Symmetric monoidal category explained

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. A ⊗ B is, in a certain strict sense, naturally isomorphic to B ⊗ 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 _{AB} : A ⊗ B \to B ⊗ A 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.

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 { 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.^[2]

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

Notes and References

1. Fong . Brendan . Spivak . David I. . 2018-10-12 . Seven Sketches in Compositionality: An Invitation to Applied Category Theory . math.CT . 1803.05316 .
2. Robert Wayne Thomason . R.W. . Thomason . Symmetric Monoidal Categories Model all Connective Spectra . Theory and Applications of Categories . 1 . 5 . 1995 . 78–118 . 10.1.1.501.2534 .