Contents

category theory

Contents

Idea

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.

Definitions

Definition

In a finitely complete category $C$, 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: $\tau_p: R \times_X R \to R$ which factors the left/right projection map $R \times_X R \to X \times X$ through $R$, i.e., the following diagram commutes

$\array{ && R \\ & {}^{\mathllap{t}}\nearrow & \downarrow \\ R \times_X R & \stackrel{(s \circ p_1,t \circ p_2)}\rightarrow & X \times X }$

where $p_1$ and $p_2$ are the projections defined by the pullback diagram

$\array{ R \times_X R & \stackrel{p_2}\rightarrow & R \\ \downarrow^{\mathrlap{p_1}} && \downarrow^{\mathrlap{s}} \\ R & \stackrel{t}\rightarrow & X }$
Remark

Since $i$ is a monomorphism, the maps $\rho$, and $\tau_p$ are necessarily unique if they exist.

Remark

Equivalently, a preordered object $X$ is an internal category with $X$ the object of objects, such that the (source,target)-map is a monomorphism.

Remark

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.