monoidal category MonoidalCategory 2006-12-24 09:20:27 2008-10-20 17:49:28 Definition unit coherence associativity coherence tensor product unit object associativity isomorphism left unit isomorphism right unit isomorphism Category monoidal category algebroids topological quantum field theories TQFT monoidal category % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics \usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A \emph{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 \emph{monoidal} if \begin{enumerate} \item 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, \item 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, \begin{itemize} \item $a_{-BC}: (-\otimes B)\otimes C \Rightarrow -\otimes(B\otimes C)$ is a natural transformation for arbitrary objects $B,C$ in $\mathcal{C}$, \item $a_{A-C}: (A\otimes -)\otimes C \Rightarrow A\otimes(-\otimes C)$ is a natural transformation for arbitrary objects $A,C$ in $\mathcal{C}$, \item $a_{AB-}: (A\otimes B)\otimes - \Rightarrow A\otimes(B\otimes -)$ is a natural transformation for arbitrary objects $A,B$ in $\mathcal{C}$, \end{itemize} \item there is an object $I$ in $\mathcal{C}$ called the \emph{unit object} (or simply the \emph{unit}), \item 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 \end{enumerate} satisfying the following commutative diagrams: \begin{itemize} \item \emph{unit coherence law} $$\xymatrix@+=2cm{(A\otimes I)\otimes B \ar[rr]^{a_{AIB}} \ar[dr]_{r_A\otimes 1_B} & & A\otimes (I\otimes B) \ar[dl]^{1_A \otimes r_B} \\ & A\otimes B & }$$ \item \emph{associativity coherence law} $$\xymatrix@+=2cm{((A\otimes B)\otimes C)\otimes D \ar[rr]^{a_{A\otimes B,C,C}} \ar[d]_{a_{ABC}\otimes 1_D} && (A\otimes B)\otimes (C\otimes D) \ar[dd]^{a_{A,B,C\otimes D}} \\ (A\otimes (B\otimes C))\otimes D \ar[d]_{a_{A,B\otimes C,D}} && \\ A\otimes ((B\otimes C)\otimes D) \ar[rr]_{1_A\otimes a_{BCD}} && A \otimes (B\otimes (C\otimes D))}$$ \end{itemize} The bifunctor $\otimes$ is called the \emph{tensor product} on $\mathcal{C}$, and the natural isomorphisms $a,l,r$ are called the \emph{associativity isomorphism}, the \emph{left unit isomorphism}, and the \emph{right unit isomorphism} respectively. Some examples of monoidal categories are \begin{itemize} \item 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. \item 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. \item The category of (left) modules over a ring $R$ is monoidal. The tensor product is the usual \PMlinkname{tensor product}{TensorProduct} of modules, and $R$ itself is the unit object. \item 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. \end{itemize} Monoidal categories play an important role in the topological quantum field theories (TQFT).