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

hom_C(X, X) = {id_X} for all objects X

hom_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

|hom_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.

References

• Robert Goldblatt (1984). Topoi, the Categorial Analysis of Logic (Studies in logic and the foundations of mathematics, 98). North-Holland. Reprinted 2006 by Dover Publications, and available online at Robert Goldblatt's homepage.