Symmetrical monoidal category
In mathematics , a symmetric monoidal category is a monoidal category (i.e., a category in which a "tensor product" is defined) whose tensor product is symmetric (i.e. one has a natural isomorphism between and for all objects and ).
A typical example is the category of vector spaces over a given body .
definition
Let it be a monoidal category with associativity isomorphism and left and right unit isomorphisms, respectively . The monoidal category is called symmetric if there are two objects from an isomorphism
which is of course in and , so that the following diagrams commute :
- Compatibility with the unitary object:
- Compatibility with the associative law:
- Reverse rule:
Examples of symmetric monoidal categories
- The category of sets with the set product as the tensor product and a one-element set as the unit object.
- The category of topological spaces with the direct product as a tensor product and a single-element space as a unitary object.
- The category of groups with the direct product as the tensor product and the trivial group as the unit object.
- The category of rings with the direct product as the tensor product and the zero ring as the unit object.
- The category of vector spaces over a given field with the direct sum as the tensor product and the zero vector space as the unit object.
- The category of vector spaces over a given body with the tensor product of vector spaces as the tensor product and the one-dimensional space as the unit object.
- For a given group, the category of representations of over a given body with the tensor product of representations as the tensor product and the trivial representation as a unit object.
Web links
- symmetric monoidal category (nLab)