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chain homotopy

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

A chain homotopy is a homotopy in Ch (𝒜)Ch_\bullet(\mathcal{A}). We first give the explicit definition, the more abstract characterization is below in prop. 1.

Definition

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

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

in 𝒜\mathcal{A} such that

f ng n= Dψ n+ψ n1 C. 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:

C n+1 f n+1g n+1 D n+1 n C ψ n n D C n f ng n D n n1 C ψ n1 n1 D C n1 f n1g n1 D n1 . \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 (C(Δ[1])) I_\bullet \coloneqq N_\bullet(C(\Delta[1]))

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

I =[00𝟙(id,id)𝟙𝟙]. 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 ψ:fg\psi : f \Rightarrow g is equivalently a commuting diagram

C f C I (f,g,ψ) D g C \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 Ch (𝒜)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 𝒜=\mathcal{A} = Ab.

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

(00(id,id)00) ( \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}\{0,1\} of 0-simplices in Δ[1]\Delta[1]. The other copy of \mathbb{Z} is the free abelian group on the single non-degenerate edge in Δ[1]\Delta[1]. All other cells of Δ[1]\Delta[1] are degenerate and hence do not contribute to the normalized chain complex. The single nontrivial differential sends 11 \in \mathbb{Z} to (1,1)(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 I C_\bullet \otimes I_\bullet is

C 1C 2C 2C 0C 1C 1C 1C 0C 0. \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 (f,g,ψ):C I D (f,g,\psi) : C_\bullet \otimes I_\bullet \to D_\bullet that restricted to the two copies of C C_\bullet is ff and gg, respectively, is characterized by a collection of commuting diagrams

C n+1C n+1C n (f n+1,g n+1,ψ n) D n D C nC nC n1 (f n,g n,ψ n1) D n1. \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 (1,0,0)(1,0,0) and (0,1,0)(0,1,0) in the top left this reduces to the chain map condition for ff and gg, respectively. On the element (0,0,1)(0,0,1) this is the equation for the chain homotopy

f ng nψ n1d C=d Dψ n. f_n - g_n - \psi_{n-1} d_C = d_D \psi_{n} \,.

Homotopy equivalence

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

Definition

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

(fg)(ψ:fg). (f \sim g) \Leftrightarrow \exists (\psi : f \Rightarrow g) \,.
Proposition

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

Proposition

Write Hom(C ,D ) Hom(C_\bullet,D_\bullet)_{\sim} for the quotient of the hom set Hom(C ,D )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 𝒦(𝒜)\mathcal{K}(\mathcal{A}) for the category whose objects are those of Ch (𝒜)Ch_\bullet(\mathcal{A}), and whose morphisms are chain homotopy classes of chain maps:

Hom 𝒦(𝒜)(C ,D )Hom Ch (𝒜)(C ,D ) . 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\mathcal{A}.

Remark

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

References

Section 1.4 of

Revised on June 5, 2014 08:26:10 by Ingo Blechschmidt (137.250.162.16)