# Limit of a group action in $\mathbb{F}-\mathbf{Vect}$

Given a group $G$, one can associate a category $\mathcal{G}$, which consists of one object $\star$ and one morphism $m_g$ for every group element $g\in G$. Furthermore, given a $\mathbb{F}$-vectorspace $V$, an action of $G$ on $V$ is just a functor $A\colon\mathcal{G}\to\text{$\mathbb{F}$-$\mathbf{Vect}$}$, which maps $\star$ to $V$, and the morphisms in $\mathcal{G}$ into the automorphism group of $V$. It is well known, that the quotient $V/G$ is just the categorical colimit of said functor, i.e. $(V/G,\pi) = \operatorname{Colim} A$, where $\pi$ is the associated quotient map.

I'm looking for the limit of $A$, which should be a tupel $(V_G, \varphi)$ where $V_G$ is a $\mathbb{F}$-vectorspace, and $\varphi\colon V_G\to V$ is a linear map, which satisfies $\varphi(v_G) = g\varphi(v_G)$ for all $v_G$ in $V_G$ and $g$ in $G$. And which is universal in the sense that for any other tuple $(\widetilde{V_G},\widetilde{\varphi})$ which satisfies that property, there exists a unique morphism $\psi\colon\widetilde{V_G}\to V_G$ such that $\widetilde{\varphi} = \varphi\circ\psi$.

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Well, what else could it be but the subspace of invariants? –  Zhen Lin Aug 16 '12 at 14:41
Uhm, yeah. I guess I overthought this, I was desperately looking for something that has the subspace of invariants as an image. But simply taking the subspace itself and the inclusionmap as morphism should be it. –  roman Aug 16 '12 at 18:29