Subobject classifier
In mathematics, especially in category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object X in the category correspond to the morphisms from X to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of X. Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values.
Introductory example
As an example, the set Ω = is a subobject classifier in the category of sets and functions: to every subset A of S defined by the inclusion function j : A → S we can assign the function χA from S to Ω that maps precisely the elements of A to 1, and the elements outside A to 0. Conversely, every function from S to Ω arises in this fashion from precisely one subset A.To be clearer, consider a subset A of S, where S is a set. The notion of being a subset can be expressed mathematically using the so-called characteristic function χA : S →, which is defined as follows:
The role of the characteristic function is to determine which elements belong to the subset A. In fact, χA is true precisely on the elements of A.
In this way, the collection of all subsets of S and the collection of all maps from S to Ω = are isomorphic.
To categorize this notion, recall that, in category theory, a subobject is actually represented by a pair consisting of an object A and a monic arrow A → S. Accordingly, true refers to the element 1, which is selected by the arrow: true: → that maps 0 to 1. The subset A of S can now be defined as the pullback of true along the characteristic function χA, shown on the following diagram:
Image:SubobjectClassifier-01.svg|center|frameless|class=skin-invert
Defined that way, χ is a morphism SubC → HomC. By definition, Ω is a subobject classifier if this morphism χ is an isomorphism.
Definition
For the general definition, we start with a category C that has a terminal object, which we denote by 1. The object Ω of C is a subobject classifier for C if there exists a morphismwith the following property:
Image:SubobjectClassifier-02.svg|center|frameless|class=skin-invert
Image:SubobjectClassifier-03.svg|center|frameless|class=skin-invert
The morphism χ j is then called the classifying morphism for the subobject represented by j.
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 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 → Ω 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 → Ω, where χ j,U is the union of all open sets V of U such that the restriction of x to V is contained in jV.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.