PlanetMath: preorder as a category

(more info)

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preorder as a category

(Example)

Every preorder has an associated structure of a category. Before describing what this category is, we first associate with a simpler structure, that of a precategory.

Let's call this $\operatorname{PreCat}(P)$ . The objects of this precategory are elements of and for every $a,b\in P$ , $\hom(a,b)$ is either a singleton if $a\le b$ , or the empty set otherwise. The category associated with is the category generated by enlarging $\operatorname{PreCat}(P)$ . For now, call this category $\operatorname{Cat}(P)$ . Then we see that the objects of $\operatorname{Cat}(P)$ are again elements of , and for every $a,b\in P$ , $\hom(a,b)$ is the set of all finite chains from to .

With this association, we see the following constructs also have the structure of a category:

a poset: here, a morphism in $\hom(a,b)$ is a finite chain from to where successive nodes are related such that the subsequent node covers the prior node
a partition of a (non-empty) set (a set with an equivalence relation): $\hom(a,b)$ is non-empty iff and belong to the same partition
a lattice: every pair of objects have a product and a coproduct
a well-ordered set, in particular an ordinal: if $\hom(a,b)$ is non-empty, it is a singleton. For example, $% latex2html id marker 281 $ \textbf{n}$$ is the category consisting of objects $0,1,\ldots,n-1$ , and if $a\le b$ , a morphism in $\hom(a,b)$ is the chain $a\to a+1 \to \cdots \to b-1 \to b$ .