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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.
Sometimes a chain homotopy is called a homotopy operator. This is the terminology common for instance in the standard proof of the Poincaré lemma.
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. 1.
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
in $\mathcal{A}$ such that
It may be useful to illustrate this with the following graphics, which however is not a commuting diagram:
Instead, a way to encode chain homotopies by genuine diagrammatics is below in prop. 1.
Let
be the normalized chain complex in $\mathcal{A}$ of the simplicial chains on the simplicial 1-simplex:
This is the standard interval in chain complexes. Indeed it is manifestly the “abelianization” of the standard interval object in sSet/Top.
A chain homotopy $\psi : f \Rightarrow g$ is equivalently a commuting diagram
in $Ch_\bullet(\mathcal{A})$, hence a genuine left homotopy with respect to the interval object in chain complexes.
For notational simplicity we discuss this in $\mathcal{A} =$ Ab.
Observe that $N_\bullet(\mathbb{Z}(\Delta[1]))$ is the chain complex
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
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
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
Let $C_\bullet, D_\bullet \in Ch_\bullet(\mathcal{A})$ be two chain complexes.
Define the relation chain homotopic on $Hom(C_\bullet, D_\bullet)$ by
Chain homotopy is an equivalence relation on $Hom(C_\bullet,D_\bullet)$.
Write $Hom(C_\bullet,D_\bullet)_{\sim}$ for the quotient of the hom set $Hom(C_\bullet,D_\bullet)$ by chain homotopy.
This quotient is compatible with composition of chain maps.
Accordingly the following category exists:
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:
This is usually called the homotopy category of chain complexes in $\mathcal{A}$.
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.
chain homotopy
Section 1.4 of