nLab classifying morphism

Idea

A classifying map or classifying morphism for a given object is a morphism into a classifying space or moduli space that classified this object.

Typically this means that the map characterizes the object (only) up to equivalence. There is also the more refined concept of morphisms into a moduli stack that characterizes the object itself. For emphasis this case might be called a modulating morphism.

Examples

• For subobjects one typically speaks of characteristic maps or characteristic functions. The corresponding classifiyng space is a subobject classifier .

• More generally, in an (infinity,1)-topos every “small” (see there) object in a slice (infinity,1)-topos is given by a classifying morphism into the object classifier;

• In dependent/homotopy type theory these classifying morphisms are the categorical semantics for functions into a type of types that classify dependent types. See at categorical model of dependent types for more on this.

Related concepts

• classifying space, classifying stack, moduli, moduli space, moduli stack, derived moduli space, modulating morphism

• display map, substitution