Incategory theory, a branch ofmathematics, asymmetric monoidal category is amonoidal category (i.e. a category in which a "tensor product"${\displaystyle \otimes }$ is defined) such that the tensor product is symmetric (i.e.${\displaystyle A\otimes B}$ is, in a certain strict sense,naturally isomorphic to${\displaystyle B\otimes A}$ for all objects${\displaystyle A}$ and${\displaystyle B}$ of the category). One of the prototypical examples of a symmetric monoidal category is thecategory of vector spaces over some fixedfieldk, using the ordinarytensor product of vector spaces.

A symmetric monoidal category is amonoidal category (C, ⊗,I) such that, for every pairA,B of objects inC, there is an isomorphism${\displaystyle s_{AB}:A\otimes B\to B\otimes A}$ called theswap map^[1] that isnatural in bothA andB and such that the following diagrams commute:

• The unit coherence:

  

• The associativity coherence:

  

• The inverse law:

  

In the diagrams above,a,l, andr are the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively.

Some examples and non-examples of symmetric monoidal categories:

• Thecategory of sets. The tensor product is the set theoreticcartesian product, and anysingleton can be fixed as the unit object.
• Thecategory of groups. The tensor product is thedirect product of groups, and thetrivial group is the unit object.
• More generally, any category with finiteproducts, that is, acartesian monoidal category, is symmetric monoidal. The tensor product is the direct product of objects, and anyterminal object (empty product) is the unit object.
• Thecategory of bimodules over aringR is monoidal (using the ordinarytensor product of modules), but not necessarily symmetric. IfR iscommutative, the category of leftR-modules is symmetric monoidal. The latter example class includes the category of all vector spaces over a given field.
• Given a fieldk and a group (or aLie algebra overk), the category of allk-linearrepresentations of the group (or of the Lie algebra) is a symmetric monoidal category. Here the standardtensor product of representations is used.
• The categories (Ste,${\displaystyle ⊛}$) and (Ste,${\displaystyle \odot }$) ofstereotype spaces over${\displaystyle \mathbb{C}}$ are symmetric monoidal, and moreover, (Ste,${\displaystyle ⊛}$) is aclosed symmetric monoidal category with theinternal hom-functor${\displaystyle \oslash }$.

Theclassifying space (geometric realization of thenerve) of a symmetric monoidal category is an${\displaystyle E_{\mathrm{\infty }}}$ space, so itsgroup completion is aninfinite loop space.^[2]

Adagger symmetric monoidal category is a symmetric monoidal category with a compatibledagger structure.

Acosmos is acomplete cocompleteclosed symmetric monoidal category.

In a symmetric monoidal category, the natural isomorphisms${\displaystyle s_{AB}:A\otimes B\to B\otimes A}$ are theirown inverses in the sense that${\displaystyle s_{BA}\circ s_{AB}=1_{A\otimes B}}$. If we abandon this requirement (but still require that${\displaystyle A\otimes B}$ be naturally isomorphic to${\displaystyle B\otimes A}$), we obtain the more general notion of abraided monoidal category.

• Symmetric monoidal category at thenLab
• This article incorporates material from Symmetric monoidal category onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

• Monoidal categories

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