Discrete category - Citizendia

In mathematics, especially category theory, a discrete category is a category whose only morphisms are the identity morphisms. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and 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. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously

Any subcategory of a discrete category is discrete. In Mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in Also, a category is discrete if and only if all of its subcategories are full. This is a glossary of properties and concepts in Category theory in Mathematics.

The limit of any functor from a discrete category into another category is called a product, while the colimit is called a coproduct. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological

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