Algebras and modules
Model category presentations
Geometry on formal duals of algebras
For a group and and two objects equipped with -action, the conjugation action on morphisms (not necessarily respecting the -action) is for
The invariants of the conjugation action are the -action homomorphism.
In the case that the -action on is trivial, this is the precomposition action and in the case that the action on is trivial this is the postcomposition action.
Given a discrete group and two -actions and on sets and , respectively, then the function set is naturally equipped with the conjugation action
which takes to
We need to show that for any three permutation representations, functions
which intertwine the -action on with the conjugation action on are in natural bijection with functions
which intertwine the diagonal action on the Cartesian product with the action on .
The condition on means that for all and it sends
This is equivalently a function of two variables which sends
Since this has to hold for all values of the variables, it has to hold when substituing with . After this substitution the above becomes
This is the intertwining condition on . Conversely, given satisfying this for all values of the variables, then running the argument backwards shows that its hom-adjunct satisfies its required intertwining condition.
The following is immediate but conceptually important:
Hence the inclusion of invariants into the conjugation action gives the inclusion of the external hom set of the category of -action into the set underlying the internal hom