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In category theory, a branch of mathematics, a **symmetric monoidal category** is a monoidal category (i.e. a category in which a "tensor product" is defined) such that the tensor product is symmetric (i.e. is, in a certain strict sense, naturally isomorphic to for all objects and of the category). One of the prototypical examples of a symmetric monoidal category is the category of vector spaces over some fixed field *k,* using the ordinary tensor product of vector spaces.

A symmetric monoidal category is a monoidal category (*C*, ⊗, *I*) such that, for every pair *A*, *B* of objects in *C*, there is an isomorphism called the *swap map*^{[1]} that is natural in both *A* and *B* and such that the following diagrams commute:

- The unit coherence:
- The associativity coherence:
- The inverse law:

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

Some examples and non-examples of symmetric monoidal categories:

- The category of sets. The tensor product is the set theoretic cartesian product, and any singleton can be fixed as the unit object.
- The category of groups. Like before, the tensor product is just the cartesian product of groups, and the trivial group is the unit object.
- More generally, any category with finite products, that is, a cartesian monoidal category, is symmetric monoidal. The tensor product is the direct product of objects, and any terminal object (empty product) is the unit object.
- The category of bimodules over a ring
*R*is monoidal (using the ordinary tensor product of modules), but not necessarily symmetric. If*R*is commutative, the category of left*R*-modules is symmetric monoidal. The latter example class includes the category of all vector spaces over a given field. - Given a field
*k*and a group (or a Lie algebra over*k*), the category of all*k*-linear representations of the group (or of the Lie algebra) is a symmetric monoidal category. Here the standard tensor product of representations is used. - The categories (
**Ste**, ) and (**Ste**, ) of stereotype spaces over are symmetric monoidal, and moreover, (**Ste**, ) is a closed symmetric monoidal category with the internal hom-functor .

The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an space, so its group completion is an infinite loop space.^{[2]}

A dagger symmetric monoidal category is a symmetric monoidal category with a compatible dagger structure.

A cosmos is a complete cocomplete closed symmetric monoidal category.

In a symmetric monoidal category, the natural isomorphisms are their *own* inverses in the sense that . If we abandon this requirement (but still require that be naturally isomorphic to ), we obtain the more general notion of a braided monoidal category.

**^**Fong, Brendan; Spivak, David I. (2018-10-12). "Seven Sketches in Compositionality: An Invitation to Applied Category Theory". arXiv:1803.05316 [math.CT].**^**Thomason, R.W. (1995). "Symmetric Monoidal Categories Model all Connective Spectra" (PDF).*Theory and Applications of Categories*.**1**(5): 78–118. CiteSeerX 10.1.1.501.2534.

- Symmetric monoidal category at the
*n*Lab *This article incorporates material from Symmetric monoidal category on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*