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Is there a standard name for a category $C$ in which coproducts exist, and in which coequalizers exist for any action of a finite group on an object of $C$? Equivalently, colimits are required to exist only for diagrams of the form $I \to C$ where $I$ is a finite groupoid.

Some examples satisfying this condition which are not cocomplete: the category of orbifolds, and Grothendieck's category of motives. The latter is closed under quotients of finite groups since it is $\mathbf Q$-linear and pseudo-abelian.

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I bet the answer is "no". If I needed that notion, I might say "finite-groupoid cocomplete", which is a bit clunky but transparent enough. –  Tom Leinster Feb 28 '12 at 18:29
    
I do recall however that Joyal considered somewhat similar cocompleteness conditions for analytic functors in the theory of species; see the appendix of his article in Springer LNM 1234. –  Todd Trimble Feb 28 '12 at 19:33
    
This is a condition that pops up in the definition of Galois categories (at least with the additional assumption of an initial object). See Dubuc and de la Vega's paper "On the Galois theory of Grothendieck", arxiv.org/abs/math/0009145 - axiom G2 in section 4.1 –  David Roberts Feb 28 '12 at 23:14

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