Editing Category of preordered sets

{{Short description|Category whose objects are preordered sets and whose morphisms are order-preserving functions}}
{{More citations needed|date=December 2024}}
In [[mathematics]], the [[category theory|category]] '''PreOrd''' has [[Preorder|preordered sets]] as [[object (category theory)|objects]] and [[order-preserving function]]s as [[morphism]]s.{{sfn|Eklund|Gutiérrez García|Höhle|Kortelainen|2018|loc=Section 1.3}}<ref>{{cite web |title=PreOrd in nLab |url=https://ncatlab.org/nlab/show/PreOrd |website=ncatlab.org}}</ref>
This is a category because the [[function composition|composition]] of two order-preserving functions is order preserving and the identity map is order preserving.

The [[monomorphism]]s in '''PreOrd''' are the [[injective]] order-preserving functions.

The [[empty set]] (considered as a preordered set) is the [[initial object]] of '''PreOrd''', and the [[terminal objects]] are precisely the [[singleton (mathematics)|singleton]] preordered sets. There are thus no [[zero object]]s in '''PreOrd'''.

The categorical [[product (category theory)|product]] in '''PreOrd''' is given by the [[product order]] on the [[cartesian product]].

We have a [[forgetful functor]] '''PreOrd''' → '''[[category of sets|Set]]''' that assigns to each preordered set the underlying [[Set (mathematics)|set]], and to each order-preserving function the underlying [[function (mathematics)|function]]. This functor is [[faithful functor|faithful]], and therefore '''PreOrd''' is a [[concrete category]]. This functor has a left [[adjoint functors|adjoint]] (sending every set to that set equipped with the equality relation) and a right adjoint (sending every set to that set equipped with the total relation).

While '''PreOrd''' is a category with different properties, the category of preordered groups, denoted '''PreOrdGrp''', presents a more complex picture, nonetheless both imply preordered connections.<ref>{{cite web |last1=Clementino |first1=Maria Manuel |last2=Martins-Ferreira |first2=Nelson |last3=Montoli |first3=Andrea |title=On the categorical behaviour of preordered groups |url=https://www.sciencedirect.com/science/article/abs/pii/S0022404919300143 |website=Journal of Pure and Applied Algebra |pages=4226–4245 |doi=10.1016/j.jpaa.2019.01.006 |date=1 October 2019}}</ref>

==2-category structure==
The set of morphisms (order-preserving functions) between two preorders actually has more structure than that of a set. It can be made into a preordered set itself by the pointwise relation:

: (''f'' ≤ ''g'') ⇔ (∀''x'' ''f''(''x'') ≤ ''g''(''x''))

This preordered set can in turn be considered as a category, which makes '''PreOrd''' a [[2-category]] (the additional axioms of a 2-category trivially hold because any equation of parallel morphisms is true in a [[posetal category]]).

With this 2-category structure, a [[pseudofunctor]] F from a category ''C'' to '''PreOrd''' is given by the same data as a 2-functor, but has the relaxed properties:

: ∀''x'' ∈ F(''A''), F(''id''<sub>''A''</sub>)(''x'') ≃ ''x'',

: ∀''x'' ∈ F(''A''), F(''g''<math>\circ</math>''f'')(''x'') ≃ F(''g'')(F(''f'')(''x'')),

where ''x'' ≃ ''y'' means ''x'' ≤ ''y'' and ''y'' ≤ ''x''.

==See also==
*[[FinOrd]]
*[[Simplex category]]

==Notes==
{{reflist}}

==References==
* {{cite book |last1=Eklund |first1=Patrik |last2=Gutiérrez García |first2=Javier |last3=Höhle |first3=Ulrich |last4=Kortelainen |first4=Jari |title=Semigroups in Complete Lattices: Quantales, Modules and Related Topics |date=2018 |publisher=Springer |isbn=978-3319789484}}

{{DEFAULTSORT:Category Of Preordered Sets}}
[[Category:Categories in category theory|Preordered sets]]