Cartesian monoidal category

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.

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 : X₁ ∐ ... ∐ X_n → X₁ × ... × 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 ${\displaystyle X=\bigoplus _{j\in {1,\ldots ,n}}X_{j}}$ isomorphic to ${\displaystyle \coprod _{j\in {1,\ldots ,n}}X_{j}}$ and ${\displaystyle \prod _{j\in {1,\ldots ,n}}X_{j}}$ 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.

• Cartesian closed category