\section{Introduction}

Structured ring spectra are spectra with extra algebraic structure encoded by the action of an operad ${ \mathcal{O} }$ in the closed symmetric monoidal category of modules $({ \mathsf{Mod} }_{{ \mathcal{R} }},{ \,\wedge\, },{ \mathcal{R} })$ over a commutative ring spectrum ${ \mathcal{R} }$.  For a fixed operad ${ \mathcal{O} }$, denote by ${ \Alg_{ \mathcal{O} } }$ the category of ${ \mathcal{O} }$-algebras. In this paper we are working with reduced operads ${ \mathcal{O} }$ (i.e., such that ${ \mathcal{O} }[0]=*$, where $*$ denotes the trivial ${ \mathcal{R} }$-module); such ${ \mathcal{O} }$-algebras are non-unital. Similar to the ordinary homology of spaces, a precisely analogous notion of homology for ${ \mathcal{O} }$-algebras is topological Quillen (${ \mathsf{TQ} }$) homology \cite{Basterra, Basterra_Mandell_thh, Ching_Harper, Goerss_Hopkins,  Harper_bar_constructions, Kuhn_Pereira, Kriz_towers}; useful introductions to notions of homology for algebraic structures appear, for instance, in \cite{Goerss_f2_algebras, Goerss_Schemmerhorn, Miller, Quillen, Quillen_rings}. It turns out that ${ \mathsf{TQ} }$-homology is weakly equivalent to stabilization $\Omega^\infty\Sigma^\infty$ in ${ \mathcal{O} }$-algebras \cite{Basterra_Mandell, Harper_Hess, Pereira_spectral_operad, Schwede_cotangent, Schwede_algebraic_theories} under appropriate connectivity conditions---this is because ${ \mathsf{TQ} }$ is 1-excisive and agrees to order 1 with the identity functor on ${ \mathcal{O} }$-algebras (via the ${ \mathsf{TQ} }$-Hurewicz map ${ \mathrm{id} }\rightarrow{ \mathsf{TQ} }$) in the sense of Goodwillie \cite[1.2]{Goodwillie_calculus_3}; see, for instance,  \cite{Ching_Harper, Harper_Hess, Kuhn}.

The ${ \mathsf{TQ} }$-local homotopy theory for ${ \mathcal{O} }$-algebras is established in \cite{Harper_Zhang}, where the upshot is that: if $X$ is a cofibrant ${ \mathcal{O} }$-algebra, then its weak ${ \mathsf{TQ} }$-fibrant replacement $X\rightarrow L_{ \mathsf{TQ} }(X)$ is the ${ \mathsf{TQ} }$-localization of $X$. By construction, the comparison map $X\rightarrow L_{ \mathsf{TQ} }(X)$ is a cofibration that is also a ${ \mathsf{TQ} }$-equivalence such that $L_{ \mathsf{TQ} }(X)$ is ${ \mathsf{TQ} }$-local (Defintion \ref{defn:TQ_local_O_algebras}). Intuitively, the ${ \mathsf{TQ} }$-localization $L_{ \mathsf{TQ} }(X)$ can be thought of as ``the part of $X$ that ${ \mathsf{TQ} }$-homology sees''.

In this short paper we attack the following question:
When is the comparison map $X \rightarrow L_{ \mathsf{TQ} }(X)$ a weak equivalence?  In other words, when is an ${ \mathcal{O} }$-algebra $X$ already ${ \mathsf{TQ} }$-local? For instance, we know from \cite{Ching_Harper_derived_Koszul_duality} that every connected ${ \mathcal{O} }$-algebra is ${ \mathsf{TQ} }$-complete and hence $X{ \ \simeq \ } L_{ \mathsf{TQ} }(X)$, but we also know from \cite{Harper_Hess} that every connected ${ \mathcal{O} }$-algebra is the homotopy limit of a tower of nilpotent ${ \mathcal{O} }$-algebras and hence $X$ is homotopy pro-nilpotent (Definition \ref{defn:nilpotent_and_homotopy_pro_nilpotent}); here, ${ \mathcal{R} },{ \mathcal{O} }$ were assumed to be $(-1)$-connected.

This leads us to one of the motivations of our work: what amounts to the ``first half'' of a conjecture of Francis-Gaitsgory \cite[3.4.5]{Francis_Gaitsgory} that (i) the natural map comparing $X$ with its ${ \mathsf{TQ} }$-completion should be a weak equivalence for every homotopy pro-nilpotent ${ \mathcal{O} }$-algebra $X$. Our main result, Theorem \ref{thm:main_result}, is that (i) is true in general, provided that in the comparison map we replace ``${ \mathsf{TQ} }$-completion'' with ``${ \mathsf{TQ} }$-localization''. Our strategy of attack is to leverage the ${ \mathsf{TQ} }$-local homotopy theory of ${ \mathcal{O} }$-algebras in \cite{Harper_Zhang} with the fact, proved in \cite{Ching_Harper_nilpotent_Whitehead}, that $M$-nilpotent ${ \mathcal{O} }$-algebras are ${ \mathsf{TQ} }|_{\mathrm{Nil}_M}$-complete.

\begin{defn}
\label{defn:nilpotent_of_order_M}
Let $X$ be an ${ \mathcal{O} }$-algebra and $M \geq 2$.  We say that $X$ is \emph{$M$-nilpotent} if all the $M$-ary and higher operations $ { \mathcal{O} }[t] \wedge X^{\wedge t} \rightarrow X$ of $X$ are trivial (i.e., if these maps factor through the trivial ${ \mathcal{R} }$-module ${*}$ for each $t\geq M$).
\end{defn}

\begin{defn}
\label{defn:nilpotent_and_homotopy_pro_nilpotent}
An ${ \mathcal{O} }$-algebra is \emph{nilpotent} (resp. \emph{homotopy pro-nilpotent}) if it is $M$-nilpotent for some $M\geq 2$ (resp. if it is weakly equivalent to the homotopy limit of a tower of nilpotent ${ \mathcal{O} }$-algebras).
\end{defn}

It is worth pointing out that homotopy pro-nilpotent ${ \mathcal{O} }$-algebras need not be nilpotent; the following describes a large class of such ${ \mathcal{O} }$-algebras.

\begin{prop}
\label{prop:connected_O_algebras_are_homotopy_pro_nilpotent}
If $X$ is a connected ${ \mathcal{O} }$-algebra and ${ \mathcal{O} },{ \mathcal{R} }$ are $(-1)$-connected, then $X$ is homotopy pro-nilpotent.
\end{prop}

\begin{proof}
This is proved in Harper-Hess \cite[1.12]{Harper_Hess} by showing that the homotopy completion tower of $X$ converges strongly to $X$.
\end{proof}

\begin{prop}[${ \mathsf{TQ} }$-local Whitehead theorem]
\label{prop:TQ_local_whitehead_theorem}
A map $X\rightarrow Y$ between ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras is a weak equivalence if and only if it is a ${ \mathsf{TQ} }$-homology equivalence.
\end{prop}

\begin{proof}
This follows from the definition of ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras (Definition \ref{defn:TQ_local_O_algebras}); see, for instance, Hirschhorn \cite[3.2.13]{Hirschhorn}.
\end{proof}

\begin{prop}[Preservation of the ${ \mathsf{TQ} }$-local property: Homotopy limits]
\label{prop:holims_of_TQ_local_objects_are_TQ_local}
The homotopy limit of a small diagram of ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras is ${ \mathsf{TQ} }$-local.
\end{prop}

\begin{proof}
This follows from the definition of ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras; see, for instance, Dror Farjoun \cite[1.A.8, 1.G]{Dror_Farjoun_LNM} and Hirschhorn \cite[19.4.4]{Hirschhorn}. Here is another, essentially equivalent, proof: It follows from Proposition \ref{prop:TQ_local_whitehead_theorem} that the homotopy limit in ${ \Alg_{ \mathcal{O} } }$ of a small diagram of ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras is weakly equivalent to its homotopy limit calculated in the ${ \mathsf{TQ} }$-local homotopy theory \cite[5.14]{Harper_Zhang}; hence, verifying that the homotopy limit in ${ \Alg_{ \mathcal{O} } }$ is ${ \mathsf{TQ} }$-local reduces to the usual fibrancy property of homotopy limits in a homotopy theory (in this case, in the ${ \mathsf{TQ} }$-local homotopy theory); see, for instance, Hirschhorn \cite[18.5.2]{Hirschhorn}, together with Ching-Harper \cite[8.9]{Ching_Harper_derived_Koszul_duality} for a discussion of homotopy limits in the context of ${ \mathcal{O} }$-algebras.
\end{proof}

For instance, consider any pullback diagram of the form
\begin{align*}
\xymatrix{
A \ar[r] \ar[d] & B \ar[d]^p \\
C \ar[r] & D
}
\end{align*}
in ${ \Alg_{ \mathcal{O} } }$. It follows from Proposition \ref{prop:holims_of_TQ_local_objects_are_TQ_local} that if $B,C,D$ are ${ \mathsf{TQ} }$-local and $p$ is a fibration, then $A$ is ${ \mathsf{TQ} }$-local. Taking $C=*$, for instance, shows that ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras play nicely with fibration sequences; this is not expected to be true, in general, if we replace ``${ \mathsf{TQ} }$-local'' with ``${ \mathsf{TQ} }$-complete'' and is one of the reasons why ${ \mathsf{TQ} }$-localization is often better behaved than ${ \mathsf{TQ} }$-completion (at the expense of a much larger construction).

\begin{prop}
\label{prop:completion_results_part1}
If $X$ is an $M$-nilpotent ${ \mathcal{O} }$-algebra (resp. $Z$ is an ${ \mathcal{O} }$-algebra) for some $M\geq 2$, then its ${ \mathsf{TQ} }|_{\mathrm{Nil}_M}$-completion $X^\wedge_{{ \mathsf{TQ} }|_{\mathrm{Nil}_M}}$ (resp. ${ \mathsf{TQ} }$-completion $Z^\wedge_{ \mathsf{TQ} }$) is ${ \mathsf{TQ} }$-local.
\end{prop}

\begin{proof}
By Proposition \ref{prop:holims_of_TQ_local_objects_are_TQ_local}, it suffices to verify that $X^\wedge_{{ \mathsf{TQ} }|_{\mathrm{Nil}_M}}$ (resp. $Z^\wedge_{ \mathsf{TQ} }$) is the homotopy limit of a small diagram of ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras---we defer the proof of this to Section \ref{sec:TQ_local_O_algebras} (see Propositions  \ref{prop:weak_TQ_fibrancy_conditions_for_TQ_Nil_M_resolution} and \ref{prop:TQ_completion_resolution_with_TQ_local_values}, respectively).
\end{proof}

\begin{prop}
\label{prop:completion_results_part2}
Let $M\geq 2$.
\begin{itemize}
\item[(a)] If $X$ is an $M$-nilpotent ${ \mathcal{O} }$-algebra, then the natural map $X{ \ \simeq \ } X^\wedge_{{ \mathsf{TQ} }|_{\mathrm{Nil}_M}}$ is a weak equivalence.
\item[(b)] If $Z$ is a connected ${ \mathcal{O} }$-algebra and ${ \mathcal{O} },{ \mathcal{R} }$ are $(-1)$-connected, then the natural map $Z{ \ \simeq \ } Z^\wedge_{ \mathsf{TQ} }$ is a weak equivalence.

\end{itemize}
\end{prop}

\begin{proof}
Part (a) is proved in Ching-Harper \cite[2.12]{Ching_Harper_nilpotent_Whitehead} and part (b) is proved in Ching-Harper \cite[1.2]{Ching_Harper_derived_Koszul_duality}.
\end{proof}

The following theorem is the main result of this paper. Our proof is embarrassingly simple---one of the significant advantages of having available the ${ \mathsf{TQ} }$-local homotopy theory framework to work within and organize our arguments in---it follows immediately from the above propositions, together with verifying that ${ \mathsf{TQ} }|_{\mathrm{Nil}_M}$-resolutions have ${ \mathsf{TQ} }$-local fibrant replacements in ${ \Alg_{ \mathcal{O} } }$; we defer the proof of this to Section \ref{sec:TQ_local_O_algebras} (see Proposition  \ref{prop:weak_TQ_fibrancy_conditions_for_TQ_Nil_M_resolution}).

\begin{thm}[Homotopy pro-nilpotent ${ \mathsf{TQ} }$-localization theorem]
\label{thm:main_result}
Let $X$ be a fibrant ${ \mathcal{O} }$-algebra.
\begin{itemize}
\item[(a)] If $X$ is nilpotent, then $X$ is ${ \mathsf{TQ} }$-local.
\item[(b)] If $X$ is homotopy pro-nilpotent, then $X$ is ${ \mathsf{TQ} }$-local.
\item[(c)] If $X$ is the homotopy limit of any small diagram of nilpotent ${ \mathcal{O} }$-algebras, then $X$ is ${ \mathsf{TQ} }$-local.
\item[(d)] If $X$ is connected and ${ \mathcal{O} },{ \mathcal{R} }$ are $(-1)$-connected, then $X$ is ${ \mathsf{TQ} }$-local.
\end{itemize}
\end{thm}

\begin{rem}
It is worth pointing out (Proposition \ref{prop: fibrant replacements both local}) that if some fibrant replacement of an ${ \mathcal{O} }$-algebra $X$ is ${ \mathsf{TQ} }$-local, then every fibrant replacment of $X$ is ${ \mathsf{TQ} }$-local.
\end{rem}

\begin{proof}[Proof of Theorem \ref{thm:main_result}]
Part (a) follows from Propositions \ref{prop:completion_results_part1} and \ref{prop:completion_results_part2}. Parts (b) and (c) follow from part (a), together with Proposition \ref{prop:holims_of_TQ_local_objects_are_TQ_local}. Part (d) follows from part (c), together with Proposition \ref{prop:connected_O_algebras_are_homotopy_pro_nilpotent}; alternately, it follows from Propositions \ref{prop:completion_results_part1} and \ref{prop:completion_results_part2}.
\end{proof}

As an application of the main result, we obtain the following homotopy pro-nilpotent ${ \mathsf{TQ} }$-Whitehead theorem that simultaneously extends the previously known connected and nilpotent ${ \mathsf{TQ} }$-Whitehead theorems.

\begin{thm}[Homotopy pro-nilpotent ${ \mathsf{TQ} }$-Whitehead theorem]
\label{thm: homotopy pro nilpotent whitehead}
A map $X\rightarrow Y$ between homotopy pro-nilpotent ${ \mathcal{O} }$-algebras is a weak equivalence if and only if it is a ${ \mathsf{TQ} }$-homology equivalence; more generally, this remains true if $X,Y$ are homotopy limits of small diagrams of nilpotent ${ \mathcal{O} }$-algebras.
\end{thm}

\begin{proof}
This follows from Theorem \ref{thm:main_result}, together with Proposition \ref{prop:TQ_local_whitehead_theorem}.
\end{proof}

We also compare ${ \mathsf{TQ} }$-localization with ${ \mathsf{TQ} }$-completion and show that ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras that are ${ \mathsf{TQ} }$-good are ${ \mathsf{TQ} }$-complete (Theorem \ref{thm:recognizing_TQ_complete_from_TQ_local}). Finally we show that ${ \mathcal{O} }$-algebras with a principally refined Postnikov tower are ${ \mathsf{TQ} }$-local, provided that mild connectivity assumptions are satisfied; here is the theorem, whose proof we defer to Section \ref{sec:postnikov_towers}.

\begin{thm}
\label{thm:principally_refined_are_TQ_local}
Let $X$ be a fibrant ${ \mathcal{O} }$-algebra. If $X,{ \mathcal{O} },{ \mathcal{R} }$ are $(-1)$-connected and $X$ has a principally refined Postnikov tower, then $X$ is ${ \mathsf{TQ} }$-local.
\end{thm}

\subsection{Conventions and notations}

We work in the category ${ \Alg_{ \mathcal{O} } }$ of algebras over an operad ${ \mathcal{O} }$ in ${ \mathsf{Mod} }_{{ \mathcal{R} }}$, the category of ${ \mathcal{R} }$-modules, where ${ \mathcal{R} }$ is a commutative monoid in the category of symmetric spectra \cite{Hovey_Shipley_Smith, Schwede_book_project}.
Throughout this paper, we assume that:  (i) ${ \mathcal{O} }$ is reduced, meaning that ${ \mathcal{O} }[0]=*$ (i.e., ${ \mathcal{O} }$-algebras are non-unital) and (ii) the natural maps ${ \mathcal{R} }\rightarrow{ \mathcal{O} }[1]$ and $*\rightarrow { \mathcal{O} }[n]$ are flat stable cofibrations \cite{Harper_Hess} in ${ \mathcal{R} }$-modules for each $n \geq 0$; see, for instance, \cite[2.1, 6.12]{Ching_Harper_derived_Koszul_duality}.
Unless otherwise specified, we work with the positive flat stable model structure on ${ \Alg_{ \mathcal{O} } }$ \cite{Harper_Hess, Schwede_book_project}. To keep this paper appropriately concise, we freely use notation from \cite{Ching_Harper_derived_Koszul_duality, Harper_Hess, Harper_Zhang}. Finally, ``connected'' is short for ``0-connected''.

\subsection*{Acknowledgments}
The author would like to thank John E. Harper and Niko Schonsheck for enlightening discussions and helpful suggestions, and Michael Ching and Mark W. Johnson for helpful conversations. The author is grateful to Oscar Randal-Williams for detailed and helpful critical comments on an early draft of this paper. The author was supported in part by the Simons Foundation: Collaboration Grants for Mathematicians \#638247.

\section{${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras and ${ \mathsf{TQ} }|_{\mathrm{Nil}_M}$-resolutions}
\label{sec:TQ_local_O_algebras}

The purpose of this section is to recall the definition of ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras, discuss their basic properties, and finally to verify that  ${ \mathsf{TQ} }$-resolutions of ${ \mathcal{O} }$-algebras and ${ \mathsf{TQ} }|_{\mathrm{Nil}_M}$-resolutions of $M$-nilpotent ${ \mathcal{O} }$-algebras have ${ \mathsf{TQ} }$-local fibrant replacements in ${ \Alg_{ \mathcal{O} } }$ (Propositions \ref{prop:TQ_completion_resolution_with_TQ_local_values} and   \ref{prop:weak_TQ_fibrancy_conditions_for_TQ_Nil_M_resolution}).

${ \mathsf{TQ} }$-homology and its relative form, topological Andre-Quillen ($\mathsf{TAQ}$) homology, first introduced in \cite{Basterra} for commutative ring spectra (see also  \cite{ Baker_Richter,Basterra_Mandell,Basterra_Mandell_thh,Goerss_Hopkins,Kriz_towers,Lazarev,Mandell_topological_andre_quillen,Schwede_cotangent}),  are defined as derived indecomposables of ${ \mathcal{O} }$-algebras analogous to Quillen homology of commutative algebras \cite{Andre,Quillen_rings}; see also \cite{Galatius_Kupers_Randal-Williams_cellular,Goerss_Schemmerhorn,Goerss_f2_algebras, Miller}.
More precisely, factoring the canonical truncation map ${ \mathcal{O} } \rightarrow \tau_1{ \mathcal{O} }$ (see \cite{Harper_Hess})  in the category of
operads as ${ \mathcal{O} } \rightarrow J \rightarrow \tau_1{ \mathcal{O} }$, a cofibration followed by a weak equivalence, we get the
corresponding change of operads adjunction
\begin{align}
\label{equation: QU adjunction}
\xymatrix{
{ \Alg_{ \mathcal{O} } } \ar@<.5ex>[r]^Q & { \mathsf{Alg} }_{J} \ar@<.5ex>[l]^U
}
\end{align}
with left adjoint on top, where $Q(X)=J\circ_{ \mathcal{O} }(X)$ and $U$ denotes the forgetful functor (or less concisely, the ``forget along the map ${ \mathcal{O} }\rightarrow J$ functor'').

\begin{defn}\label{def: tq homology}
Let $X$ be an ${ \mathcal{O} }$-algebra. The \emph{topological Quillen homology} (or ${ \mathsf{TQ} }$-homology, for short) of $X$ is
$${ \mathsf{TQ} }(X):={ \mathsf{R} } U({ \mathsf{L} } Q(X))$$
the ${ \mathcal{O} }$-algebra defined via the indicated composite of total right and left derived functors.  If $X$ is cofibrant, then ${ \mathsf{TQ} }(X) \simeq UQ(X)$ and the unit of the $(Q,U)$ adjunction in \eqref{equation: QU adjunction} is the ${ \mathsf{TQ} }$-Hurewicz map $X \rightarrow{UQX}$ of the form $X \rightarrow{ \mathsf{TQ} }(X)$.
\end{defn}

${ \mathsf{TQ} }$-homology has been shown to enjoy several properties analogous to the ordinary homology of spaces; see, for instance, \cite{Basterra, Basterra_Mandell, Ching_Harper, Ching_Harper_derived_Koszul_duality, Harper_Hess, Kuhn}. Furthermore, it turns out that ${ \mathsf{TQ} }$-homology is weakly equivalent to stabilization in the category of ${ \mathcal{O} }$-algebras, provided that ${ \mathcal{O} },{ \mathcal{R} }$ are $(-1)$-connected; see, for instance,  \cite{Basterra_Mandell, Kuhn, Pereira_spectral_operad, Schwede_cotangent, Schwede_algebraic_theories}; a simple proof using Goodwillie's functor calculus \cite{Goodwillie_calculus_3} is given in \cite[1.14]{Harper_Hess}.

Localization has been proved to be a useful method in algebraic topology.  The idea is to focus on the information one cares about and ignore everything else, hence simplifying problems accordingly.   See \cite{Artin_Mazur_etale, Bousfield_localization_spaces, Bousfield_Kan, Dwyer_localizations,  Hilton_Mislin_Roitberg, Sullivan_mit_notes} for a discussion of localization methods.   Also see, for useful developments and ideas, \cite{Baues_combinatorial, Dror_Farjoun_LNM, Dwyer_localizations, Hirschhorn, Jardine_local_homotopy_theory, May_Ponto}.

In this paper we consider localization of ${ \mathcal{O} }$-algebras with respect to ${ \mathsf{TQ} }$-homology. Recall the following from \cite{Harper_Zhang}.

\begin{defn}
\label{defn:TQ_local_O_algebras}
Let $\function{f}{A}{B}$ be a map in ${ \Alg_{ \mathcal{O} } }$. We say that $f$ is a \emph{${ \mathsf{TQ} }$-equivalence} if $f$ induces a weak equivalence ${ \mathsf{TQ} }(A){ \ \simeq \ }{ \mathsf{TQ} }(B)$ on ${ \mathsf{TQ} }$-homology. We say that $f$ is a \emph{${ \mathsf{TQ} }$-acyclic strong cofibration} if $f$ is a cofibration between cofibrant objects   which is also a ${ \mathsf{TQ} }$-equivalence. An ${ \mathcal{O} }$-algebra $X$ is called \emph{${ \mathsf{TQ} }$-local} if (i) $X$ is fibrant in ${ \Alg_{ \mathcal{O} } }$ and (ii) every ${ \mathsf{TQ} }$-acyclic strong cofibration $A\rightarrow B$ induces a weak equivalence
\begin{align*}
\Hombold(A,X)\xleftarrow{{ \ \simeq \ }}\Hombold(B,X)
\end{align*}
on mapping spaces in ${ \mathsf{sSet} }$; here we are using the simplicial model structure on ${ \Alg_{ \mathcal{O} } }$ (see, for instance, \cite{Ching_Harper_derived_Koszul_duality, EKMM,  Goerss_Hopkins_moduli_problems, Goerss_Jardine, Harper_Hess}).
\end{defn}

It is useful to note that the collection of ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras is exactly the collection of  weak ${ \mathsf{TQ} }$-fibrant objects in the ${ \mathsf{TQ} }$-local homotopy theory \cite{Harper_Zhang}. Intuitively, a ${ \mathsf{TQ} }$-local object in ${ \Alg_{ \mathcal{O} } }$ is an ${ \mathcal{O} }$-algebra $X$ which only sees ${ \mathsf{TQ} }$-homology information.  The following proposition will be useful for detecting ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras.

\begin{prop}
\label{prop: local by lifting}
Let $X$ be a fibrant ${ \mathcal{O} }$-algebra. Then $X$ is ${ \mathsf{TQ} }$-local if and only if $X\rightarrow *$ satisfies the right lifting property with respect to every ${ \mathsf{TQ} }$-acyclic strong cofibration $A\rightarrow B$ in ${ \mathsf{Alg} }_{{ \mathcal{O} }}$.
\end{prop}

\begin{proof}
This is proved in \cite[3.12]{Harper_Zhang}.
\end{proof}

\begin{prop}\label{prop:u_local}
Let $Y$ be a fibrant object in ${ \mathsf{Alg} }_{J}$.  Then $UY \in  { \mathsf{Alg} }_{{ \mathcal{O} }}$  is ${ \mathsf{TQ} }$-local.
\end{prop}

\begin{proof}
This follows from Proposition \ref{prop: local by lifting} by using the $(Q,U)$ adjunction \eqref{equation: QU adjunction}.
\end{proof}

Next we observe that the ${ \mathsf{TQ} }$-local property is preserved by weak equivalences between fibrant ${ \mathcal{O} }$-algebras.

\begin{prop}\label{prop: we local}
Let $X \rightarrow Y$ be a weak equivalence between fibrant objects in ${ \Alg_{ \mathcal{O} } }$.  Then $X$ is ${ \mathsf{TQ} }$-local if and only if  $Y$ is ${ \mathsf{TQ} }$-local.
\end{prop}

\begin{proof}
Let $A \rightarrow B$ be a  ${ \mathsf{TQ} }$-acyclic strong cofibration. Consider the commutative diagram of mapping spaces of the form
\begin{align*}
\xymatrix{
\Hombold(A,X)\ar[d]^{\sim} & \Hombold(B,X)\ar[d]^{\sim}\ar[l] \\
 \Hombold(A,Y) & \Hombold(B,Y)\ar[l]
}
\end{align*}
in ${ \mathsf{sSet} }$. Since the vertical maps are weak equivalences, it follows that the top map is a weak equivalence if and only if the bottom map is a weak equivalence.
\end{proof}

This observation generalizes as follows.

\begin{prop}
\label{prop: generalized we local}
Consider any weak equivalence $X \rightarrow Y$ in ${ \Alg_{ \mathcal{O} } }$. Let $X', Y'$ be fibrant replacements of $X,Y$, respectively, in ${ \Alg_{ \mathcal{O} } }$. Then $X'$ is ${ \mathsf{TQ} }$-local if and only if  $Y'$ is ${ \mathsf{TQ} }$-local.
\end{prop}

\begin{proof}
By assumption, the comparison map $X \rightarrow X'$ is an acyclic cofibration and $Y'$ is fibrant.  Then it follows immediately (via lifting) that there exists a map $\xi$ that makes the diagram
\begin{align*}
\xymatrix{
X \ar[r]^{\sim} \ar[d]^{\sim} & Y \ar[d]^{\sim} \\
X'  \ar@{.>}[r]^-{\xi} & Y'
}
\end{align*}
in ${ \Alg_{ \mathcal{O} } }$ commute. By Proposition \ref{prop: we local}, $X'$ is ${ \mathsf{TQ} }$-local if and only if  $Y'$ is ${ \mathsf{TQ} }$-local since $\xi$ is a weak equivalence between fibrant objects.
\end{proof}

\begin{prop}\label{prop: fibrant replacements both local}
Let $X$ be an ${ \mathcal{O} }$-algebra and suppose $X',X''$ are fibrant replacements of $X$ in ${ \Alg_{ \mathcal{O} } }$. Then $X'$ is ${ \mathsf{TQ} }$-local if and only if  $X''$ is ${ \mathsf{TQ} }$-local.
\end{prop}

\begin{proof}
This follows from Proposition \ref{prop: generalized we local}.
\end{proof}

The following generalization of Proposition \ref{prop:u_local} will be used in our proof of Propositions \ref{prop:TQ_completion_resolution_with_TQ_local_values} and \ref{prop:weak_TQ_fibrancy_conditions_for_TQ_Nil_M_resolution}.

\begin{prop}
\label{prop:fibrant_replacements_of_UY_are_TQ_local}
Let $Y$ be a (not necessarily fibrant) object in ${ \mathsf{Alg} }_{J}$, then every fibrant replacement of $UY$ in ${ \Alg_{ \mathcal{O} } }$ is ${ \mathsf{TQ} }$-local.
\end{prop}

\begin{proof}
Let $UY \rightarrow  \widetilde{UY}$ be a fibrant replacement of $UY$ in ${ \Alg_{ \mathcal{O} } }$; in particular, $UY \rightarrow  \widetilde{UY}$ is an acyclic cofibration.  Let $Y'$ be a fibrant replacement of $Y$ in ${ \Alg_J }$.  Then it follows immediately (via lifting) that there exists a map $\xi$ that makes the diagram
\begin{align*}
\xymatrix{
UY  \ar[r]^\sim \ar[d]_\sim & UY'\ar[d] \\
\widetilde{UY}\ar@{.>}[ur]^(0.45){\xi}\ar[r] & {*}
}
\end{align*}
in ${ \Alg_{ \mathcal{O} } }$ commute.  Since $UY'$ is  ${ \mathsf{TQ} }$-local (Proposition \ref{prop:u_local}) and $\xi$ is a weak equivalence between fibrant objects, $\widetilde{UY}$ is ${ \mathsf{TQ} }$-local by Proposition \ref{prop: we local}.
\end{proof}

Now we are in a good position to prove Propositions \ref{prop:TQ_completion_resolution_with_TQ_local_values} and \ref{prop:weak_TQ_fibrancy_conditions_for_TQ_Nil_M_resolution}. First recall the ${ \mathsf{TQ} }$-completion construction. Let  $Z$ be a cofibrant ${ \mathcal{O} }$-algebra and consider the cosimplicial resolution of $Z$ with respect to ${ \mathsf{TQ} }$-homology of the form
\begin{align*}
\xymatrix{
Z \ar[r] &
(UQ) Z \ar@<0.5ex>[r] \ar@<-0.5ex>[r]  &
(UQ)^2 Z \ar@<-1.5ex>[l]   \ar@<1.0ex>[r] \ar[r] \ar@<-1.0ex>[r] &
(UQ)^3 Z \ar@<-2.0ex>[l] \ar@<-3.0ex>[l] \cdots
}
\end{align*}
in ${ \Alg_{ \mathcal{O} } }$, denoted $Z \rightarrow \mathbf{C} (Z)$, with coface maps obtained by iterating the ${ \mathsf{TQ} }$-Hurewicz map ${ \mathrm{id} } \rightarrow UQ$ (Definition \ref{def: tq homology}) and codegeneracy maps built from the counit map of the adjunction $(Q,U)$ in the usual way. Taking the homotopy limit (over $\Delta$) gives the ${ \mathsf{TQ} }$-completion map \cite{Ching_Harper_derived_Koszul_duality,Harper_Hess} of the form
\begin{align}
\label{eq:def_of_tq_completion}
  Z \rightarrow
  Z^{\wedge}_{{ \mathsf{TQ} }}=\holim\nolimits_\Delta \mathbf{C}(Z)
  { \ \simeq \ }\holim\nolimits_\Delta\widetilde{\mathbf{C}(Z)}
\end{align}
in ${ \Alg_{ \mathcal{O} } }$, where $\widetilde{\mathbf{C}(Z)}$ denotes any functorial fibrant replacement functor $\widetilde{(-)}$ on ${ \Alg_{ \mathcal{O} } }$ (obtained, for instance,  by running the small object argument with respect to the generating acyclic cofibrations in ${ \Alg_{ \mathcal{O} } }$) applied to the cosimplicial ${ \mathcal{O} }$-algebra $\mathbf{C}(Z)$.

The following proposition verifies that $\widetilde{\mathbf{C}(Z)}$ is an objectwise ${ \mathsf{TQ} }$-local diagram---this amounts to checking the weak ${ \mathsf{TQ} }$-fibrancy condition in the ${ \mathsf{TQ} }$-local homotopy theory \cite[5.14]{Harper_Zhang} using the above technical propositions.

\begin{prop}
\label{prop:TQ_completion_resolution_with_TQ_local_values}
If $Z$ is an ${ \mathcal{O} }$-algebra, then the ${ \mathsf{TQ} }$-completion $Z^\wedge_{ \mathsf{TQ} }$ of $Z$ is the homotopy limit of a small diagram of ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras.
\end{prop}

\begin{proof}
It suffices to consider the case when $Z$ is a cofibrant ${ \mathcal{O} }$-algebra; e.g., otherwise, work with its cofibrant replacement in ${ \Alg_{ \mathcal{O} } }$. We want to show that the $\Delta$-shaped diagram $\widetilde{\mathbf{C}(Z)}$ in \eqref{eq:def_of_tq_completion} is objectwise ${ \mathsf{TQ} }$-local; i.e., that $\widetilde{\mathbf{C}(Z)^s}$ is ${ \mathsf{TQ} }$-local for each $s\geq 0$. This follows  from Proposition \ref{prop:fibrant_replacements_of_UY_are_TQ_local}. In more detail: Consider the case $s=0$. Let $Y:=QZ\in{ \Alg_J }$. Then $UY=(UQ)Z$, hence it suffices to verify that $\widetilde{UY}$ is ${ \mathsf{TQ} }$-local which is true by Proposition \ref{prop:fibrant_replacements_of_UY_are_TQ_local}. Similarly, consider the case $s\geq 1$. Let $Y:=Q(UQ)^s Z\in{ \Alg_J }$. Then $UY=(UQ)^{s+1}Z$, hence it suffices to verify that $\widetilde{UY}$ is ${ \mathsf{TQ} }$-local and Proposition \ref{prop:fibrant_replacements_of_UY_are_TQ_local} completes the proof.
\end{proof}

Recall the ${ \mathsf{TQ} }|_{\mathrm{Nil}_M}$-completion construction from \cite{Ching_Harper_nilpotent_Whitehead}. For each $n\geq 1$, $\tau_n{ \mathcal{O} }$ is the operad associated to ${ \mathcal{O} }$ where
\begin{align*}
  (\tau_n { \mathcal{O} })[t] :=
    \begin{cases}
      { \mathcal{O} }[t] & \text{for $t \leq n$}\\
      * &  \text{otherwise}
    \end{cases}
\end{align*}
and consider the associated commutative diagram of operad maps
\cite{Ching_Harper_nilpotent_Whitehead}
\begin{align*}
\xymatrix{
{ \mathcal{O} } \ar@/_1pc/[dr]  \ar[r] &
J_n \ar[d]^\sim  \ar[r]  &
J_1 \ar[d]^\sim\ar@{=}[r]  & J\\
& \tau_n { \mathcal{O} } \ar[r] &
\tau_1 { \mathcal{O} }
}
\end{align*}
 where the upper horizontal maps
 are cofibrations of operads, the left-hand and bottom horizontal maps are the natural truncations, and the vertical maps are weak equivalences of operads; for notational simplicity, here we take $J=J_1$. The corresponding change of operad adjunctions have the form
\begin{align*}
\xymatrix{
  { \Alg_{ \mathcal{O} } } \ar@<.5ex>[r]^{R_n}  &
  { \mathsf{Alg} }_{J_n} \ar@<.5ex>[l]^{V_n} \ar@<.5ex>[r]^{Q_n}  &
  { \mathsf{Alg} }_{J} \ar@<.5ex>[l]^{U_n}
}\quad\quad
\xymatrix{
  { \Alg_{ \mathcal{O} } } \ar@<.5ex>[r]^{Q}  &
  { \mathsf{Alg} }_{J} \ar@<.5ex>[l]^{U}
}
\end{align*}
with left adjoints on top, where $R_n=J_n\circ_{ \mathcal{O} }(-)$, $Q_n=J\circ_{J_n}(-)$, $Q=J\circ_{ \mathcal{O} }(-)$, and $V_n,U_n,U$ denote the indicated forgetful functors; in particular, the adjunction on the right is the composite of the adjunctions on the left.

Next recall the ${ \mathsf{TQ} }|_{\mathrm{Nil}_M}$-completion construction. Let $n\geq 1$ and define $M:=n+1$. Let $X$ be a cofibrant $J_n$-algebra and consider the cosimplicial resolution of $X$ with respect to ${ \mathsf{TQ} }|_{\mathrm{Nil}_M}$-homology of the form
\begin{align*}
\xymatrix{
  X \ar[r] &  (U_n Q_n) X \ar@<0.5ex>[r] \ar@<-0.5ex>[r]  &  (U_n Q_n)^2 X  \ar@<-1.5ex>[l]   \ar@<1.0ex>[r] \ar[r] \ar@<-1.0ex>[r] &  (U_n Q_n)^3 X \ar@<-2.0ex>[l] \ar@<-3.0ex>[l] ~ \cdots
}
\end{align*}
in ${ \mathsf{Alg} }_{J_n}$, denoted $X\rightarrow \mathbf{N}(X)$, with coface maps obtained by iterating the ${ \mathsf{TQ} }|_{\mathrm{Nil}_M}$-Hurewicz map ${ \mathrm{id} } \rightarrow U_nQ_n$ and codegeneracy maps built from the counit map of the adjunction $(Q_n,U_n)$ in the usual way; see Ching-Harper \cite{Ching_Harper}. Applying the forgetful functor $V_n$ gives the diagram $V_n X \rightarrow V_n \mathbf{N} (X)$ of the form
\begin{align*}
\xymatrix{
  V_n X \ar[r] &
  V_n(U_nQ_n) X\ar@<0.5ex>[r]\ar@<-0.5ex>[r] &
  V_n(U_nQ_n)^2 X\ar@<-1.5ex>[l]\ar@<1.0ex>[r]
  \ar[r]\ar@<-1.0ex>[r] &
  V_n (U_nQ_n)^3 X\ar@<-2.0ex>[l]\ar@<-3.0ex>[l]\cdots
}
\end{align*}
in ${ \Alg_{ \mathcal{O} } }$.  Taking the homotopy limit (over $\Delta$) gives the ${ \mathsf{TQ} }|_{\mathrm{Nil}_M}$-completion map of the form
\begin{align}
\label{eq:defn_of_TQ_Nil_M_completion}
  X\rightarrow
  X^\wedge_{{ \mathsf{TQ} }|_{\mathrm{Nil}_M}}=
  \holim\nolimits_\Delta V_n\mathbf{N}(X)
  { \ \simeq \ }\holim\nolimits_\Delta\widetilde{V_n\mathbf{N}(X)}
\end{align}
in ${ \Alg_{ \mathcal{O} } }$, where $\widetilde{V_n\mathbf{N}(X)}$ denotes any functorial fibrant replacement functor $\widetilde{(-)}$ on ${ \Alg_{ \mathcal{O} } }$ applied to the cosimplicial ${ \mathcal{O} }$-algebra $V_n\mathbf{N}(X)$.

\begin{prop}
\label{prop:weak_TQ_fibrancy_conditions_for_TQ_Nil_M_resolution}
If $X$ is a $J_n$-algebra, then the ${ \mathsf{TQ} }|_{\mathrm{Nil}_M}$-completion $X^\wedge_{{ \mathsf{TQ} }|_{\mathrm{Nil}_M}}$ of $X$ is the homotopy limit of a small diagram of ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras.
\end{prop}

\begin{proof}
It suffices to consider the case when $X$ is a cofibrant $J_n$-algebra; e.g., otherwise, work with its cofibrant replacement in ${ \mathsf{Alg} }_{J_n}$. We want to show that the $\Delta$-shaped diagram $\widetilde{V_n\mathbf{N}(X)}$ in \eqref{eq:defn_of_TQ_Nil_M_completion} is objectwise ${ \mathsf{TQ} }$-local; i.e., that $\widetilde{V_n\mathbf{N}(X)^s}$ is ${ \mathsf{TQ} }$-local for each $s\geq 0$. This follows  from Proposition \ref{prop:fibrant_replacements_of_UY_are_TQ_local}. In more detail: Consider the case $s=0$. Let $Y:=Q_nX\in{ \Alg_J }$, then $UY=V_n(U_nQ_n)X$ and hence it suffices to verify that $\widetilde{UY}$ is ${ \mathsf{TQ} }$-local; this is true by Proposition \ref{prop:fibrant_replacements_of_UY_are_TQ_local}. Similarly, consider the case $s\geq 1$. Let $Y:=Q_n(U_nQ_n)^s X\in{ \Alg_J }$. Then $UY=V_n(U_nQ_n)^{s+1}X$ and hence it suffices to verify that $\widetilde{UY}$ is ${ \mathsf{TQ} }$-local; this is true by Proposition \ref{prop:fibrant_replacements_of_UY_are_TQ_local} which completes the proof.
\end{proof}

\section{Comparing ${ \mathsf{TQ} }$-localization with ${ \mathsf{TQ} }$-completion}

In this section we discuss the relation between ${ \mathsf{TQ} }$-localization and ${ \mathsf{TQ} }$-completion. Let $X$ be a cofibrant ${ \mathcal{O} }$-algebra.  One can construct the ${ \mathsf{TQ} }$-localization map $X\rightarrow L_{ \mathsf{TQ} }(X)$ (see \cite{Harper_Zhang}) by running the small object argument.  By construction, $L_{ \mathsf{TQ} }(X)$ is ${ \mathsf{TQ} }$-local and the ${ \mathsf{TQ} }$-localization map $X\rightarrow L_{ \mathsf{TQ} }(X)$ is a ${ \mathsf{TQ} }$-acyclic strong cofibration (Definition \ref{defn:TQ_local_O_algebras}).

The ${ \mathsf{TQ} }$-completion map  $X \rightarrow X^{\wedge}_{{ \mathsf{TQ} }}$ can be thought of as an approximation of the ${ \mathsf{TQ} }$-localization map.   For instance, we know that the ${ \mathsf{TQ} }$-completion $X^{\wedge}_{{ \mathsf{TQ} }}$ of $X$ is always ${ \mathsf{TQ} }$-local (Proposition \ref{prop:completion_results_part1}).

\begin{thm}[Recognizing when ${ \mathsf{TQ} }$-local ${ \mathcal{O} }$-algebras are ${ \mathsf{TQ} }$-complete]
\label{thm:recognizing_TQ_complete_from_TQ_local}
Let $X$ be a cofibrant ${ \mathcal{O} }$-algebra.  Then the ${ \mathsf{TQ} }$-completion map  $\function{c}{X}{X^{\wedge}_{{ \mathsf{TQ} }}}$ factors through the ${ \mathsf{TQ} }$-localization map $\function{l}{X}{L_{ \mathsf{TQ} }(X)}$ via a commutative diagram of the form
\begin{align*}
\xymatrix{
X \ar[r]^c \ar[d]_l & X^{\wedge}_{{ \mathsf{TQ} }} \ar[d] \\
L_{ \mathsf{TQ} }(X)  \ar@{.>}[ur]^-{\xi} \ar[r] & {*}
}
\end{align*}
in ${ \Alg_{ \mathcal{O} } }$. Furthermore, if $X$ is ${ \mathsf{TQ} }$-local, then the following are equivalent:
\begin{itemize}
\item[(i)] The natural map $X\rightarrow X^{\wedge}_{{ \mathsf{TQ} }}$ is a ${ \mathsf{TQ} }$-equivalence; i.e., $X$ is \emph{${ \mathsf{TQ} }$-good}.
\item[(ii)] The natural map $X{ \ \simeq \ } X^{\wedge}_{{ \mathsf{TQ} }}$ is a weak equivalence; i.e., $X$ is ${ \mathsf{TQ} }$-complete.
\item[(iii)] The comparison map $\xi$ is a weak equivalence.
\end{itemize}
\end{thm}

\begin{proof}
This is analogous to the Bousfield-Kan completion of spaces \cite{Bousfield_Kan}.  Since $X^{\wedge}_{{ \mathsf{TQ} }}$ is ${ \mathsf{TQ} }$-local and $\function{l}{X}{L_{ \mathsf{TQ} }(X)}$ is a ${ \mathsf{TQ} }$-acyclic strong cofibration,  there exists a lift $\xi$ that makes the diagram commute (Proposition \ref{prop: local by lifting}) in ${ \Alg_{ \mathcal{O} } }$.  Suppose $X$ is ${ \mathsf{TQ} }$-local, then $l$ is a ${ \mathsf{TQ} }$-equivalence between ${ \mathsf{TQ} }$-local objects, hence a weak equivalence by the ${ \mathsf{TQ} }$-local Whitehead theorem (Proposition \ref{prop:TQ_local_whitehead_theorem}).  Therefore $\xi$ is a weak equivalence if and only if $c$ is a weak equivalence.  This verifies (ii) $\Leftrightarrow$ (iii).  Since $X, X^{\wedge}_{{ \mathsf{TQ} }}$ are ${ \mathsf{TQ} }$-local, $c$ is a ${ \mathsf{TQ} }$-equivalence if and only if $c$ is a weak equivalence by the ${ \mathsf{TQ} }$-local Whitehead theorem. This verifies (i) $\Leftrightarrow$ (ii).
\end{proof}

It is worth pointing out the following two propositions.

\begin{prop}
A map $\function{f}{X}{Y}$ between ${ \mathcal{O} }$-algebras is a ${ \mathsf{TQ} }$-homology equivalence if and only if the induced map $\function{f^{\wedge}_{{ \mathsf{TQ} }}}{X^{\wedge}_{{ \mathsf{TQ} }}}{Y^{\wedge}_{{ \mathsf{TQ} }}}$ is a weak equivalence.
\end{prop}

\begin{proof}
This is proved by arguing exactly as in \cite[I.5]{Bousfield_Kan}, but here is the basic idea: The ``if'' direction is proved using retract argument and the ``only if'' direction is because $\holim_\Delta$ preserves weak equivalences.
\end{proof}

\begin{prop}
Let $X$ be an ${ \mathcal{O} }$-algebra, then the following are equivalent:
\begin{itemize}
\item[(i)]  $X$ is \emph{${ \mathsf{TQ} }$-good}.
\item[(ii)]  $X^{\wedge}_{{ \mathsf{TQ} }}$ is ${ \mathsf{TQ} }$-complete.
\item[(iii)]  $X^{\wedge}_{{ \mathsf{TQ} }}$ is ${ \mathsf{TQ} }$-good.
\end{itemize}
\end{prop}

\begin{proof}
This follows from exactly the same argument as in \cite[I.5]{Bousfield_Kan}.
\end{proof}

\section{Postnikov towers and ${ \mathsf{TQ} }$-localization}
\label{sec:postnikov_towers}

In this section we assume that ${ \mathcal{O} },{ \mathcal{R} }$ are $(-1)$-connected.    We show that a $(-1)$-connected ${ \mathcal{O} }$-algebra is ${ \mathsf{TQ} }$-local if it has a principally refined Postnikov tower.

\begin{prop}\label{prop: -1 has postnikov tower}

Let $X$ be a $(-1)$-connected cofibrant ${ \mathcal{O} }$-algebra.  Then there exists a coaugmented tower $\{X\}\rightarrow \{X_n\}$ (the Postnikov tower of $X$) of the form
\begin{align*}
\xymatrix{
  X\ar@<0.8ex>[d]\ar@/^1ex/[dr]\ar@/^1.5ex/[drr]\ar@/^2ex/[drrr]\\
  {*}=X_{-1} & X_0\ar[l] & X_1\ar[l] & X_2\ar[l] & \cdots\ar[l]
}
\end{align*}
in ${ \Alg_{ \mathcal{O} } }$ such that for each $n\geq -1$:
\begin{itemize}
  \item[(a)] $X_n$ is a cofibrant and fibrant ${ \mathcal{O} }$-algebra.
  \item[(b)] the structure map $X \rightarrow X_n$ is $(n+1)$-connected and  $\pi_k X_n = *$ for all $k \geq n+1$.
  \item[(c)] the structure map $X_{n+1} \rightarrow X_{n}$ is a fibration.
\end{itemize}
\end{prop}

\begin{proof}
The Postnikov tower can be constructed using small object arguments analogous to the arguments in \cite{Ching_Harper, Dugger_Shipley, EKMM, Schwede_book_project}. In more detail: Let $I_n$ be the set of $n$-connected generating cofibrations in ${ \Alg_{ \mathcal{O} } }$ and let $J$ be the set of generating acyclic cofibrations in ${ \Alg_{ \mathcal{O} } }$ (see, for instance, \cite{Harper_symmetric_spectra}). Start by setting $X_{-1} = *$.  For each $n \geq 0$, we inductively run the small object argument with respect to $I_{n+1}  \bigcup   J$ to factor the map $X \rightarrow X_{n-1}$ in ${ \Alg_{ \mathcal{O} } }$ as $X \rightarrow X_{n} \rightarrow X_{n-1}$.  Then $X \rightarrow X_{n}$ is a cofibration, $X_n \rightarrow X_{n-1}$ is a fibration and $\pi_k X_n = * $ for all $k \geq n+1$.
By assumption, ${ \mathcal{O} },{ \mathcal{R} }$ are $(-1)$-connected, hence $X \rightarrow X_n$ is $(n+1)$-connected by the small object argument construction.
\end{proof}

Analogous to the definition for spaces, principal Postnikov towers and principally refined Postnikov towers are defined as follows.

\begin{defn}
Let $X$ be a $(-1)$-connected ${ \mathcal{O} }$-algebra.  We say that a Postnikov tower $\{X_n\}$  of $X$ is \emph{principal} if for each $n \geq 0$, the structure map $X_{n}\rightarrow X_{n-1}$ fits into a homotopy pullback diagram of the left-hand form
\begin{align}
\label{eq:homotopy_pullback_diagrams_for_principal_and_principally_refined}
\xymatrix{
  X_n\ar[r]\ar[d] & {*}^\mathrm{fat}\ar[d]\\
  X_{n-1}\ar[r] & K(\pi_n X, n+1)
}\quad\quad
\xymatrix{
  M_{i}\ar[r]\ar[d] & {*}^\mathrm{fat}\ar[d]\\
  M_{i-1}\ar[r] & K(G_i, n+1)
}
\end{align}
in ${ \Alg_{ \mathcal{O} } }$, where ${*}^\mathrm{fat}$ is an ${ \mathcal{O} }$-algebra that is weakly equivalent to ${*}$ (i.e., a ``fat point'' in ${ \Alg_{ \mathcal{O} } }$) and $K(\pi_n X, n+1)$ is an object in ${ \mathsf{Alg} }_{{ \mathcal{O} }}$ with $\pi_n X$ as the only nontrivial homotopy group concentrated at level $n+1$.

We say that $\{X_n\}$ is \emph{principally refined} if, for each $n \geq 0$, the structure map $X_{n} \rightarrow X_{n-1}$ can be factored as a finite composite $X_{n} = M_{t_n} \rightarrow \cdots  \rightarrow M_2 \rightarrow M_1 \rightarrow M_0 = X_{n-1}$ of maps such that, for each $t_n \geq i \geq 1$, the map $M_{i}\rightarrow M_{i-1}$ fits into a homotopy pullback diagram of the right-hand form \eqref{eq:homotopy_pullback_diagrams_for_principal_and_principally_refined} in ${ \Alg_{ \mathcal{O} } }$, where the $G_i$'s are abelian groups and $K(G_i, n+1)$ is an object in ${ \mathsf{Alg} }_{{ \mathcal{O} }}$ with $G_i$ as the only nontrivial homotopy group concentrated at level $n+1$. In particular, every principal Postnikov tower is  principally refined.
\end{defn}

\begin{proof}[Proof of Theorem \ref{thm:principally_refined_are_TQ_local}]
We know that every $0$-connected fibrant ${ \mathcal{O} }$-algebra is ${ \mathsf{TQ} }$-local (Theorem \ref{thm:main_result}), hence, in particular, each  Eilenberg-MacLane object $K(G_i, n+1)$ appearing in the principally refined Postnikov tower of $X$ has ${ \mathsf{TQ} }$-local fibrant replacements in ${ \Alg_{ \mathcal{O} } }$.
By inducting up the principally refined Postnikov tower, it follows that  each $X_n$ is ${ \mathsf{TQ} }$-local (Proposition \ref{prop:holims_of_TQ_local_objects_are_TQ_local}).  Since $X$ is the homotopy limit of its Postnikov tower $\{X_n\}$, which is objectwise ${ \mathsf{TQ} }$-local, it follows that $X$ is ${ \mathsf{TQ} }$-local (Proposition \ref{prop:holims_of_TQ_local_objects_are_TQ_local}) which completes the proof.
\end{proof}

We provide some examples of $(-1)$-connected algebras which admit principally refined Postnikov towers.

\begin{itemize}
\item[(i)] Let $X$ be a cofibrant $0$-connected ${ \mathcal{O} }$-algebra.    Analogous to results in \cite{Basterra, Basterra_multiplication, Dugger_Shipley, Hatcher_algebraic}, one can show that the Postnikov tower of  $X$ is principal.

\item[(ii)] Consider $\Omega X$ for any $0$-connected cofibrant ${ \mathcal{O} }$-algebra $X$.  Since the loop functor $\Omega$ commutes with homotopy pullbacks in ${ \Alg_{ \mathcal{O} } }$, $\Omega X$ has a principal Postnikov tower by applying $\Omega$ to the principal Postnikov tower of $X$.

\item[(iii)] Consider $UY$ for any $(-1)$-connected cofibrant $J$-algebra $Y$.  The category ${ \mathsf{Alg} }_{J}$ is
Quillen equivalent to ${ \mathsf{Alg} }_{\tau_1{ \mathcal{O} }} \cong { \mathsf{Mod} }_{{ \mathcal{O} }[1]}$ \cite[7.21]{Harper_Hess}, hence the homotopy category of ${ \mathsf{Alg} }_{J}$ is stable.   Therefore, the Postnikov tower of  $Y$ in ${ \mathsf{Alg} }_{J}$  is already principal.  Applying forgetful functor $U$ induces principal Postnikov tower for $UY$ in ${ \Alg_{ \mathcal{O} } }$.

\item[(iv)] One can construct additional examples by pulling back quotient towers along cocellular maps  as described in \cite[3.3]{May_Ponto}.\end{itemize}

\bibliographystyle{plain}


\section{Introduction}
The word order between source and target languages significantly influences the translation quality in statistical machine translation (SMT) \cite{Tillmann:2004:UOM:1613984.1614010,hayashi-EtAl:2013:EMNLP,nakagawa2015}.
Models that adjust orders of translated phrases in decoding have been proposed to solve this problem \cite{Tillmann:2004:UOM:1613984.1614010, Koehnetal:2005edinburgh, Nagata:2006:CGP:1220175.1220265}.
However, such reordering models do not perform well for long-distance reordering. In addition, their computational costs are expensive.
To address these problems, preordering \cite{xia-mccord:2004:COLING, wang-collins-koehn:2007:EMNLP-CoNLL2007, xu-EtAl:2009:NAACLHLT09, isozaki10hfe, gojunfraser2012, nakagawa2015} and post-ordering \cite{goto-utiyama-sumita:2012:ACL2012short,Goto:2013:PPI:2523057.2518100,hayashi-EtAl:2013:EMNLP} models have been proposed. Preordering reorders source sentences before translation, while post-ordering reorders sentences translated without considering the word order after translation. In particular, preordering effectively improves the translation quality because it solves long-distance reordering and computational complexity issues \cite{jehl-EtAl:2014:EACL,nakagawa2015}.

Rule-based preordering methods either manually create reordering rules \cite{wang-collins-koehn:2007:EMNLP-CoNLL2007, xu-EtAl:2009:NAACLHLT09, isozaki10hfe, gojunfraser2012} or extract reordering rules from a corpus \cite{xia-mccord:2004:COLING, genzel:2010:PAPERS}. On the other hand, studies in \cite{neubig-watanabe-mori:2012:EMNLP-CoNLL, lerner-petrov:2013:EMNLP, hoshino-EtAl:2015:ACL-IJCNLP, nakagawa2015} apply machine learning to the preordering problem. \citet{hoshino-EtAl:2015:ACL-IJCNLP} proposed a method that learns whether child nodes should be swapped at each node of a syntax tree. \citet{neubig-watanabe-mori:2012:EMNLP-CoNLL} and \citet{nakagawa2015} proposed methods that construct a binary tree and reordering simultaneously from a source sentence. These methods require a manual feature design for every language pair, which makes language dependent design costly.
To overcome this challenge, methods based on feed forward neural networks that do not require a manual feature design have been proposed \cite{degispert-iglesias-byrne:2015:NAACL-HLT, botha-EtAl:2017:EMNLP2017}. However, these methods decide whether to reorder child nodes without considering the sub-trees, which contains important information for reordering.

As a preordering method that is free of manual feature design and makes use of information in sub-trees, we propose a preordering method with a recursive neural network (RvNN). RvNN calculates reordering in a bottom-up manner (from the leaf nodes to the root) on a source syntax tree. Thus, preordering is performed considering the entire sub-trees.
Specifically, RvNN learns whether to reorder nodes of a syntax tree\footnote{In this paper, we used binary syntax trees.}  with a vector representation of sub-trees and syntactic categories. We evaluate the proposed method for English-to-Japanese translations using both phrase-based SMT (PBSMT) and neural MT (NMT). The results confirm that the proposed method achieves comparable translation quality to the state-of-the-art preordering method \cite{nakagawa2015} that requires a manual feature design.

\section{Preordering with a Recursive Neural Network}
We explain our design of the RvNN to conduct preordering after describing how to obtain gold-standard labels for preordering.
\subsection{Gold-Standard Labels for Preordering}
\label{sec:rank_eval}
We created training data for preordering by labeling whether each node of the source-side syntax tree has reordered child nodes against a target-side sentence. The label is determined based on Kendall's $\tau$ \cite{kendall1938measure} as in \cite{nakagawa2015}, which is calculated by Equation (\ref{eq:kendall_tau}).
\begin{eqnarray}
\label{eq:kendall_tau}
\tau&=&\frac{4\sum^{|{\bf y}|-1}_{i=1}\sum^{|{\bf y}|}_{j=i+1} \delta ({\bf y}_i<{\bf y}_j)}{|{\bf y}|(|{\bf y}|-1)} - 1, \ \ \ \\
\delta(x)&=&\begin{cases}
1 & (x\ {\rm is\ true}), \\
0 & ({\rm otherwise}),
\end{cases} \nonumber
\end{eqnarray}
where ${\bf y}$ is a vector of target word indexes that are aligned with source words. The value of Kendall's $\tau$ is in $[-1,1]$. When it is $1$, it means the sequence of ${\bf y}$ is in a complete ascending order, {\em i.e.}, target sentence has the same word order with the source in terms of word alignment. At each node, if Kendall's $\tau$ increases by reordering child nodes, an ``Inverted'' label is assigned; otherwise, a ``Straight'' label, which means the child nodes do not need to be reordered, is assigned. When a source word of a child node does not have an alignment, a ``Straight'' label is assigned.

\subsection{Preordering Model}
\begin{figure}[t!]
\centering
\includegraphics[width=0.35\textwidth, bb=0 0 263 194]{rec_example.pdf}
\caption{Preordering an English sentence ``My parents live in London'' with RvNN (Nodes with a horizontal line mean ``Inverted'').}
\label{fig:rec_example}
\end{figure}

RvNN is constructed given a binary syntax tree. It predicts the label determined in Section \ref{sec:rank_eval} at each node. RvNN decides whether to reorder the child nodes by considering the sub-tree. The vector of the sub-tree is calculated in a bottom-up manner from the leaf nodes.
Figure \ref{fig:rec_example} shows an example of preordering of an English sentence ``My parents live in London.''
At the VP node corresponding to ``live in London,'' the vector of the node is calculated by Equation (\ref{eq:concatnodes}), considering its child nodes correspond to ``live'' and ``in London.''
\begin{eqnarray}
\label{eq:concatnodes}
{\bf p} &=& f([{\bf p_l}; {\bf p_r}]W + {\bf b}),\\
\label{eq:labelfunc}
{\bf s} &=& {\bf p}W_s + {\bf b}_s,
\end{eqnarray}
where $f$ is a rectifier, $W \in \mathbb{R}^{2\lambda \times \lambda}$ is a weight matrix, ${\bf p_l}$ and ${\bf p_r}$ are vector representations of the left and right child nodes, respectively. $[\cdot;\cdot]$ denotes the concatenation of two vectors. $W_s \in \mathbb{R}^{\lambda \times 2}$ is a weight matrix for the output layer, and ${\bf b}, {\bf b}_s \in \mathbb{R}^{\lambda}$ are the biases. ${\bf s}\in \mathbb{R}^2$ calculated by Equation (\ref{eq:labelfunc}) is a weight vector for each  label, which is fed into a softmax function to calculate the probabilities of the ``Straight'' and ``Inverted'' labels.

At a leaf node, a word embedding calculated by Equations (\ref{eq:embedding}) and (\ref{eq:leaf_node}) is fed into Equation (\ref{eq:concatnodes}).
\begin{eqnarray}
\label{eq:embedding}
{\bf e} &=& {\bm x}W_E, \\
\label{eq:leaf_node}
{\bf p_e} &=& f({\bf e}W_l + {\bf b}_l),
\end{eqnarray}
where ${\bm x} \in \mathbb{R}^{N}$ is a one-hot vector of an input word with a vocabulary size of $N$, $W_E \in \mathbb{R}^{N \times \lambda}$ is an embedding matrix, and ${\bf b}_l \in \mathbb{R}^\lambda$ is the bias.
The loss function is the cross entropy defined by Equation (\ref{eq:loss_func}).
\begin{eqnarray}
\label{eq:loss_func}
L(\theta) &=& -\frac{1}{K}\sum^K_{k=1} \sum_{n\in \mathcal{T}} l^n_k\log p(l^n_k; \theta),
\end{eqnarray}
where $\theta$ is the parameters of the model, $n$ is the node of a syntax tree $\mathcal{T}$, $K$ is a mini batch size, and $l^n_k$ is the label of the $n$-th node in the $k$-th syntax tree in the mini batch.

With the model using POS tags and syntactic categories, we use Equation (\ref{eq:concatnodes_pos}) instead of Equation (\ref{eq:concatnodes}).
 \begin{eqnarray}
\label{eq:concatnodes_pos}
{\bf p} &=& f([{\bf p_l}; {\bf p_r}; {\bf e}_t]W_t + {\bf b}_t),
\end{eqnarray}
where ${\bf e}_t$ represents a vector of POS tags or syntactic categories, $W_t \in \mathbb{R}^{3\lambda \times \lambda}$ is a weight matrix, and ${\bf b}_t \in \mathbb{R}^\lambda$ is the bias. ${\bf e}_t$ is calculated in the same manner as Equations (\ref{eq:embedding}) and (\ref{eq:leaf_node}), but the input is a one-hot vector of the POS tags or syntactic categories at each node. $\lambda$ is tuned on a development set, whose effects are investigated in Section \ref{sec:result}.

\section{Experiments}
\subsection{Settings}
We conducted English-to-Japanese translation experiments using the ASPEC corpus \cite{NAKAZAWA16.621}. This corpus provides $3$M sentence pairs as training data, $1,790$ sentence pairs as development data, and $1,812$ sentence pairs as test data.
We used Stanford CoreNLP\footnote{http://stanfordnlp.github.io/CoreNLP/} for tokenization and POS tagging, Enju\footnote{http://www.nactem.ac.uk/enju/} for parsing of English, and MeCab\footnote{http://taku910.github.io/mecab/} for tokenization of Japanese. For word alignment, we used MGIZA.\footnote{http://github.com/moses-smt/giza-pp} Source-to-target and target-to-source word alignments were calculated using IBM model $1$ and hidden Markov model, and they were combined with the intersection heuristic following \cite{nakagawa2015}.

We implemented our RvNN preordering model with Chainer.\footnote{http://chainer.org/} The ASPEC corpus was created using the sentence alignment method proposed in \citep{Utiyama07ajapanese-english} and was sorted based on the alignment confidence scores. In this paper, we used $100$k sentences sampled from the top $500$k sentences as training data for preordering. The vocabulary size $N$ was set to $50$k.
We used Adam \cite{DBLP:journals/corr/KingmaB14} with a weight decay and gradient clipping for optimization. The mini batch size $K$ was set to $500$.

We compared our model with the state-of-the-art preordering method proposed in \cite{nakagawa2015}, which is hereafter referred to as {\em BTG}. We used its publicly available implementation,\footnote{http://github.com/google/topdown-btg-preordering} and trained it on the same $100$k sentences as our model.

We used the $1.8$M source and target sentences as training data for MT. We excluded part of the sentence pairs whose lengths were longer than $50$ words or the source to target length ratio exceeded $9$.
For SMT, we used Moses.\footnote{http://www.statmt.org/moses/} We trained the $5$-gram language model on the target side of the training corpus with KenLM.\footnote{http://github.com/kpu/kenlm}
Tuning was performed by minimum error rate training \cite{och:2003:ACL}. We repeated tuning and testing of each model $3$ times and reported the average of scores.
For NMT, we used the attention-based encoder-decoder model of \cite{luong-pham-manning:2015:EMNLP} with $2$-layer LSTM implemented in OpenNMT.\footnote{http://opennmt.net/} The sizes of the vocabulary, word embedding, and hidden layer were set to $50$k, $500$, and $500$, respectively. The batch size was set to $64$, and the number of epochs was set to $13$. The translation quality was evaluated using BLEU \cite{papineni-EtAl:2002:ACL} and RIBES \cite{isozaki-EtAl:2010:EMNLP} using the bootstrap resampling method \cite{koehn:2004:EMNLP} for the significance test.

\begin{figure}[!t]
\centering
\includegraphics[width=0.48\textwidth, bb=0 0 720 576]{loss_curve_enja.pdf}
\vspace{-8truemm}
\caption{Learning curve of our preordering model.}
\label{fig:loss_curve_enja}
\end{figure}

\begin{table}[t!]
\centering
\begin{tabular}{@{}c|c|c|c@{}}\hline
Node dimensions & $100$ & $200$ & $500$ \\\hline\hline
w/o preordering & \multicolumn{3}{c}{$22.73$} \\\hline
w/o tags and categories & $24.63$ & $24.95$ & $25.02$ \\\hline
w/ tags and categories& $25.22$ & $25.41$ & $25.38$ \\\hline
\end{tabular}
\caption{BLEU scores with preordering by our model and without preordering under different $\lambda$ settings (trained on a $500$k subset of the training data).}
\label{tab:rec_bleu}
\vspace{-3truemm}
\end{table}

\subsection{Results}
\setcounter{figure}{3}
\begin{figure*}[!t]
\centering
\includegraphics[width=0.75\textwidth, bb=0 0 802 217]{parse_ex.pdf}
\caption{Example of a syntax tree with a parse-error (the phrase ``(1811)'' was divided in two phrases by mistake). Our preordering result was affected by such parse-errors. (Nodes with a horizontal line means ``Inverted''.)}
\label{fig:parse_ex}
\end{figure*}
\label{sec:result}
Figure \ref{fig:loss_curve_enja} shows the learning curve of our preordering model with $\lambda=200$.\footnote{The learning curve behaves similarly for different $\lambda$ values.}
Both the training and the development losses decreased until $2$ epochs. However, the development loss started to increase after $3$ epochs. Therefore, the number of epochs was set up to $5$, and we chose the model with the lowest development loss. The source sentences in the translation evaluation were preordered with this model.

\begin{table}[t!]
\centering
{\small
\begin{tabular}{@{}l||c|c|c|c@{}}\hline
\multicolumn{1}{c||}{} & \multicolumn{2}{c|}{PBSMT} & \multicolumn{2}{c}{NMT} \\\hline
 & BLEU & RIBES & BLEU & RIBES \\\hline\hline
w/o preordering &  22.88 & 64.07 &  {\bf 32.68} &  {\bf 81.68} \\\hline
w/ BTG & {\bf 29.51} & {\bf 77.20} & 28.91 & {\bf 79.58} \\\hline
w/ RvNN & {\bf 29.16} & {\bf 76.39} & 29.01 & {\bf 79.63} \\\hline
\end{tabular}}
\caption{BLEU and RIBES scores on the test set. (All models are trained on the entire training corpus of $1.8$M sentence pairs.) Numbers in {\bf bold} indicate the best systems and the systems that are statistically insignificant at $p < 0.05$ from the best systems.}
\label{tab:eval_bleu}
\end{table}

\setcounter{figure}{2}
\begin{figure}[t!]
\centering
\includegraphics[width=0.48\textwidth, bb=0 0 460 345]{tau_distribution.pdf}
\caption{Distribution of Kendall's $\tau$ in the training data without preordering, preordering by BTG, and preordering by our RvNN.}
\label{fig:tau_bleu}
\end{figure}
\setcounter{figure}{4}

\begin{table*}[t!]
\centering
{\small
\begin{tabular}{@{}l|l@{}}
\hline
\multicolumn{2}{c}{Preordered examples} \\\hline
Source sentence & because of the embedding heterostructure, current leakage around the threshold was minimal. \\
BTG & of the embedding heterostructure because, the threshold around current leakage minimal was. \\
RvNN & embedding heterostructure the of because, around threshold the current leakage minimal was. \\\hline\hline
\multicolumn{2}{c}{Translation examples by PBSMT} \\\hline
Reference & 埋込みヘテロ構造のため、しきい値近くでの漏れ電流は非常に小さかった。 \\
 & (embedding heterostructure of because, threshold around leakage very minimal.) \\
w/o preordering & 埋込みヘテロ構造のため、漏れ電流のしきい値付近では最低であった。 \\
 & (embedding heterostructure of because, leakage threshold around minimal.) \\
BTG　 & の埋込みヘテロ構造のため、このしきい値付近での漏れ電流の最小であった。 \\
 & (of embedding heterostructure of because, the threshold around leakage minimal.) \\
RvNN & 埋込みヘテロ構造のため、周辺のしきい値の電流漏れは認められなかった。 \\
 & (embedding heterostructure of because, around threshold leakage recognized not.) \\\hline
\end{tabular}}
\caption{Example where preordering improves translation. (Literal translations are given in the parenthesis under the Japanese sentences.)}
\label{tab:preorder_success}
\end{table*}

\begin{table*}[t!]
\centering
{\small
\begin{tabular}{l|l}
\hline
\multicolumn{2}{c}{Preordered examples} \\\hline
Source sentence & avogadro's hypothesis (1811) contributed to the development in since then. \\
BTG & avogadro's hypothesis (1811) the then since in development to contributed . \\
RvNN & avogadro's hypothesis (1811 then since in to development the contributed).  \\\hline\hline
\multicolumn{2}{c}{Translation examples by PBSMT} \\\hline
Reference & Avogadroの仮説 (1811)は，以後の発展に貢献した。 \\
& (Avogadro's hypothesis (1811), since then development to contributed.) \\
w/o preordering & Avogadroの仮説 (1811)の開発に貢献し以後である。 \\
& (Avogadro's hypothesis (1811) development to contributed since then.) \\
BTG & Avogadroの仮説 (1811)以後の発展に貢献した。 \\
& (Avogadro's hypothesis (1811) since then development to contributed.) \\
RvNN & Avogadroの仮説 (1811以降のこれらの開発に貢献した。　\\
& (Avogadro's hypothesis (1811 since then these development to contributed.) \\\hline
\end{tabular}}
\caption{Example of a parse-error disturbed preordering in our method. (Literal translations are given in the parenthesis under the Japanese sentences.)}
\label{tab:translation_ex}
\end{table*}

Next, we investigated the effect of $\lambda$. Table \ref{tab:rec_bleu} shows the BLEU scores with different $\lambda$ values, as well as the BLEU score without preordering. In this experiment, PBSMT was trained with a $500$k subset of training data, and the distortion limit was set to $6$.
Our RvNNs consistently outperformed the plain PBSMT without preordering. The BLEU score improved as $\lambda$ increased when only word embedding was considered. In addition, RvNNs involving POS tags and syntactic categories achieved even higher BLEU scores. This result shows the effectiveness of POS tags and syntactic categories in reordering. For these models, setting $\lambda$ larger than $200$ did not contribute to the translation quality. Based on these, we further evaluated the RvNN with POS tags and syntactic categories where $\lambda=200$.

Table \ref{tab:eval_bleu} shows BLEU and RIBES scores of the test set on PBSMT and NMT trained on the entire training data of $1.8$M sentence pairs. The distortion limit of SMT systems trained using preordered sentences by RvNN and BTG was set to $0$, while that without preordering was set to $6$.
Compared to the plain PBSMT without preordering, both BLEU and RIBES increased significantly with preordering by RvNN and BTG. These scores were comparable (statistically insignificant at $p<0.05$)  between RvNN and BTG,\footnote{The $p$-value for BLEU and RIBES were $0.068$ and $0.226$, respectively.} indicating that the proposed method achieves a translation quality comparable to BTG.
In contrast to the case of PBSMT, NMT without preordering achieved a significantly higher BLEU score than NMT models with preordering by RvNN and BTG. This is the same phenomenon in the Chinese-to-Japanese translation experiment reported in \cite{sudoh-nagata:2016:WAT2016}. We assume that one reason is the isolation between preordering and NMT models, where both models are trained using independent optimization functions.
In the future, we will investigate this problem and consider a model that unifies preordering and translation in a single model.

Figure \ref{fig:tau_bleu} shows the distribution of Kendall's $\tau$ in the original training data as well as the distributions after preordering by RvNN and BTG. The ratio of high Kendall's $\tau$ largely increased in the case of RvNN, suggesting that the proposed method learns preordering properly. Furthermore, the ratio of high Kendall's $\tau$ by RvNN is more than that of BTG, implying that preordering by RvNN is better than that by BTG.

We also manually investigated the preordering and translation results. We found that our model improved both.
Table \ref{tab:preorder_success} shows a successful preordering and translation example on PBSMT.
The word order is notably different between source and reference sentences.
After preordering, the word order between the source and reference sentences became the same.
Because RvNN depends on parsing, sentences with a parse-error tended to fail in preordering. For example, the phrase ``(1811)'' in Figure \ref{fig:parse_ex} was divided in two phrases by mistake. Consequently, preordering failed.
Table \ref{tab:translation_ex} shows preordering and translation examples for the sentence in Figure \ref{fig:parse_ex}.
Compared to the translation without preordering, the translation quality after preordering was improved to deliver correct meaning.

\section{Conclusion}
In this paper, we proposed a preordering method without a manual feature design for MT. The experiments confirmed that the proposed method achieved a translation quality comparable to the state-of-the-art preordering method that requires a manual feature design. As a future work, we plan to develop a model that jointly parses and preorders a source sentence. In addition, we plan to integrate preordering into the NMT model.

\section*{Acknowledgement}
\noindent This work was supported by NTT communication science laboratories and Grant-in-Aid for Research Activity Start-up \#17H06822, JSPS.