Symmetric Monoidal ∞-category

In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" $\otimes$ is defined) such that the tensor product is symmetric (i.e. $A\otimes B$ is, in a certain strict sense, naturally isomorphic to $B\otimes A$ for all objects $A$ and $B$ 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.

Definition

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 $s_: A \otimes B \to B \otimes A$ 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. 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,$\circledast$) and (Ste,$\odot$) of stereotype spaces over $$ 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 $s_: A \otimes B \to B \otimes A$ are their ''own'' inverses in the sense that $s_\circ s_=1_$. 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.

References

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{{Category theory

Monoidal categories