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. .

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