Enriched Adjunctions
When I started in on adjoint functors, I gave the definition in terms of a bijection of hom-sets. Then I showed that we can also specify it in terms of its unit and counit. Both approaches (and their relationship) generalize to the enriched setting.
Given a functor 
and another 
, an adjunction is given by natural transformations 
and 
. These transformations must satisfy the equations 
and 
. By the weak Yoneda Lemma, this is equivalent to giving a 
-natural isomorphism 
.
Indeed, a -natural transformation in this direction must be of the form 
, and one in the other direction must be of the form 
. The equations 
and 
are equivalent, by the weak Yoneda Lemma, to the equations satisfied by the unit and counit of an adjunction.
The 
-functor 
that sends an enriched category to its underlying ordinary category sends an enriched adjunction to an ordinary adjunction. The function underlying the -natural isomorphism 
is the bijection of this underlying adjunction.
As we saw before, a -functor 
has a left adjoint 
if and only if 
is representable for each 
. Also, an enriched equivalence is an enriched adjunction whose unit and counit are both -natural isomorphisms. Just as for ordinary adjunctions, we have transformations between enriched adjunctions, a category of enriched adjunctions between two enriched categories, enriched adjunctions with parameters, and so on.
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September 4, 2007 -
Posted by John Armstrong |
Category theory
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