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Concept
Discrete category
Summary
In mathematics, in the field of , a discrete category is a category whose only morphisms are the identity morphisms:
homC(X, X) = {idX} for all objects X
homC(X, Y) = ∅ for all objects X ≠ Y
Since by axioms, there is always the identity morphism between the same object, we can express the above as condition on the cardinality of the hom-set
| homC(X, Y) | is 1 when X = Y and 0 when X is not equal to Y.
Some authors prefer a weaker notion, where a discrete category merely needs to be equivalent to such a category.
Any class of objects defines a discrete category when augmented with identity maps.
Any of a discrete category is discrete. Also, a category is discrete if and only if all of its subcategories are .
The of any functor from a discrete category into another category is called a , while the colimit is called a coproduct. Thus, for example, the discrete category with just two objects can be used as a or diagonal functor to define a product or coproduct of two objects. Alternately, for a general category C and the discrete category 2, one can consider the C2. The diagrams of 2 in this category are pairs of objects, and the limit of the diagram is the product.
The functor from Set to Cat that sends a set to the corresponding discrete category is left adjoint to the functor sending a small category to its set of objects. (For the right adjoint, see .
Official source
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