Given a groupoid (in the category of sets) with the set of objects and the set of morphisms , one defines its inertia groupoid as the groupoid whose set of objects is the set of loops, i.e. the equalizer of the source and target maps ; and whose set of maps from to consists of the commutative squares with the same vertical maps of the form
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 of is the Morita equivalence class of the (proper étale) action groupoid for the action of by conjugation on the subspace 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.
Convolution algebra and Relation to Drinfeld double
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)