# nLab free loop orbifold

Contents

### Context

Ingredients

Concepts

Constructions

Examples

Theorems

#### Mapping space

internal hom/mapping space

# Contents

## 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:

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 $\mathcal{X}$.

(a Fréchet–Lie groupoid presentation)