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 $\mathcal{C}$ is monoidal if
 1.

2.
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,

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${a}_{BC}:(\otimes B)\otimes C\Rightarrow \otimes (B\otimes C)$ is a natural transformation for arbitrary objects $B,C$ in $\mathcal{C}$,

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${a}_{AC}:(A\otimes )\otimes C\Rightarrow A\otimes (\otimes C)$ is a natural transformation for arbitrary objects $A,C$ in $\mathcal{C}$,

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${a}_{AB}:(A\otimes B)\otimes \Rightarrow A\otimes (B\otimes )$ is a natural transformation for arbitrary objects $A,B$ in $\mathcal{C}$,

–

3.
there is an object $I$ in $\mathcal{C}$ called the unit object (or simply the unit),

4.
for any object $A$ in $\mathcal{C}$, there are isomorphisms:
$${l}_{A}:I\otimes A\cong A\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}{r}_{A}:A\otimes I\cong A,$$ 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^{}:

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unit coherence law
$$\text{xymatrix}\mathrm{@}+=2cm(A\otimes I)\otimes B\text{ar}{[rr]}^{{a}_{AIB}}\text{ar}{[dr]}_{{r}_{A}\otimes {1}_{B}}\mathrm{\&}\mathrm{\&}A\otimes (I\otimes B)\text{ar}{[dl]}^{{1}_{A}\otimes {r}_{B}}\mathrm{\&}A\otimes B\mathrm{\&}$$ 
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associativity coherence law
$$\text{xymatrix}\mathrm{@}+=2cm((A\otimes B)\otimes C)\otimes D\text{ar}{[rr]}^{{a}_{A\otimes B,C,C}}\text{ar}{[d]}_{{a}_{ABC}\otimes {1}_{D}}\mathrm{\&}\mathrm{\&}(A\otimes B)\otimes (C\otimes D)\text{ar}{[dd]}^{{a}_{A,B,C\otimes D}}(A\otimes (B\otimes C))\otimes D\text{ar}{[d]}_{{a}_{A,B\otimes C,D}}\mathrm{\&}\mathrm{\&}A\otimes ((B\otimes C)\otimes D)\text{ar}{[rr]}_{{1}_{A}\otimes {a}_{BCD}}\mathrm{\&}\mathrm{\&}A\otimes (B\otimes (C\otimes D))$$
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

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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.

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The category of sets is monoidal. The tensor product here is just the settheoretic cartesian product^{}, and any singleton can be used as the unit object.

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The category of (left) modules over a ring $R$ is monoidal. The tensor product is the usual tensor product (http://planetmath.org/TensorProduct) of modules, and $R$ itself is the unit object.

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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).
Title  monoidal category 
Canonical name  MonoidalCategory 
Date of creation  20130322 16:30:21 
Last modified on  20130322 16:30:21 
Owner  juanman (12619) 
Last modified by  juanman (12619) 
Numerical id  14 
Author  juanman (12619) 
Entry type  Definition 
Classification  msc 8100 
Classification  msc 1800 
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 