# Discrete 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

homC(X, X) = {idX} for all objects X
homC(X, Y) = ∅ for all objects XY

Since by axioms, there is always the identity morphism between the same object, the above is equivalent to saying

|homC(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.

### Other articles related to "discrete category":

Diagram (category Theory) - Examples
... If J is a (small) discrete category, then a diagram of type J is essentially just an indexed family of objects in C (indexed by J) ... So, for example, when J is the discrete category with two objects, the resulting limit is just the binary product ... B and the two arrows B → A, B → C, the resulting diagram would simply be the discrete category with the two objects A and C, and the colimit would simply be the ...

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