mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically in the field known as

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a

monoidal category In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and r ...

where the monoidal ("tensor") product is the

categorical product In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or ring ...

is called a cartesian monoidal category. Any

category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the

terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

is the monoidal unit.

Dually Dually may refer to: *Dualla, County Tipperary, a village in Ireland *A pickup truck with dual wheels on the rear axle * DUALLy, s platform for architectural languages interoperability * Dual-processor See also * Dual (disambiguation) Dual or ...

, a monoidal finite coproduct category with the monoidal structure given by the

coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...

and unit the

initial object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

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 In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...

that is an

adjoint functor In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...

to the product are called

Cartesian closed categories In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in ma ...

Properties

Cartesian monoidal categories have a number of special and important properties, such as the existence of

diagonal map In category theory, a branch of mathematics, for any object a in any category \mathcal where the product a\times a exists, there exists the diagonal morphism :\delta_a : a \rightarrow a \times a satisfying :\pi_k \circ \delta_a = \operatornam ...

s Δ_''x'' : ''x'' → ''x'' ⊗ ''x'' and augmentations ''e''_''x'' : ''x'' → ''I'' for any

object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...

''x''. In applications to

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

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 In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of m ...

with the

singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...

serving as the unit. *Cat, the bicategory of small categories with the

product category In the mathematical field of category theory, the product of two categories ''C'' and ''D'', denoted and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifun ...

, where the category with one object and only its identity map is the unit. Cocartesian monoidal categories: *Vect, the

category of vector spaces In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring o ...

over a given

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

, can be made cocartesian monoidal with the monoidal product given by the

direct sum of vector spaces In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, ma ...

and the trivial vector space as unit. *Ab, the

category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab is ...

, with the

direct sum of abelian groups The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...

as monoidal product and the

trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...

as unit. * More generally, the category ''R''-Mod of (left)

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

s over a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

''R'' (

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

or not) becomes a cocartesian monoidal category with the

direct sum of modules In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, ma ...

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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

''n'', in the event that the map ''f'' is an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

, we say that a

biproduct In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for ...

for the objects ''X''_''j'' is an object

X = \bigoplus_ X_j

isomorphic to

\coprod_X_j

and

\prod_X_j

together with maps ''i''_''j'' : ''X''_''j'' → ''X'' and ''p''_''j'' : ''X'' → ''X''_''j'' such that the pair (''X'', ) is a coproduct diagram for the objects ''X''_''j'' and the pair (''X'', ) is a product diagram for the objects ''X''_''j'' , and where ''p''_''j'' ∘ ''i''_''j'' = id_{''X''_''j''}. If, in addition, the category in question has a

zero object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

, 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 In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...

, 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.

References

{{Reflist Category theory Monoidal categories