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.

Notes

1. 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.

Further reading

• https://mathoverflow.net/questions/122537/what-is-the-intuition-behind-the-inertia-orbifold-or-stack
• https://ncatlab.org/nlab/show/inertia+orbifold
• https://thehighergeometer.wordpress.com/2024/09/15/abelian-differentiable-gerbes-recap/
• http://pantodon.jp/index.rb?body=inertia_groupoid in Japanese