Now that we know how to transform adjoints, we can talk about whole families of adjoints parametrized by some other category. That is, for each object 
of the parametrizing category 
we’ll have an adjunction 
, and for each morphism of we’ll have a transformation of the adjunctions.
Let’s actually approach this from a slightly different angle. Say we have a functor 
, and that for each 
the functor 
has a right adjoint 
. Then I claim that there is a unique way to make 
into a functor from 
to 
so that the bijection 
is natural in all three variables. Note that must be contravariant in here to make the composite functors have the same variance in .
If we hold fixed, the bijection is already natural in 
and 
. Let’s hold and fixed and see how to make it natural in . The components 
are already given in the setup, so we can’t change them. What we need are functions 
and 
for each arrow 
.
For naturality to hold, we need 
. But from what we saw last time this just means that the pair of natural transformations 
forms a conjugate pair from 
to 
. And this lets us define 
uniquely in terms of 
, the counit 
of , and the unit 
of by using the first of the four listed equalities.
From here, it’s straightforward to show that this definition of how acts on morphisms of makes it functorial in both variables, proving the claim. We can also flip back to the original viewpoint to define an adjunction between categories and 
parametrized by the category as a functor from to the category 
of adjunctions between those two categories.
July 31, 2007
Posted by John Armstrong |
Category theory |
4 Comments
