81-00-SymmetricMonoidalCategory.tex

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\begin{document}
A monoidal category $\mathcal{C}$ with tensor product $\otimes$ is said to be \emph{symmetric} if for every pair $A,B$ of objects in $\mathcal{C}$, there is an isomorphism $$s_{AB}:A\otimes B\cong B\otimes A$$ that is natural in both $A$ and $B$ such that the following diagrams are commutative
\begin{enumerate}
\item (\emph{unit coherence for $s$}):
$$\xymatrix@+=2cm{A\otimes I \ar[rr]^{s_{AI}} \ar[dr]_{r_A} & & I\otimes A \ar[dl]^{l_A} \\ & A &}$$
\item (\emph{associativity coherence for $s$}):
$$\xymatrix@+=2cm{ (A\otimes B)\otimes C \ar[rr]^{s_{AB}\otimes 1_C} \ar[d]_{a_{ABC}} & & (B\otimes A)\otimes C \ar[d]^{a_{BAC}} \\ A\otimes (B\otimes C) \ar[d]_{s_{A,B\otimes C}} & & B\otimes (A\otimes C) \ar[d]^{1_B\otimes s_{AC}} \\ (B\otimes C)\otimes A \ar[rr]_{a_{BCA}} & & B\otimes(C\otimes A)
}$$
\item (inverse law):
$$\xymatrix@+=2cm{& B\otimes A \ar[dr]^{s_{BA}} & \\ A\otimes B \ar[ur]^{s_{AB}} \ar@{=}[rr]_{1_{A\otimes B}} && A\otimes B }$$
\end{enumerate}
In the diagrams above, $a,l,r$ are the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively.

Some examples and non-examples of symmetric monoidal categories:
\begin{itemize}
\item The category of sets.  The tensor product is the set theoretic cartesian product, and any singleton can be fixed as the unit object.
\item The category of groups.  Like before, the tensor product is just the cartesian product of groups, and the trivial group is the unit object.
\item More generally, a category with finite products is symmetric monoidal.  The tensor product is the direct product of objects, and any terminal object (empty product) is the unit object.
\item The category of bimodules over a ring $R$ is monoidal.  However, this category is only symmetric monoidal if $R$ is commutative.
\end{itemize}

\textbf{Remark}.  A symmetric monoidal category is a braided monoidal category such that the inverse law: $s_{BA}\circ s_{AB}=1_{A\otimes B}$ holds.
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