nLab
Pos

Context

Categories of categories

Contents

Definition

PosPos (or PosetPoset or PosetsPosets) is the category whose objects are posets and whose morphisms are monotone (weakly increasing) maps.

Since posets can be identified with (0,1)-categories, PosPos can be identified with the full subcategory of Cat spanned by these; thus it might also be called (0,1)Cat(0,1)Cat.

Properties

The hom-sets of PosPos themselves have the structure of posets, because PosPos is a cartesian closed category. This means PosPos is a 2-poset (aka (1,2)(1,2)-category) or locally posetal 2-category. If Set is the primordial example of a category and Cat is the primordial example of a 2-category, then PosPos is the primordial example of a 22-poset.

The category PosPos is a locally presentable category. This implies it is complete and cocomplete. This can also be seen by viewing PosPos as a reflective subcategory of PreOrd, which is topological over SetSet and therefore cocomplete.

We may illustrate the latter point of view by showing how to construct coequalizers in PosPos. Concretely, to form the coequalizer of a pair of poset maps f,g:XYf, g: X \rightrightarrows Y, one first takes their coequalizer q:YZq: Y \to Z in SetSet, and endows ZZ with the smallest reflexive transitive relation that makes qq order-preserving. This makes ZZ a preorder; it is the coequalizer of f,gf, g in PreOrdPreOrd. Then one passes to the poset reflection of this preorder (i.e., identifying zz and zz' in ZZ if zzz \leq z' and zzz' \leq z) to form the coequalizer in PosPos.

category: category

Revised on September 21, 2017 14:24:32 by Todd Trimble (47.18.191.229)