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One usually considers the category of complex linear group representations for a fixed group $G$. It is defined as the category whose objects are group morphisms $G \rightarrow GL(V)$ where $V$ is a complex vector space and whose morphisms are $G$-equivariant linear maps (equivalently, it is the category of functors $Vect_\mathbb{C}^G$).

My question is: can we consider a category of group representations where the group is allowed to change?

I would be tempted to define it as follows. Objects would be triples $(V, G, \pi: G \rightarrow GL(V))$ and morphisms $\big(V, G, \pi: G \rightarrow GL(V)\big) \xrightarrow{\Phi} \big(V', G', \pi': G' \rightarrow GL(V')\big)$ would be pairs $\Phi:=(\phi, \alpha)$ where $\phi$ is a linear map from $V$ to $V'$ and $\alpha$ is a group morphism from $G$ to $G'$ such that the following property is satisfied: $$\forall g \in G, \pi'(\alpha(g)) \circ \phi= \phi \circ \pi(g)$$

Does this definition make sense? If yes, why is this category never considered?

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    $\begingroup$ This is an example of the Grothendieck construction. $\endgroup$ – Zhen Lin Dec 21 '15 at 11:23
  • $\begingroup$ @ZhenLin, could you develop a little bit more (as I am not a mathematician and I am not familiar with Grothendieck's work)? Which construction are you talking about? Do you have a reference I could read about it? $\endgroup$ – FZalamea Dec 21 '15 at 11:41
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    $\begingroup$ It is only known as the Grothendieck construction, that is the problem. $\endgroup$ – Zhen Lin Dec 21 '15 at 11:44
  • $\begingroup$ The fact that the normally considered category is equivalent to functors $\operatorname{Vect}^G_\mathbb{C}$ - do you have any reference to a source where this is stated/proved? (And I'm sorry to post this as an answer, I wanted to comment, but I am new and not allowed yet :-( ) $\endgroup$ – P. Schulze Dec 13 '18 at 13:55
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The Grothendieck construction (see e. g. wikipedia for start) is the following:

Let $C$ be any category and $F \colon C \to \def\Cat{\mathsf{Cat}}\Cat$ a functor. The Grothendieck construction for $F$ is the following category $\Gamma(F)$:

  • Objects of $\Gamma(F)$ are pairs $(A, x)$, where $A \in \def\Ob{\mathrm{Ob}}\Ob(C)$ and $x \in \Ob\bigl(F(A)\bigr)$.
  • $\def\Hom{\mathrm{Hom}}$Morphisms $f\colon (A,x) \to (B,y)$ are pairs $f = (f_0, f_1)$ where $f_0 \in \Hom_C(A,B)$ and $f_1 \in \Hom_{F(B)}(F(f_0)x, y)$.
  • Composition of $f \in \Hom{\Gamma(F)}((A,x), (B,y))$ and $g \in \Hom_{\Gamma(F)}((B,y), (D,z))$ is given by $$ (g_0, g_1) \circ (f_0, f_1) = (g_0 \circ_C f_0, g_1 \circ_{F(D)} F(g_0)f_1\bigr) $$

Your construction is the Grothendieck construction for the functor $F \colon \mathsf{Group}^{\rm op} \to \mathsf{Cat}$, $F(G) = [G, \def\V{\mathsf{Vect}_{\mathrm C}}\V]$, which sends a group to the category of its representations.

A morphism $f \colon (G, \pi) \to (G', \pi')$ in the Grothendieck construction is a pair $(\alpha, \tau)$, where $\alpha \colon G \to G'$ is a group homomorphism and $\tau \colon F(\alpha)\pi \to \pi'$ is a natural transformation of functors $G' \to \V$. Such a natural transformation is (as $G'$ only has one object), a $\V$-morphism, that is a linear map.

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