*Note: as usual, I will be playing free and loose with category theory in this post. Apologies to those who know better.*

One way to define a subgroup of a group is as the image of a homomorphism into . Given the inclusion map , the functor in the category of groups acts contravariantly to give a map called restriction. More concretely, the restricted representation of a representation is defined simply by . Hence there is a functorial way to pass from a representation of a group to one of a subgroup .

It is not obvious, however, whether there is a functorial way to pass from a representation of back to one of . There is such a construction, which goes by the name of induction, and we will need it later. Today we’ll discuss the general category-theoretic context in which induction is understood, where it is called an **adjoint** functor. For more about adjoints, see (in no particular order) posts at Concrete Nonsense, the Unapologetic Mathematician, and Topological Musings.