Editing Moduli space (section)

===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''.