I don’t believe that the left adjoint to the forgetful functor is the “trivial action functor”. For example, let’s G be the two-element group, E be the regular G-set and X be a singleton. Then, GSet(TX, E) is empty, because a G-map sends fixed points to fixed points, and E has none, whereas Set(X,FE) has two elements.

What I think is true is that we have two chains of adjunctions L/F/E and pi0/T/Fix, where F, T and pi0 are the functors you defined, L is the functor sending a set X to |X| disjoint copies of the regular G-set, E is the functor sending a set X to the G-set X^G and Fix is the functor sending a G-set to its set of fixed points.

I don’t know how that effects the big-picture message you’re trying to convey.

]]>There is also a nice article by Philip Heath on applications of groupoids to Nielsen Fixed Point Theory at Topology and its Applications 181 (2015) 3–33.

See also the article arXiv:1207.6404 “Groupoids and Faa di Bruno formulae for Green functions in bialgebras of trees” for uses of fibrations of groupoids.

T&G is still the only topology text in English to use the fundamental groupoid on a set of base points, although this notion was published by me to give a generalisation of the Seifert-van Kampen Theorem, and so calculate the fundamental group of the circle, for example, in 1967. .

]]>Yes, the thing I actually want to do is more like remembering the structure of as an “essentially small large setoid” (a class with an equivalence relation, here isomorphism, which has a set’s worth of equivalence classes) and take the coproduct over with this structure in mind (so identifying isomorphic objects in the coproduct). But it seemed like too much of a distraction to spell this out explicitly. Maybe I should’ve used sets of objects after all.

]]>On families of objects: although thinking of them as full subcategories gets around the problem of invariance under equivalence, it seems to me that the coproduct operation you describe is evil, or at least non-functorial. After all, it involves splitting a surjection. I don’t really know how to fix this – maybe it would be better to think of families of objects as sets after all.

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