A monoidal category is a category which has the structure of a monoid, that is, among the objects there is a binary operation which is associative and has an unique neutral or unit element. Specifically, a category is monoidal if
there is an object in called the unit object (or simply the unit),
for any object in , there are isomorphisms:
such that and are natural in : both and are natural transformations
satisfying the following commutative diagrams:
unit coherence law
associativity coherence law
Some examples of monoidal categories are
The category of (left) modules over a ring is monoidal. The tensor product is the usual tensor product (http://planetmath.org/TensorProduct) of modules, and itself is the unit object.
The category of bimodules over a ring is monoidal. The tensor product and the unit object are the same as in the previous example.
Monoidal categories play an important role in the topological quantum field theories (TQFT).
|Date of creation||2013-03-22 16:30:21|
|Last modified on||2013-03-22 16:30:21|
|Last modified by||juanman (12619)|
|Defines||left unit isomorphism|
|Defines||right unit isomorphism|