# nLab inertia orbifold

Contents

### Context

Ingredients

Concepts

Constructions

Examples

Theorems

#### Mapping space

internal hom/mapping space

# Contents

## Idea

Inertia orbifold is a particular model for the free loop space object of an orbifold $X$ (or plain groupoid or smooth groupoid/stack etc.): the smooth groupoid whose objects are automorphisms in $X$ and whose morphisms are conjugation of automorphisms by morphisms in $X$. 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 $\mathcal{G} \coloneqq (\mathcal{G}_1 \rightrightarrows \mathcal{G}_0)$ (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:

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

$S \;\coloneqq\; \Delta/\partial\Delta \;\;\; \in \; sSet$

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

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

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

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)

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

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).

Original articles:

Review and further development: