nLab free loop orbifold

Idea

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):

$$\mathcal{L} \mathcal{X} \;\coloneqq\; \big[ S^1, \, \mathcal{X} \big] \,.$$

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

(1)$$\mathcal{X} \simeq X \!\sslash\! G$$

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:

(2)$$\mathbf{H} \big( \mathbb{R}^n,\, \mathcal{L}\mathcal{X} \big) \;\simeq\; LieGroupoids \big( \mathcal{U}_{S^1} \times \mathbb{R}^n, \, (X \times G \rightrightarrows X) \big) \,.$$

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 $G$.

Remark

(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

  $$ʃS^1 \;\simeq\; \mathbf{B}\mathbb{Z} \;\;\; \in Groupoids \xrightarrow{ \; Disc\; } SmoothGroupoids \; \,.$$

• $[-,-]$ denote the mapping stack-construction.

Then we have:

1. The cohesive free loop orbifold of $\mathcal{X}$ is

   $$\mathcal{L}\mathcal{X} \;=\; \big[ S^1, \, \mathcal{X} \big] \,.$$

2. The inertia orbifold of $\mathcal{X}$ is

   $$\Lambda \mathcal{X} \;\simeq\; \big[ ʃS^1, \, \mathcal{X} \big] \;\simeq\; \big[ \mathbf{B}\mathbb{Z}, \, \mathcal{X} \big] \;\simeq\; \mathcal{X} \times^h_{\mathcal{X} \times \mathcal{X}} \mathcal{X} \,,$$

   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

$$\Lambda \mathcal{X} \;=\; \big[ ʃS^1, \, \mathcal{X} \big] \xrightarrow{ \; [\eta_{S^1},\mathcal{X}]\; } \big[ S^1, \, \mathcal{X} \big] \;=\; \mathcal{L} \mathcal{X} \,.$$

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 $\mathcal{X}$.

Related concepts

• inertia orbifold, Huan's inertia orbifold

• cyclic loop stack

• free loop space, derived loop space

• free loop space of classifying space

References

• 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)