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the old entry representation contained an old query box with some discussion.
I am hereby moving this old discussion from there to here:
+– {: .query} I don't agree with this $D \coloneqq Aut(V)$ business. A $k$-linear representation of a group $G$ is a functor from $\mathbf{B}G$ to $k Vect$, period. Because $\mathbf{B}G$ has one object (or is pointed), we can pick out an object $V$ of $k Vect$, and it was remiss of me not to mention this (and the language ‘on $V$’ vs ‘in $D$’. But we usually don't want $D$ to actually be $Aut(V)$ instead of $k Vect$; when doing representation theory, we fix $G$ and fix $k$ (or fix $D$ in some other way), but we don't fix $V$. —Toby
If you look at the textbooks of representation of groups, then they start with representation of groups as homomorphisms of groups, that is just functors. Then they say, that usually the target groups are groups of automorphisms of some other objects. And at the end they say that one usually restricts just to linear automorphisms of linear objects when linearizing the general problematics to the linear one. Now the fact that in some special case there is a category which expresses the same fact does not extend to other symmetry objects, like for representations of vertex operator algebras, pseudotensor categories etc. I mean End(something) or Aut(something) is just inner end in some setup like in closed monoidal category, but there are symmetries in mathematics which have a notion of End of Aut for a single object but do not have good notion of category one level up which has inner homs leading to the same End or Aut. Conceptually actions are about endosymmetries or symmetries (automorphisms) being reducable to categorical ones but not necessarily, I think. In a way you say that you are sure that any symmetry of another object can be expressed internally in some sort of a higher category of such objects, what is to large extent true, but I am sure not for absolutely all examples.
(for “on” terminology:) Ross Street uses monads in a 2-category and monads on a 1-category and I know of no objects in category theory.
Another important thing is that the endomorphisms are by definitions often equipped with some additional (e.g. topological) structure which is not necessarily coming from some enrichement of the category of objects. –Zoran
(Zoran on word “classical representation” being just for groups: so the representations of associative algebras, Lie algebras, Leibniz algebras, topological groups, quivers, are not classical ??).
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