nLab
preordered object

Contents

Context

Relations

Category theory

(0,1)(0,1)-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 CC, a preordered object XX is an object with an internal preorder on XX: a subobject of the product R(s,t)X×XR\stackrel{(s,t)}\hookrightarrow X \times X equipped with the following morphisms:

  • internal reflexivity: ρ:XR\rho \colon X \to R which is a section both of ss and of tt, i.e., sρ=tρ=1 Xs \circ \rho = t \circ \rho = 1_X;

  • internal transitivity: τ p:R× XRR\tau_p: R \times_X R \to R which factors the left/right projection map R× XRX×XR \times_X R \to X \times X through RR, i.e., the following diagram commutes

    R t R× XR (sp 1,tp 2) X×X\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 1p_1 and p 2p_2 are the projections defined by the pullback diagram

    R× XR p 2 R p 1 s R t X\array{ R \times_X R & \stackrel{p_2}\rightarrow & R \\ \downarrow^{\mathrlap{p_1}} && \downarrow^{\mathrlap{s}} \\ R & \stackrel{t}\rightarrow & X }
Remark

Since ii is a monomorphism, the maps ρ\rho, and τ p\tau_p are necessarily unique if they exist.

Remark

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

Remark

We can equivalently define an internal preorder RR as (a representing object of) a representable sub-presheaf of hom(,X×X)\hom(-, X \times X) so that for each object YY, the composite of R(Y)hom(Y,X×X)hom(Y,X)×hom(Y,X)R(Y) \hookrightarrow \hom(Y, X \times X) \cong \hom(Y, X) \times \hom(Y, X) exhibits R(Y)R(Y) as an preorder on the set hom(Y,X)\hom(Y, X). The upshot of this definition is that it makes sense even when CC is not finitely complete.

See also

Last revised on May 14, 2022 at 10:58:56. See the history of this page for a list of all contributions to it.