Subobject classifier · Lablypedia

Introductory example

As an example, the set Ω = {0,1} 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 χ_A from S to Ω that maps precisely the elements of A to 1 (see characteristic function). Every function from S to Ω arises in this fashion from precisely one subset A.

To be clearer, consider a subset A of S (A ⊆ S), where S is a set. The notion of being a subset can be expressed mathematically using the so-called characteristic function χ_A : S → {0,1}, which is defined as follows:

χ_{A} (x) = {\begin{cases} 0, & if x \notin A \\ 1, & if x \in A \end{cases}

(Here we interpret 1 as true and 0 as false.) The role of the characteristic function is to determine which elements belong to the subset A. In fact, χ_A 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 = {0,1} are isomorphic.

To categorize this notion, recall that, in category theory, a subobject is actually a pair consisting of an object and a monic arrow (interpreted as the inclusion into another object). Accordingly, true refers to the element 1, which is selected by the arrow: true: {0} → {0, 1} that maps 0 to 1. The subset A of S can now be defined as the pullback of true along the characteristic function χ_A, shown on the following diagram:

Defined that way, χ is a morphism Sub_C(S) → Hom_C(S, Ω). By definition, Ω is a subobject classifier if this morphism is an isomorphism.

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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 morphism

1 → Ω

with the following property:

For each monomorphism j: U → X there is a unique morphism χ_j: X → Ω such that the following commutative diagram

is a pullback diagram—that is, U is the limit of the diagram:

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(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 η_U : 1(U) → Ω(U) defined by η_U(*)=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 j_V(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

Given a small category $C$ , the category of presheaves ${S e t}^{C^{o p}}$ (i.e. the functor category consisting of all contravariant functors from $C$ to $S e t$ ) has a subobject classifer given by the functor sending any $c \in C$ to the set of sieves on $c$ . 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.

A quasitopos has an object that is almost a subobject classifier; it only classifies strong subobjects.