# 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. 1.

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,

2. 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,

• $a_{-BC}:(-\otimes B)\otimes C\Rightarrow-\otimes(B\otimes C)$ 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 $\mathcal{C}$,

3. 3.

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

4. 4.

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

 $\xymatrix@+=2cm{(A\otimes I)\otimes B\ar[rr]^{a_{AIB}}\ar[dr]_{r_{A}\otimes 1_% {B}}&&A\otimes(I\otimes B)\ar[dl]^{1_{A}\otimes r_{B}}\\ &A\otimes B&}$
• associativity coherence law

 $\xymatrix@+=2cm{((A\otimes B)\otimes C)\otimes D\ar[rr]^{a_{A\otimes B,C,C}}% \ar[d]_{a_{ABC}\otimes 1_{D}}&&(A\otimes B)\otimes(C\otimes D)\ar[dd]^{a_{A,B,% C\otimes D}}\\ (A\otimes(B\otimes C))\otimes D\ar[d]_{a_{A,B\otimes C,D}}&&\\ A\otimes((B\otimes C)\otimes D)\ar[rr]_{1_{A}\otimes a_{BCD}}&&A\otimes(B% \otimes(C\otimes D))}$

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

• 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 (http://planetmath.org/TensorProduct) 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).

 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