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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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  • $\begingroup$ 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. $\endgroup$
    – Tom Leinster
    Commented Feb 28, 2012 at 18:29
  • $\begingroup$ 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. $\endgroup$
    – Todd Trimble
    Commented Feb 28, 2012 at 19:33
  • $\begingroup$ 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 $\endgroup$
    – David Roberts ♦
    Commented Feb 28, 2012 at 23:14
  • $\begingroup$ Certain colimits of this form, called quasi-coproducts were studied in Hu–Tholen's Quasi-coproducts and accessible categories with wide pullbacks. $\endgroup$
    – varkor
    Commented Jul 6, 2022 at 18:42

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