category theory

# Contents

## Idea

The concept of internal relation is an internalization of the concept of a relation from Set to more general categories and it is often called just a relation in the category $C$.

If $C$ is a regular category, then its category of internal binary relations is an allegory. The objects of an allegory may, but do not need to be, internal relations in some ambient category.

## Definitions

Let $C$ be a category. An internal binary relation from an object $X$ to an object $Y$ is an object $R$ and a pair of maps $d_0: R \to X$ and $d_1: R \to Y$ that are jointly monic, that is such that, given any object $G$ and morphism $e, e': G \to R$, if $d_0 \circ e = d_0 \circ e'$ and $d_1 \circ e = d_1 \circ e'$, then it must be that $e = e'$.

Now suppose that $C$ has binary products. Then we can simplify the definition; an internal binary relation from $X$ to $Y$ is simply a subobject $r: R\hookrightarrow X\times Y$. The $d_0$ and $d_1$ above may be recovered as the composites

$d_0 = p_X\circ r : R\hookrightarrow X\times Y\to X,\quad d_1 = p_Y\circ r : T\hookrightarrow X\times Y\to Y ,$

called the projections of the relation $R$.

A relation from $X$ to $X$ is also said to be an internal binary relation on $X$.

As with relations in general, we can extend from binary to arbitrary internal relations by generalising from the pair $(X,Y)$ to an arbitrary family of objects. If this family has a product in $C$, then the internal relation is simply a subobject of that product; in general, the internal relation is given by a jointly monic family of morphisms.

## Kinds of internal relations

The various kinds of relations described at relation can often be interpreted internally.

For example, an internal relation $R$ on $X$ is said to be reflexive if it contains the diagonal subobject $X\hookrightarrow X\times X$ of $X$; this can even be stated if $X \times X$ does not exist in the category.

An internal equivalence relation is often called a congruence.

Last revised on May 1, 2015 at 15:13:24. See the history of this page for a list of all contributions to it.