# Quotient of a category by a group

Let $G$ be a group acting on a category $\mathcal{C}$; that is, a monoidal functor $G\rightarrow \mathrm{Aut}(\mathcal{C})$, where the latter is the 2-group of auto-equivalences of $G$. Explicitly, this means for every $g\in G$ we have an functor $F_g : \mathcal{C}\rightarrow \mathcal{C}$ such that $F_1$ is the identity functor and the functor $F_{g^{-1}}$ is the quasi-inverse to $F_g$. Moreover, for every pair $g,h\in G$, we have a fixed isomorphism of functors $i_{g,h}:F_{gh} \rightarrow F_g \circ F_h$, so that for for any triple $g,h,k\in G$, we have the equality $i_{h,k} i_{g,hk}=i_{g,h}i_{gh,k}$ of isomorphisms $F_{ghk} \rightarrow F_g\circ F_h \circ F_k$.

Question 1: How does one define the quotient category $\mathcal{C}/G$?

Presumably, giving an action of $G$ on $\mathcal{C}$ in the above sense is equivalent to giving a pseudo-functor $BG\rightarrow \mathbf{Cat}$ where $BG$ denotes the one object groupoid corresponding to $G$ and $\mathbf{Cat}$ denotes the 2-category of all small categories. Therefore, the quotient I am asking about should be the (pseudo) 2-colimit of this functor. There are general constructions of 2-limits discussed for example, here: http://ncatlab.org/nlab/show/2-limit. I guess my question boils down to this: how does one make this 2-colimit explicit in the case at hand?

Note that if the functors $F_g$ are automorphisms, then this question is discussed here Quotient of a category by a group action. However, it seems like the references mentioned there do not discuss the more general case considered in this post. A related question is this:

Question 2: Suppose $\mathcal{C}$ is an abelian (resp. dg) category equipped with an action of $G$ in the above sense, but now we assume that the functors $F_g$'s are exact (resp. dg). How does one define $\mathcal{C}/G$ as an abelian (resp. dg) category?

Remark: Here are the "classical" analogues of these questions. If $G$ acts on a set $S$, then we have the quotient set $S/G$. If $G$ acts on a vector space $V$, then the quotient space is simply the coinvariants $V_G$ of the action. In both cases, this is the colimit of the functor $BG\rightarrow \mathbf{Sets}$ (resp. $BG\rightarrow \mathbf{Vect}$), but to work with these quotients, one needs an explicit description of this colimit. Of course, we have the explicit descriptions in these classical cases.

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Perhaps via an 'action 2-category' - this could be seen as a sort of weak quotient. If you want a category out of this, take isomorphism classes of 1-arrows in this 2-category. – David Roberts Oct 15 '13 at 0:24
But see also mathoverflow.net/questions/22686/… – David Roberts Oct 15 '13 at 0:25
Taking isomorphisms classes of 1-arrows in the action 2-category does indeed seem reasonable. I wonder if there is a reference where this is done and where it is checked that this indeed does give the colimit of the functor $BG\rightarrow \mathbf{Cat}$. – Dr. Evil Oct 15 '13 at 0:52
I'm not aware of one, but that's not to say it hasn't been done (e.g. in SGA or similar) – David Roberts Oct 15 '13 at 1:25
I doubt that there will be a nice description, although it is clear what the quotient should "do", namely it should add isomorphisms $x \cong F_g(x)$ to our category. How does this work for preorders? – Martin Brandenburg Oct 15 '13 at 8:25