Given a monoid$A$ with multiplication $\mu$ in a symmetric monoidal category$C$ with symmetry $\tau$ its opposite $A^{op}$ is the same underlying object $A$ with multiplication $\mu\circ\tau$. The enveloping monoid$A^e$ of $A$ is the monoid whose underlying object is $A\otimes A^{op}$ and for which the multiplication is given by

where $\tau=\tau_{A,A}$ is the symmetry $A\otimes A\to A\otimes A$.

The enveloping monoid is sometimes called the enveloping algebra, especially if the monoidal category is a category of vector spaces. The left $A^e$-modules in $C$ are in 1-1 correspondence with the $A$-$A$-bimodules in $C$.

Last revised on May 20, 2016 at 18:10:37.
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