Consider the theorem at the end of chapter six:
Theorem: Let be a small category and a category with all limits of shape . Then defines a functor which is right-adjoint to the diagonal functor .
We wish interpret this theorem in the special case that and is a group.
First note that is the category of -sets – that is, groups equipped with a -action . In this scenario, the diagonal functor takes a set to , where is now equipped with the trivial -action where for all and (alternatively, for all ).
To find out what does, we first need to consider what constitutes a cone over a -set. Given a -set , a cone consists of a set and a single map such that for all . To rephrase, that means that for all and . This means that every is stabilized by the whole group . So a cone is a pair where must take each element to a -invariant element. So a universal cone (limit cone) can be computed by the usual construction giving arbitrary limits in . We find that the limit is precisely the subset of consisting of -invariant elements.
We are finally ready to interpret the adjunction. The adjunction means that for any set and -set . Indeed, one can observe that a -equivariant map from to can only map onto -invariant elements of , so of course such a map is determined completely by a map , which is precisely the subset of -invariant elements.
The dual of the preceding theorem says that colimits are left-adjoint to the diagonal functor. In the present context, let’s interpet the functor. Consider some -set . We can use the general construction for colimits in set and find that the colimit is , where is the precisely the equivalence relation induced by the -action (orbits correspond to equivalence classes). So, the adjunction says that . Indeed, its easy to see that for a -equivariant map , all members of an orbit must map on to the same element, so of course such a map is determined by a mapping from the orbits of to .