Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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 :
1 to 9 of 9