Rightarrow g$ is equivalently a commuting diagram

$$\array{ C_\bullet \\ \downarrow & \searrow^{\mathrlap{f}} \\ I_\bullet\otimes C_\bullet &\stackrel{(f,g,\psi)}{\to}& D_\bullet \\ \uparrow & \nearrow_{\mathrlap{g}} \\ C_\bullet }$$

in $Ch_\bullet(\mathcal{A})$, hence a genuine left homotopy with respect to the interval object in chain complexes.

Proof

For notational simplicity we discuss this in $\mathcal{A} =$ Ab.

Observe that $N_\bullet(\mathbb{Z}(\Delta[1]))$ is the chain complex

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

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

It follows that the tensor product of chain complexes $C_\bullet \otimes I_\bullet$ is

$$\cdots \to C_2 \oplus C_{2} \oplus C_1 \to C_1 \oplus C_{1} \oplus C_0 \to C_0 \oplus C_0 \oplus C_{-1} \to \cdots \,.$$

Therefore a chain map $(f,g,\psi) : C_\bullet \otimes I_\bullet \to D_\bullet$ that restricted to the two copies of $C_\bullet$ is $f$ and $g$, respectively, is characterized by a collection of commuting diagrams

$$\array{ C_{n+1}\oplus C_{n+1} \oplus C_{n} &\stackrel{(f_{n+1},g_{n+1}, \psi_n)}{\to}& D_{n+1} \\ {}^{\mathllap{}}\downarrow && \downarrow^{\mathrlap{\partial^D}} \\ C_{n} \oplus C_{n} \oplus C_{n-1} &\stackrel{(f_n,g_n,\psi_{n-1})}{\to} & D_n } \,.$$

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

$$f_n - g_n - \psi_{n-1} d_C = d_D \psi_{n} \,.$$

Homotopy equivalence

Let $C_\bullet, D_\bullet \in Ch_\bullet(\mathcal{A})$ be two chain complexes.

Definition

Define the relation chain homotopic on $Hom(C_\bullet, D_\bullet)$ by

$$(f \sim g) \Leftrightarrow \exists (\psi : f \Rightarrow g) \,.$$

Proposition

Chain homotopy is an equivalence relation on $Hom(C_\bullet,D_\bullet)$.

Proposition

Write $Hom(C_\bullet,D_\bullet)_{\sim}$ for the quotient of the hom set $Hom(C_\bullet,D_\bullet)$ by chain homotopy.

Proposition

This quotient is compatible with composition of chain maps.

Accordingly the following category exists:

Definition

Write $\mathcal{K}(\mathcal{A})$ for the category whose objects are those of $Ch_\bullet(\mathcal{A})$, and whose morphisms are chain homotopy classes of chain maps:

$$Hom_{\mathcal{K}(\mathcal{A})}(C_\bullet, D_\bullet) \coloneqq Hom_{Ch_\bullet(\mathcal{A})}(C_\bullet, D_\bullet)_\sim \,.$$

This is usually called the homotopy category of chain complexes in $\mathcal{A}$.

Remark

Beware, as discussed there, that another category that would deserve to carry this name instead is called the derived category of $\mathcal{A}$. 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.

Related concepts

• null homotopy

• category of chain complexes

  • chain complex

  • chain map, quasi-isomorphism

  • chain homotopy

• homotopy category of chain complexes

References

Historical articles:

• Tibor Rado , A Remark on Chain-Homotopy, Proceedings of the American Mathematical Society Vol. 2, No. 3 (Jun., 1951), pp. 458-463 (jstor:2031776, doi:10.2307/2031776)

Review:

• Charles Weibel, Section 1.4 of: An Introduction to Homological Algebra