Symmetric monoidal category

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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In algebra, given a ring R, the category of left modules over R is the whose are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the . The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring R but that category is equivalent to the category of left (or right) modules over the enveloping algebra of R (or over the opposite of that).

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In mathematics, the category of rings, denoted by Ring, is the whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is , meaning that the class of all rings is proper. The category Ring is a meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure.

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