# Relations In Category Theory

Probably a silly question. Suppose that $C$ is a category that does not have finite Cartesian products. So we cannot define a relation on some objects to be a sub object of their Cartesian product (a monic arrow into their Cartesian product). Is there some other natural notion that we can use $inside$ the category to generalise the notion of `relation'? I'm not interested in using the concretisation, so let's suppose $C$ is not concrete.

You could describe a relation between $X$ and $Y$ to be a pair of maps $f\colon R\to X$, $g\colon R\to Y$, so that the family of maps $\{f,g\}$ is monic (meaning, if $fh=fh'$ and $gh=gh'$, then $h=h'$.)
• One also says "$f$ and $g$ are jointly monic"... – მამუკა ჯიბლაძე Oct 14 '14 at 13:38
• All this being said, it's hard to do much with relations in a category $C$ to simulate the usual sort of calculus of relations, unless one assumes more of $C$. For example, to get a half-decent notion of composition of relations, one typically assumes that $C$ is a regular category. – Todd Trimble Oct 14 '14 at 14:28
• @ToddTrimble, but, actually, you do not have to assume that $C$ has products to define associative compositions (it suffices to assume that $C$ has pullbacks and stable images). – Michal R. Przybylek Oct 14 '14 at 16:15