I’d like to step aside from homology because I’m on the road and I can throw off a post about weak 2-categories in my sleep by now.
When we were talking about enriched categories we mentioned the case of 2-categories, where between each pair of objects we have a hom-category and so on. We also mentioned that if a 2-category has only one object it’s the same thing as a strict monoidal category. But monoidal categories aren’t generally strict! That is, the monoidal product isn’t associative “on the nose”, but only up to a natural isomorphism. And so we should look for the same sort of thing to happen in the case of more general 2-categories.
So, a (weak) 2-category will have a collection of objects. Between each pair of objects there will be a category whose objects we call 1-morphisms from to . Between two such 1-morphisms we have a set of “2-morphisms” .
We can “vertically” compose 2-morphisms using the composition from a given hom-category. That is, given and we have . Of course, there is an “identity” 2-morphism on any 1-morphism , and the composition is associative. At this top level, everything is just like any other category.
Down at the level of 1-morphisms, things get hairier. For each triple of objects we have a functor . This functor is not required to be associative in the usual sense, nor to have identities. Instead, for every triple of 1-morphisms , , and , there is a 2-isomorphism which replaces the associative law. Similarly, for each object we have a 1-morphism and for each 1-morphism we have 2-morphisms and to replace the left and right unit laws.
Now because is a functor, it also acts on 2-morphisms. That is, if we have 1-morphisms and , and 2-morphisms and , then we have a “horizontal” composition .
Functoriality says that this horizontal composition has to preserve the vertical composition inside each hom-category. So let’s take 1-morphisms and . Then take 2-morphisms , , , and . We can vertically compose to get and . These can then be horizontally composed to get . On the other hand we could have composed horizontally before vertically and obtained . Functoriality tells us that these two 2-morphisms must be the same. We call this equation the “exchange identity”.
As an exercise, write all this out in the case where we just have one object. Then there’s only one hom-category to worry about. Verify that this restates the definition of a (weak) monoidal category, and show what the exchange identity means in this case.