# Glossary of stack theory

This is a glossary of stack theory. For the definition of a stack, see stack.

## F

fiber product
A stack $F \times_G H$ given for $f: F \to G, g:H \to G$: an object over B is a triple (x, y, ψ), x in F(B), y in H(B), ψ an isomorphism $f(x) \to g(y)$ in G(B); an arrow from (x, y, ψ) to (x', y', ψ') is a pair of morphisms $\alpha: x \to x', \beta: y \to y'$ such that $\psi' \circ f(\alpha) = g(\beta) \circ \psi$. The resulting square with obvious projections does not commute; rather, it commutes up to natural isomorphism; i.e., it 2-commutes.

inertia stack

## M

morphism
A morphism $f: F \to G$ of stacks (over, say, the category of S-schemes) is a functor such that $f \circ P_G = P_F$ where $P_F, P_G$ are structures maps to the base category.

## P

property P
Let P be a property of a scheme that is stable under base change (finite-type, proper, smooth, étale, etc.). Then a representable morphism $f: F \to G$ is said to have propert P if, for any $B \to G$ with B a scheme, the base change $F \times_G B \to B$ has property P.

## Q

quotient stack
Usually denoted by [X/G], a quotient stack generalizes a quotient of a scheme or variety.

## R

representable morphism
A morphism $F \to G$ of stacks such that, for any morphism $B \to G$ from a scheme B, the base change $F \times_G B$ is a scheme. (Some authors impose algebraic space conditions.)