Paths and cylinders
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 Ab be the abelian category of abelian groups. Write for the category of chain complexes in .
A chain homotopy is a homotopy in . We first give the explicit definition, the more abstract characterization is below in prop. 1.
A chain homotopy between two chain maps in is a sequence of morphisms
in such that
Instead, a way to encode chain homotopies by genuine diagrammatics is below in prop. 1.
In terms of general homotopy
be the normalized chain complex in 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 is equivalently a commuting diagram
in , hence a genuine left homotopy with respect to the interval object in chain complexes.
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 and the other is 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
Write for the quotient of the hom set by chain homotopy.
Accordingly the following category exists:
Write for the category whose objects are those of , and whose morphisms are chain homotopy classes of chain maps:
This is usually called the homotopy category of chain complexes in .
Section 1.4 of