nLab preordered object




Category theory

(0,1)(0,1)-Category theory



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 CC with pullbacks, 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:R× XRR\tau: 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 τ R× XR (sp 1,tp 2) X×X\array{ && R \\ & {}^{\mathllap{\tau}}\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 }

Since i=(s,t)i = (s,t) is a monomorphism, the maps ρ\rho and τ\tau are necessarily unique if they exist.


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


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 February 6, 2024 at 04:57:15. See the history of this page for a list of all contributions to it.