(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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A stack $A$ is called a moduli stack for certain structures, if for any other object $X$ the groupoid of morphisms $X \to A$ into $A$ 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 gauge equivalences/ automorphism groups of these structures.
There is an evident generalization of the concept of moduli stacks in the more general context of higher topos theory, to moduli ∞-stacks.
Notice that every stack is the moduli stack of something and in fact in general of different things at the same time (see below). So to some extent saying “moduli stack” is redundant. It is usually used to indicate, roughly, that there are some spaces/stacks that one is working on/over, and then there are apart from this stacks, the moduli stacks, that one is mapping into.
This distinction however easily disappears. For instance a historically famous moduli stack is the moduli stack of elliptic curves which started out as an object used to classify bundles of elliptic curves over other spaces. Later in the study of elliptic cohomology and tmf the “moduli stack” of elliptic curves came to be regarded as a space interesting in itself for the geometry on it, specifically since this is naturally a derived algebraic geometry.
Analogous comments apply to other moduli stacks. For instance for $G$ a topological group, the moduli stack $\mathbf{B}G \simeq \ast //G$ for topological $G$-principal bundles is itself interesting for its own geometry. Notably it is the base stack of the universal principal bundle which as such may be equipped with differential geometry such as a connection on a bundle etc.
Generally, what one needs for a stack to classify bundles in this way is a universal bundle over it, for then what the stack modulates are precisely the pullbacks of this universal bundle. A stack with a prescribed universal bundle over it may be regarded as a stack equipped with an atlas.
In conclusion then “moduli stack” pretty much means “stack” or more precisely “stack with specified universal bundle over it or atlas into it”, with the implicit implication that we say “moduli stack” to indicate that we care about pulling back that atlas/bundle along maps into the stack.
(Compare this to how one says “presheaf” for what is really just a functor in order to indicate a certain attitude, namely that one will be interested in asking which presheaves are sheaves.)
Let $\mathbf{H} := Sh_\infty(SmthMnfd)$ 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 a group object in $\mathbf{H}$ – a “group stack” – and its delooping in $\mathbf{H}$ produces a stack denoted $\mathbf{B}G$. 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
are the same as smooth $G$-principal bundles over $X$. More precisely the groupoid $G Bund(X)$ of smooth $G$-principal bundles and smooth gauge transformations between them is canonically equivalent to the hom-groupoid of maps from $X$ to $\mathbf{B}G$:
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.
A famous moduli stack is that of elliptic curves. See moduli stack of elliptic curves for more on this.