nLab
inertia orbifold

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Context

Higher Geometry

Mapping space

Contents

Idea

Inertia orbifold is a particular model for the free loop space object of an orbifold XX (or plain groupoid or smooth groupoid/stack etc.): the smooth groupoid whose objects are automorphisms in XX and whose morphisms are conjugation of automorphisms by morphisms in XX. In fact for each fibered category one can construct another fibered category, its inertia and the inertia stacks are a special case of this construction.

Definition

For bare groupoids

Definition

(inertia groupoid)
Given a groupoid 𝒢(𝒢 1𝒢 0)\mathcal{G} \coloneqq (\mathcal{G}_1 \rightrightarrows \mathcal{G}_0) (internal to Sets) with

one defines its inertia groupoid Λ𝒢\Lambda \mathcal{G} as the groupoid whose

  • set of objects is the set of automorphisms in 𝒢\mathcal{G}, i.e. the equalizer of the source and target maps s,t:G 1G 0s,t \colon G_1\to G_0:

    (Λ𝒢) 0x𝒢 0Aut 𝒢(x) (\Lambda \mathcal{G})_0 \;\coloneqq\; \underset{x \in \mathcal{G}_0}{\coprod} Aut_{\mathcal{G}}(x)
  • whose hom-set from f:aaf \colon a \to a to g:bbg\colon b \to b consists of the commutative squares with the same vertical maps of the form

a f a u u b g b \array{ a &\stackrel{f}{\longrightarrow}& a \\ {}^{\mathllap{u}} \big\downarrow && \big\downarrow {}^{\mathrlap{u}} \\ b &\stackrel{g}{\longrightarrow}& b }

i.e. of the morphisms u:abu \colon a\to b in 𝒢 1\mathcal{G}_1 such that u 1gu=fu^{-1}\circ g\circ u = f.

Proposition

The inertia groupoid (Def. ) is equivalent (in fact isomorphic as groupoid objects) to the functor groupoid

Λ𝒢[B,𝒢], \Lambda \mathcal{G} \;\simeq\; \big[ \mathbf{B}\mathbb{Z}, \mathcal{G} \big] \,,

where

B(*) \mathbf{B}\mathbb{Z} \;\coloneqq\; \big( \mathbb{Z} \rightrightarrows \ast \big)

denotes the delooping groupoid of the additive group of integers, i.e. the free groupoid on a single object with a single automorphism.

Proposition

The inertia groupoid (Def. ) is also equivalent to the free loop space object of 𝒢\mathcal{G} in the (2,1)-category of groupoids.

Λ𝒢𝒢𝒢× 𝒢×𝒢 h𝒢. \Lambda \mathcal{G} \;\simeq\; \mathcal{L} \mathcal{G} \;\coloneqq\; \mathcal{G} \times^h_{\mathcal{G} \times \mathcal{G}} \mathcal{G} \,.

For internal groupoids

The same construction can be performed for groupoid objects internal to any finitely complete category, or more generally whenever the relevant limits exist.

For orbifolds

If a (differential, topological or algebraic) stack (or, in particular, an orbifold) is represented by a groupoid, then the inertia groupoid of that groupoid represents its inertia stack. In particular, an orbifold corresponds to a Morita equivalence class of a proper étale groupoid. The inertia groupoid ΛG\Lambda G of GG is the Morita equivalence class of the (proper étale) action groupoid for the conjugation action of 𝒢 1\mathcal{G}_1 on the subspace SG 1S\subset G_1 of closed loops.

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}.

Remark

For quantum field theory on orbifolds, or rather string theory on orbifolds, the inertia orbifold is related to so called twisted sectors of the corresponding QFT. (…)

Properties

Skeleton for global quotient orbifolds

Proposition

(skeleton of inertia orbifold for proper good orbifolds)
If 𝒳XG\mathcal{X} \,\simeq\, X \!\sslash\! G is a good orbifold presented as the global quotient orbifold of a smooth manifold with smooth proper group action by a discrete group GG, then its inertia orbifold is equivalent to the following disjoint union of global quotient orbifolds

Λ(XG)[g]ConjCl(G)X gC g, \Lambda \big( X \!\sslash\! G \big) \;\simeq\; \underset{ [g] \in ConjCl(G) }{\coprod} X^g \!\sslash\! C_g \,,

where

Examples

Intertia groupoid of a delooping groupoid

For GG a finite group (most of the following holds more generally for discrete groups), we discuss the inertia groupoid ΛBG\Lambda \mathbf{B}G of the delooping groupoid BG=(G*)\mathbf{B}G \,=\, (G \rightrightarrows \ast).

Remark

(delooping groupoid and simplicial classifying space of finite group)
The nerve of the delooping groupoid of a discrete group GG is isomorphic to the simplicial classifying space of GG (see this Example):

N(G*)W¯GsSet. N \big( G \rightrightarrows \ast \big) \;\simeq\; \overline{W} G \;\;\; \in \; sSet \,.

For notational brevity we will be referring to W¯G\overline{W}G in the following, but it may be helpful to keep thinking of the nerve of the delooping groupoid. From that perspective, an n-simplex in W¯G\overline{W}G, which is an n-tuple of group elements, is suggestively denoted as a sequence of composable arrows:

(W¯G) n={g n1g 1g 0|g iG}. \big(\overline{W}G\big)_{n} \;\; = \;\; \Big\{ \bullet \xrightarrow{g_{n-1}} \bullet \xrightarrow{\;} \cdots \bullet \xrightarrow{\;} \bullet \xrightarrow{g_1} \bullet \xrightarrow{g_0} \bullet \;\big\vert\; g_i \in G \Big\} \,.

Proposition

The inertia groupoid ΛBG\Lambda \mathbf{B} G is isomorphic to the action groupoid of the adjoint action of GG on itself:

ΛBGG adG=(G×GAd ()()pr 2G) \Lambda \mathbf{B}G \;\simeq\; G_{ad} \sslash G \;=\; \left( G \times G \underoverset {Ad_{(-)}(-)} {pr_2} {\rightrightarrows} G \right)

This follows by immediate inspection. For more discussion see at free loop space of a classifying space the section Examples – For finite groups.

Proposition

The groupoid convolution algebra of the inertia groupoid of the delooping groupoid BG\mathbf{B}G is the Drinfeld double of the group convolution algebra of GG.

Definition

(minimal simplicial circle)
Write

SΔ[1]/Δ[1]sSet S \;\coloneqq\; \Delta[1]/\partial\Delta[1] \;\;\; \in \; sSet

for the simplicial set with exactly two non-degenerate cells,

  • one of which in degree 0, which we denote by [0]=[1][0] = [1],

  • and one in degree 1, which we denote by [[0],[1]]\big[ [0], [1] \big].

The following proposition follows on abstract grounds, but the explicit component-based proof we give is necessary in order to understand the transgression-formula for cocycles in the group cohomology of GG to cocycles on the inertia groupoid.

Proposition

The nerve of the inertia groupoid of a delooping groupoid of a finite group GG is isomorphic to the simplicial hom complex out of the minimal simplicial circle SS (Def. ) into the simplicial classifying space W¯G\overline{W}G (Rem. ):

N(ΛBG) [S,W¯G] . N\big( \Lambda \mathbf{B}G\big)_\bullet \;\simeq\; [S,\overline{W}G]_\bullet \,.

Proof

We claim that the isomorphism is given by sending, for each nn \in \mathbb{N}, any n-simplex (γ,g n1,,g 1,g 0)(\gamma, g_{n-1}, \cdots, g_1, g_0) of N(Func(B,BG))N\big( Func( \mathbf{B}\mathbb{Z}, \, \mathbf{B}G )\big), being a sequence of natural transformations of the form

g n1 g n2 g nj1 g nj2 g nj3 g 0 γ g n1 1γg n1 (g n1g 0) 1γ(g n1g 0) g n1 g n2 g nj1 g nj2 g nj3 g 0 , \array{ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j-1}}& \bullet &\xrightarrow{g_{n-j-2}}& \bullet &\xrightarrow{g_{n-j-3}}& \cdots &\xrightarrow{g_{0}}& \bullet \\ \big\downarrow {}^{\mathrlap{ \gamma }} && \big\downarrow {}^{\mathrlap{ g_{n-1}^{-1} \cdot \gamma \cdot g_{n-1} }} && && \big\downarrow && \big\downarrow && && \big\downarrow {}^{\mathrlap{ ( g_{n-1} \cdots g_{0} )^{-1} \cdot \gamma \cdot ( g_{n-1} \cdots g_{0} ) }} \\ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j-1}}& \bullet &\xrightarrow{g_{n-j-2}}& \bullet &\xrightarrow{g_{n-j-3}}& \cdots &\xrightarrow{g_{0}}& \bullet \mathrlap{\,,} }

to the homomorphism of simplicial sets

Δ[n]×SW¯G, \Delta[n] \times S \xrightarrow{\;\;} \overline{W}G \,,

which, in turn, sends a non-degenerate (n+1)(n+1)-simplex in Δ[n]×S\Delta[n] \times S of the form (in the path notation discussed at product of simplices)

(0,[0]) (1,[0]) (j,[0]) (j,[1]) (j+1,[1]) (n,[1]) \array{ (0,[0]) &\to& (1,[0]) &\to& \cdots &\to& (j,[0]) \\ && && && \big\downarrow \\ && && && (j,[1]) &\to& (j+1,[1]) &\to& \cdots &\to& (n,[1]) }

to the n+1n+1-simplex in W¯G\overline{W}G (Rem. ) of the form

g n1 g n2 g nj (g n1g nj) 1γ(g n1g nj) g nj g nj1 g 0 \array{ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j}}& \bullet \\ && && && \big\downarrow {}^{\mathrlap{ ( g_{n-1} \cdots g_{n-j} )^{-1} \cdot \gamma \cdot ( g_{n-1} \cdots g_{n-j} ) }} \\ && && && \bullet &\xrightarrow{g_{n-j}}& \bullet &\xrightarrow{g_{n-j-1}}& \cdots &\xrightarrow{g_{0}}& \bullet }

As a consequence:

Proposition

The evaluation map

[S,W¯G]×SW¯G [S,\overline{W}G] \times S \xrightarrow{\;\;} \overline{W}G

(out of the product of the simplicial hom complex out of SS with SS) takes any non-degenerate n+1n+1-simplex of [S,W¯G]×S[S,\overline{W}G] \times S of the form (still in the path notation discussed at product of simplices)

(γ,[0]) (g n1,id) (Ad n1(γ),[0]) (g n2,id) (g j,id) (Ad j(γ),[0]) (e,[[0],[1]]) (Ad j(γ),[1]) (g nj,id) (g 0,id) (Ad 0(γ),[1]), \array{ (\gamma, [0]) &\xrightarrow{ (g_{n-1}, id) }& ( Ad_{n-1}(\gamma), [0] ) &\xrightarrow{ (g_{n-2}, id) }& \cdots &\xrightarrow{ (g_{j}, id) }& \big( Ad_{j}(\gamma), [0] \big) \\ && && && \big\downarrow {}^{\mathrlap{ (e, [ [0],[1] ]) }} \\ && && && (Ad_j(\gamma), [1]) &\xrightarrow{ (g_{n-j},id) } & \cdots &\xrightarrow{ (g_0,id) }& (Ad_0(\gamma), [1]) \mathrlap{\,,} }

where we are abbreviating

Ad j(γ) Ad (g n1g j)(γ) (g n1g j) 1γ(g n1g j), \begin{aligned} Ad_j(\gamma) & \;\coloneqq\; Ad_{(g_{n-1} \cdots g_j)}(\gamma) \\ & \;\coloneqq\; (g_{n-1} \cdots g_j)^{-1} \cdot \gamma \cdot (g_{n-1} \cdots g_j) \,, \end{aligned}

to the following n+1n+1 simplex of W¯G\overline{W}G:

g n1 g n2 g j Ad j(γ) g nj g 0 . \array{ \bullet &\xrightarrow{ g_{n-1} }& \bullet &\xrightarrow{ g_{n-2} }& \cdots &\xrightarrow{ g_{j} }& \bullet \\ && && && \big\downarrow {}^{\mathrlap{ Ad_j(\gamma) }} \\ && && && \bullet &\xrightarrow{ g_{n-j} } & \cdots & \xrightarrow{ g_0 } & \bullet \mathrlap{\,.} }

References

Original articles:

Review and further development:

See also:

In relation to Drinfeld doubles:

In view of the Chern character for twisted orbifold K-theory:

  • Jean-Louis Tu, Ping Xu, Chern character for twisted K-theory of orbifolds, Advances in Mathematics 207 (2006) 455–483, pdf (cf. sec. 2.3)

See also the references at free loop orbifold.

On transgression in group cohomology, for discrete groups, to groupoid cohomology of inertia groupoids:

category: Lie theory

Last revised on July 12, 2021 at 14:26:13. See the history of this page for a list of all contributions to it.