Monoidal category where A ⊗ B is naturally equivalent to B ⊗ A
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.
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
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
called the swap map
[1] that is
natural in both A and B and such that the following diagrams commute:
- The unit coherence:
-
![](https://upload.wikimedia.org/wikipedia/commons/9/90/Symmetric_monoidal_unit_coherence.png)
- The associativity coherence:
-
![](https://upload.wikimedia.org/wikipedia/commons/5/51/Symmetric_monoidal_associativity_coherence.png)
- The inverse law:
-
![](https://upload.wikimedia.org/wikipedia/commons/7/7a/Symmetric_monoidal_inverse_law.png)
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,
) and (Ste,
) of
stereotype spaces over
are symmetric monoidal, and moreover, (Ste,
) is a
closed symmetric monoidal category with the internal hom-functor
.
Properties
The
classifying space (geometric realization of the
nerve) of a symmetric monoidal category is an
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
are their own inverses in the sense that
. If we abandon this requirement (but still require that
be naturally isomorphic to
), we obtain the more general notion of a
braided monoidal category.
References