monoidal category
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
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there is an isomorphism , for arbitrary objects in , such that is natural in and . In other words,
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is a natural transformation for arbitrary objects in ,
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is a natural transformation for arbitrary objects in ,
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is a natural transformation for arbitrary objects in ,
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there is an object in called the unit object (or simply the unit),
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for any object in , there are isomorphisms:
such that and are natural in : both and are natural transformations
satisfying the following commutative diagrams:
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unit coherence law
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associativity coherence law
The bifunctor is called the tensor product on , and the natural isomorphisms are called the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively.
Some examples of monoidal categories are
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A prototype is the category of isomorphism classes of vector spaces over a field , herein the tensor product is the associative operation and the field itself is the unit element.
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The category of sets is monoidal. The tensor product here is just the set-theoretic cartesian product, and any singleton can be used as the unit object.
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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.
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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).
Title | monoidal category |
Canonical name | MonoidalCategory |
Date of creation | 2013-03-22 16:30:21 |
Last modified on | 2013-03-22 16:30:21 |
Owner | juanman (12619) |
Last modified by | juanman (12619) |
Numerical id | 14 |
Author | juanman (12619) |
Entry type | Definition |
Classification | msc 81-00 |
Classification | msc 18-00 |
Classification | msc 18D10 |
Synonym | monoid |
Related topic | Category |
Related topic | Algebroids |
Related topic | Monoid |
Related topic | StateOnTheTetrahedron |
Defines | unit coherence |
Defines | associativity coherence |
Defines | tensor product |
Defines | unit object |
Defines | associativity isomorphism |
Defines | left unit isomorphism |
Defines | right unit isomorphism |