Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
The notion of a preordered object is the generalization of that of preordered sets as one passes from the ambient category of sets into more general ambient categories with suitable properties.
In a category $C$ with pullbacks, a preordered object $X$ is an object with an internal preorder on $X$: a subobject of the product $R\stackrel{(s,t)}\hookrightarrow X \times X$ equipped with the following morphisms:
internal reflexivity: $\rho \colon X \to R$ which is a section both of $s$ and of $t$, i.e., $s \circ \rho = t \circ \rho = 1_X$;
internal transitivity: $\tau: R \times_X R \to R$ which factors the left/right projection map $R \times_X R \to X \times X$ through $R$, i.e., the following diagram commutes
where $p_1$ and $p_2$ are the projections defined by the pullback diagram
Since $i = (s,t)$ is a monomorphism, the maps $\rho$ and $\tau$ are necessarily unique if they exist.
Equivalently, a preordered object $X$ is an internal category with $X$ the object of objects, such that the (source,target)-map is a monomorphism.
We can equivalently define an internal preorder $R$ as (a representing object of) a representable sub-presheaf of $\hom(-, X \times X)$ so that for each object $Y$, the composite of $R(Y) \hookrightarrow \hom(Y, X \times X) \cong \hom(Y, X) \times \hom(Y, X)$ exhibits $R(Y)$ as an preorder on the set $\hom(Y, X)$. The upshot of this definition is that it makes sense even when $C$ is not finitely complete.
Last revised on February 6, 2024 at 04:57:15. See the history of this page for a list of all contributions to it.