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 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.

Cocartesian monoidal categories:

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 ∐ ... ∐ 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 (X, {ij}) is a coproduct diagram for the objects Xj and the pair (X, {pj}) 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.

See also

References

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