symmetric monoidal (∞,1)-category of spectra
For a group and pair of objects equipped with -action, the conjugation action of on a morphism (not necessarily respecting the -action) is for given by
The invariants (fixed points) of the conjugation action are the maps which do respect the -action, hence the homomorphisms for the -action, hence the -equivariant maps.
In the case that the -action on is trivial, the conjugation action becomes the precomposition action and in the case that the action on is trivial it becomes 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
The conjugation action construction of def. is the internal hom in the category of actions.
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:
The invariants of the conjugation action on is the set of action homomorphisms/intertwiners.
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
Regarding the conjugation action as the internal hom of actions immediately gives the generalization of this concept to more general kinds of actions, notably to infinity-actions in general (infinity,1)-toposes. See at infinity-action – Conjugation action for more on this.
Last revised on August 26, 2024 at 05:40:14. See the history of this page for a list of all contributions to it.