Cartesian monoidal category
In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal product is the categorical product is called a cartesian monoidal category. Any category with finite products can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the terminal object is the monoidal unit. Dually, a monoidal finite coproduct category with the monoidal structure given by the coproduct and unit the initial object is called a cocartesian monoidal category, and any finite coproduct category can be thought of as a cocartesian monoidal category.
Cartesian categories with an internal Hom functor that is an adjoint functor to the product are called Cartesian closed categories.
Properties
Cartesian monoidal categories have a number of special and important properties, such as the existence of diagonal maps Δx : x → x ⊗ x and augmentations ex : x → I for any object x. In applications to computer science we can think of Δ as "duplicating data" and e as "deleting data". These maps make any object into a comonoid. In fact, any object in a cartesian monoidal category becomes a comonoid in a unique way.Examples
Cartesian monoidal categories:- Set, the category of sets with the singleton set serving as the unit.
- Cat, the bicategory of small categories with the product category, where the category with one object and only its identity map is the unit.
- Vect, the category of vector spaces over a given field, can be made cocartesian monoidal with the monoidal product given by the direct sum of vector spaces and the trivial vector space as unit.
- Ab, the category of abelian groups, with the direct sum of abelian groups as monoidal product and the trivial group as unit.
- More generally, the category R-Mod of modules over a ring R becomes a cocartesian monoidal category with the direct sum of modules as tensor product and the trivial module as unit.