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 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.

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:
Cocartesian monoidal categories:
In each of these categories of modules equipped with a cocartesian monoidal structure, finite products and coproducts coincide. Or more formally, if f : X1 ∐... ∐ Xn → X1 × ... × Xn is the "canonical" map from the n-ary coproduct of objects Xj 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 Xj is an object isomorphic to and together with maps ij : Xj → X and pj : X → Xj such that the pair is a coproduct diagram for the objects Xj and the pair is a product diagram for the objects Xj, and where pj ∘ ij = idXj. If, in addition, the category in question has a zero object, so that for any objects A and B there is a unique map 0A,B : A → 0 → B, it often follows that pk ∘ ij = : δij, the Kronecker delta, where we interpret 0 and 1 as the 0 maps and identity maps of the objects Xj and Xk, respectively. See pre-additive category for more.