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\begin{document}

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\section*{bimonoid}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{bimonoids}{}\section*{{Bimonoids}}\label{bimonoids}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition_in_symmetric_monoidal_categories}{Definition in symmetric monoidal categories}\dotfill \pageref*{definition_in_symmetric_monoidal_categories} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{MonoidalStructureOnModules}{Monoidal structure on modules}\dotfill \pageref*{MonoidalStructureOnModules} \linebreak
\noindent\hyperlink{nonsymmetric_variants}{Non-symmetric variants}\dotfill \pageref*{nonsymmetric_variants} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A bimonoid is something which is both a monoid and a comonoid in a compatible way. The compatibility is easy to formulate in symmetric monoidal categories and much harder in a nonsymmetric setup.

\hypertarget{definition_in_symmetric_monoidal_categories}{}\subsection*{{Definition in symmetric monoidal categories}}\label{definition_in_symmetric_monoidal_categories}

In a [[symmetric monoidal category]], a \textbf{bimonoid} (or \textbf{bimonoid object}) is an object $B$ equipped with a structure of a [[monoid]] and a [[comonoid]] which are compatible in one of two equivalent ways: the comultiplication and the counit are morphisms of monoids or the multiplication and the unit are morphisms of comonoids. The symmetry of the monoidal structure is involved in the definition of the tensor product $B\otimes B$ as monoids and as comonoids. In terms of [[string diagrams]], the equations expressing the compatibility of the monoidal and comonoidal structures on $B$ may be represented as follows:

A bimonoid which additionally has an [[antipode]] (a map $s:B\to B$ satisfying an axiom that makes it act like ``inverses'' in a group) is called a [[Hopf monoid]]; a Hopf monoid in [[Vect]] is a [[Hopf algebra]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item A bimonoid in [[Vect]] (with its usual [[tensor product]]) is generally called a \textbf{[[bialgebra]]}.


\item In a category with [[biproducts]], with the biproduct as the monoidal product, every object is a bimonoid in a unique way (\href{http://golem.ph.utexas.edu/category/2010/09/bimonoids_from_biproducts.html}{Caf\'e{} post}).


\item More generally, in a [[cartesian monoidal category]], every monoid object is a bimonoid in a unique way, with comultiplication being the diagonal map. Dually, every comonoid object in a cocartesian monoidal category is a bimonoid in a unique way.



\end{itemize}
\hypertarget{MonoidalStructureOnModules}{}\subsection*{{Monoidal structure on modules}}\label{MonoidalStructureOnModules}

If $B$ is a bimonoid in $\mathcal{C}$, then the category $Mod_B$ of $B$-[[modules]] inherits a [[monoidal category|monoidal structure]] such that the [[forgetful functor]] $Mod_B \to \mathcal{C}$ (the ``[[fiber functor]]'') is a [[strong monoidal functor]]. For $B$-modules $M$ and $N$, we equip the [[tensor product]] $M\otimes N$ in $\mathcal{C}$ with the $B$-action given by

\begin{displaymath}
B\otimes (M\otimes N) \xrightarrow{\Delta}
(B\otimes B) \otimes (M\otimes N) \xrightarrow{\cong}
(B\otimes M) \otimes (B\otimes N) \xrightarrow{act \otimes act}
M\otimes N
\end{displaymath}
where $\Delta$ denotes the comultiplication of $B$ as a comonoid. The bimonoid compatibility axioms are exactly what is needed to make this a $B$-module structure, and coassociativity makes it associative. Similarly, we use the counit $B\to I$ of $B$ to give the unit object $I$ a $B$-module structure.

If $B$ moreover a [[Hopf monoid]], then $Mod_B$ also inherits a [[closed monoidal category|closed]] monoidal structure if $\mathcal{C}$ has one; see at \emph{[[Hopf monoid]]}.

These relations are known as \emph{[[Tannaka duality]]} \href{Tannaka%20duality#ForAlgebraModules}{for monoids/algebras}, see at \emph{[[structure on algebras and their module categories - table]]}.

\hypertarget{nonsymmetric_variants}{}\subsection*{{Non-symmetric variants}}\label{nonsymmetric_variants}

It is interesting how to generalize this notion in various nonsymmetric situations, for example involving [[braided monoidal category|braidings]], or more generally in [[duoidal categories]], or in relative situations over noncommutative rings (e.g. Takeuchi bialgebroids). In the completely noncommutative situation of the monoidal category of endofunctors, one can look at various compatibilities between [[monad]]s and comonads or monads and tensor products, for example involving distributive laws.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Hopf monoid]], [[Frobenius monoid]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

As far as compatibility with tensor product is concerned, there is a notion of a bimonad and involves a version of [[distributive law]]s, hence it is related to a lifting problem:

\begin{itemize}%
\item [[Ieke Moerdijk]], \emph{Monads on tensor categories}, Category theory 1999 (Coimbra), J. Pure Appl. Algebra 168 (2002), no. 2-3, 189--208 (MR2003e:18012)


\item Korn\'e{}l Szlach\'a{}nyi, \emph{The monoidal Eilenberg--Moore construction and bialgebroids}, J. Pure Appl. Algebra 182, no. 2--3 (2003) 287--315; \emph{Adjointable monoidal functors and quantum groupoids}, \href{http://arxiv.org/abs/math/0301253}{math.QA/0301253}



\end{itemize}
Szlach\'a{}nyi uses earlier analysis of

\begin{itemize}%
\item P. Schauenburg, \emph{Bialgebras over noncommutative rings, and a structure theorem for Hopf bimodules}, Applied Categorical Structures \textbf{6}, 193-222 (1998) \href{http://dx.doi.org/10.1023/A:1008608028634}{doi}


\item H.E. Porst, \emph{On categories of monoids, comonoids and bimonoids}, Quaest. Math. \textbf{31} (2008) 127-139 \href{http://www.ams.org/mathscinet-getitem?mr=2529129}{MR2010d:18010} \href{http://dx.doi.org/10.2989/QM.2008.31.2.2.474}{doi}



\end{itemize}
For a dual version see

\begin{itemize}%
\item P. Mac Crudden, Opmonoidal monads, Theory Appl. Cat., Vol. 10, 2002, No. 19, pp 469-485, \href{http://www.tac.mta.ca/tac/volumes/10/19/10-19abs.html}{link}

\end{itemize}
There is also a more subtle notion also called Hopf monad in

\begin{itemize}%
\item R. Wisbauer, B. Mesablishvili, Bimonads and Hopf monads on categories, \href{http://arxiv.org/abs/0710.1163}{arXiv:0710.1163}

\end{itemize}
In some of these generalized cases, one does not have a good notion of of antipode, so that the difference between bimonoids and Hopf monoids has to be stated in different terms. A similar case is in the case of [[Hopf algebroid]]s over a noncommutative base:

\begin{itemize}%
\item [[Brian Day]], [[Ross Street]], \emph{Monoidal bicategories and Hopf algebroids}, Advances in Math. \textbf{129}, 1 (1997) 99--157


\item [[Gabi Böhm]], \emph{Hopf algebroids}, (a chapter of) Handbook of algebra, \href{http://arxiv.org/abs/0805.3806}{arxiv:math.RA/0805.3806}; \emph{An alternative notion of Hopf algebroid}; in ``Hopf algebras in noncommutative geometry and physics'', 31--53, Lec. Notes in Pure and Appl. Math. \textbf{239}, Dekker, New York 2005; \href{http://arxiv.org/abs/math.QA/0301169}{math.QA/0301169}



\end{itemize}
[[!redirects bimonoid]] [[!redirects bimonoids]] [[!redirects bimonoid object]] [[!redirects bimonoid objects]]



\end{document}