Symmetric monoidal category

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 s_{AB}: A \otimes B \to B \otimes A called the swap map 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. ==Examples==

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. The tensor product is the direct 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,\circledast) and (Ste,\odot) of stereotype spaces over {\mathbb C} are symmetric monoidal, and moreover, (Ste,\circledast) is a closed symmetric monoidal category with the internal hom-functor \oslash. == Properties ==

The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an E_\infty space, so its group completion is an infinite loop space. == Specializations ==

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. == Generalizations ==

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