Chain Homotopies

We’ve defined chain complexes in an abelian category, and chain maps between them, to form an -category . Today, we define chain homotopies between chain maps, which gives us a 2-category.

First, we say that a chain map given by is “null-homotopic” if we have arrows such that . Here’s the picture:

In particular, the zero chain map with for all is null-homotopic — just pick .

Now we say that chain maps and are homotopic if is null-homotopic. That is, . We call the collection a chain homotopy from to . Then a chain map is null-homotopic if and only if it is homotopic to the zero chain map. We can easily check that this is an equivalence relation. Any chain map is homotopic to itself because and the zero chain map is null-homotopic. If and are homotopic by a chain homotopy , then is a chain homotopy from to . Finally, if is null-homotopic by and is null-homotopic by , then is null-homotopic by .

Another way to look at this is to note that we have an abelian group of chain maps from to , and the null-homotopic maps form a subgroup. Then two chain maps are homotopic if and only if they differ by a null-homotopic chain map, which leads us to consider the quotient of by this subgroup. We will be interested in properties of chain maps which are invariant under chain homotopies — properties that only depend on this quotient group.

In the language of category theory, the homotopies are 2-morphisms. Given 1-morphisms (chain maps) , , and from to , and homotopies and , we compose them by simply adding the corresponding components to get .

On the other hand, if we have 1-morphisms and from to , 1-morphisms and from to , and 2-morphisms and , then we can “horizontally” compose these chain homotopies to get with components . Indeed, we calculate

We could also have used and done a similar calculation. In fact, it turns out that and are themselves homotopic in a sense, and so we consider them to be equivalent. If we pay attention to this homotopy between homotopies, we get a structure analogous to the tensorator. I’ll leave you to verify the exchange identity on your own, which will establish the 2-categorical structure of .

One thing about this structure that’s important to note is that every 2-morphism is an isomorphism. That is, if two chain maps are homotopic, they are isomorphic as 1-morphisms. Thus if we decategorify this structure by replacing 1-morphisms by isomorphism classes of 1-morphisms, we are just passing from chain maps to homotopy classes of chain maps. In other words, we pass from the abelian group of chain maps to its quotient by the null-homotopic subgroup.