Cartesian monoidal category
# 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 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 $Δ_x$ : 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 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 (left) modules over a ring R (commutative or not) becomes a cocartesian monoidal category with the direct sum of modules as tensor product and the trivial module as unit.

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 $X=\bigoplus _{j\in {1,\ldots ,n}}X_{j}$ isomorphic to $\coprod _{j\in {1,\ldots ,n}}X_{j}$ and $\prod _{j\in {1,\ldots ,n}}X_{j}$ 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.

