In 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
MorC(X, X) = {idX} for all objects X
MorC(X, Y) = ∅ for all objects X ≠ Y
Clearly, any class of objects defines a discrete category when augmented with identity maps.
Any subcategory of a discrete category is discrete.
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces.
In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point).
A category with all finite biproducts is known as an additive category.
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