Discrete category

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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

Clearly, any class of objects defines a discrete category when augmented with identity maps.

If a category has only one object, then the category is discrete. For example, when a group is considered as a category of one object, then the group is discrete.

Any subcategory of a discrete category is discrete.

The limit of any functor from a discrete category into another category is called a product, while the colimit is called a coproduct.