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\begin{document}

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\section*{free monad}

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

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{free_monads}{}\section*{{Free monads}}\label{free_monads}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{free_finitary_monads}{Free finitary monads}\dotfill \pageref*{free_finitary_monads} \linebreak
\noindent\hyperlink{algebraicallyfree_monads}{Algebraically-free monads}\dotfill \pageref*{algebraicallyfree_monads} \linebreak
\noindent\hyperlink{constructions}{Constructions}\dotfill \pageref*{constructions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{free monad} is a [[free object]] relative to a [[forgetful functor]] whose domain is a [[category]] of [[monads]].

This general concept has many different specific incarnations, since there are potentially many different such forgetful functors. Suppose $C$ is a [[category]], and write $Mnd(C)$ for the category whose objects are monads on $C$ and whose morphisms are natural transformations commuting with the monad structure maps; i.e. it is the category of [[monoids]] in the [[monoidal category]] of endofunctors with composition. Then we have a string of forgetful functors:

\begin{displaymath}
Mnd(C) \to PtEnd(C) \to End(C) \to [ob(C),C]
\end{displaymath}
where $End(C)$ denotes the category of [[endofunctors]] of $C$, and $PtEnd(C)$ denotes the category of [[pointed endofunctors]], i.e. endofunctors $F$ equipped with a natural transformation $Id\to F$. A \emph{free monad} can then be considered as a free object relative to any one of these forgetful functors.

\hypertarget{free_finitary_monads}{}\subsection*{{Free finitary monads}}\label{free_finitary_monads}

In general, these forgetful functors cannot be expected to have [[left adjoints]], i.e. there will not be a [[free functor|``free monad functor'']], but individual objects can often be shown to generate free monads. One general case in which this is true is when $C$ is [[locally presentable category|locally presentable]] and we consider monads and endofunctors which are [[accessible functor|accessible]], i.e. preserve sufficiently highly [[filtered colimits]]. Suppose for the sake of argument that $C$ is locally \emph{finitely} presentable (the higher-ary case is analogous). Then we can restrict the above string of forgetful functors to the [[finitary monads]], i.e. those preserving filtered colimits, to obtain:

\begin{displaymath}
Mnd_f(C) \to PtEnd_f(C) \to End_f(C) \to [ob(C)_f,C]
\end{displaymath}
where the subscript $f$ denotes restriction to finitary things, and $ob(C)_f$ is the set of [[compact objects]] of $C$. In this case, all these forgetful functors do have left adjoints, and moreover at least the functors $Mnd_f(C) \to End_f(C)$ and $Mnd_f(C) \to [ob(C)_f,C]$ are monadic. (This is shown in the papers cited below.) The construction is by a convergent [[transfinite composition]].

For example, the left adjoint to $Mnd_f(C) \to End_f(C)$, shows that there exists a ``free finitary monad'' on any finitary endofunctor. Note, though, that this does not \emph{automatically} imply that the ``free finitary monad'' on a finitary endofunctor is also a ``free monad'' on that endofunctor, i.e. that as a free object it satisfies the requisite universal property relative to all objects of $Mnd(C)$, not merely those lying in $Mnd_f(C)$. It is, however, generally true that this is the case: free finitary monads are also free monads.

\hypertarget{algebraicallyfree_monads}{}\subsection*{{Algebraically-free monads}}\label{algebraicallyfree_monads}

We say that a monad $T$ is \textbf{algebraically-free} on an endofunctor $F$ if the category $T Alg_{mnd}$ of $T$-algebras (in the sense of [[algebras for a monad]]) is equivalent to the category $F Alg_{endo}$ of $F$-algebras (in the sense of [[algebras for an endofunctor]]). \textbf{N.B.}: Any such equivalence must be an isomorphism $T Alg_{mnd} \cong F Alg_{endo}$, because the underlying functors from the categories of algebras in each case are [[amnestic functor|amnestic]] [[isofibrations]]. See remarks at the article [[monadicity theorem]] on monadicity \emph{vis-\`a{}-vis} strict monadicity.

\emph{A priori}, being algebraically free is different from being free. However, one can show the following.

\begin{theorem}
\label{}\hypertarget{}{}
Any algebraically-free monad is free.

\end{theorem}
\begin{proof}
First observe that for a (perhaps pointed) endofunctor $F$ and a monad $T$, to give a functor $T Alg_{mnd} \to F Alg_{endo}$ over $C$ is equivalent to giving a (pointed) transformation $F\to T$, and if $F$ is a monad then such a functor takes values in $F Alg_{mnd}$ iff the transformation $F\to T$ is a monad morphism. Thus, if $F Alg_{endo} \cong T Alg_{mnd}$, then for any other monad $S$, (pointed) transformations $F\to S$ correspond to maps $S Alg_{mnd} \to F Alg_{endo} \cong T Alg_{mnd}$ and hence to monad morphisms $T\to S$, i.e. $T$ is free on $F$.

\end{proof}
\begin{theorem}
\label{}\hypertarget{}{}
If $C$ is [[locally small category|locally small]] and [[complete category|complete]], then any free monad is algebraically-free.

\end{theorem}
\begin{proof}
For any object $x\in C$, the assumptions ensure that the [[codensity monad]] of $x$ exists. This is the right [[Kan extension]] of $x\colon 1\to C$ along itself, which we write as $\langle x,x\rangle$. The universal property of Kan extensions means that for any endofunctor $F$, to give a map $F x \to x$ (i.e. to make $x$ into an $F$-algebra) is the same as to give a natural transformation $F\to \langle x,x\rangle$. Moreover, one can check that if $F$ is a pointed endofunctor (resp. a monad), then the map $F x \to x$ is a pointed (resp. monad) algebra iff the corresponding transformation $F\to \langle x,x\rangle$ is a morphism of pointed endofunctors (resp. of monads). Therefore, if $T$ is the free monad on $F$, then applying its universal property in $Mnd(C)$ to the monad $\langle x,x\rangle$, we see that it is also algebraically-free.

\end{proof}
Notice that this second proof relies crucially on the fact that free monads have a universal property relative to a forgetful functor whose domain is all of $Mnd(C)$, not just some subcategory of finitary or accessible monads, since $\langle x,x\rangle$ will not in general be finitary or accessible.

\hypertarget{constructions}{}\subsection*{{Constructions}}\label{constructions}

Perhaps the most general [[set theory|set-theoretically]] based construction of (algebraically) free monads is the [[transfinite construction of free algebras]]. (In [[type theory]], it is natural to use instead [[higher inductive types]].)

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

\begin{itemize}%
\item The monadicity of the above adjunctions can be used to give [[presentation]]s of monads in terms of [[generators and relations]]. This has close connections with [[Lawvere theories]] and related ideas.


\item Free monads on pointed endofunctors play an important role in the construction of cofibrantly generated [[algebraic weak factorization systems]].



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

\begin{itemize}%
\item The [[algebra over a monad]] over a free monad on an endofunctor is an \emph{[[algebra over an endofunctor]]}.

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

\begin{itemize}%
\item [[Max Kelly]], \href{http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=4759448}{``A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on''}


\item [[Max Kelly]] and [[John Power]], \href{http://www.sciencedirect.com/science/article/pii/0022404993900928}{``Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads''}


\item [[Steve Lack]], \href{http://www.maths.usyd.edu.au/res/Catecomb/Lack/1997-29.html}{``On the monadicity of finitary monads''}


\item [[Nicola Gambino]], [[Martin Hyland]], section 6 of \emph{Wellfounded trees and dependent polynomial functors}. In Types for proofs and programs, volume 3085 of Lecture Notes in Comput. Sci., pages 210--225. Springer-Verlag, Berlin, 2004 (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.98.4534}{web})



\end{itemize}
[[!redirects free monads]] [[!redirects algebraically-free monad]] [[!redirects algebraically-free monads]] [[!redirects algebraically free monad]] [[!redirects algebraically free monads]]



\end{document}