Discrete category
Encyclopedia
In mathematics
, especially category theory
, a discrete category is a category whose only morphism
s are the identity morphisms. It is the simplest kind of category. Specifically a category C is discrete if
Since by axioms, there is always the identity morphism between the same object, the above is equivalent to saying
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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a discrete category is a category whose only morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s are the identity morphisms. It is the simplest kind of category. Specifically a category C is discrete if
- homC(X, X) = {idX} for all objects X
- homC(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
- |homC(X, Y)| is 1 when X = Y and 0 when X is not equal to Y.
Clearly, any class
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
of objects defines a discrete category when augmented with identity maps.
Any subcategory
Subcategory
In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and...
of a discrete category is discrete. Also, a category is discrete if and only if all of its subcategories are full.
The limit
Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
of any functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
from a discrete category into another category is called a product
Product (category theory)
In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
, while the colimit is called a coproduct
Coproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
.