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 and transitions between them. It is commonly denoted as and is defined as inertia groupoids as charts. The notion often appears in particular as an inertia orbifold.

Inertia groupoid

Let be a groupoid. Then the inertia groupoid is a groupoid where

the objects are the automorphism groups:
the morphisms from x to y are conjugations by invertible morphisms ; that is, an automorphism is sent to
the composition is that of morphisms in.

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