\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{mathtools} \usepackage{graphicx} \usepackage{color} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{xparse} \usepackage{hyperref} %----Macros---------- % % Unresolved issues: % % \righttoleftarrow % \lefttorightarrow % % \color{} with HTML colorspec % \bgcolor % \array with options (without options, it's equivalent to the matrix environment) % Of the standard HTML named colors, white, black, red, green, blue and yellow % are predefined in the color package. Here are the rest. \definecolor{aqua}{rgb}{0, 1.0, 1.0} \definecolor{fuschia}{rgb}{1.0, 0, 1.0} \definecolor{gray}{rgb}{0.502, 0.502, 0.502} \definecolor{lime}{rgb}{0, 1.0, 0} \definecolor{maroon}{rgb}{0.502, 0, 0} \definecolor{navy}{rgb}{0, 0, 0.502} \definecolor{olive}{rgb}{0.502, 0.502, 0} \definecolor{purple}{rgb}{0.502, 0, 0.502} \definecolor{silver}{rgb}{0.753, 0.753, 0.753} \definecolor{teal}{rgb}{0, 0.502, 0.502} % Because of conflicts, \space and \mathop are converted to % \itexspace and \operatorname during preprocessing. % itex: \space{ht}{dp}{wd} % % Height and baseline depth measurements are in units of tenths of an ex while % the width is measured in tenths of an em. \makeatletter \newdimen\itex@wd% \newdimen\itex@dp% \newdimen\itex@thd% \def\itexspace#1#2#3{\itex@wd=#3em% \itex@wd=0.1\itex@wd% \itex@dp=#2ex% \itex@dp=0.1\itex@dp% \itex@thd=#1ex% \itex@thd=0.1\itex@thd% \advance\itex@thd\the\itex@dp% \makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}} \makeatother % \tensor and \multiscript \makeatletter \newif\if@sup \newtoks\@sups \def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}% \def\reset@sup{\@supfalse\@sups={}}% \def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else% \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}% \else \append@sup#2 \@suptrue \fi% \expandafter\mk@scripts\fi} \def\tensor#1#2{\reset@sup#1\mk@scripts#2_/} \def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2% \reset@sup\mk@scripts#3_/} \makeatother % \slash \makeatletter \newbox\slashbox \setbox\slashbox=\hbox{$/$} \def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$} \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa \copy\slashbox \kern-\@tempdima \box\@tempboxa} \def\slash{\protect\itex@pslash} \makeatother % math-mode versions of \rlap, etc % from Alexander Perlis, "A complement to \smash, \llap, and lap" % http://math.arizona.edu/~aprl/publications/mathclap/ \def\clap#1{\hbox to 0pt{\hss#1\hss}} \def\mathllap{\mathpalette\mathllapinternal} \def\mathrlap{\mathpalette\mathrlapinternal} \def\mathclap{\mathpalette\mathclapinternal} \def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}} \def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}} \def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}} % Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2} \let\oldroot\root \def\root#1#2{\oldroot #1 \of{#2}} \renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}} % Manually declare the txfonts symbolsC font \DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n} \SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n} \DeclareFontSubstitution{U}{txsyc}{m}{n} % Manually declare the stmaryrd font \DeclareSymbolFont{stmry}{U}{stmry}{m}{n} \SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n} % Manually declare the MnSymbolE font \DeclareFontFamily{OMX}{MnSymbolE}{} \DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n} \SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n} \DeclareFontShape{OMX}{MnSymbolE}{m}{n}{ <-6> MnSymbolE5 <6-7> MnSymbolE6 <7-8> MnSymbolE7 <8-9> MnSymbolE8 <9-10> MnSymbolE9 <10-12> MnSymbolE10 <12-> MnSymbolE12}{} % Declare specific arrows from txfonts without loading the full package \makeatletter \def\re@DeclareMathSymbol#1#2#3#4{% \let#1=\undefined \DeclareMathSymbol{#1}{#2}{#3}{#4}} \re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46} \re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12} \re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64} \re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6} \re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77} \re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77} \makeatother % \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE \makeatletter \def\Decl@Mn@Delim#1#2#3#4{% \if\relax\noexpand#1% \let#1\undefined \fi \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}} \def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}} \def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}} \Decl@Mn@Open{\llangle}{mnomx}{'164} \Decl@Mn@Close{\rrangle}{mnomx}{'171} \Decl@Mn@Open{\lmoustache}{mnomx}{'245} \Decl@Mn@Close{\rmoustache}{mnomx}{'244} \makeatother % Widecheck \makeatletter \DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}} \def\@widecheck#1#2{% \setbox\z@\hbox{\m@th$#1#2$}% \setbox\tw@\hbox{\m@th$#1% \widehat{% \vrule\@width\z@\@height\ht\z@ \vrule\@height\z@\@width\wd\z@}$}% \dp\tw@-\ht\z@ \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@ \setbox\tw@\hbox{% \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box \tw@}}}% {\ooalign{\box\tw@ \cr \box\z@}}} \makeatother % \mathraisebox{voffset}[height][depth]{something} \makeatletter \NewDocumentCommand\mathraisebox{moom}{% \IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{% \IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}% }{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}% \mathpalette\@temp{#4}} \makeatletter % udots (taken from yhmath) \makeatletter \def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.} \mkern2mu\raise4\p@\hbox{.}\mkern1mu \raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}} \makeatother %% Fix array \newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}} %% \itexnum is a noop \newcommand{\itexnum}[1]{#1} %% Renaming existing commands \newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}} \newcommand{\widevec}{\overrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\nearr}{\nearrow} \newcommand{\nwarr}{\nwarrow} \newcommand{\searr}{\searrow} \newcommand{\swarr}{\swarrow} \newcommand{\curvearrowbotright}{\curvearrowright} \newcommand{\uparr}{\uparrow} \newcommand{\downuparrow}{\updownarrow} \newcommand{\duparr}{\updownarrow} \newcommand{\updarr}{\updownarrow} \newcommand{\gt}{>} \newcommand{\lt}{<} \newcommand{\map}{\mapsto} \newcommand{\embedsin}{\hookrightarrow} \newcommand{\Alpha}{A} \newcommand{\Beta}{B} \newcommand{\Zeta}{Z} \newcommand{\Eta}{H} \newcommand{\Iota}{I} \newcommand{\Kappa}{K} \newcommand{\Mu}{M} \newcommand{\Nu}{N} \newcommand{\Rho}{P} \newcommand{\Tau}{T} \newcommand{\Upsi}{\Upsilon} \newcommand{\omicron}{o} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\Union}{\bigcup} \newcommand{\Intersection}{\bigcap} \newcommand{\Oplus}{\bigoplus} \newcommand{\Otimes}{\bigotimes} \newcommand{\Wedge}{\bigwedge} \newcommand{\Vee}{\bigvee} \newcommand{\coproduct}{\coprod} \newcommand{\product}{\prod} \newcommand{\closure}{\overline} \newcommand{\integral}{\int} \newcommand{\doubleintegral}{\iint} \newcommand{\tripleintegral}{\iiint} \newcommand{\quadrupleintegral}{\iiiint} \newcommand{\conint}{\oint} \newcommand{\contourintegral}{\oint} \newcommand{\infinity}{\infty} \newcommand{\bottom}{\bot} \newcommand{\minusb}{\boxminus} \newcommand{\plusb}{\boxplus} \newcommand{\timesb}{\boxtimes} \newcommand{\intersection}{\cap} \newcommand{\union}{\cup} \newcommand{\Del}{\nabla} \newcommand{\odash}{\circleddash} \newcommand{\negspace}{\!} \newcommand{\widebar}{\overline} \newcommand{\textsize}{\normalsize} \renewcommand{\scriptsize}{\scriptstyle} \newcommand{\scriptscriptsize}{\scriptscriptstyle} \newcommand{\mathfr}{\mathfrak} \newcommand{\statusline}[2]{#2} \newcommand{\tooltip}[2]{#2} \newcommand{\toggle}[2]{#2} % Theorem Environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \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*{idempotent monad} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{idempotents}{}\paragraph*{{Idempotents}}\label{idempotents} [[!include idempotents - contents]] \hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra} [[!include higher algebra - contents]] \hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory} [[!include 2-category theory - contents]] \hypertarget{modalities_closure_and_reflection}{}\paragraph*{{Modalities, Closure and Reflection}}\label{modalities_closure_and_reflection} [[!include modalities - contents]] \hypertarget{idempotent_monads}{}\section*{{Idempotent monads}}\label{idempotent_monads} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak \noindent\hyperlink{AlgebrasForAnIdempotentMonad}{Algebras for an idempotent monad and Localization}\dotfill \pageref*{AlgebrasForAnIdempotentMonad} \linebreak \noindent\hyperlink{AssociatedIdemopotentMonad}{The associated idempotent monad of a monad}\dotfill \pageref*{AssociatedIdemopotentMonad} \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} An \emph{idempotent monad} is a [[monad]] that ``squares to itself'' in the evident [[category theory|category-theoretic]] sense. Idempotent monads hence serve as [[categorification|categorified]] [[projection]] operators, in that they encode [[reflective subcategories]] and the reflection/[[localization]] onto these. In terms of [[type theory]] idempotent monads interpret (co-)[[modal operators]]; see [[modal type theory]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{IdempotentMonad}\hypertarget{IdempotentMonad}{} An \textbf{idempotent monad} is a [[monad]] $(T,\mu,\eta)$ on a [[category]] $C$ such that one (hence all) of the following equivalent statements are true: \begin{enumerate}% \item $\mu\colon T T \to T$ is a [[natural isomorphism]]. \item All components of $\mu \,\colon\, T T \to T$ are [[monomorphisms]]. \item The maps $T\eta, \eta T \,\colon\, T \to T T$ are equal. \item For every $T$-[[algebra over a monad|algebra]] ($T$-module) $(M,u)$, the corresponding $T$-[[action]] $u\colon T M \to M$ is an [[isomorphism]]. \item The [[forgetful functor]] $C^T \to C$ (where $C^T$ is the [[Eilenberg-Moore category]] of $T$-algebras) is a [[full and faithful functor]]. \item There exists a pair of [[adjoint functors]] $F\dashv U$ such that the induced monad $(UF, U\epsilon F)$ is isomorphic to $(T,\mu)$ and $U$ is a [[full and faithful functor]]. \end{enumerate} \end{defn} e.g. (\hyperlink{Borceux}{Borceux, prop. 4.2.3}). \begin{prop} \label{EquivalentConditions}\hypertarget{EquivalentConditions}{} The conditons in Def. \ref{IdempotentMonad} are indeed equivalent. \end{prop} \begin{proof} $1\Rightarrow 2$ is trivial. $2\Rightarrow 3$ Compositions $\mu\circ T\eta$ and $\mu\circ\eta T$ are always the identity (unit axioms for the monad), and in particular agree; if $\mu$ has all components monic, this implies $T\eta = \eta T$. $3\Rightarrow 4$ Compatibility of action and unit is $u \circ \eta_M = id_M$, hence also $T(u)\circ T(\eta_M) = id_{T M}$. If $T\eta = \eta T$ then this implies $id_{T M} = T(u)\circ \eta_{T M} = \eta_M\circ u$, where the naturality of $\eta$ is used in the second equality. Therefore we exhibited $\eta_M$ both as a left and a right inverse of $u$. $4\Rightarrow 1$ If every action is iso, then the components of multiplication $\mu_M\colon T T M\to T M$ are isos as a special case, namely of the free action on $T M$. $4\Rightarrow 5$ For any monad $T$, the forgetful functor from Eilenberg-Moore category $C^T$ to $C$ is faithful: a morphism of $T$-algebras is always a morphism of underlying objects in $C$. To show that it is also full, we consider any pair $(M,u)$, $(M',u')$ in $C^T$ and must show that any $f\colon M\to M'$ is actually a map $f\colon (M,u)\to (M',u')$; i.e. $u'\circ T f = f\circ u$. But we know that $\eta_M, \eta_{M'}$ are inverses of $u,u'$ respectively and the naturality for $\eta$ says $\eta_{M'}\circ f = T f \circ \eta_M$. Compose that equation with $u$ on the right and $u'$ on the left with the result (notice that we used just the invertibility of $u$). $5\Rightarrow 6$ Trivial, because the Eilenberg-Moore construction induces the original monad by the standard recipe. $6\Rightarrow 3$ By $6$ the counit $\epsilon$ is iso, hence $U\epsilon F$ has a unique 2-sided inverse; by triangle identities, $T\eta$ and $\eta T$ are both right inverses of $U\epsilon F$, hence 2-sided inverses, hence they are equal. $6\Rightarrow 1$ If $F\dashv U$ is an adjunction with $U$ fully faithful, then the counit $\epsilon$ is iso. Since $D(FU X,Y)\simeq C(UX,UY)\simeq D(X,Y)$ where the last equivalence holds since $U$ is full and faithful; hence by essential unicity of the representing object there is an iso $FUX\stackrel{\sim}{\to} X$ ; let $X=Y$ then the adjoint of this identity is the counit of the adjunction; since the hom objects correspond bijectively, the counit is an isomorphism. Hence the multiplication of the induced monad $\mu = U\epsilon F$ is also an iso. \end{proof} Part 5 means that in such a case $C^T$ is, up to equivalence a full [[reflective subcategory]] of $C$. Conversely, the monad induced by any reflective subcategory is idempotent, so giving an idempotent monad on $C$ is equivalent to giving a reflective subcategory of $C$. In the language of [[stuff, structure, property]], an idempotent monad may be said to equip its algebras with \emph{properties only} (since $C^T\to C$ is fully faithful), unlike an arbitrary monad, which equips its algebras with \emph{at most structure} (since $C^T\to C$ is, in general, faithful but not full). If $T$ is idempotent, then it follows in particular that an object of $C$ admits at most one structure of $T$-algebra, that this happens precisely when the unit $\eta_X\colon X\to T X$ is an isomorphism, and in this case the $T$-algebra structure map is $\eta_X^{-1}\colon T X \to X$. However, it is possible to have a non-idempotent monad for which any object of $C$ admits at most one structure of $T$-algebra, in which case $T$ can be said to equip objects of $C$ with [[property-like structure]]; an easy example is the monad on [[semigroups]] whose algebras are [[monoids]]. \begin{remark} \label{}\hypertarget{}{} Let us be in a $2$-category $K$. Part of the structure of an idempotent monad $(C,T,\eta,\mu)$ in $K$ is of course an idempotent morphism $T:C\to C$. More precisely (Definition 1.1.9) considers $\mu$ as part of the structure such that an idempotent 1-cell has a 2-isomorphism $\mu:TT\to T$ such that $\mu T=T\mu$. Equivalently an idempotent morphism is a normalized pseudofunctor from the two object monoid $\{*,e\}$ with $e^2=e$ to $K$. Recall that a \emph{splitting of an idempotent} $(T,\mu)$ consists of a pair of 1-cells $I:D\to C$ and $R:C\to D$ and a pair of 2-isomorphisms $a:R I\to id_D$ and $b:T\to I R$ such that $\mu=b^{-1}(I\circ s\circ R)(b\circ b)$ where $\circ$ denotes horizontal composition of 2-cells. Equivalently an splitting of an idempotent is a limit or a colimit of the defining pseudofunctor. If $K$ has equalizers or coequalizers, then all its idempotents split. Now if $(I,R,a,b)$ is a splitting of an idempotemt monad, then $R\dashv I$ are adjoint. And in this case the splitting of an idempotent is equivalently an Eilenberg-Moore object for the monad $(C,T,\eta,\mu)$. In this case $D$ is called an \emph{adjoint retract of $C$}. \end{remark} ([[Sketches of an Elephant|Johnstone, B 1.1.9, p.248-249]]) \begin{remark} \label{}\hypertarget{}{} Equivalences (resp. cores) in an allegory are precisely those symmetric idempotents which are idempotent monads (resp. comonads). In an allegory the following statements are equivalent: all symmetric idempotents split, idempotent monads split, idempotent comonads split. A similar statement holds at least for some 2-categories. \end{remark} \begin{remark} \label{}\hypertarget{}{} Given an algebra $(X, u:TX\to X)$ , by (1) and (4) the action $u$ yields an isomorphism in $C^T$ between the free algebra $(TX, \mu_X)$ and $(X,u)$ i.e. for an idempotent monad the Eilenberg-Moore and the Kleisli categories coincide. \end{remark} \hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties} \hypertarget{AlgebrasForAnIdempotentMonad}{}\subsubsection*{{Algebras for an idempotent monad and Localization}}\label{AlgebrasForAnIdempotentMonad} \begin{prop} \label{}\hypertarget{}{} Let $(T, \eta, \mu)$ be an idempotent monad on a category $E$. The following conditions on an object $e$ of $E$ are equivalent: \begin{enumerate}% \item The object $e$ carries an $T$-[[algebra for a monad|algebra structure]]. \item The [[unit of a monad|unit]] $\eta e\colon e \to T e$ is a [[split monomorphism]]. \item The [[unit of a monad|unit]] $\eta e$ is an [[isomorphism]]. \end{enumerate} (It follows from 3. that there is at most one algebra structure on $e$, given by $\xi = (\eta e)^{-1}\colon T e \to e$.) \end{prop} \begin{proof} The implication 1. $\Rightarrow$ 2. is immediate. Next, if $\xi\colon M e \to e$ is any retraction of $\eta e$, we have both $\xi \circ \eta e = 1_e$ and \begin{displaymath} \itexarray{ \eta e \circ \xi & = & (M \xi)(\eta M e) & & \text{naturality of}\, \eta \\ & = & (M \xi)(M \eta e) & & \text{see definitions above} \\ & = & M(\xi \circ \eta e) & & \text{functoriality} \\ & = & 1_{M e} & & } \end{displaymath} so 2. implies 3. Finally, if $\eta e$ is an isomorphism, put $\xi = (\eta e)^{-1}$. Then $\xi \circ \eta e = 1_e$ (unit condition), and the associativity condition for $\xi$, \begin{displaymath} \xi \circ \mu e = \xi \circ M \xi, \end{displaymath} follows by inverting the naturality equation $\eta M e \circ \eta e = M \eta e \circ \eta e$. Thus 3. implies 1. \end{proof} \begin{remark} \label{ReflectiveSubcategoryOfIdempotentMonad}\hypertarget{ReflectiveSubcategoryOfIdempotentMonad}{} This means that the [[Eilenberg-Moore category]] of an [[idempotent monad]] is equivalently the [[reflective subcategory]] (a ``[[localization]]'' of the ambient category) whose embedding-reflection [[adjunction]] gives the idempotent monad. \end{remark} See also (\hyperlink{Borceux}{Borceux, volume 2, corollary 4.2.4}). \begin{remark} \label{}\hypertarget{}{} Hence [[duality|dually]] the co-algebras over an idempotent [[comonad]] form a [[coreflective subcategory]], hence a ``co-localization'' of the ambient category. \end{remark} \begin{remark} \label{}\hypertarget{}{} In [[modal type theory]] one thinks of a (idempotent) (co-)monad as a (co-)[[modal operator]] and of its algebras as (co-)[[modal types]]. In this terminology the above says that categories of (co-)modal types are precisely the (co-)reflective [[localizations]] of the ambient type system. \end{remark} \hypertarget{AssociatedIdemopotentMonad}{}\subsubsection*{{The associated idempotent monad of a monad}}\label{AssociatedIdemopotentMonad} We discuss here how under suitable conditions, for every [[monad]] $T$ there is a ``completion'' to an idempotent monad $\tilde T$ in that the completion construction is [[right adjoint]] to the inclusion of idempotent monads into all monads, exhibiting idempotent monads as a [[coreflective subcategory]]. Here $\tilde T$ inverts the same morphisms that $T$ does and hence exhibits the [[localization]]([[reflective subcategory]]) at the $T$-equivalences, and in fact the factorization of any [[adjunction]] inducing $T$ through that localization (\hyperlink{Fakir70}{Fakir 70}, \hyperlink{ApplegateTierney70}{Applegate-Tierney 70}, \hyperlink{Day74}{Day 74} \hyperlink{CasacubertaFrei99}{Casacuberta-Frei 99}. \href{Lucyshyn-Wright14}{Lucyshyn-Wright 14}). \begin{theorem} \label{ExistenceOfIdempotentCore}\hypertarget{ExistenceOfIdempotentCore}{} Let $C$ be a [[complete category|complete]], [[well-powered category]], and let $M\colon C \to C$ be a [[monad]] with [[unit of a monad|unit]] $u\colon 1 \to M$ and multiplication $m\colon M M \to M$. Then there is a universal idempotent monad, giving a [[right adjoint]] to the inclusion \begin{displaymath} IdempotentMonad(C) \hookrightarrow Monad(C) \end{displaymath} \end{theorem} \begin{proof} Given a [[monad]] $M$, define a [[functor]] $M'$ as the [[equalizer]] of $M u$ and $u M$: \begin{displaymath} M' \hookrightarrow M \stackrel{\overset{u M}{\longrightarrow}}{\underset{M u}{\longrightarrow}} M M. \end{displaymath} This $M'$ acquires a monad structure (see this \href{http://mathoverflow.net/questions/147264/regarding-a-difficulty-in-the-fakir-article-about-associated-idempotent-triple/147272#147272}{MathOverflow thread} for some detailed discussion). It might not be an idempotent monad (although it will be if $M$ is [[left exact functor|left exact]]). However we can apply the process again, and continue transfinitely. Define $M_0 = M$, and if $M_\alpha$ has been defined, put $M_{\alpha+1} = M_{\alpha}'$; at limit ordinals $\beta$, define $M_\beta$ to be the inverse limit of the chain \begin{displaymath} \ldots \hookrightarrow M_{\alpha} \hookrightarrow \ldots \hookrightarrow M \end{displaymath} where $\alpha$ ranges over ordinals less than $\beta$. This defines the monad $M_\alpha$ inductively; below, we let $u_\alpha$ denote the [[unit of a monad|unit]] of this monad. Since $C$ is [[well-powered category|well-powered]] (i.e., since each object has only a small number of [[subobjects]]), the large limit \begin{displaymath} E(M)(c) = \underset{\alpha \in Ord}{\lim} M_\alpha(c) \end{displaymath} exists for each $c$. Hence the large limit $E(M) = \underset{\alpha \in Ord}{\lim} M_\alpha$ exists as an endofunctor. The underlying functor \begin{displaymath} Monad(C) \to Endo(C) \end{displaymath} reflects limits (irrespective of size), so $E = E(M)$ acquires a monad structure defined by the limit. Let $\eta\colon 1 \to E$ be the unit and $\mu\colon E E \to E$ the multiplication of $E$. For each $\alpha$, there is a monad map $\pi_\alpha\colon E \to M_\alpha$ defined by the limit projection. \end{proof} \begin{lemma} \label{}\hypertarget{}{} $E$ is idempotent. \end{lemma} \begin{proof} For this it suffices to check that $\eta E = E \eta\colon E \to E E$. This may be checked objectwise. So fix an object $c$, and for that particular $c$, choose $\alpha$ so large that $\pi_\alpha (c)\colon E(c) \to M_\alpha(c)$ and $\pi_\alpha E(c)\colon E E(c) \to M_{\alpha} E(c)$ are isomorphisms. In particular, $\pi_\alpha \pi_\alpha(c)\colon E E (c) \to M_\alpha M_\alpha(c)$ is invertible. Now $u_\alpha M_\alpha(c) = M_{\alpha} u_{\alpha} c$, since $\pi_\alpha\colon E \to M_\alpha$ factors through the equalizer $M_{\alpha + 1} \hookrightarrow M_\alpha$. Because $\pi_\alpha$ is a monad morphism, we have \begin{displaymath} \itexarray{ \eta E(c) & = & (\pi_\alpha \pi_\alpha (c))^{-1} (u_\alpha M_\alpha(c))\pi_\alpha(c) \\ & = & (\pi_\alpha \pi_\alpha (c))^{-1} (M_\alpha u_\alpha(c))\pi_\alpha(c) \\ & = & E \eta(c) } \end{displaymath} as required. Finally we must check that $M \mapsto E(M)$ satisfies the appropriate universal property. Suppose $T$ is an idempotent monad with unit $v$, and let $\phi\colon T \to M$ be a monad map. We define $T \to M_\alpha$ by induction: given $\phi_\alpha\colon T \to M_\alpha$, we have \begin{displaymath} (u_\alpha M_\alpha)\phi_\alpha = \phi_\alpha \phi_\alpha (v T) = \phi_\alpha \phi_\alpha (T v) = (M_\alpha u_{\alpha})\phi_\alpha \end{displaymath} so that $\phi_{\alpha}$ factors uniquely through the inclusion $M_{\alpha + 1} \hookrightarrow M_\alpha$. This defines $\phi_{\alpha + 1}\colon T \to M_{\alpha + 1}$; this is a monad map. The definition of $\phi_\alpha$ at limit ordinals, where $M_\alpha$ is a limit monad, is clear. Hence $T \to M$ factors (uniquely) through the inclusion $E(M) \hookrightarrow M$, as was to be shown. \end{proof} \begin{theorem} \label{FactorizationOfIdempotentCore}\hypertarget{FactorizationOfIdempotentCore}{} For $(L \dashv R) \;\colon\; \mathcal{C} \stackrel{\overset{L}{\longrightarrow}}{\underset{R}{\longleftarrow}} \mathcal{D}$ a pair of [[adjoint functors]] with induced [[monad]] $T = R\circ L$ on the complete and well-powered category $\mathcal{C}$, then the idempotent monad $\tilde T$ of theorem \ref{ExistenceOfIdempotentCore} corresponds via remark \ref{ReflectiveSubcategoryOfIdempotentMonad} to a [[reflective subcategory]] inclusion $\mathcal{C}_T \stackrel{i}{\hookrightarrow} \mathcal{C}$ which factors the original adjunction \begin{displaymath} (L\dashv R) \;\colon\; \mathcal{C} \stackrel{\overset{}{\longrightarrow}}{\underset{i}{\longleftarrow}} \mathcal{C}_T \stackrel{\overset{L'}{\longrightarrow}}{\underset{}{\longleftarrow}} \mathcal{D} \end{displaymath} such that $L'$ is a [[conservative functor]]. \end{theorem} (\hyperlink{LucyshynWright14}{Lucyshyn-Wright 14, theorem 4.15}) \begin{remark} \label{IdempotentFactorizationAndBousfieldLocalization}\hypertarget{IdempotentFactorizationAndBousfieldLocalization}{} The factorization in theorem \ref{FactorizationOfIdempotentCore} has its analog in [[homotopy theory]] in the concept of [[Bousfield localization of model categories]]: given a [[Quillen adjunction]] \begin{displaymath} (L \dashv R) \;\colon\; \mathcal{C} \stackrel{\longrightarrow}{\longleftarrow} \mathcal{D} \end{displaymath} then (if it exists) the [[Bousfield localization of model categories|Bousfield localized]] [[model category]] structure $\mathcal{C}_W$ obtained from $\mathcal{C}$ by adding the $L$-weak equivalences factors this into two consecutive Quillen adjunctions of the form \begin{displaymath} \mathcal{C} \stackrel{\overset{id}{\longrightarrow}}{\underset{id}{\longleftarrow}} \mathcal{C}_{W} \stackrel{\overset{L}{\longrightarrow}}{\underset{R}{\longleftarrow}} \mathcal{D} \,. \end{displaymath} On the [[(∞,1)-categories]] [[presentable (∞,1)-category|presented]] by these model categories this gives a factorization of the [[derived functor|derived]] [[(∞,1)-adjunction]] through [[localization of an (∞,1)-category|localization]] onto a [[reflective sub-(∞,1)-category]] followed by a [[conservative (∞,1)-functor]]. \end{remark} \begin{example} \label{}\hypertarget{}{} Let $A$ be a commutative ring, and let $f\colon A \to B$ be a flat (commutative) $A$-algebra. Then the [[forgetful functor]] \begin{displaymath} f^\ast = Ab^f\colon Ab^B \to Ab^A \end{displaymath} from $B$-modules to $A$-modules has a [[left exact functor|left exact]] [[left adjoint]] $f_! = B \otimes_A -$. The induced monad $f^\ast f_!$ on the category of $B$-modules preserves [[equalizers]], and so its associated idempotent monad $T$ may be formed by taking the equalizer \begin{displaymath} T(M) \to B \otimes_A M \stackrel{\overset{f^\ast f_! \eta M}{\longrightarrow}}{\underset{\eta f^\ast f_! M}{\longrightarrow}} B \otimes_A B \otimes_A M \end{displaymath} \end{example} (To be continued. This example is based on how Joyal and Tierney introduce effective descent for commutative ring homomorphisms, in An Extension of the Galois Theory of Grothendieck. I would like to consult that before going further -- Todd.) [[Mike Shulman]]: How about some examples of monads and their associated idempotent monads? Do 2-monads have associated lax-, colax-, or pseudo-idempotent 2-monads? \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[reflective subcategory]] \item [[closure operator]], [[modality]] \item [[lax-idempotent 2-monad]] \item [[idempotent (∞,1)-monad]] \item The analog in [[model category]] theory of the localization at idempotent monad is the content of the [[Bousfield-Friedlander theorem]] (``[[Quillen idempotent monad]]''). \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} General discussion includes \begin{itemize}% \item [[Francis Borceux]], \emph{[[Handbook of Categorical Algebra]]}, vol.2, p. 196. \item [[Pierre Gabriel]] and [[Michel Zisman]], \emph{Calculus of Fractions and Homotopy Theory} \item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}, A.4.3.11, p.194, B1.1.9, p.249 \end{itemize} The idempotent monad which exhibits the [[localization]] at the $T$-equivalences for a given monad $T$ is discussed in \begin{itemize}% \item H. Applegate and [[Myles Tierney]], \emph{Iterated cotriples}, Lecture Notes in Math. 137 (1970), 56-99 \item S. Fakir, \emph{Monade idempotente associ\'e{}e \`a{} une monade}, C. R. Acad. Sci. Paris Ser. A-B 270 (1970), A99-A101. (\href{http://gallica.bnf.fr/ark:/12148/bpt6k480298g/f103.image}{gallica}) \item [[Brian Day]], \emph{On adjoint-functor factorisation}, Lecture Notes in Math. 420 (1974), 1-19. \item [[Carles Casacuberta]], Armin Frei, \emph{Localizations as idempotent approximations to completions}, Journal of Pure and Applied Algebra 142 (1999), 25-33 (\href{http://atlas.mat.ub.es/personals/casac/articles/cfre1.pdf}{pdf}) \end{itemize} and for [[enriched category theory]] in \begin{itemize}% \item [[Rory Lucyshyn-Wright]], \emph{Completion, closure, and density relative to a monad, with examples in functional analysis and sheaf theory} . (\href{http://arxiv.org/abs/1406.2361}{arXiv:1406.2361}) \end{itemize} Extension of idempotent monads along subcategory inclusions is discussed in \begin{itemize}% \item [[Carles Casacuberta]], Armin Frei, Tan Geok Choo, \emph{Extending localization functors} , Journal of Pure and Applied Algebra 103 (1995), 149-165. (\href{http://atlas.mat.ub.es/personals/casac/articles/cft.pdf}{pdf}) \item A. Deleanu, A. Frei, [[Peter Hilton|P. Hilton]], \emph{Idempotent triples and completion} , Math. Z. \textbf{143} (1975) pp.91-104. (\href{http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&&action=pdf&metsFile=PPN266833020_0143&divID=LOG_0014&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN266833020_0143%7C&targetFileName=PPN266833020_0143_LOG_0014.pdf&}{pdf}) \end{itemize} [[!redirects idempotent monad]] [[!redirects idempotent monads]] [[!redirects idempotent comonad]] [[!redirects idempotent comonads]] \end{document}