Distributive category
Encyclopedia
In mathematics
, a category is distributive if it has finite product
s and finite coproducts such that for every choice of objects
, the canonical map
is an isomorphism, and for all objects
, the canonical map 
is an isomorphism. Equivalently. if for every object 
the functor 
preserves coproducts up to isomorphisms 
. It follows that 
and aforementioned canonical maps are equal for each choice of objects.
For example, Set
is distributive, while Grp
is not.
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...
, a category is distributive if it has finite 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...
s and finite coproducts such that for every choice of objects
, the canonical map
is an isomorphism, and for all objects
, the canonical map is an isomorphism. Equivalently. if for every object the functor preserves coproducts up to isomorphisms . It follows that and aforementioned canonical maps are equal for each choice of objects.For example, Set
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
is distributive, while Grp
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
is not.