# Cartesian monoidal category

In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a **cartesian monoidal category**. Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the terminal object is the tensor 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.[1]

## Properties

Cartesian monoidal categories have a number of special and important properties, such as the existence of diagonal maps Δ_{x} : *x* → *x* ⊗ *x* and augmentations *e*_{x} : *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.

Cocartesian monoidal categories:

**Vect**, the category of vector spaces over a given field, can be made cocartesian monoidal with the "tensor 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
of (left) modules on a ring (commutative or not) becomes a cocartesian monoidal category with the direct sum of modules as tensor product and the trivial module as unit.*R*-mod

In each of these categories of modules equipped with a cocartesian monoidal structure, finite products and coproducts coincide (in the sense that the product and coproduct of finitely many objects are isomorphic). Or more formally if *f* : *X*_{1} ∐ ... ∐ *X*_{n} → *X*_{1} × ... × *X*_{n} is the "canonical" map from the *n*-ary coproduct of objects *X*_{j} to their product, for a natural number *n*, in the event that the map *f* is an isomorphism, we say that a biproduct for the objects *X*_{j} is an object isomorphic to and together with maps *i*_{j} : *X*_{j} → *X* and *p*_{j} : *X* → *X*_{j} such that the pair (*X*, {*i*_{j}}) is a coproduct diagram for the objects *X*_{j} and the pair (*X*, {*p*_{j}}) is a product diagram for the objects *X*_{j} , and where *p*_{j} ∘ *i*_{j} = id_{Xj}. If in addition the category in question has a zero object, so that for any objects *A* and *B* there is a unique map 0_{A,B} : *A* → 0 → *B*, it often follows that *p*_{k} ∘ *i*_{j} = : *δ*_{ij}, the Kronecker delta, where we interpret 0 and 1 as the 0 maps and identity maps of the objects *X*_{j} and *X*_{k}, respectively. See pre-additive category for more.