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 $\Lambda X$ and is defined as inertia groupoids as charts. The notion often appears in particular as an inertia orbifold.

Inertia groupoid

Let $U=(U_1\rightrightarrows U_0)$ be a groupoid. Then the inertia groupoid $\Lambda U$ is a groupoid (= a category whose morphisms are all invertible) where

• the objects are the automorphism groups: $\mathrm{Aut}(x),x\in U_0,$
• 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 $\Lambda U$ keeps track of the changes of base points.

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