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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*{adjoint lifting 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{sketch_of_proof}{Sketch of proof}\dotfill \pageref*{sketch_of_proof} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{forgetful_functors_between_varieties_of_algebras}{Forgetful functors between varieties of algebras}\dotfill \pageref*{forgetful_functors_between_varieties_of_algebras} \linebreak \noindent\hyperlink{sufficient_conditions_for_cocompleteness_of_monadic_categories}{Sufficient conditions for cocompleteness of monadic categories}\dotfill \pageref*{sufficient_conditions_for_cocompleteness_of_monadic_categories} \linebreak \noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{statement}{}\subsection*{{Statement}}\label{statement} \begin{utheorem} \textbf{(The adjoint lifting theorem)}. Consider the following [[commutative diagram|commutative square]] of [[functor]]s: \begin{displaymath} \begin{array}{cccc}\mathcal{A} & \overset{Q}{\to} & \mathcal{B} \\ ^{U}\downarrow & & \downarrow^{V} \\ \mathcal{C} & \underset{R}{\to} & \mathcal{D} \end{array} \end{displaymath} and suppose that \begin{itemize}% \item $U$ and $V$ are [[monadic functor|monadic]], and \item $\mathcal{A}$ has [[coequalizer]]s of reflexive pairs. \end{itemize} Then, if $R$ has a [[left adjoint]], then $Q$ also has a [[left adjoint]]. \end{utheorem} A detailed proof may be found in Sec. 4.5 of Vol. 2 of \hyperlink{Borceux}{Borceux} (see especially Theorem 4.5.6 on p. 226 and Ex. 4.8.6 on p. 252). Also (\hyperlink{Johnstone}{Johnstone, prop. 1.1.3}) For a sketch of proof, see ahead. \begin{ucor} If the bottom functor $R$ of the above square is the identity arrow (so that $U=V\circ Q$), if $U$ and $V$ are monadic, and if $\mathcal{A}$ has coequalizers of reflexive pairs, then $Q$ is monadic. \end{ucor} \begin{proof} The adjoint lifting theorem implies the existence of a left adjoint, and the rest is a straightforward application of the [[monadicity theorem]]. \end{proof} \hypertarget{sketch_of_proof}{}\subsection*{{Sketch of proof}}\label{sketch_of_proof} We may assume the situation of the following diagram (with $V Q = R U$): \begin{displaymath} \begin{array}{cccc}\mathcal{C}^\mathbb{T} & \underoverset{Q}{K(?)}{\leftrightarrows} & \mathcal{D}^\mathbb{S} \\ ^{U}\downarrow \uparrow^{F} & & ^{G}\uparrow\downarrow^{V} \\ \mathcal{C} & \underoverset{R}{L}{\leftrightarrows} & \mathcal{D} \end{array} \end{displaymath} where $\mathbb{T}=\langle T,\varepsilon\colon 1_{\mathcal{C}}\Rightarrow T,\mu \rangle$ is a monad on $\mathcal{C}$, $\mathbb{S}=\langle S,\zeta\colon 1_{\mathcal{D}}\Rightarrow S,\eta\rangle$ is a monad on $\mathcal{D}$, $U$ and $V$ are the forgetful functors, and $F$ and $G$ are the free algebra functors. Let us write $\tau\colon F U\Rightarrow 1_{\mathcal{C}^{\mathbb{T}}}$ for the counit of the adjunction $F\dashv U$ and $\sigma\colon G V\Rightarrow 1_{\mathcal{D}^{\mathbb{S}}}$ for the counit of the adjunction $G\dashv V$. As usual, we have $T = U F$, $S = V G$, $\mu=U\tau F$, and $\eta=V\sigma G$. Finally, let $L$ be a left adjoint to $R$ (which exists by assumption), and let $\alpha\colon 1_{\mathcal{D}}\Rightarrow R L$ and $\beta\colon LR\Rightarrow 1_{\mathcal{C}}$ be the unit and counit (respectively) of the adjunction $L\dashv R$. We would like to construct a functor $K\colon \mathcal{D}^{\mathbb{S}}\to \mathcal{C}^{\mathbb{T}}$. To get a hint of what $K$ should look like, let us assume for a moment that such a $K$ already exists. In this case, we have $K G\dashv V Q(=R U)$. But $F L$ is also left adjoint to $R U$, and by the uniqueness of a left adjoint we must have $K G = F L$ (at least up to a natural isomorphism). From this, we already know how to define $K$ on free algebras. Also, being a left adjoint, $K$ preserves in particular all coequalizers. But every $\mathbb{S}$-algebra $\langle D,\xi\colon SD\to D\rangle$ is the (object part) of a reflexive coequalizer, namely, the [[canonical presentation]] \begin{displaymath} G S(D)\underoverset{G\xi}{\sigma_{G D}}{\rightrightarrows}G(D)\overset{\sigma_{\langle D,\xi\rangle}=\xi}{\rightarrow}\langle D,\xi \rangle \end{displaymath} Applying $K$ and using $K G=F L$, we see that $K\langle D,\xi\rangle$ should be the object part of a reflexive coequalizer in $\mathcal{C}^{\mathbb{T}}$ of the form \begin{displaymath} F L S(D)\underoverset{F L\xi}{?}{\rightrightarrows}F L(D)\overset{x}{\rightarrow}K\langle D,\xi \rangle \end{displaymath} (recall that we assume that $\mathcal{C}^{\mathbb{T}}$ has coequalizers of reflexive pairs). To eventually define $K\langle D,\xi \rangle$ as a coequalizer (as above), we first need some reasonable guess for the ∞-arrow. For this, we will need a lemma. \begin{ulemma} There exists a natural transformation $\lambda\colon S R \Rightarrow R T$ for which the following diagram of functors and natural transformations is commutative: \begin{displaymath} \begin{array}{ccccccc} R & \overset{\zeta R}{\to} & S R & & \overset{\eta R}{\leftarrow}& & S S R \\ & ^{R\varepsilon}\searrow & ^{\lambda}\downarrow & & & & ^{S\lambda}\downarrow\\ & & R T & \overset{R\mu}{\leftarrow} & R T T & \overset{\lambda T}{\leftarrow} & S R T \end{array} \end{displaymath} \end{ulemma} \begin{proof} Define $\lambda := V\sigma Q F \circ V G R\varepsilon$, so that \begin{displaymath} \lambda \colon S R = V G R\overset{V G R\varepsilon}{\to}V G R T = V G R U F = V G V Q F\overset{V\sigma Q F}{\to} V Q F = R U F = R T. \end{displaymath} The required commutativity may be verified by using the commutative diagrams in the definitions of a monad and an EM-algebra, naturality, and the triangular identities. For details, see the proof of Lemma 4.5.1 of \hyperlink{Borceux}{Borceux}, pp. 222-223. (Note that this lemma does not depend neither on the existence of a left adjoint for the bottom horizontal arrow, nor on the existence of coequalizers. Only the commutativity is required.) \end{proof} We may now return to our task of defining the ∞-arrow in the diagram preceding the lemma. We would like to get from $F L S(D)$ to $F L(D)$, and for this, we will construct a natural transformation $F L S\Rightarrow F L$ in the following way. First, we have \begin{displaymath} S\overset{S\alpha}{\to} S R L \overset{\lambda L}{\to} R T L. \end{displaymath} Applying $L$ and composing with $\beta T L$, we get \begin{displaymath} LS\overset{L\lambda L\circ LS\alpha}{\longrightarrow}L R T L \overset{\beta TL}{\to} TL. \end{displaymath} Applying $F$ and composing with $\tau F L$, we finally get \begin{displaymath} F L S\overset{F\beta T L\circ F L\lambda L\circ F L S\alpha}{\longrightarrow}F T L = F U F L\overset{\tau F L}{\to} F L \end{displaymath} Let us call the resulting natural transformation $\omega$, that is, \begin{displaymath} \omega:=\tau F L \circ F\beta T L\circ F L\lambda L\circ F L S\alpha. \end{displaymath} Now we take the sought for ∞-arrow to be $\omega_D$, and \emph{define} $K\langle D,\xi\rangle$ as the object of some fixed coequalizer of $\omega_D$ and $F L\xi$: \begin{displaymath} F L S(D)\underoverset{F L\xi}{\omega_D}{\rightrightarrows}F L(D)\overset{x}{\rightarrow}K\langle D,\xi \rangle. \end{displaymath} In order to do this, we must first verify that the parallel arrows above have a common section (since we only assume that $\mathcal{C}^{\mathbb{T}}$ has coequalizers of reflexive pairs). To find a guess for a common section, note that the common section for the parallel pair in the above canonical presentation in $\mathcal{D}^{\mathbb{S}}$ is $G\zeta_{V\langle D,\xi\rangle}=G\zeta_D$, and if $K$ exists, then applying $K$ gives $K G\zeta_D=F L\zeta_D$. Having this guess, it is now straightforward to verify that $F L\zeta_D$ is indeed a common section, as required. So, we have defined an object function of a would be left adjoint $K$. To make it into a functor left adjoint to $Q$, we will build a universal arrow from $\langle D,\xi\rangle$ to $Q$, whose object part is $K\langle D,\xi\rangle$ (Theorem IV.1.2(ii) of [[Categories Work]]). To get an arrow $\langle D,\xi\rangle\to Q K\langle D,\xi\rangle$ , suppose for a moment that we have a natural transformation $\varphi\colon G\Rightarrow Q F L$ such that $\varphi\circ \sigma G = Q\omega\circ \varphi S$. Then the left square in the following diagram commutes: \begin{displaymath} \begin{array}{ccccc}G S(D)& \underoverset{G\xi}{\sigma_{G D}}{\rightrightarrows} & G(D) &\overset{\sigma_{\langle D,\xi\rangle}=\xi}{\rightarrow}&\langle D,\xi \rangle \\ ^{\varphi_{S D}}\downarrow && ^{\varphi_D}\downarrow && ^{\chi}\downarrow\\ Q F L S D &\underoverset{Q F L\xi}{Q\omega_D}{\rightrightarrows} & Q F L D &\overset{Q x}{\rightarrow}&Q K\langle D,\xi\rangle \end{array} \end{displaymath} Since both rows are forks, it follows that $Q x\circ \varphi_D$ has the same composition with the arrows of the upper\newline parallel pair, and hence there exists a unique arrow $\chi\colon \langle D,\xi\rangle\to Q K\langle D,\xi\rangle$ making the right square commutative (recall that the upper row is a coequalizer). It is now possible to prove that the pair $\langle K\langle D,\xi\rangle,\chi \rangle$ is a universal arrow from $\langle D,\xi\rangle$ to $Q$, showing that $K$ is indeed (the object function) of a left adjoint (for details, see the proof of Theorem 4.5.6, pp. 226-227 in \hyperlink{Borceux}{Borceux}). But we still have to prove the existence of a natural transformation $\varphi\colon G\Rightarrow Q F L$ such that $\varphi\circ \sigma G = Q\omega\circ \varphi S$. For this, we define $\varphi:=\sigma Q F L \circ G R\varepsilon L \circ G\alpha$. Since $V$ is faithful, to prove the required property of $\varphi$, it is enough to prove that $V\varphi \circ V\sigma G = V Q\omega\circ V\varphi S$, and this is a long, yet straightforward, computation (noting that $V\varphi = \lambda L \circ VG\alpha$ and using the commutative diagram from Lemma 1; see Lemma 4.5.3, p. 224 of \hyperlink{Borceux}{Borceux}). \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{forgetful_functors_between_varieties_of_algebras}{}\subsubsection*{{Forgetful functors between varieties of algebras}}\label{forgetful_functors_between_varieties_of_algebras} Since varieties of algebras are [[cocompleteness of varieties of algebras|cocomplete]] and monadic over $\mathbf{Set}$, the corollary implies that forgetful functors between varieties of algebras (e.g., the forgetful functor $\mathbf{Rng}\to\mathbf{Ab}$) are monadic. \hypertarget{sufficient_conditions_for_cocompleteness_of_monadic_categories}{}\subsubsection*{{Sufficient conditions for cocompleteness of monadic categories}}\label{sufficient_conditions_for_cocompleteness_of_monadic_categories} Let $\mathcal{J}$ be an arbitrary category, and consider the commutative diagram \begin{displaymath} \begin{array}{ccccc}\mathcal{A} & \overset{\Delta}{\to} & \mathcal{A}^\mathcal{J} \\ ^{U}\downarrow & & \downarrow^{U^{\mathcal{J}}} \\ \mathcal{C} & \underset{\Delta}{\to} & \mathcal{C}^\mathcal{J} \end{array} \end{displaymath} where $U$ is monadic, $\Delta$ is the [[diagonal functor]] and $U^{\mathcal{J}}=U\circ -$. If $F$ is left adjoint \newline to $U$, then $F^\mathcal{J}$ is left adjoint to $U^{\mathcal{J}}$ (using the unit and counit of the original adjunction, one can construct appropriate natural transformations that satisfy the triangular identities, see, e.g., p. 119 of [[Categories Work]]). Also, the conditions of the monadicity theorem for $U$ imply those for $U^\mathcal{J}$ (basically because the definition of a split fork involves only compositions and identities, and because natural transformations are composed componentwise). Now, if $\mathcal{C}$ is $\mathcal{J}$-cocomplete (so that the bottom horizontal functor has a left adjoint) and $\mathcal{A}$ has coequalizers of reflexive pairs, then the adjoint lifting theorem implies that $\mathcal{A}$ is $\mathcal{J}$-cocomplete. In particular, if $\mathcal{A}$ has coequalizers of reflexive pairs and $\mathcal{C}$ is small-cocomplete, then $\mathcal{A}$ is small cocomplete. \hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages} \begin{itemize}% \item The adjoint lifting theorem is a corollary of the [[adjoint triangle theorem]]. \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Michael Barr]], [[Charles Wells]], \emph{Toposes, Triples and Theories} , Springer Heidelberg 1985. (Reprinted as \href{http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html}{TAC reprint no.12} (2005); section 3.7, pp.131ff) \item [[Francis Borceux]], \emph{[[Handbook of Categorical Algebra]] II} , Cambridge UP 1994. (section 4.5, pp.221ff) \end{itemize} \begin{itemize}% \item [[Peter Johnstone]], \emph{Adjoint lifting theorems for categories of algebras} , Bull. London Math. Soc. \textbf{7} (1975) pp.294-297. \item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]] I} , Oxford UP 2002. (section A1.1, p.5) \end{itemize} The dual theorem for comonads is also in \begin{itemize}% \item William F. Keigher, \emph{Adjunctions and comonads in differential algebra}, Pacific J. Math. 59, n. 1 (1975) 99-112 \href{http://projecteuclid.org/euclid.pjm/1102905501}{euclid} \item [[John Power]], \emph{A unified approach to the lifting of adjoints} , Cah. Top. G\'e{}om. Diff. Cat. \textbf{XXIX} no.1 (1988) pp.67-77. (\href{http://www.numdam.org/item/CTGDC_1988__29_1_67_0}{numdam}) \end{itemize} \end{document}