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Added reference:
A Fréchet–Lie groupoid presenting the cohesive free loop space-orbifold is given in
- David Michael Roberts, Raymond Vozzo, Smooth loop stacks of differentiable stacks and gerbes, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol LIX no 2 (2018) pp 95-141 journal version, arXiv:1602.07973.
We also claim in the announcement
- David Michael Roberts, Raymond Vozzo, The smooth Hom-stack of an orbifold, In: Wood D., de Gier J., Praeger C., Tao T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1 (2018) doi:10.1007/978-3-319-72299-3_3, arXiv:1610.05904, MATRIX hosted version
that the Lie groupoid presenting the resulting infinite-dimensional differentiable stack is even proper étale, hence an orbifold groupoid, though the full write-up of this is still on the back-burner.
How about making an entry loop orbifold or loop stack where this would fit?
It has become tradition to say that the inertia orbifold is the -fixed locus in a groupoid presentation of the free loop stack (e.g. Lupercio & Uribe 2001, Thm 3.6.4). However, it seems to me that this statement is not homotopy-meaningful:
The identification of the smooth free loop stack with the co-free -action induced by (as seen in the diagram here) shows at once that its homotopy-fixed locus is just :
where
(here for )
It seems to me that in order to relate the inertia and the smooth loop stack in a homotopy-meaningful way, one needs to invoke cohesion.
I have made explicit (here) the following basic fact, which, for what it’s worth, becomes a triviality only through appeal to the path-notation for the Eilenberg-Zilber decomposition:
The nerve of the inertia groupoid of a delooping groupoid of a finite group is isomorphic to the simplicial hom complex out of the minimal simplicial circle (Def. \ref{MinimalSimplicialCircle}) into the simplicial classifying space :
Under this isomorphism, an n-simplex of , being a sequence of natural transformations of the form
is sent to the homomorphism
which, in turn, sends a non-degenerate -simplex in of the form (in the path notation discussed at product of simplices)
to the -simplex in of the form
added (here) the consequence for the form of the simplicial evaluation map:
The evaluation map
(out of the product of the simplicial hom complex out of with ) takes any non-degenerate -simplex of of the form (still in the path notation discussed at product of simplices)
where we are abbreviating
to the following simplex of :
7: let me start understanding your picture. Lupercio and Uribe in 2001 paper Theorem 3.6.4 and Proposition (in fact Corollary) 3.6.6 (pdf) say that their loop groupoid is Morita equivalent to their inertia groupoid. Now the inertia orbifold entry says that in cohesive picture there is a comparison morphism which is in good cases faithful inclusion. So Lupercio-Uribe 3.6.4/3.6.6 does not survive in your cohesive version but is weakened in some sense ? By groupoid presentation you mean choosing in Morita class ?
I spent some time in 1999-2003 understanding works of Lupercio and Uribe (including listening their seminars in Wisconsin) and somehow like comparison to their picture.
We are finally writing this up, preliminary but readable notes are at cyclic loop spaces 2022 (schreiber)
Great, do you plan to include remarks on various results around crepant resolution conjecture ?
11: You do not use cyclic nerve anywhere in your constructions planned for the writeup ?
A couple of typoses.
Page 5 formula 11: cyclified cocyle (green in the diagram at the top)
Page 19 formula 19, also cocyle in blue
Why do you say “the Cyc-adjunction” rather than the adjunction Cyc, it is not a modifier but a name.
Presumably the name of this concept owes its origins to ’inertia group’ and ’inertia field’ in number theory, e.g. as here. I’d add something to that effect if I knew anything about it.
So it’s coming from the idea of prime ideals remaining prime through field extensions, Inertial prime number.
I don’t know the historical origin of the terminology “inertia orbifold”. But, yes, it probably refers to the “inert actions” in the sense of “stabilizer subgroups” which make up the points of such an orbifold.
See also the first paragraph in section 3.4 of Noohi’s Foundations of Topological Stacks (p. 15).
Thanks!
Re #18, four names for the same concept!
Other synonyms for inertia group are stabilizer group, isotropy group and ramification group.
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