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| In [[mathematics]], a '''monoidal category''' (or '''tensor category''') is a [[category (mathematics)|category]] '''C''' equipped with a [[bifunctor]]
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| :⊗ : '''C''' × '''C''' → '''C'''
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| which is [[associative]], [[up to]] a [[natural isomorphism]], and an object ''I'' which is both a [[left identity|left]] and [[right identity]] for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain [[coherence condition]]s which ensure that all the relevant diagrams commute.
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| In a monoidal category, analogs of usual [[monoid]]s from [[abstract algebra]] can be defined using the same commutative diagrams. In fact, usual monoids are exactly the [[monoid object]]s in the monoidal category of sets with Cartesian product.
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| The ordinary [[tensor product]] makes [[vector space]]s, [[abelian group]]s, [[module (mathematics)|''R''-modules]], or [[algebra (ring theory)|''R''-algebras]] into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples.
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| In [[category theory]], monoidal categories can be used to define the concept of a [[monoid object]] and an associated action on the objects of the category. They are also used in the definition of an [[enriched category]].
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| Monoidal categories have numerous applications outside of category theory proper. They are used to define models for the multiplicative fragment of intuitionistic [[linear logic]]. They also form the mathematical foundation for the [[topological order]] in condensed matter. [[Braided monoidal categories]] have applications in [[quantum field theory]] and [[string theory]].
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| ==Formal definition==
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| A '''monoidal category''' is a category <math>\mathbf C</math> equipped with
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| *a [[bifunctor]] <math>\otimes \colon \mathbf C\times\mathbf C\to\mathbf C</math> called the ''[[tensor product]]'' or ''monoidal product'',
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| *an object <math>I</math> called the ''unit object'' or ''identity object'',
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| *three [[natural isomorphism]]s subject to certain [[coherence condition]]s expressing the fact that the tensor operation
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| **is associative: there is a natural isomorphism <math>\alpha</math>, called ''associator'', with components <math>\alpha_{A,B,C} \colon (A\otimes B)\otimes C \cong A\otimes(B\otimes C)</math>,
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| **has <math>I</math> as left and right identity: there are two natural isomorphisms <math>\lambda</math> and <math>\rho</math>, respectively called ''left'' and ''right unitor'', with components <math>\lambda_A \colon I\otimes A\cong A</math> and <math>\rho_A \colon A\otimes I\cong A</math>.
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| :
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| The coherence conditions for these natural transformations are:
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| * for all <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> in <math>\mathbf C</math>, the pentagon [[diagram (category theory)|diagram]]
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| :[[Image:monoidal-category-pentagon.png]]
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| [[Commutative diagram|commutes]];
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| * for all <math>A</math> and <math>B</math> in <math>\mathbf C</math>, the triangle [[diagram (category theory)|diagram]]
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| :[[Image:monoidal-category-triangle.png]]
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| [[Commutative diagram|commutes]];
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| It follows from these three conditions that ''a large class'' of such diagrams (i.e. diagrams whose morphisms are built using <math>\alpha</math>, <math>\lambda</math>, <math>\rho</math>, identities and tensor product) commute: this is [[Saunders Mac Lane|Mac Lane's]] "[[coherence theorem]]". It is sometimes inaccurately stated that ''all'' such diagrams commute.
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| A '''strict monoidal category''' is one for which the natural isomorphisms α, λ and ρ are identities. Every monoidal category is monoidally [[equivalence of categories|equivalent]] to a strict monoidal category.
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| ==Examples==
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| *Any category with finite [[product (category theory)|product]]s is monoidal with the product as the monoidal product and the [[terminal object]] as the unit. Such a category is sometimes called a '''cartesian monoidal category'''. For example:
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| **'''Set''', the category of sets with the Cartesian product, one-element sets serving as the unit.
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| *Any category with finite [[coproduct]]s is monoidal with the coproduct as the monoidal product and the [[initial object]] as the unit.
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| *'''''R''-Mod''', the [[category of modules]] over a [[commutative ring]] ''R'', is a monoidal category with the [[tensor product of modules]] ⊗<sub>''R''</sub> serving as the monoidal product and the ring ''R'' (thought of as a module over itself) serving as the unit. As special cases one has:
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| **'''''K''-Vect''', the [[category of vector spaces]] over a [[field (mathematics)|field]] ''K'', with the one-dimensional vector space ''K'' serving as the unit.
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| **'''Ab''', the [[category of abelian groups]], with the group of [[integer]]s '''Z''' serving as the unit.
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| *For any commutative ring ''R'', the category of [[R-algebra|''R''-algebras]] is monoidal with the [[tensor product of algebras]] as the product and ''R'' as the unit.
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| *The [[category of pointed spaces]] is monoidal with the [[smash product]] serving as the product and the pointed [[0-sphere]] (a two-point discrete space) serving as the unit.
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| *The category of all [[endofunctor]]s on a category '''C''' is a strict monoidal category with the composition of functors as the product and the identity functor as the unit.
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| *Just like for any category '''E''', the [[Subcategory#Embeddings|full subcategory]] spanned by any given object is a monoid, it is the case that for any [[2-category]] '''E''', and any object '''C'''∈Ob('''E'''), the full 2-subcategory of '''E''' spanned by {'''C'''} is a monoidal category. In the case '''E'''='''Cat''', we get the [[endofunctor]]s example above.
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| * [[Semilattice|Bounded-above meet semilattices]] are strict [[symmetric monoidal category|symmetric monoidal categories]]: the product is meet and the identity is the top element.
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| == Free strict monoidal category ==
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| For every category '''C''', the [[free category|free]] strict monoidal category Σ('''C''') can be constructed as follows:
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| * its objects are lists (finite sequences) ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> of objects of '''C''';
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| * there are arrows between two objects ''A''<sub>1</sub>, ..., ''A''<sub>''m''</sub> and ''B''<sub>1</sub>, ..., ''B''<sub>''n''</sub> only if ''m'' = ''n'', and then the arrows are lists (finite sequences) of arrows ''f''<sub>1</sub>: ''A''<sub>1</sub> → ''B''<sub>1</sub>, ..., ''f''<sub>''n''</sub>: ''A''<sub>''n''</sub> → ''B''<sub>''n''</sub> of '''C''';
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| * the tensor product of two objects ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> and ''B''<sub>1</sub>, ..., ''B''<sub>''m''</sub> is the concatenation ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub>, ''B''<sub>1</sub>, ..., ''B''<sub>''m''</sub> of the two lists, and, similarly, the tensor product of two morphisms is given by the concatenation of lists.
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| This operation Σ mapping category '''C''' to Σ('''C''') can be extended to a strict 2-monad on '''Cat'''.
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| == See also ==
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| * Many monoidal categories have additional structure such as [[braided monoidal category|braiding]], [[symmetric monoidal category|symmetry]] or [[closed monoidal category|closure]]: the references describe this in detail.
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| * [[Monoidal functor]]s are the functors between monoidal categories which preserve the tensor product and [[monoidal natural transformation]]s are the natural transformations, between those functors, which are "compatible" with the tensor product.
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| * There is a general notion of [[monoid object]] in a monoidal category, which generalizes the ordinary notion of [[monoid]]. In particular, a strict monoidal category can be seen as a monoid object in the category of categories '''Cat''' (equipped with the monoidal structure induced by the cartesian product).
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| * A monoidal category can also be seen as the category '''B'''(□, □) of a [[bicategory]] '''B''' with only one object, denoted □.
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| * [[Rigid category|Rigid categories]] are monoidal categories in which duals with nice properties exist.
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| * [[Autonomous category|Autonomous categories]] (or [[compact closed category|compact closed categories]]) are monoidal categories in which inverses exist; they abstract the idea of '''FdVect''', finite-dimensional vector spaces.
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| * [[Dagger symmetric monoidal category|Dagger symmetric monoidal categories]], equipped with an extra dagger functor, abstracting the idea of '''FdHilb''', finite-dimensional Hilbert spaces. These include the [[dagger compact category|dagger compact categories]].
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| * A category '''C''' [[Enriched category|enriched]] in a monoidal category '''M''' replaces the notion of a set of morphisms between pairs of objects in '''C''' with the notion of an '''M'''-object of morphisms between every two objects in '''C'''.
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| * [[Tannakian category|Tannakian categories]] are monoidal categories enriched over a field which are very similar to representation categories of linear algebraic groups.
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| * [[Spherical category]]
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| == References ==
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| * [[André Joyal|Joyal, André]]; [[Ross Street|Street, Ross]] (1993). "Braided Tensor Categories". ''Advances in Mathematics'' ''102'', 20–78.
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| * [[Max Kelly|Kelly, G. Max]] (1964). "On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc." ''Journal of Algebra'' ''1'', 397–402
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| * {{cite book | last = Kelly | first = G. Max | url = http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf | title = Basic Concepts of Enriched Category Theory | series = London Mathematical Society Lecture Note Series No. 64 | publisher = Cambridge University Press | year = 1982}}
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| * [[Saunders Mac Lane|Mac Lane, Saunders]] (1963). "Natural Associativity and Commutativity". ''Rice University Studies'' ''49'', 28–46.
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| * Mac Lane, Saunders (1998), ''[[Categories for the Working Mathematician]]'' (2nd ed.). New York: Springer-Verlag.
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| *{{nlab|id=monoidal+category|title=Monoidal category}}
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| {{Portal|Category theory}}
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| [[Category:Monoidal categories| ]]
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