symmetric monoidal (∞,1)-category of spectra
For $G$ a group and $V_1$ and $V_2$ two objects equipped with $G$-action, the conjugation action on morphisms $f : V_1 \to V_2$ (not necessarily respecting the $G$-action) is for $g \in G$
The invariants of the conjugation action are the $G$-action homomorphism.
In the case that the $G$-action on $V_2$ is trivial, this is the precomposition action and in the case that the action on $V_1$ is trivial this is the postcomposition action.
In matrix calculus conjugation actions are also known as similarity transformations.
Given a discrete group $G$ and two $G$-actions $\rho_1$ and $\rho_2$ on sets $S_1$ and $S_2$, respectively, then the function set $[S_1, S_2]$ is naturally equipped with the conjugation action
which takes $((S_1 \stackrel{f}{\to} S_2), g)$ to
The conjugation action construction of def. 1 is the internal hom in the category of actions.
We need to show that for any three permutation representations, functions
which intertwine the $G$-action on $S_3$ with the conjugation action on $[S_1,S_2]$ are in natural bijection with functions
which intertwine the diagonal action on the Cartesian product $S_3 \times S_1$ with the action on $S_2$.
The condition on $\phi$ means that for all $g\in G$ and $s_3 \in S_3$ it sends
This is equivalently a function $\tilde \phi$ of two variables which sends
Since this has to hold for all values of the variables, it has to hold when substituing $s_1$ with $\rho_1(s_1)(g)$. After this substitution the above becomes
This is the intertwining condition on $\tilde \phi$. Conversely, given $\tilde \phi$ satisfying this for all values of the variables, then running the argument backwards shows that its hom-adjunct $\phi$ satisfies its required intertwining condition.
The following is immediate but conceptually important:
The invariants of the conjugation action on $[S_1,S_2]$ 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 $G$-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.