nLab
free loop orbifold

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Higher Geometry

Mapping space

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Idea

For 𝒳\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 1S^1 (the latter regarded as a smooth manifold and hence as an orbifold/differentiable stack in the canonical way):

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

For 𝒳=X\mathcal{X} = X a smooth manifold, this reduces to the smooth loop space of XX, which is still a Fréchet manifold.

More generally, for

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

a good orbifold equivalent to a global quotient orbifold of XX be a discrete group action, the free loop orbifold combines the properties of the smooth loop space of XX with the properties of the inertia orbifold of 𝒳\mathcal{X} (see Remark ).

Concretely, for 𝒰 S 1S 1\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 n\mathbb{R}^n given by the following hom-groupoid of Lie groupoids:

(2)H( n,𝒳)LieGroupoids(𝒰 S 1× n,(X×GX)). \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 H\mathbf{H} denotes the correct (2,1)-category of differentiable stacks (see at Smooth∞Groupoid), while LieGroupoidsLieGroupoids is its presentation by Lie groupoids (groupoid objects internal to SmoothManifolds, where Morita morphisms are not inverted).

Notice here how:

A general plot (2) is a circular sequence of smooth paths in XX whose endpoints are cyclically related by the group action of GG.

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 𝒳\mathcal{X} is

    𝒳=[S 1,𝒳]. \mathcal{L}\mathcal{X} \;=\; \big[ S^1, \, \mathcal{X} \big] \,.
  2. The inertia orbifold of 𝒳\mathcal{X} is

    Λ𝒳[ʃS 1,𝒳][B,𝒳]𝒳× 𝒳×𝒳 h𝒳, \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 𝒜η 𝒜ʃ𝒜\mathcal{A} \xrightarrow{ \eta_{\mathcal{A}}} ʃ \mathcal{A} induces a canonical comparison morphism between the two

Λ𝒳=[ʃS 1,𝒳][η S 1,𝒳][S 1,𝒳]=𝒳. \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 𝒳XG\mathcal{X} \simeq X \!\sslash\! G is a global quotient orbifold of a smooth manifold XX (for instance for a good orbifold, but XX 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 𝒳\mathcal{X}.

References

(a Fréchet–Lie groupoid presentation)

Last revised on July 11, 2021 at 15:24:13. See the history of this page for a list of all contributions to it.