In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. As the name suggests, what a subobject classifier does is to identify/classify subobjects of a given object according to which elements belong to the subobject in question. Because of this role, the subobject classifier is also referred to as the truth value object. In fact the way in which the subobject classifier classifies subobjects of a given object, is by assigning the values true to elements belonging to the subobject in question, and false to elements not belonging to the subobject. This is way the subobject classifier is widely used in the categorical description of logic. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In category theory, there is a general definition of subobject extending the idea of subset and subgroup. ...

Introductory example

As an example, the set Ω = {0,1} is a subobject classifier in the category of sets and functions: to every subset j:U → X we can assign the function χ_j from X to Ω that maps precisely the elements of U to 1 (see characteristic function). Every function from X to Ω arises in this fashion from precisely one subset U. In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ...

To render this example more clear let us 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→{0,1}, which is defined as follows: A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...

(Here we interpret 1 as true and 0 as false.) The role of the characteristic function is to determine which elements belong or not to a certain subset. Since in any category subobjects are identified as monic arrows, we identify the value true with the arrow: true: {0} → {0, 1} which maps 0 to 1. Given this definition it can be easily seen that the subset A can be uniquely defined through the characteristic function A=χ_A^-1(1). Therefore the diagram In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ...

is a pullback. Image File history File links No higher resolution available. ... In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ...

The above example of subobject classifier in Set is very useful because it enables us to easily prove the following axiom:

Axiom: Given a category C, then there exists an isomorphism, In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

y: Sub_C(X) ≅ Hom_C(X, Ω) ∀ X ∈ C

In Set this axiom can be restated as follows:

Axiom: The collection of all subsets of S denoted by , and the collection of all maps from S to the set {0, 1}=2 denoted by 2^S are isomorphic i.e. the function , which in terms of single elements of is A → χ_A, is a bijection. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... A bijective function. ...

The above axiom implies the alternative definition of a subobject classifier:

Definition: Ω is a subobject classifier iff there is a one to one correspondence between subobject of X and morphisms from X to Ω. In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ...

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 In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C...

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. In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... Image File history File links No higher resolution available. ... In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ... In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ... Image File history File links No higher resolution available. ...

Further examples

Every topos has a subobject classifier. For the topos of sheaves of sets on a topological space X, it can be described in these terms: take the disjoint union Ω of all the open sets U of X, and its natural mapping π to X coming from all the inclusion maps. Then π is a local homeomorphism, and the corresponding sheaf is the required subobject classifier (in other words the construction of Ω is by means of its espace étalé). One can also consider Ω to be, in a (tautological) sense, the graph of the membership relation between points x and open sets U of X. In mathematics, a topos (plural topoi or toposes) is a type of category that behaves like the category of sheaves of sets on a topological space. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, inclusion is a partial order on sets. ... In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces. ... In mathematics, a sheaf is the basic tool for expressing relationships between small regions of a space and large regions. ...

Let us consider an example of a subobject classifer in the Topos of presheafs . The formal definition goes as follows

Definition: A Subobject Classifier Ω is a presheaf such that to each object there corresponds an object which represents the set of all sieves (see sieve). In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In general, a sieve separates wanted/desired elements from unwanted material using a tool such as a mesh, net or other filtration or distillation methods. ...

References

• Robert Goldblatt: Topoi, the Categorial Analysis of Logic. North-Holland, New York, 1984. (Studies in logic and the foundations of mathematics, 98.). A good start.

This book has been reprinted by Dover Publications, Inc (2006). The book can also be accessed freely on Robert Goldblatt's homepage: Topoi, the Categorial Analysis of Logic.

• Topos-physics: An explanation of Topos theory and its implementation in Physics

Topos-physics, Where Geometry meets Dynamics