Modulius

Modulius is a fictional mathematical concept used to denote a universal moduli space that parameterizes isomorphism classes of objects under a specified moduli problem. The term is sometimes employed in speculative expositions and science fiction settings to discuss the idea of a single space that encodes all possible deformations and families of a given type of geometric or algebraic object.

In formal language, a moduli problem assigns to each scheme S a groupoid Mod(S) of families of objects over S. A moduli space M would represent this functor if Hom(S, M) ≅ Mod(S) for all S; often the best available object is a moduli stack rather than a scheme, reflecting automorphisms. A hypothetical modulius would be a universal object from which all moduli representations can be obtained by pullback along maps from a base S to Modulius, and would carry a universal family U → Modulius.

In practice, representability can fail; familiar examples include moduli of curves which are represented by algebraic stacks M_g; coarse moduli spaces M_g^{coarse} exist under certain conditions. A modulius would generalize such ideas, subsuming classical moduli spaces and their stacks into a single framework.

Variants and structure: The modulius could be stratified by invariants such as genus, degree, stability, polarization; its geometry would reflect deformation theory and obstruction spaces.

Uses: In mathematics, a notional tool for thinking about universality of moduli problems; in fiction, a device for exploring classification and consistency of physical theories.

See also: Moduli space, Moduli stack, Deformation theory, Algebraic geometry, Universal property.