In the mathematical field of category theory, a subobject classifier of a category such that, informally, the subobjects. This provides an analogue of the set of Booleans in categories other than the category of sets.

The main use of subobject classifiers is in topos theory, where an elementary topos is defined as a category with a subobject classifier and certain additional requirements. In the internal language of an elementary topos, the subobject classifier is used to interpret truth values, hence the alternative name “object of truth values”.

Introduction

can be equivalently described by its indicator function

of a category does in the category of sets in a “purely categorical way”, one can take a pullback

is the inclusion map, equipped with the inclusion map.

Definition

be a finitely complete category (we denote the terminal object). A subobject classifier together with a monomorphism., called the characteristic function or classifying map, such that the following diagram is a pullback square):

is locally small which represents the subobject functor. That is, there exists a bijection, natural.

Further examples

Sheaves of sets

The category of sheaves of sets on a topological space X has a subobject classifier Ω which can be described as follows: For any open set U of X, Ω(U) is the set of all open subsets of U. The terminal object is the sheaf 1 which assigns the singleton {*} to every open set U of X. The morphism η:1 → Ω is given by the family of maps ηU : 1(U) → Ω(U) defined by ηU(*)=U for every open set U of X. Given a sheaf F on X and a sub-sheaf j: G → F, the classifying morphism χ j : F → Ω is given by the family of maps χ j,U : F(U) → Ω(U), where χ j,U(x) is the union of all open sets V of U such that the restriction of x to V (in the sense of sheaves) is contained in jV(G(V)).

Roughly speaking an assertion inside this topos is variably true or false, and its truth value from the viewpoint of an open subset U is the open subset of U where the assertion is true.

Presheaves

, the category of presheaves (i.e. the functor category to the set of sieves. The classifying morphisms are constructed quite similarly to the ones in the sheaves-of-sets example above.

Elementary topoi

Both examples above are subsumed by the following general fact: every elementary topos, defined as a category with finite limits and power objects, necessarily has a subobject classifier. The two examples above are Grothendieck topoi, and every Grothendieck topos is an elementary topos.

See also

• Quasitopos, a type of category with a strong subobject classifier, which only classifies strong subobjects

Notes

Citations

References