PlanetMath: discrete category

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discrete category

(Definition)

A category $\mathcal{C}$ is said to be a discrete category if the only morphisms in $\mathcal{C}$ are the identity morphisms associated with each of the objects in $\mathcal{C}$ .

A discrete category with one object is called a trivial category. A discrete category with no objects is called the empty category.

Remark. Given any category $\mathcal{C}$ , the smallest subcategory consisting of all objects in $\mathcal{C}$ is discrete, which is also the largest discrete subcategory in $\mathcal{C}$ (largest in the sense that it contains all objects of $\mathcal{C}$ ). For every object $X\in \mathcal{C}$ , we can associate the trivial category $\mathcal{C}_X$ consisting of one object, , and one morphism .

"discrete category" is owned by CWoo.

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Also defines:

trivial category, empty category

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Cross-references: associate, contains, discrete, subcategory, objects, identity, morphisms, category
There are 5 references to this entry.

This is version 2 of discrete category, born on 2006-09-16, modified 2007-10-24.
Object id is 8357, canonical name is DiscreteCategory.
Accessed 865 times total.

Classification:

AMS MSC:

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

Pending Errata and Addenda

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