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monoidal category
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(Definition)
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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 $\mathcal{C}$ is monoidal if
- there is a bifunctor $\otimes: \mathcal{C}\times\mathcal{C}\to \mathcal{C}$ , where the images of object $(A,B)$ and morphism $(f,g)$ are written $A\otimes B$ and $f\otimes g$ respectively,
- there is an isomorphism $a_{ABC}: (A\otimes B)\otimes C \cong A\otimes (B\otimes C)$ , for arbitrary objects $A,B,C$ in $\mathcal{C}$ , such that $a_{ABC}$ is natural in $A,B$ and $C$ . In other words,
- $a_{-BC}: (-\otimes B)\otimes C \Rightarrow -\otimes(B\otimes C)$ is a natural transformation for arbitrary objects $B,C$ in $\mathcal{C}$ ,
- $a_{A-C}: (A\otimes -)\otimes C \Rightarrow A\otimes(-\otimes C)$ is a natural transformation for arbitrary objects $A,C$ in $\mathcal{C}$ ,
- $a_{AB-}: (A\otimes B)\otimes - \Rightarrow A\otimes(B\otimes -)$ is a natural transformation for arbitrary objects $A,B$ in $\mathcal{C}$ ,
- there is an object $I$ in $\mathcal{C}$ called the unit object (or simply the unit),
- for any object $A$ in $\mathcal{C}$ , there are isomorphisms: $$l_A: I\otimes A\cong A \qquad \mbox{and} \qquad r_A: A\otimes I\cong A,$$ such that $l_A$ and $r_A$ are natural in $A$ : both $l: I\otimes - \Rightarrow -$ and $r: -\otimes I\Rightarrow - $ are natural transformations
satisfying the following commutative diagrams:
- unit coherence law
- associativity coherence law
The bifunctor $\otimes$ is called the tensor product on $\mathcal{C}$ , and the natural isomorphisms $a,l,r$ are called the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively.
Some examples of monoidal categories are
- A prototype is the category of isomorphism classes of vector spaces over a field $\mathbb{K}$ , herein the tensor product is the associative operation and the field $\mathbb{K}$ itself is the unit element.
- 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.
- The category of (left) modules over a ring $R$ is monoidal. The tensor product is the usual tensor product of modules, and $R$ itself is the unit object.
- The category of bimodules over a ring $R$ 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).
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"monoidal category" is owned by juanman. [ full author list (3) ]
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See Also: category, algebroid structures and extended symmetries, monoid, quantum field state on the tetrahedron
| Also defines: |
unit coherence, associativity coherence, tensor product, unit object, associativity isomorphism, left unit isomorphism, right unit isomorphism |
| Keywords: |
Category, monoidal category, algebroids, topological quantum field theories, TQFT, monoidal category |
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Cross-references: TQFT, quantum field theories, bimodules, ring, modules, singleton, Cartesian product, category of sets, operation, field, vector spaces, classes, natural isomorphisms, commutative diagrams, natural transformation, isomorphism, morphism, images, bifunctor, unit, associative, binary operation, objects, structure, category
There are 35 references to this entry.
This is version 10 of monoidal category, born on 2006-12-24, modified 2008-10-20.
Object id is 8681, canonical name is MonoidalCategory.
Accessed 3748 times total.
Classification:
| AMS MSC: | 18D10 (Category theory; homological algebra :: Categories with structure :: Monoidal categories , symmetric monoidal categories, braided categories) | | | 18-00 (Category theory; homological algebra :: General reference works ) | | | 81-00 (Quantum theory :: General reference works ) |
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Pending Errata and Addenda
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![$\displaystyle \xymatrix@+=2cm{(A\otimes I)\otimes B \ar[rr]^{a_{AIB}} \ar[dr]_{... ...es 1_B} & & A\otimes (I\otimes B) \ar[dl]^{1_A \otimes r_B} \\ & A\otimes B & }$](http://images.planetmath.org:8080/cache/objects/8681/js/img1.png "$\displaystyle \xymatrix@+=2cm{(A\otimes I)\otimes B \ar[rr]^{a_{AIB}} \ar[dr]_{... ...es 1_B} & & A\otimes (I\otimes B) \ar[dl]^{1_A \otimes r_B} \\ & A\otimes B & }$")
![$\displaystyle \xymatrix@+=2cm{((A\otimes B)\otimes C)\otimes D \ar[rr]^{a_{A\ot... ...C)\otimes D) \ar[rr]_{1_A\otimes a_{BCD}} && A \otimes (B\otimes (C\otimes D))}$](http://images.planetmath.org:8080/cache/objects/8681/js/img2.png "$\displaystyle \xymatrix@+=2cm{((A\otimes B)\otimes C)\otimes D \ar[rr]^{a_{A\ot... ...C)\otimes D) \ar[rr]_{1_A\otimes a_{BCD}} && A \otimes (B\otimes (C\otimes D))}$")