Discrete category

In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms:

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, we can express the above as condition on the cardinality of the hom-set

| hom_C(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.