Limits are Adjoints
When considering limits, we started by talking about the diagonal functor . This assigns to an object
the “constant” functor
that sends each object of
to
and each morphism of
to
.
Then towards the end of our treatment of limits we showed that taking limits is a functor. That is, if each functor from
to
has a limit
, then
is a functor from
to
. Dually, if every such functor has a colimit
, then
is also a functor.
And now we can fit these into the language of adjoints: when it exists, the limit functor is right adjoint to the diagonal functor. Dually, the colimit functor is left adjoint to the diagonal functor when it exists. I’ll handle directly the case of colimits, but the limit statements and proofs are straightforward dualizations.
So we definitely have a well-defined functor . By assumption we have for each functor
an object
. If we look at the third entry in our list of ways to specify an adjunction, all we need now is a universal arrow
. But this is exactly how we defined limits! Now the machinery we set up yesterday takes over and promotes this collection of universal arrows into the unit of an adjunction
.
For thoroughness’ sake: the unit of this adjunction is the colimiting cocone, considered as a natural transformation from
to the constant functor on the colimiting object. The counit of this adjunction
is just the identity arrow on
because the colimit of the constant functor is just the constant value. The “quasi-inverse” conditions state that
is the identity natural isomorphism on
, and that
is the identity natural isomorphism on
, both of which are readily checked.
And our original definition of an adjoint here reads that . That is, for each cocone to
on
(one of the natural transformations on the right) there is a unique arrow from the colimiting object of
to
.