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.
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$\begingroup$ By googling, I have found Is there a standard notation for binary relations in category theory?. $\endgroup$– Philippe GaucherOct 14, 2014 at 13:05
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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 $h=h'$.)
answered Oct 14, 2014 at 13:00
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4$\begingroup$ One also says "$f$ and $g$ are jointly monic"... $\endgroup$ Oct 14, 2014 at 13:38
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$\begingroup$ In fact, the above is the standard definition of an internal relation: an internal relation is a span of morphisms that are jointly mono. In case the category has binary products such relations can be represented as single monomorphisms into cartesian products. The latter definition is a bit less convenient to work with, but it is much easier to generalize it to non-canonical notions of subobjects, therefore some authors use it as their primary definition. $\endgroup$ Oct 14, 2014 at 14:24
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2$\begingroup$ 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. $\endgroup$– Todd Trimble ♦Oct 14, 2014 at 14:28
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$\begingroup$ @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). $\endgroup$ Oct 14, 2014 at 16:15
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