Arrow Categories
One very useful example of a category is the category of arrows of a given category .
We start with any category with objects
and morphisms
. From this we build a new category called
, for reasons that I’ll explain later. The objects of
are just the morphisms of
. The morphisms of this new category are where things start getting interesting.
Let’s take two objects of — that is, two morphisms of
— and lay them side-by-side:
Now we want something that transforms one into the other. What we’ll do is connect each of the objects on the left to the corresponding object on the right by an arrow:
and require that the resulting square commute: as morphisms in
. This is a morphism from
to
. Sometimes we’ll write
, and sometimes we’ll name the square and write
.
If we have three morphisms ,
, and
in
, and commuting squares
and
then we can get a commuting square
. We check that this square commutes:
. This gives a composition of commuting squares. It’s easily checked that this is associative.
Given any morphism in
we can just apply the identity arrows to each of
and
to get a commuting square
between
and itself. It is clear that this square serves as the identity arrow on the object
in
, completing our proof that arrows and commuting squares in
do form a category.
Thanks for this — nice crisp explanation. Would be even more helpful if it had a couple of simple, finite examples. Posets maybe?
Well, these sorts of things show up in all sorts of other categorical constructions I’ve discussed, although I haven’t explicitly linked back here. Look at categories of natural transformations, or of representations, or (coming eventually) bundles.
How do you show that $hom(f,g)$ and $hom(f’,g’)$ are disjoint?
Tony, this is probably a bit late for you but they are disjoint by construction. We take all the morphisms as objects and add arrows wherever there are commuting squares – at this point it’s just a graph – and as it turns out we can define associative composition of the arrows by the method given above and we get identity arrows out of it too, so it’s a category.