nLab
moduli stack

Idea

A stack $A$ is called a moduli stack for certain structures, if for any other object $X$ the groupoid of morphisms $X\to A$ is equivalent to the groupoid of these kinds of structures on $X$.

This is in contrast to the notion of moduli space, which is only about equivalence classes of structures and loses the information about the automorphism groups of these structures.

There is an evident generalization of the concept of moduli stacks further up in higher topos theory, to moduli $\mathrm{\infty }$-stacks.

Notice, however, that every stack is the moduli stack of something .

Examples

Of smooth principal bundles

Let $H:={\mathrm{Sh}}_{\mathrm{\infty }}(\mathrm{SmthMnfd})$ be collection of differentiable stacks and generally of stacks and ∞-stacks over the site of smooth manifolds (see Smooth∞Grpd for details).

Then every Lie group $G$ is canonically an group object in $H$ – a “group stack” – and its delooping in $H$ produces a stack denoted $BG$. This is simply the (stackification of) the Lie groupoid $*//G$ with a single object and $G$ worth of automorphisms on this object.

Let the $X$ be any smooth manifold, also regarded as a stack, via the Yoneda embedding. Then one finds that morphisms of stacks

$$X\to BG$$X \to \mathbf{B}G

are the same as smooth $G$-principal bundles over $X$. More precisely the groupoid $G\mathrm{Bund}(X)$ of smooth $G$-principal bundles and smooth gauge transformations between them is canonically equivalent to the hom-groupoid of maps from $X$ to $BG$:

$$G\mathrm{Bund}(X)\simeq H(X,BG).$$G Bund(X) \simeq \mathbf{H}(X, \mathbf{B}G) \,.

This is discussed in some detail at principal bundle.

The statement immediately generalizes to higher degrees and to other notions of (higher) geometry. This is discussed at principal ∞-bundle.

Of elliptic curves

A famous moduli stack is that of elliptic curves. See moduli stack of elliptic curves for more on this.

Related concepts

• classifying space, moduli space, derived moduli space, geometric invariant theory

• rigidification of a stack

• Yoneda lemma, (∞,1)-Yoneda lemma