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What is the canonical name for the category whose objects are all pairs $(X, \rho)$, where $X$ is a set and $\rho$ is a binary relation on $X$, and whose objects are relation-preserving maps? That is, a morphism $(X, \rho) \overset{f}{\to} (Y, \sigma)$ is a map $f : X \to Y$ that satisfies $x \mathbin{\rho} x' \implies f(x) \mathbin{\sigma} f(x')$.

Abstract and Concrete Categories calls this category $\mathbf{Rel}$ (in fact the above definition was lifted from Page 22), but Wikipedia defines $\mathbf{Rel}$ as the category whose objects are sets and whose morphisms are relations between them. Why is there such a conflict of definition, and which definition is accepted?

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up vote 2 down vote accepted

The wikipedia definition and terminology is the accepted one and corresponds to MacLane's CWM definition (2nd ed. page 26). It has the nice property of being the Kleisli category of the covariant powerset monad.

On the other end, the category defined in "Abstract and Concrete categories" (ACC) is shown to be $\mathbf{Spa}(S^2)$ on page 76.

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You seem to be missing a few words after Kleisli category. Perhaps you mean something like "the Kleisli category of the covariant powerset monad"? – Zhen Lin Nov 28 '12 at 17:00
thank you @ZhenLin. Fixed. – magma Nov 28 '12 at 17:32

As long as you define your notation then it is not a problem to call this category $\mathbf{Rel}$. If you want to give all of your categories extremely short names which reflect one property or feature of the category they refer to, then you're going to have some conflicts, as illustrated here.

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