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EspañolDISCRETE CATEGORY
In mathematics, especially category theory, a 'discrete category' is a category whose only morphisms are the identity morphisms. It is the simplest kind of category. Specifically a category ''C'' is discrete if
:Mor''C''(''X'', ''X'') = {id''X''} for all objects ''X''
:Mor''C''(''X'', ''Y'') = ∅ for all objects ''X'' ≠ ''Y''
Since by axioms, there is always the identity morphism between the same object, the above is equivalent to saying
:|Mor''C''(''X'', ''Y'')| is 1 when X = Y and 0 when X is not equal to Y.
Clearly, any class of objects defines a discrete category when augmented with identity maps.
Any subcategory of a discrete category is discrete. Also, a category is discrete if and only if all of its subcategories are full.
The limit of any functor from a discrete category into another category is called a product, while the colimit is called a coproduct.
:Mor''C''(''X'', ''X'') = {id''X''} for all objects ''X''
:Mor''C''(''X'', ''Y'') = ∅ for all objects ''X'' ≠ ''Y''
Since by axioms, there is always the identity morphism between the same object, the above is equivalent to saying
:|Mor''C''(''X'', ''Y'')| is 1 when X = Y and 0 when X is not equal to Y.
Clearly, any class of objects defines a discrete category when augmented with identity maps.
Any subcategory of a discrete category is discrete. Also, a category is discrete if and only if all of its subcategories are full.
The limit of any functor from a discrete category into another category is called a product, while the colimit is called a coproduct.
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