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preorder as a category
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(Example)
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Every preorder $P$ has an associated structure of a category. Before describing what this category is, we first associate $P$ with a simpler structure, that of a precategory.
Let's call this $\operatorname{PreCat}(P)$ The objects of this precategory are elements of $P$ 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 $P$ 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 $P$ and for every $a,b\in P$ $\hom(a,b)$ is the set of all finite chains $f$ from $a$ to $b$
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 $a$ to $b$ 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 $a$ and $b$ 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, ${n}$ 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 $a\to a+1 \to \cdots \to b-1 \to b$
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"preorder as a category" is owned by CWoo. [ full author list (2) ]
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| Also defines: |
preorder category, ordinal category, poset category, lattice category |
This object's parent.
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Cross-references: chain, ordinal, well-ordered set, coproduct, product, lattice, iff, equivalence relation, partition, covers, nodes, morphism, poset, finite chains, generated by, empty set, singleton, objects, precategory, associate, category, structure, preorder
There is 1 reference to this entry.
This is version 5 of preorder as a category, born on 2007-02-24, modified 2007-02-25.
Object id is 8971, canonical name is PreorderAsACategory.
Accessed 2953 times total.
Classification:
| AMS MSC: | 18B35 (Category theory; homological algebra :: Special categories :: Preorders, orders and lattices ) |
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Pending Errata and Addenda
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