{{Short description| Concept in set theory}} {{More citations needed|date=December 2024}} In mathematics, the category '''PreOrd''' has preordered sets as objects and order-preserving functions as morphisms.{{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 composition of two order-preserving functions is order preserving and the identity map is order preserving.
The monomorphisms 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 preordered sets. There are thus no zero objects in '''PreOrd'''.
The categorical product in '''PreOrd''' is given by the product order on the cartesian product.
We have a forgetful functor '''PreOrd''' → '''Set''' that assigns to each preordered set the underlying set, and to each order-preserving function the underlying function. This functor is faithful, and therefore '''PreOrd''' is a concrete category. This functor has a left 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}} Preordered sets