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.
In matrix calculus conjugation actions are also known as similarity transformations.
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