Inertia stack

In mathematics, especially in differential and algebraic geometries, an inertia stack of a groupoid X is a stack that parametrizes automorphism groups on $X$ and transitions between them. It is commonly denoted as $Λ X$ and is defined as inertia groupoids as charts. The notion often appears in particular as an inertia orbifold.

Inertia groupoid

Let $U = (U_{1} ⇉ U_{0})$ be a groupoid. Then the inertia groupoid $Λ U$ is a groupoid (= a category whose morphisms are all invertible) where

the objects are the automorphism groups: $Aut (x), x \in U_{0},$
the morphisms from x to y are conjugations by invertible morphisms $f : x \to y$ ; that is, an automorphism $g : x \to x$ is sent to $f \circ g \circ f^{- 1} : y \to y,$
the composition is that of morphisms in $U$ .

For example, if U is a fundamental groupoid, then $Λ U$ keeps track of the changes of base points.

Notes

↑ Adem, Ruan & Zhang 2008, Definition 2.6.

References

Farsi, Carla; Seaton, Christopher (2009). "Nonvanishing vector fields on orbifolds". Transactions of the American Mathematical Society. 362: 509–535. arXiv:0807.2738. doi:10.1090/S0002-9947-09-04938-1.
Adem, Alejandro; Ruan, Yongbin; Zhang, Bin (2008). "A Stringy Product on Twisted Orbifold K-theory". arXiv:math/0605534.

Inertia stack

Inertia groupoid

Notes

References

Further reading