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 ${\displaystyle X}$ and transitions between them. It is commonly denoted as ${\displaystyle \Lambda X}$ and is defined as inertia groupoids as charts. The notion often appears in particular as an inertia orbifold.

Inertia groupoid

Let ${\displaystyle U=(U_{1}\rightrightarrows U_{0})}$ be a groupoid. Then the inertia groupoid ${\displaystyle \Lambda U}$ is a grouoiud (= a category whose morphisms are all invertible) where

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

For example, if U is a fundamental groupoid, then ${\displaystyle \Lambda U}$ keeps track of the changes of base points.

References

1. Adem, Ruan & Zhang, Definition 2.6. harvnb error: no target: CITEREFAdemRuanZhang (help)

• Carla Farsi and Christopher Seaton. Nonvanishing vector fields on orbifolds. arXiv: 0807.2738.
• Alejandro Adem, Yongbin Ruan, and Bin Zhang. A Stringy Product on Twisted Orbifold K-theory. arXiv: math/0605534.

Further reading

• https://mathoverflow.net/questions/122537/what-is-the-intuition-behind-the-inertia-orbifold-or-stack
• https://ncatlab.org/nlab/show/inertia+orbifold
• http://pantodon.jp/index.rb?body=inertia_groupoid in Japanese