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[parent] category associated to a partial order (Example)

Let $ S$ be a collection of objects, and let $ \leq$ be a partial order on $ S$. Then we can construct a category $ C$ as follows. Let the objects of $ C$ be exactly $ S$. For a pair of objects $ A$ and $ B$ in $ S$, construct a single arrow from $ A$ to $ B$ if $ A\leq B$; otherwise there are no arrows from $ A$ to $ B$. Then $ C$ is a category.

Categories of this form are of interest because they provide some justification for using categories as indices for direct limits and inverse limits. This is exactly the construciton one goes through to construct the Zariski site, for example, and a stalk on that site is easily recognizable as a limit over a category of this form.



"category associated to a partial order" is owned by archibal.
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See Also: partial order, inverse limit, limit of a functor


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Cross-references: limit, site, stalk, Zariski site, direct limits, indices, arrow, category, partial order, objects, collection
There are 2 references to this entry.

This is version 3 of category associated to a partial order, born on 2004-02-24, modified 2004-02-24.
Object id is 5618, canonical name is CategoryAssociatedToAPartialOrder.
Accessed 1809 times total.

Classification:
AMS MSC: 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)
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