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\section*{Hopf monad}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{monoidal_monads}{Monoidal monads\ldots{}}\dotfill \pageref*{monoidal_monads} \linebreak
\noindent\hyperlink{opmonoidal_monads_with_invertible_fusion}{Opmonoidal monads with invertible fusion}\dotfill \pageref*{opmonoidal_monads_with_invertible_fusion} \linebreak
\noindent\hyperlink{distributing_monad_and_comonad_structures}{Distributing monad and comonad structures}\dotfill \pageref*{distributing_monad_and_comonad_structures} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \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}

One can consider various compatibilities between a ([[comonad|co]])[[monad]] on a [[monoidal category]] and the underlying monoidal product; they are variants of the idea of a [[distributive law]]. Similarly, one can look at a compatibility between an action of a monoidal category and a (co)monad on the same category. A \emph{Hopf monad} satisfies conditions analogous to that of a \emph{[[Hopf monoid]]} (\hyperlink{Bruguieres}{Brugui\`e{}res 06}).

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Warning: more than one notion comes up with the names of [[bimonad]] and Hopf monad; there are also related notions of monoidal monad and opmonoidal monad as well of a [[strong monad]].

\hypertarget{monoidal_monads}{}\subsubsection*{{Monoidal monads\ldots{}}}\label{monoidal_monads}

A \textbf{monoidal monad} on a monoidal category is a [[monad]] whose underlying [[endofunctor]] is a [[lax monoidal functor]] and such that the unit and multiplication are monoidal natural transformations. Consequently, the [[Kleisli category]] of a monoidal monad has a canonical monoidal structure such that the [[forgetful functor]] is strict monoidal.

(\ldots{})

\hypertarget{opmonoidal_monads_with_invertible_fusion}{}\subsubsection*{{Opmonoidal monads with invertible fusion}}\label{opmonoidal_monads_with_invertible_fusion}

An \textbf{opmonoidal monad} on a monoidal category is a [[monad]] whose underlying [[endofunctor]] is a [[colax monoidal functor]], and such that the unit and multiplication are monoidal natural transformations. An opmonoidal monad is also called a \textbf{bimonad}.

For an opmonoidal monad $T$, one can define the \textbf{left fusion operator} to be the natural transformation

\begin{displaymath}
T(X\otimes T Y) \to T X \otimes T^2 Y \to T X \otimes T Y
\end{displaymath}
of the opmonoidal constraint of $T$ and its monad multiplication. Similarly, we have the \textbf{right fusion operator}

\begin{displaymath}
T(T X\otimes Y) \to T^2 X \otimes T Y \to T X \otimes T Y
\end{displaymath}
The fusion operators satisfy certain axioms. In fact, an opmonoidal monad structure on $T$ is uniquely determined by a fusion operator (of either sort) along with the monad unit $Id\to T$ and the opmonoidal unit constraint $T I \to I$, satisfying appropriate axioms.

An opmonoidal monad is called a \textbf{Hopf monad} if both of its fusion operators are invertible. More generally we have \emph{left} and \emph{right} Hopf monads where only one fusion operator is invertible. This definition is in Brugui\`e{}res-Lack-Virelizier.

\hypertarget{distributing_monad_and_comonad_structures}{}\subsubsection*{{Distributing monad and comonad structures}}\label{distributing_monad_and_comonad_structures}

Alternatively, one might be tempted to define a Hopf monad to be a [[Hopf monoid]] in the monoidal category of endofunctors, but that monoidal category is not [[symmetric monoidal category|symmetric]] or even [[braided monoidal category|braided]] or [[duoidal category|duoidal]], so this doesn't make sense. However, we can instead ask for a ``local'' braiding in the form of a mixed [[distributive law]].

Thus, we might define a \textbf{bimonad} to be an endofunctor $H$ equipped with the structure of both a monad and a comonad, along with a [[distributive law]] $\lambda: H H \to H H$ satisfying suitable axioms analogous to those of a bimonoid. A bimonad in this sense is a \textbf{Hopf monad} if it has an antipode $s:H\to H$ making the same diagrams commute as for a [[Hopf monoid]]. This definition is in Mesablishvili-Wisbauer.

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

\begin{itemize}%
\item If $H$ is a [[bimonoid]] in a [[braided monoidal category]], then the monad $T_H X =H\otimes X$ (whose algebras are $H$-[[modules]]) is opmonoidal, with constraints induced by the comultiplication and counit of $H$. If $H$ is moreover a [[Hopf monoid]], then $T_H$ is a (Brugui\`e{}res-Lack-Virelizier) Hopf monad.


\item In fact, if $H$ is a bimonoid as before, then $T_H$ is \emph{also} a bimonad in the sense of Mesablishvili-Wisbauer, with monad and comonad structures induced by the monoid and comonoid structures of $H$. Moreover, if $H$ is a Hopf monoid, then $T_H$ is a Hopf monad in their sense as well.



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

If $T$ is a (Brugui\`e{}res-Lack-Virelizier) Hopf monad on a [[closed monoidal category]], then its category of algebras is also closed and the monadic forgetful functor preserves internal-homs.

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

\begin{itemize}%
\item [[Hopf algebra]]


\item [[Hopf monoid]]


\item [[Hopf monoidal category]]



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

\begin{itemize}%
\item wikipedia \href{http://en.wikipedia.org/wiki/Monoidal_monad}{opmonoidal monad}


\item [[Kornél Szlachányi]], \emph{The monoidal Eilenberg--Moore construction and bialgebroids}, J. Pure Appl. Algebra \textbf{182}, no. 2--3 (2003) 287--315


\item R. Wisbauer, \emph{Bimonads and Hopf monads on categories}, \href{http://www.math.uni-duesseldorf.de/~wisbauer/Hopfmonad.pdf}{pdf}


\item Bachuki Mesablishvili, [[Robert Wisbauer]], \emph{Notes on bimonads and Hopf monads}, Theory Appl. Cat. \textbf{26}:10, 2012, 281-303, \href{http://www.tac.mta.ca/tac/volumes/26/10/26-10abs.html}{abs} \href{http://www.tac.mta.ca/tac/volumes/26/10/26-10.pdf}{pdf} \href{http://arxiv.org/abs/1010.3628}{arxiv/1010.3628}


\item D. Chikhladze, S. Lack, [[R. Street]], \emph{Hopf monoidal comonads}, Theory and Appl. of Cat. \textbf{24} (2010) No. 19, 554-563.\href{http://www.tac.mta.ca/tac/volumes/24/19/24-19abs.html}{tac}


\item A. Brugui\`e{}res, \emph{Hopf monads}, \href{http://arxiv.org/abs/math/0604180}{math.QA/0604180}; \emph{Hopf monads: an introduction}, an expos\`e{}, \href{http://www.cirm.univ-mrs.fr/videos/2006/exposes/23/Bruguieres.pdf}{pdf}; \emph{Hopf monads, tensor categories and quantum invariants}, an expos\`e{}, \href{http://www.cirm.univ-mrs.fr/videos/2008/exposes/307/Bruguiere.pdf}{pdf}; \emph{Galois-Grothendieck duality, Tannaka duality and Hopf (co)monads}. talk at workshop ``Hopf algebras and tensor categories'', University of Almer\'i{}a, July 4-8, 2011, \href{http://www.ual.es/Congresos/hopf2010/charlas/bruguiertalk.pdf}{pdf}



\end{itemize}
\begin{itemize}%
\item Alain Brugui\`e{}res, [[Steve Lack]], Alexis Virelizier, \emph{Hopf monads on monoidal categories}, \href{http://arxiv.org/abs/1003.1920}{arxiv/1003.1920}
\item A. Brugui\`e{}res, A. Virelizier, \emph{The double of a Hopf monad}, Adv. Math. \textbf{227} No. 2, June 2011, pp 745--800, \href{http://arxiv.org/abs/0812.2443}{arxiv/0812.2443}
\item Group = Hopf algebra, blog discussion, \href{http://sbseminar.wordpress.com/2007/10/07/group-hopf-algebra}{sbseminar}
\item Marek Zawadowski, \emph{The formal theory of monoidal monads}, \href{http://arxiv.org/abs/1012.0547}{arxiv/1012.0547}
\item M. Zawadowski, \emph{coMalcev monads}, slides from a talk, \href{http://duch.mimuw.edu.pl/~zawado/Talks/coMalcev_talk.pdf}{pdf}
\item [[Ieke Moerdijk]], \emph{Monads on tensor categories}, J. Pure. Appl. Alg. \textbf{168}, 2-3, (2002), 189-208, 
\item [[Gabriella Böhm]], [[Stephen Lack]], [[Ross Street]], \emph{Weak bimonads and weak Hopf monads}, J. Alg. \textbf{328}, n. 1, Feb 2011, 1-30, \href{http://dx.doi.org/10.1016/j.jalgebra.2010.07.032}{doi}, \href{http://arxiv.org/abs/1002.4493}{arxiv/1002.4493} [[G. Böhm]], \emph{Weak bimonads and weak Hopf monads}, conference slides, 2010, \href{http://www.ual.es/congresos/hopf2010/charlas/bohm.pdf}{pdf}
\item S. Willerton, \emph{A diagrammatic approach to Hopf monads}, \href{http://arxiv.org/abs/0807.0658}{arxiv/0807.0658}
\item K. Dosen, Z. Petric, \emph{Coherence for monoidal monads and comonads}, \href{http://arxiv.org/abs/0907.2199}{arxiv/0907.2199}
\item [[Anders Kock]], \emph{Strong functors and monoidal monads}, \href{http://home.imf.au.dk/kock/SFMM.pdf}{pdf}, Archiv der Math. \textbf{23}: 113--120, (1972) \href{http://dx.doi.org/10.1007%2FBF01304852}{doi}; \emph{Monads on symmetric monoidal categories}, \href{http://home.imf.au.dk/kock/MSMCC.pdf}{pdf}
\item B. J. Day, \emph{Note on monoidal monads}, J. Austral. Math. Soc. A \textbf{23}, 292-311 (1977) \href{http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=4897004}{pdf}
\item Alain Brugui\`e{}res, Sonia Natale, \emph{Exact sequences of tensor categories}, \href{http://arxiv.org/abs/1006.0569}{arXiv:math.QA/1006.0569} (generalizes Schneider's work on [[exact sequences of Hopf algebras]]; uses Hopf monads)

\end{itemize}
[[!redirects Hopf monads]] [[!redirects bimonad]] [[!redirects opmonoidal monad]]



\end{document}