# nLab inertia orbifold

Contents

### Context

#### $(\infty,1)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# 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

Given a groupoid $G$ (in the category of sets) with the set of objects $G_0$ and the set of morphisms $G_1$, one defines its inertia groupoid as the groupoid whose set $S$ of objects is the set of loops, i.e. the equalizer of the source and target maps $s,t: G_1\to G_0$; and whose set of maps 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

$\array{a &\stackrel{f}\rightarrow& a\\ u \downarrow && \downarrow u\\ b &\stackrel{g}\rightarrow& b }$

i.e. of the morphisms $u \colon a\to b$ in $G_1$ such that $u^{-1}\circ g\circ u = f$.

This is isomorphic to the functor category $[S^1,G]$, where $S^1$ denotes the free groupoid on a single object with a single automorphism (equivalently, the delooping $B\mathbb{Z}$ of the integers). It is equivalent to the free loop space object of $G$ in the (2,1)-category of groupoids.

The same construction can be performed for a groupoid internal to any finitely complete category, or more generally whenever the relevant limits exist. 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 action of $G_1$ by conjugation on the subspace $S\subset G_1$ of closed loops.

For quantum field theory on orbifolds the inertia orbifold is related to so called twisted sectors of the corresponding QFT. One can also consider more generally twisted multisectors.

## Properties

### Convolution algebra and Relation to Drinfeld double

At least for a finite group $G$, the groupoid convolution algebra of the inertia groupoid of $\mathbf{B}G$ is the Drinfeld double of the group convolution algebra of $G$.

• Stacks Project: Inertia, more, more, The inertia stack
• Ernesto Lupercio, Bernardo Uribe, Inertia orbifolds, configuration spaces and the ghost loop space, Quarterly Journal of Mathematics 55, Issue 2, pp. 185-201, arxiv/math.AT/0210222; Loop groupoids, gerbes, and twisted sectors on orbifolds, in: Orbifolds in Mathematics and

Physics, Madison, WI, 2001, in: Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 163–184, MR2004c:58043, math.AT/0110207

• T. Kawasaki, The signature theorem for V-manifolds, Topology 17 (1978), no. 1, 75–83.
• V. Hinich, Drinfeld double for orbifolds, Contemporary Math. 433, AMS Providence, 2007, 251-265,

arXiv:math.QA/0511476

• L. Dixon, J. A. Harvey, C. Vafa, E. Witten, Strings on orbifolds, Nuclear Phys. B 261 (1985), no. 4, 678–686. MR87k:81104a, doi; Strings on orbifolds. II, Nuclear Phys. B 274 (1986), no. 2, 285–314, MR87k:81104b, doi
• 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)
category: Lie theory