Editing Moduli space (section)

==Definitions==
There are several related notions of things we could call moduli spaces.  Each of these definitions formalizes a different notion of what it means for the points of space ''M'' to represent geometric objects.

===Fine moduli space===
This is the standard concept.  Heuristically, if we have a space ''M'' for which each point ''m'' ∊ ''M'' corresponds to an algebro-geometric object ''U<sub>m</sub>'', then we can assemble these objects into a [[tautological bundle|tautological]] family ''U'' over ''M''.  (For example, the Grassmannian '''G'''(''k'', ''V'') carries a rank ''k'' bundle whose fiber at any point [''L''] ∊ '''G'''(''k'', ''V'') is simply the linear subspace ''L'' ⊂ ''V''.) ''M'' is called a '''base space''' of the family ''U''. We say that [[universal bundle | such a family]] is '''universal''' if any family of algebro-geometric objects ''T'' over any base space ''B'' is the [[Pullback (category theory)|pullback]] of ''U'' along a unique map ''B'' → ''M''. A fine moduli space is a space ''M'' which is the base of a universal family.

More precisely, suppose that we have a functor ''F'' from schemes to sets, which assigns to a scheme ''B'' the set of all suitable families of objects with base ''B''. A space ''M'' is a '''fine moduli space''' for the functor ''F'' if ''M'' [[representable functor|represents]] ''F'', i.e., there is a natural isomorphism
τ : ''F'' → '''Hom'''(−, ''M''), where '''Hom'''(−, ''M'') is the functor of points. This implies that ''M'' carries a universal family; this family is the family on ''M'' corresponding to the identity map '''1'''<sub>''M''</sub> ∊ '''Hom'''(''M'', ''M'').

===Coarse moduli space===
Fine moduli spaces are desirable, but they do not always exist and are frequently difficult to construct, so mathematicians sometimes use a weaker notion, the idea of a coarse moduli space. A space ''M'' is a '''coarse moduli space''' for the functor ''F'' if there exists a natural transformation τ : ''F'' → '''Hom'''(−, ''M'') and τ is universal among such natural transformations. More concretely, ''M'' is a coarse moduli space for ''F'' if any family ''T'' over a base ''B'' gives rise to a map φ<sub>''T''</sub> : ''B'' → ''M'' and any two objects ''V'' and ''W'' (regarded as families over a point) correspond to the same point of ''M'' if and only if ''V'' and ''W'' are isomorphic. Thus, ''M'' is a space which has a point for every object that could appear in a family, and whose geometry reflects the ways objects can vary in families. Note, however, that a coarse moduli space does not necessarily carry any family of appropriate objects, let alone a universal one.

In other words, a fine moduli space includes ''both'' a base space ''M'' and universal family ''U'' → ''M'', while a coarse moduli space only has the base space ''M''.

===Moduli stack<!--'Moduli stacks' redirects here-->===
It is frequently the case that interesting geometric objects come equipped with many natural [[automorphism]]s. This in particular makes the existence of a fine moduli space impossible (intuitively, the idea is that if ''L'' is some geometric object, the trivial family ''L'' × [0,1] can be made into a twisted family on the circle '''S'''<sup>1</sup> by identifying ''L'' × {0} with ''L'' × {1} via a nontrivial automorphism. Now if a fine moduli space ''X'' existed, the map '''S'''<sup>1</sup> → ''X'' should not be constant, but would have to be constant on any proper open set by triviality), one can still sometimes obtain a coarse moduli space. However, this approach is not ideal, as such spaces are not guaranteed to exist, they are frequently singular when they do exist, and miss details about some non-trivial families of objects they classify.

A more sophisticated approach is to enrich the classification by remembering the isomorphisms. More precisely, on any base ''B'' one can consider the category of families on ''B'' with only isomorphisms between families taken as morphisms. One then considers the [[fibred category]] which assigns to any space ''B'' the groupoid of families over ''B''. The use of these ''categories fibred in groupoids'' to describe a moduli problem goes back to Grothendieck (1960/61). In general, they cannot be represented by schemes or even [[algebraic space]]s, but in many cases, they have a natural structure of an [[algebraic stack]].

Algebraic stacks and their use to analyze moduli problems appeared in Deligne-Mumford (1969) as a tool to prove the irreducibility of the (coarse) [[moduli of algebraic curves|moduli space of curves]] of a given genus. The language of algebraic stacks essentially provides a systematic way to view the fibred category that constitutes the moduli problem as a "space", and the '''moduli stack'''<!--boldface per WP:R#PLA--> of many moduli problems is better-behaved (such as smooth) than the corresponding coarse moduli space.