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There was an old entry faithful representation. I have edited and expanded it a little. While I was was at it, I have added the definition of faithful infinity-actions in an infinity-topos.
(Nothing non-trivial here, just for completeness.)
Are $V$ and $G$ wrongly swapped in some of the exposition?
morphism $\mathbf{B}G \longrightarrow \mathbf{B}\mathbf{Aut}(V) \hookrightarrow Obj_\kappa$
$\tilde \rho \colon G \longrightarrow \mathbf{Aut}(V) \,.$
Shouldn’t these be
morphism $\mathbf{B}G \longrightarrow \mathbf{B}\mathbf{Aut}(G) \hookrightarrow Obj_\kappa$
$\tilde \rho \colon V \longrightarrow \mathbf{Aut}(G) \,.$
or am I missing some of the mysteries of deloopling?
In the pull back diagram
$\array{ V &\longrightarrow& V/G &\longrightarrow& &\longrightarrow& \widehat{Obj}_\kappa \\ && \downarrow && \downarrow && \downarrow \\ && \mathbf{B}G &\longrightarrow& \mathbf{B}\mathbf{Aut}(V) &\hookrightarrow& Obj_\kappa } \,.$
shouldn’t the blank spot be filled in as
$\array{ V &\longrightarrow& V/G &\longrightarrow&Aut(G) &\longrightarrow& \widehat{Obj}_\kappa \\ && \downarrow && \downarrow && \downarrow \\ && \mathbf{B}G &\longrightarrow& \mathbf{B}\mathbf{Aut}(G) &\hookrightarrow& Obj_\kappa } \,.$
or was that spot left blank for a reason?
And should $Aut(G) \longrightarrow \widehat{Obj}_\kappa$ be $Aut(G) \hookrightarrow \widehat{Obj}_\kappa$?
No. Wait, $G$ is the group that acts. Why would you consider $Aut(G)$?
This issue has nothing to do with delooping: an action of a group $G$ on some $V$ is equivalently a group homomorphism $G \to Aut(V)$.
The name of what used to be a blank spot is $V/\mathbf{Aut}(V)$. I have filled it in.
Sorry. I’m the one with a definite swapping problem. But at least I got you to fill in the blank. :)
From the way I might understand things now, maybe the exposition could be clearer by being a bit more explicit and stating the goal up front.
start from action: $\rho \;\colon\; G \otimes V \longrightarrow V$
want “currying”: $\tilde \rho \colon G \longrightarrow \mathbf{Aut}(V)$
$\tilde \rho$ turns out to be the adjoint in some sense of something in the pullback diagram (I’m guessing $V/G \longrightarrow \mathbf{BAut}(V)$).
Is this adjoint special in any way? Or is there no way to derive $\tilde \rho$ from an adjoint in the diagram, and it is only in some sense the adjoint of the whole thing.
Also would it be possible to label the arrows in the pullback to indicate which are normally epis (e) and monos( m) (there are no LaTex up/down hook arrows for monos and no split-tail arrows for epis) and which are (E) and (M) when $\rho$ is faithful?
Rod, let’s focus these questions of yours first on the section with the traditional definition.
The adjunction is question is the tensor-hom adjunction $(V \otimes -) \dashv [V,-]$. Under this adjunction, $\tilde \rho \colon G \to [V,V]$ is the adjunct of $\rho \colon G \otimes V \to V$. This is the operation that is also called currying (the entry currying almost says this, somebody should edit it such as to say this more explicitly).
I have made this more explicit in the “traditional”-section.
I have started a section Properties with the brief statement that algebraic groups have finite-dimensional faithful representations, and that every other finite-dimensional representation is a subquotient of any of these.
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