In mathematics, especially in differential and algebraic geometries, an inertia stack of a groupoid X is a stack that parametrizes automorphism groups on X {\displaystyle X} and transitions between them. It is commonly denoted as Λ X {\displaystyle \Lambda 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 ) {\displaystyle U=(U_{1}\rightrightarrows U_{0})} be a groupoid. Then the inertia groupoid Λ U {\displaystyle \Lambda U} is a groupoid (= a category whose morphisms are all invertible) where
the objects are the automorphism groups: Aut ( x ) , x ∈ U 0 , {\displaystyle \operatorname {Aut} (x),x\in U_{0},}
the morphisms from x to y are conjugations by invertible morphisms f : x → y {\displaystyle f:x\to y} ; that is, an automorphism g : x → x {\displaystyle g:x\to x} is sent to f ∘ g ∘ f − 1 : y → y , {\displaystyle f\circ g\circ f^{-1}:y\to y,}
the composition is that of morphisms in U {\displaystyle U} . For example, if U is a fundamental groupoid, then Λ U {\displaystyle \Lambda U} keeps track of the changes of base points.
Notes
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