internal relation



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 CC.

If CC 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.


Let CC be a category. An internal binary relation from an object XX to an object YY is an object RR and a pair of maps d 0:RXd_0: R \to X and d 1:RYd_1: R \to Y that are jointly monic, that is such that, given any object GG and morphism e,e:GRe, e': G \to R, if d 0e=d 0ed_0 \circ e = d_0 \circ e' and d 1e=d 1ed_1 \circ e = d_1 \circ e', then it must be that e=ee = e'.

Now suppose that CC has binary products. Then we can simplify the definition; an internal binary relation from XX to YY is simply a subobject r:RX×Yr: R\hookrightarrow X\times Y. The d 0d_0 and d 1d_1 above may be recovered as the composites

d 0=p Xr:RX×YX,d 1=p Yr:TX×YY, 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 RR.

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

As with relations in general, we can extend from binary to arbitrary internal relations by generalising from the pair (X,Y)(X,Y) to an arbitrary family of objects. If this family has a product in CC, 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 RR on XX is said to be reflexive if it contains the diagonal subobject XX×XX\hookrightarrow X\times X of XX; this can even be stated if X×XX \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.