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loop space object, free loop space object, derived loop space
For $\mathcal{X}$ an orbifold, (or, more generally, any differentiable stack or yet more generally a smooth ∞-groupoid), its free loop orbifold (free loop stack) is the mapping stack into it out of the circle $S^1$ (the latter regarded as a smooth manifold and hence as an orbifold/differentiable stack in the canonical way):
For $\mathcal{X} = X$ a smooth manifold, this reduces to the smooth loop space of $X$, which is still a Fréchet manifold.
More generally, for
a good orbifold equivalent to a global quotient orbifold of $X$ be a discrete group action, the free loop orbifold combines the properties of the smooth loop space of $X$ with the properties of the inertia orbifold of $\mathcal{X}$ (see Remark ).
Concretely, for $\mathcal{U}_{S^1} \xrightarrow{\;\;\simeq\;\;} S^1$ the Cech groupoid of a good open cover of the circle, the free loop orbifold of a good orbifold (1) has plots of shape $\mathbb{R}^n$ given by the following hom-groupoid of Lie groupoids:
Here $\mathbf{H}$ denotes the correct (2,1)-category of differentiable stacks (see at Smooth∞Groupoid), while $LieGroupoids$ is its presentation by Lie groupoids (groupoid objects internal to SmoothManifolds, where Morita morphisms are not inverted).
Notice here how:
the morphisms in the Cech groupoid detect the non-trivial morphisms in $\mathcal{X}$ as for an inertia orbifold,
while the cohesive smooth structure on the space of objects of the Cech groupoid detects smooth paths in $X$.
A general plot (2) is a circular sequence of smooth paths in $X$ whose endpoints are cyclically related by the group action of $G$.
(different notions of loop orbifolds)
For the case of orbifolds, the cohesion of the loops leads to a distinction between various in-equivalent notions of “free loop spaces” of orbifolds:
Let:
$\mathcal{X} \,\in\, SmoothGroupoids_\infty$ be an orbifold, regarded as a smooth groupoid, regarded as a differentiable stack.
$S^1 \,\in\, SmoothManifolds \hookrightarrow SmoothGroupoids_\infty$ be the circle with its standard cohesive structure as a smooth manifold, and hence as a differentiable stack.
Notice that the shape of $S^1$ (in the cohesive) (2,1)-topos of smooth ∞-groupoids is the delooping groupoid of the integers, regarded as a discrete smooth groupoid
$[-,-]$ denote the mapping stack-construction.
Then we have:
The cohesive free loop orbifold of $\mathcal{X}$ is
The inertia orbifold of $\mathcal{X}$ is
which is the actual free loop space object formed in smooth groupoids.
The shape modality-unit $\mathcal{A} \xrightarrow{ \eta_{\mathcal{A}}} ʃ \mathcal{A}$ induces a canonical comparison morphism between the two
When $\mathcal{X} \simeq X \!\sslash\! G$ is a global quotient orbifold of a smooth manifold $X$ (for instance for a good orbifold, but $X$ could more generally be a diffeological space for the present discussion), then this inclusion is the faithful inclusion of the cohesively constant loops, namely those that map to points in the naive quotient space of $\mathcal{X}$.
Ernesto Lupercio, Bernardo Uribe, Loop groupoids, gerbes, and twisted sectors on orbifolds, in: Alejandro Adem, Jack Morava, Yongbin Ruan (eds.), Orbifolds in Mathematics and Physics, Madison, WI, 2001, in: Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 163–184, (math.AT/0110207, ISBN:978-0-8218-2990-5, MR2004c:58043)
Kai Behrend, Grégory Ginot, Behrang Noohi, Ping Xu, Section 5 of: String topology for stacks, Astérisque, no. 343 (2012) , 183 p. (arXiv:0712.3857, numdam:AST_2012__343__R1_0)
(with an eye towards string topology of orbidolds)
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)
(a Fréchet–Lie groupoid presentation)
Zhen Huan, Section 2 of: Quasi-Elliptic Cohomology I, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 (arXiv:1805.06305, doi:10.1016/j.aim.2018.08.007)
(in view of equivariant elliptic cohomology via Tate K-theory)
Last revised on July 11, 2021 at 19:24:13. See the history of this page for a list of all contributions to it.