Fix a group . We are asked to find interesting functors between and (the category of left -sets). Here, instead of considering the objects of to be functors , simply consider them to be pairs where satisfies .
An obvious start would be the forgetful functor that takes a -set to its underlying set. We ask if this functor has an adjoint. Indeed, let be defined as follows: for a set , where . For a map of sets , is the map , which is easily verified to be -equivariant. Notice this is essentially a ‘free’ functor, taking any set to a -set in the most general arbitrary way.
Let’s make this functor more precise for the sake of pedantry. Let be a -set and a set. If is a map of sets, we define as . In the other direction, if is a -equivariant map, define where iff .
This idea also works for the categories of vector fields over and of -linear representations of . Again let be the forgetful functor. This has a left adjoint where, for a -vector space with basis and assuming , we have which has basis , equipped with the representation . This is the so-called ‘induced representation’ from representation theory in the special case that the sub-representation is the trivial representation of the trivial subgroup. Again, the ‘induced’ representation captures the idea of the ‘most general’ representation that extends another.