Symmetric monoidal category

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. 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,${\displaystyle ⊛}$) and (Ste,${\displaystyle \odot }$) of stereotype spaces over ${\displaystyle \mathbb{C}}$ are symmetric monoidal, and moreover, (Ste,${\displaystyle ⊛}$) is a closed symmetric monoidal category with the internal hom-functor ${\displaystyle \oslash }$.

Properties

The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an ${\displaystyle E_{\mathrm{\infty }}}$ 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 ${\displaystyle s_{AB}:A\otimes B\to B\otimes A}$ are their own 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 a braided monoidal category.