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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{monadicity theorem} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory} [[!include category theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak \noindent\hyperlink{variations}{Variations}\dotfill \pageref*{variations} \linebreak \noindent\hyperlink{the_crude_monadicity_theorem}{The crude monadicity theorem}\dotfill \pageref*{the_crude_monadicity_theorem} \linebreak \noindent\hyperlink{duskins_monadicity_theorem}{Duskin's monadicity theorem}\dotfill \pageref*{duskins_monadicity_theorem} \linebreak \noindent\hyperlink{monadicity_over_set}{Monadicity over Set}\dotfill \pageref*{monadicity_over_set} \linebreak \noindent\hyperlink{strict_monadicity}{Strict monadicity}\dotfill \pageref*{strict_monadicity} \linebreak \noindent\hyperlink{examples_and_applications}{Examples and Applications}\dotfill \pageref*{examples_and_applications} \linebreak \noindent\hyperlink{groups_over_sets}{Groups over sets}\dotfill \pageref*{groups_over_sets} \linebreak \noindent\hyperlink{categories_over_computads}{Categories over computads}\dotfill \pageref*{categories_over_computads} \linebreak \noindent\hyperlink{monadic_descent}{Monadic descent}\dotfill \pageref*{monadic_descent} \linebreak \noindent\hyperlink{InInfinityCategories}{In $(\infty,1)$-categories}\dotfill \pageref*{InInfinityCategories} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{statement}{}\subsection*{{Statement}}\label{statement} \begin{defn} \label{CreationOfCoequalizersOfuSplitPairs}\hypertarget{CreationOfCoequalizersOfuSplitPairs}{} Given a [[functor]] $U : D \rightarrow C$, then a [[parallel pair]] $f,g : a \rightarrow b$ in $D$ is called \textbf{$U$-split} if the pair $U f, U g$ has a [[split coequalizer]] in $C$. Specifically, this means that there is a diagram in $C$: \begin{displaymath} U a \;\underoverset{U f}{U g}{\rightrightarrows}\; U b \;\overset{h}{\rightarrow}\; c \end{displaymath} such that $h \cdot U f = h \cdot U g$, and $h$ and $U f$ have respective [[sections]] $s$ and $t$ satisfying $U g \cdot t = s \cdot h$. This implies that the arrow $h$ is necessarily a coequalizer of $U f$ and $U g$. The functor $U$ is said to \emph{create coequalizers of $U$-split pairs} if for any such $U$-split pair, there exists a coequalizer $e$ of $f,g$ in $D$ which is preserved by $U$, and moreover any [[fork]] in $D$ whose image in $C$ is a split coequalizer must itself be a coequalizer (not necessarily split). \end{defn} \begin{theorem} \label{}\hypertarget{}{} \textbf{(Beck's monadicity theorem}, \textbf{tripleability theorem)} A [[functor]] $U : D \rightarrow C$ is [[monadic]] (tripleable) if and only if \begin{enumerate}% \item $U$ has a [[left adjoint]], and \item $U$ [[created limit|creates]] [[coequalizers]] of $U$-split pairs, def. \ref{CreationOfCoequalizersOfuSplitPairs}. \end{enumerate} \end{theorem} The proof is reproduced for instance in (\hyperlink{Borceux}{Borceux, vol 2, theorem 4.4.4}, \hyperlink{MacLane}{MacLane, p. 147-150}). An equivalent, and sometimes easier, way to state these conditions is to say that \begin{theorem} \label{}\hypertarget{}{} A functor $U : D \to C$ is monadic precisely if \begin{enumerate}% \item $U$ has a [[left adjoint]], \item $U$ reflects [[isomorphisms]] (i.e. it is [[conservative functor|conservative]]), and \item $D$ has, and $U$ [[preserved limit|preserves]], coequalizers of $U$-split pairs. \end{enumerate} \end{theorem} This is equivalent because a conservative functor [[reflected limit|reflects]] any limits or colimits which exist in its domain and which it [[preserved limit|preserves]], while monadic functors are always conservative. \hypertarget{variations}{}\subsection*{{Variations}}\label{variations} \hypertarget{the_crude_monadicity_theorem}{}\subsubsection*{{The crude monadicity theorem}}\label{the_crude_monadicity_theorem} The \textbf{crude monadicity theorem} gives a \emph{sufficient}, but not necessary, condition for a functor to be monadic. It states that a functor $U : D \rightarrow C$ is [[monadic]] if \begin{enumerate}% \item $U$ has a left adjoint \item $U$ reflects isomorphisms \item $D$ has and $U$ preserves coequalizers of [[reflexive coequalizer|reflexive pairs]]. \end{enumerate} (Recall that a parallel pair $f,g : a \rightarrow b$ is \textbf{reflexive} if $f$ and $g$ have a common [[section]].) This sufficient, but not necessary, condition is sometimes easier to verify in practice. In contrast to the crude monadicity theorem, the necessary and sufficient condition above is sometimes called the \textbf{precise monadicity theorem}. \hypertarget{duskins_monadicity_theorem}{}\subsubsection*{{Duskin's monadicity theorem}}\label{duskins_monadicity_theorem} Duskin's monadicity theorem gives a different sufficient, but not necessary, condition which refers only to quotients of [[congruences]]. It says that a functor $U \colon D \to C$ is monadic if \begin{enumerate}% \item $U$ has a left adjoint \item $D$ and $C$ are [[finitely complete category|finitely complete]] \item $U$ [[created limit|creates]] coequalizers for [[congruences]] in $D$ whose images in $C$ have split coequalizers. \end{enumerate} We can weaken the hypothesis a bit further to obtain the theorem: \begin{itemize}% \item A right adjoint between finitely complete categories is monadic if it creates [[quotients]] for [[congruences]]. \end{itemize} As usual, we can also modify it by replacing reflection of limits by reflection of isomorphisms. \begin{itemize}% \item A conservative right adjoint $U\colon D \to C$ between finitely complete categories is monadic if any congruence in $D$ which has a quotient in $C$ already has a quotient in $D$, and that quotient that is preserved by $U$. \end{itemize} If we view the objects of $D$ as underlying $C$-objects with structure, this says that any congruence in $D$ induces a $D$-structure on its quotient in $C$. As with the crude monadicity theorem, this condition is sometimes easier to verify since quotients of congruences are often better-behaved than arbitrary coequalizers. This is the case in many ``algebraic'' situations. Duskin actually gave a slightly more precise version only assuming the categories $C$ and $D$ to have particular finite limits, rather than all of them. \hypertarget{monadicity_over_set}{}\subsubsection*{{Monadicity over Set}}\label{monadicity_over_set} In the case when the base category $C$ is [[Set]], one can further refine the requisite conditions. Linton proved that a functor $U\colon D\to Set$ is monadic if and only if \begin{enumerate}% \item $U$ has a left adjoint, \item $D$ admits [[kernel pairs]] and [[coequalizers]], \item A [[parallel pair]] $R \rightrightarrows S$ in $D$ is a kernel pair if and only if its image under $U$ is so, and \item A morphism $A\to B$ in $D$ is a [[regular epimorphism]] if and only if its image under $U$ is so. \end{enumerate} There are other versions of this theorem, including generalizations to monadicity over [[presheaf categories]], which can be viewed as analogues of [[Giraud's theorem]]. \hypertarget{strict_monadicity}{}\subsubsection*{{Strict monadicity}}\label{strict_monadicity} The version of the monadicity theorem given in [[Categories Work]] uses a notion of ``creation of limits'' which fails to observe the [[principle of equivalence]], concluding that the comparison functor is an \emph{isomorphism} of categories, rather than merely an equivalence. But the versions mentioned above can be found in the exercises. Note however that if $U: D \to C$ is an [[amnestic functor|amnestic]] [[isofibration]], then $U$ is monadic iff it is strictly monadic. For an application of this observation, see for example the discussion of algebraically free monads at [[free monad]]. \hypertarget{examples_and_applications}{}\subsection*{{Examples and Applications}}\label{examples_and_applications} \hypertarget{groups_over_sets}{}\subsubsection*{{Groups over sets}}\label{groups_over_sets} We will use Duskin's variant to prove that the [[forgetful functor]] $U\colon$[[Grp]]$\to$[[Set]] is monadic. Of course, this is also easy to show by explicit computation, but it serves as a useful example of how to use a monadicity theorem. We first need it to have a left adjoint: this is easy to show by a direct construction of [[free groups]], but we could also invoke the [[adjoint functor theorem]]. It is also easy to show that it is conservative (a bijective group homomorphism is a group isomorphism), so it remains to consider congruences. Since limits in $Grp$ are created in $Set$, a [[congruence]] in $Grp$ on a group $G$ is an [[equivalence relation]] on $G$ which is also a [[subgroup]] of $G\times G$. This latter condition means that if $g_1\sim g_2$ and $h_1\sim h_2$, then also $g_1^{-1}\sim g_2^{-1}$ and $g_1 h_1 \sim g_2 h_2$. Since $g\sim g$ for all $g$, it follows that $g\sim h$ if and only if $1\sim h g^{-1}$, so $\sim$ is determined by the subset $H\subseteq G$ of those $h\in G$ such that $1\sim h$. This $H$ is clearly a subgroup of $G$, and moreover a [[normal subgroup]], since if $h\in H$ and $g\in G$ we have $1 = g^{-1} g \sim g^{-1} h g$, so $g^{-1} h g\in H$. Conversely, it is easy to construct a congruence from any normal subgroup, so the two notions are equivalent. It remains only to observe that the quotient of a group by a normal subgroup is, in fact, a quotient of its associated congruence in $Grp$, which is preserved by $U$. Thus, by Duskin's monadicity theorem, $U$ is monadic. \hypertarget{categories_over_computads}{}\subsubsection*{{Categories over computads}}\label{categories_over_computads} The monadicity theorem becomes more important when the base category $C$ is more complicated and harder to work with explicitly, and when the objects of $D$ are not obviously defined as ``objects of $C$ with extra structure.'' For instance, the category of [[strict 2-categories]] is monadic over the category of 2-[[globular sets]], essentially by definition, but it is much less trivial to show that it is \emph{also} monadic over the category of [[2-computads]]. This latter fact can, however, be proven using the monadicity theorem. \hypertarget{monadic_descent}{}\subsubsection*{{Monadic descent}}\label{monadic_descent} The monadicity theorem also plays a central role in [[monadic descent]]. \hypertarget{InInfinityCategories}{}\subsection*{{In $(\infty,1)$-categories}}\label{InInfinityCategories} There is a version of the monadicity theorem for [[(∞,1)-monad]]s in \href{http://arxiv.org/PS_cache/math/pdf/0702/0702299v5.pdf#page=107}{section 3.4} of \begin{itemize}% \item [[Jacob Lurie]], \emph{Noncommutative algebra} (\href{http://arxiv.org/abs/math/0702299}{arXiv}) \end{itemize} There is also a 2-categorical approach using quasicategories in \begin{itemize}% \item [[Emily Riehl]], [[Dominic Verity]] \emph{The 2-category theory of quasi-categories} (\href{http://arxiv.org/abs/1306.5144}{arXiv}), \emph{Homotopy coherent adjunctions and the formal theory of monads} (\href{http://arxiv.org/abs/1310.8279}{arXiv}) \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[monadic functor]], [[comonadic functor]] \item [[functor of descent type]] \item [[monadic decomposition]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Canonical textbook references include \begin{itemize}% \item [[Francis Borceux]], Section 4 in volume 2 of \emph{[[Handbook of Categorical Algebra]]}, in 3 vols. \item [[Saunders MacLane]], Section VI.7 of \emph{[[Categories for the Working Mathematician]]}. \item Chapter 3 of [[Michael Barr]], [[Charles Wells]], \emph{Triples, toposes, and theories} , Grundlehren der math. Wissenschaften 278, Springer-Verlag 1983, \href{ftp://ftp.math.mcgill.ca/pub/barr/ttt}{ftp}, \href{http://www.cwru.edu/artsci/math/wells/pub/ttt.html}{web}, \href{http://www.case.edu/artsci/math/wells/pub/pdf/ttt.pdf}{pdf} \end{itemize} Other references include: \begin{itemize}% \item [[descent]], [[FGA explained]], [[SGA 1]], [[Benabou-Roubaud theorem| Bénabou-Roubaud's theorem]] \item [[Jean Bénabou]], [[Jacques Roubaud]], \emph{Monades et descente} , C. R. Acad. Sc. Paris, t. 270 (12 Janvier 1970), Serie A, 96--98 \item Du\v{s}ko Pavlovi, Categorical interpolation: descent and the Beck-Chevalley condition without direct images, Category theory Como 1990, pp. 306--325, Lecture Notes in Mathematics 1488, Springer 1991 \item [[Pierre Deligne]], \emph{[[Catégories Tannakiennes]]} , Grothendieck Festschrift, vol. II, Birkh\"a{}user Progress in Math. 87 (1990) pp. 111-195. \item wikipedia: \href{http://en.wikipedia.org/wiki/Beck%27s_monadicity_theorem}{monadicity theorem} \item John Bourke, \emph{Two dimensional monadicity}, \href{http://arxiv.org/abs/1212.5123}{arxiv/1212.5123} \end{itemize} There is a version for [[Morita context]]s instead of monads: \begin{itemize}% \item [[Tomasz Brzezi?ski]], Adrian Vazquez Marquez, Joost Vercruysse, \emph{The Eilenberg-Moore category and a Beck-type theorem for a Morita context}, Appl. Categ. Structures \textbf{19} (2011), no. 5, 821--858 \href{http://www.ams.org/mathscinet-getitem?mr=2836546}{MR2836546} \href{http://dx.doi.org/10.1007/s10485-009-9217-0}{doi} \end{itemize} Discussion for [[(infinity,1)-monads]] is in \begin{itemize}% \item [[Jacob Lurie]], section 6.2. of \emph{[[Higher Algebra]]} \end{itemize} and realized in the context of [[quasi-categories]] in \begin{itemize}% \item [[Emily Riehl]], [[Dominic Verity]], section 7.2 of \emph{Homotopy coherent adjunctions and the formal theory of monads} (\href{http://arxiv.org/abs/1310.8279}{arXiv:1310.8279}) \end{itemize} [[!redirects Beck's Monadicity Theorem]] [[!redirects Beck's Monadicity Theorem]] [[!redirects Beck Monadicity Theorem]] [[!redirects Barr-Beck theorem]] [[!redirects monadicity theorem]] [[!redirects Barr-Beck monadicity theorem]] [[!redirects Beck's monadicity theorem]] [[!redirects Beck's monadicity theorem]] [[!redirects Beck monadicity theorem]] [[!redirects tripleability theorem]] [[!redirects Beck's tripleability theorem]] [[!redirects Beck's tripleability theorem]] [[!redirects Beck tripleability theorem]] [[!redirects Beck's theorem]] [[!redirects Beck's theorem]] [[!redirects Beck theorem]] [[!redirects crude monadicity theorem]] [[!redirects crude tripleability theorem]] [[!redirects monadicity]] \end{document}