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Concept
Symmetric monoidal category
Summary
In , a branch of mathematics, a symmetric monoidal category is a (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 over some fixed field k, using the ordinary tensor product of vector spaces.
A symmetric monoidal category is a (C, ⊗, I) such that, for every pair A, B of objects in C, there is an isomorphism 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.
Some examples and non-examples of symmetric monoidal categories:
The . The tensor product is the set theoretic cartesian product, and any singleton can be fixed as the unit object.
The . 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 , is symmetric monoidal. The tensor product is the direct product of objects, and any terminal object (empty product) is the unit object.
The 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 symmetric monoidal category with the internal hom-functor .
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