A chain homotopy between two chain maps in is a sequence of morphisms
in 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.
A chain homotopy is equivalently a commuting diagram
For notational simplicity we discuss this in Ab.
Observe that is the chain complex
where the term is in degree 0: this is the free abelian group on the set of 0-simplices in . The other copy of is the free abelian group on the single non-degenerate edge in . All other cells of are degenerate and hence do not contribute to the normalized chain complex. The single nontrivial differential sends to , 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 is
Therefore a chain map that restricted to the two copies of is and , respectively, is characterized by a collection of commuting diagrams
On the elements and in the top left this reduces to the chain map condition for and , respectively. On the element this is the equation for the chain homotopy
Let be two chain complexes.
Define the relation chain homotopic on by
Chain homotopy is an equivalence relation on .
Accordingly the following category exists:
This is usually called the homotopy category of chain complexes in .
Beware, as discussed there, that another category that would deserve to carry this name instead is called the derived category of . In the derived category one also quotients out chain homotopy, but one allows that first the domain of the two chain maps and is refined along a quasi-isomorphism.
Section 1.4 of