nLab inertia orbifold

Idea

An inertia orbifold $\Lambda \mathcal{X}$ is (a particular representative of) the free loop space object of an orbifold $\mathcal{X}$ (or of a plain groupoid or smooth groupoid/differentiable stack etc.): the (smooth) groupoid whose

• objects are endomorphisms (necessarily automorphisms) in $\mathcal{X}$

• morphisms are conjugations of such automorphisms $g$ by morphisms (necessarily: isomorphisms) $f$ in $\mathcal{X}$:

In the special “global” case where $\mathcal{X} \simeq X\sslash G$ is a quotient stack (action groupoid) of a group action, then the points of the inertia orbifold $\Lambda (X \sslash G)$ are the “inert” actions, consisting of elements of stabilizer subgroups $Stab_G(x)$ (also “isotropy groups”) for points $x$ in $X$ (compare also the terminology “inertia group” for stabilizer subgroups used in number theory, e.g here).

Definition

For bare groupoids

Definition

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

• set of objects $\mathcal{G}_0$

• set of morphisms $\mathcal{G}_1$,

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

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

  $$(\Lambda \mathcal{G})_0 \;\coloneqq\; \underset{x \in \mathcal{G}_0}{\coprod} Aut_{\mathcal{G}}(x)$$

• whose hom-set from $f \colon a \to a$ to $g\colon b \to b$ consists of the commutative squares with the same vertical maps of the form

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

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

Proposition

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

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

where

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

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 $\mathcal{G}$ in the (2,1)-category of groupoids.

$$\Lambda \mathcal{G} \;\simeq\; \mathcal{L} \mathcal{G} \;\coloneqq\; \mathcal{G} \times^h_{\mathcal{G} \times \mathcal{G}} \mathcal{G} \,.$$

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 $\Lambda G$ of $G$ is the Morita equivalence class of the (proper étale) action groupoid for the conjugation action of $\mathcal{G}_1$ on the subspace $S\subset G_1$ 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:

• $\mathcal{X} \,\in\, SmoothGroupoids_\infty$ 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 $\mathcal{X}$.

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

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

where

• $ConjCl(G) \coloneqq G /_{ad} G$ is the set of conjugacy classes of $G$;

• $X^g \coloneqq X^{\langle g\rangle}$ is the fixed locus of the action of (the cyclic group generated by) $g \in G$ (which is again a smooth manifold by the discussion here);

• $C_g \subset G$ is the centralizer of $\{g\} \subset G$.

Examples

Intertia groupoid of a delooping groupoid

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

Remark

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

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

For notational brevity we will be referring to $\overline{W}G$ 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 $\overline{W}G$, which is an n-tuple of group elements, is suggestively denoted as a sequence of composable arrows:

$$\big(\overline{W}G\big)_{n} \;\; = \;\; \Big\{ \bullet \xrightarrow{g_{n-1}} \bullet \xrightarrow{\;} \cdots \bullet \xrightarrow{\;} \bullet \xrightarrow{g_1} \bullet \xrightarrow{g_0} \bullet \;\big\vert\; g_i \in G \Big\} \,.$$

Proposition

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

$$\Lambda \mathbf{B}G \;\simeq\; G_{ad} \sslash G \;=\; \left( G \times G \underoverset {Ad_{(-)}(-)} {pr_2} {\rightrightarrows} G \right)$$

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 $\mathbf{B}G$ is the Drinfeld double of the group convolution algebra of $G$.

Definition

(minimal simplicial circle)
Write

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

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

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

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

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 $G$ to cocycles on the inertia groupoid.

Proposition

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

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

Proof

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

$$\array{ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j-1}}& \bullet &\xrightarrow{g_{n-j-2}}& \bullet &\xrightarrow{g_{n-j-3}}& \cdots &\xrightarrow{g_{0}}& \bullet \\ \big\downarrow {}^{\mathrlap{ \gamma }} && \big\downarrow {}^{\mathrlap{ g_{n-1}^{-1} \cdot \gamma \cdot g_{n-1} }} && && \big\downarrow && \big\downarrow && && \big\downarrow {}^{\mathrlap{ ( g_{n-1} \cdots g_{0} )^{-1} \cdot \gamma \cdot ( g_{n-1} \cdots g_{0} ) }} \\ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j-1}}& \bullet &\xrightarrow{g_{n-j-2}}& \bullet &\xrightarrow{g_{n-j-3}}& \cdots &\xrightarrow{g_{0}}& \bullet \mathrlap{\,,} }$$

to the homomorphism of simplicial sets

$$\Delta[n] \times S \xrightarrow{\;\;} \overline{W}G \,,$$

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

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

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

$$\array{ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j}}& \bullet \\ && && && \big\downarrow {}^{\mathrlap{ ( g_{n-1} \cdots g_{n-j} )^{-1} \cdot \gamma \cdot ( g_{n-1} \cdots g_{n-j} ) }} \\ && && && \bullet &\xrightarrow{g_{n-j}}& \bullet &\xrightarrow{g_{n-j-1}}& \cdots &\xrightarrow{g_{0}}& \bullet }$$

As a consequence, the evaluation map on the inertia groupoid has essentially this same expression, too, which explains the traditional formula for transgression in group cohomology (see the details there).

Related concepts

• free loop orbifold

• Huan's inertia orbifold

• free loop space, derived loop space

• transgression in group cohomology

References

Original articles:

• Tetsuro Kawasaki, The signature theorem for V-manifolds, Topology 17 (1978), no. 1, 75–83 (doi:10.1016/0040-9383(78)90013-7)

Review and further development:

• Ernesto Lupercio, Bernardo Uribe, Section 4 of: Inertia orbifolds, configuration spaces and the ghost loop space, Quarterly Journal of Mathematics 55 2 (2004) 185-201 (arxiv:math.AT/0210222, doi:10.1093/qmath/hag053)

• Ieke Moerdijk, Section 6 in: Orbifolds as Groupoids: an Introduction, in: Alejandro Adem, Jack Morava, Yongbin Ruan (eds.) Orbifolds in Mathematics and Physics, Contemporary Math 310 , AMS (2002), 205–222 (arXiv:math.DG/0203100)

• Ulrich Bunke, Markus Spitzweck, Thomas Schick, Inertia and delocalized twisted cohomology, Homotopy, Homology and Applications 10(1) 129–180 (2008)math.KT/0609576

  doi

See also:

• The Stacks Project, The inertia stack (more, more)

In relation to Drinfeld doubles:

• Vladimir Hinich, Drinfeld double for orbifolds, Contemporary Math. 433, AMS Providence, 2007, 251-265,

  arXiv:math.QA/0511476

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:

• Simon Willerton, Section 1 of: The twisted Drinfeld double of a finite group via gerbes and finite groupoids, Algebr. Geom. Topol. 8 (2008) 1419-1457 (arXiv:math/0503266)

• Jean-Louis Tu, Ping Xu, Section 3 of: The ring structure for equivariant twisted K-theory, J. Reine Angew. Math. 635 (2009), 97–148 (arXiv:math/0604160, doi:10.1515/CRELLE.2009.077)

• Alejandro Adem, Yongbin Ruan, Bin Zhang, Section 4 of: A Stringy Product on Twisted Orbifold K-theory, Morfismos (10th Anniversary Issue), Vol. 11, No 2 (2007), 33-64. (arXiv:math/0605534, Morfismos pdf)

category: Lie theory