Last time we took an adjunction and came up with two natural transformations, weakened versions of the natural isomorphisms defining an equivalence. Today we’ll see how to go back the other way.
So let’s say we have an adjunction 
given by natural isomorphism 
. Remember that we defined the unit and counit by 
and 
. We can take either one of these and reverse-engineer it. For instance, given an arrow 
in 
we can calculate
so once we know the unit of the adjunction we can calculate 
from it. Notice how we use the naturality of in the second equality.
Dually, we can determine 
in terms of the counit. Given 
in 
, we calculate:
so we can also determine the natural isomorphism of hom-sets in terms of the counit.
Of course, since we can determine the same isomorphism (technically the isomorphism and its inverse) from either the unit or the counit, they must be related. So what do these equations really tell us?
For this we have to go back to the way we compose natural transformations. The obvious way is where we have natural transformations 
and 
between three functors from to . We put them together to get 
.
Less obviously, we can consider functors 
and 
from to , functors 
and 
from to 
, and natural transformations 
and 
. We can put these together to get 
, defined by 
or 
(exercise: show that these two composites are equal).
Now what we need here is this “horizontal” composite. Let’s go back to the adjunction and take the natural transformations 
and 
. The components of their horizontal composite 
is then given by 
. Similarly, if we take the natural transformations 
and 
, their horizontal composite has components given by 
. Now the “vertical” composite of these two 
has components 
. And the above formula for the adjunction isomorphism in terms of the unit tells us that this is 
.
To put it at a bit of a higher level, if we start with the functor 
, use the unit to turn it into the functor 
, then use the counit to move back to , the composite natural transformation is just the identity transformation on . Similarly, we can show that the composite 
taking 
to 
and back to is the identity transformation on .
Inherent in this is also the converse statement. If we have natural transformations and satisfying these two identities, then we can use the above formulae to define a natural isomorphism 
in terms of 
and its inverse in terms of 
. Thus an adjunction is determined by a unit and a counit satisfying these “quasi-inverse” relations.
If you’re up to it, try to see where we’ve seen these quasi-inverse relations before in a completely different context. I’ll be coming back to this later.
July 18, 2007
Posted by John Armstrong |
Category theory |
1 Comment
