text:
\begin{document}

\address{Department of Mathematics, University of Pennsylvania,
209 South 33rd Street,
Philadelphia, PA, 19104-6395, USA}
    \email{mperoux@sas.upenn.edu}

\subjclass [2010] {16T15, 18D10, 18N40, 18N70, 55P42, 55P43}

\keywords{homotopy, spectrum, coalgebra, $\infty$-category, rigidification, Dold-Kan Correspondence}

\begin{abstract}
We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories.
The result will induce a Dold-Kan correspondence of coalgebras in $\infty$-categories. Moreover it shows that Shipley's zig-zag of Quillen equivalences provides an explicit symmetric monoidal equivalence of $\infty$-categories for the stable Dold-Kan correspondence.
We study homotopy coherent coalgebras associated to a monoidal model category and we show examples when these coalgebras cannot be rigidified. That is, their $\infty$-categories are not equivalent to the Dwyer-Kan localizations of strict coalgebras in the usual monoidal model categories of spectra and of connective discrete $R$-modules.
\end{abstract}

\maketitle

\section{Introduction}

Let $\M$ be a model category and $\W$ its morphism class of weak equivalences.
Recall that the homotopy category $\mathsf{Ho}(\M)$, associated to $\M$, is an ordinary category obtained by inverting all weak equivalences, and can also be denoted $\M[ \W^{-1}]$, see \cite[1.2.1, 1.2.10]{hovey}. However, the higher homotopy information is lost in $\mathsf{Ho}(\M)$. Dwyer and Kan, in \cite{dwyer-kan}, suggested instead a simplicial category $\mathsf{L}^H(\M, \W)$ sometimes called \emph{the hammock localization} of $\M$, that retains the higher information.
The idea has been translated into $\infty$-categories by Lurie in \cite[1.3.4.1, 1.3.4.15]{lurie1}. Following \cite{hin}, we shall prefer the term of\emph{ Dwyer-Kan localization} instead of \emph{underlying $\infty$-category} of a model category (see motivation by Remark \ref{rem: why call it DK loc?} below). If the model category is endowed with a symmetric monoidal structure compatible with the model structure, then the Dwyer-Kan localization is symmetric monoidal with respect to the derived tensor product.

The main result of this paper is Theorem \ref{thm: Weak monoidal eq imply strong in infinity} which shows that weak monoidal Quillen equivalences (as in \cite{monmodSS}) lift to equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localizations.

Any $\ai$-ring spectrum is homotopic to a strictly unital and associative ring spectrum, in some monoidal model category representing spectra, say symmetric spectra.
Similarly, any $\ei$-ring spectrum is homotopic to a strictly unital, associative and commutative ring spectrum. See \cite{SS}, \cite{MMSS}, \cite{MM} and \cite{EKMM}.
Associative and commutative algebras in the Dwyer-Kan localization of a symmetric monoidal model category $\M$ are precisely the $\ai$-algebras and $\ei$-algebras of $\M$, see  \cite[4.1.8.4, 4.5.4.7]{lurie1}.

In this paper, we answer the following question: can $\ai$-coalgebras and $\ei$-coalgebras  in spectra be homotopic to strictly counital, coassociative and cocommutative coalgebras over the sphere spectrum? In other words, can we \emph{rigidify} the comultiplication in spectra?
We show in Corollary \ref{cor: non rigi  for coassociative} that $\ai$-coalgebras cannot be rigidified to strictly coassociative and counital coalgebras in spectra, following a previous result in \cite{perouxshipley}.

A consequence of Theorem \ref{thm: Weak monoidal eq imply strong in infinity} is that it provides a Dold-Kan correspondence for $\ai$ and $\ei$-coalgebras in the Dwyer-Kan localization of  simplicial modules and non-negatively graded chain complexes. At the level of model categories, it was shown in \cite{sore3} that the Dold-Kan correspondence does not lift to a Quillen equivalence between simplicial coalgebras and chain coalgebras.
Therefore, we show in Corollary \ref{cor: failure of rigi} that $\ai$-coalgebras in connective modules over an Eilenberg-Mac Lane spectrum of a commutative ring do not correspond to their strict analogue in either simplicial modules or non-negatively graded chain complexes. Moreover, we show in Corollary \ref{cor: stable dold-kan eq} that the zig-zag of Quillen equivalences from \cite{hzalgshipley} provides Lurie's equivalence in the stable Dold-Kan correspondence {\cite[7.1.2.13]{lurie1}}.

In \cite{connectivecomod}, we are interested in comodules in the Dwyer-Kan localization of non-negatively graded chain complexes over a finite product of fields and show that homotopy coherent comodules can be rigidified to strict comodules.

The paper is constructed as follows. In Section \ref{section: dwyer kan} we recall the various definitions of Dwyer-Kan localizations in the literature in anticipation of our main result Theorem \ref{thm: Weak monoidal eq imply strong in infinity}.
In Section \ref{sec: rigidification setting}, we provide a comparison maps between $\ai$ and $\ei$ coalgebras and their strict analogues. We also provide a simple example where these maps are equivalences in Proposition \ref{prop: works for cartesian}.
In Section \ref{section: DK loc for coalgebras}, we apply Theorem \ref{thm: Weak monoidal eq imply strong in infinity} to the weak monoidal Quillen equivalences of \cite{monmodSS} and \cite{hzalgshipley} and show the failure of rigidification for connective discrete modules in Corollary \ref{cor: failure of rigi}.
In Section \ref{sec: non rigi for spectra}, we provide a model structure for strict coalgebras in symmetric spectra following \cite{left2} and show in Theorem \ref{thm: perouxshipley result} that its Dwyer-Kan localization is not equivalent to the $\infty$-category of $\ai$-coalgebras in spectra.

\subsection*{Acknowledgment}The results here are part of my PhD thesis \cite{phd}, and as such, I would like to express my gratitude to my advisor Brooke Shipley for her help and guidance throughout the years. I would also like to thank Ben Antieau and Tasos Moulinos for helpful discussions in the early process of writing this paper.

\section{The Dwyer-Kan Localization of a Model Category}
\label{section: dwyer kan}

We show here our main result which is Theorem \ref{thm: Weak monoidal eq imply strong in infinity}. We recall the definition of Dwyer-Kan localization following \cite{hin}, \cite{tch} and \cite{lurie1}.

\subsection{The General Definition}\label{DKloc: general}
We begin with the general definition of the Dwyer-Kan localization of an $\infty$-category.

\begin{defi}[{\cite[1.3.4.1]{lurie1}}]\label{def: dk loc C[W-1] of any inf cat}
Let $\Cinf$ be an $\infty$-category and fix a collection  $\mathcal{W}\subseteq \mathsf{Hom}_{\mathsf{sSet}}(\Delta^1, \Cinf)$ of morphisms in $\Cinf$. The \emph{Dwyer-Kan localization of $\Cinf$ with respect to the collection $\mathcal{W}$} is an $\infty$-category, denoted $\Cinf [\mathcal{W}^{-1}]$, together with a functor $f:\Cinf\rightarrow \Cinf [\mathcal{W}^{-1}]$ that respects the following universal property.
\begin{enumerate}
\item[(U)] For any other $\infty$-category $\mathcal{D}$, the functor $f$ induces an equivalence of $\infty$-categories:
\[
\begin{tikzcd}
\mathcal{Fun}( \Cinf [\mathcal{W}^{-1}] , \mathcal{D}) \ar{r}{\simeq} & \mathcal{Fun}^\mathcal{W}(\Cinf, \mathcal{D}),
\end{tikzcd}
\]
where $\mathcal{Fun}^\mathcal{W}(\Cinf, \mathcal{D})$ is the full subcategory of functors $\Cinf \rightarrow \mathcal{D}$ that sends morphisms in $\mathcal{W}$ to equivalences in $\mathcal{D}$.
\end{enumerate}
\end{defi}

The Dywer-Kan localization $\Cinf[\mathcal{W}^{-1}]$ always exists, for any choice of $\Cinf$ and $\mathcal{W}$, see \cite[1.3.4.2]{lurie1}, and is unique up to a contractible choice.
We shall be interested in the case when $\Cinf$ equals $\N(\M)$, the nerve of a model category $\M$.

\begin{defi}[{\cite[1.3.4.15]{lurie1}}]
Let $\M$ be a model category and $\W$ its class of weak equivalences.
We call $\N(\M)[\W^{-1}]$ the \emph{Dwyer-Kan localization of $\M$ with respect to $\W$} as in Definition \ref{def: dk loc C[W-1] of any inf cat}, where we abuse notation and let $\W$ denote the induced class of morphisms in the nerve $\N(\M)$.
\end{defi}

Notice that the homotopy category of $\N(\M)[{\W}^{-1}]$ is precisely the homotopy category $\mathsf{Ho}(\M)$ associated to $\M$ as in \cite[1.2.1, 1.2.10]{hovey}.

\begin{rem}\label{rem: why call it DK loc?}
We do not define the hammock localization $\mathsf{L}^H(\M, \W)$, but invite the interested reader to look for the explicit definition in \cite[2.1]{dwyer-kan}.
Since simplicial categories represent  $\infty$-categories,  a hammock localized simplicial category $\mathsf{L}^H(\M, \W)$ is a model for the Dwyer-Kan localization $\N(\M)[\W^{-1}]$.
More precisely, by \cite[2.2.5.1]{htt}, there is a Quillen equivalence between the category of simplicial sets $\mathsf{sSet}$ endowed with the Joyal model structure and the category of simplicial categories $\mathsf{sCat}$ endowed with the Bergner model structure (see \cite{bergner}):
\[
\begin{tikzcd}[column sep= huge]
\mathsf{sSet} \ar[shift left=2]{r}{\mathfrak{C}}[swap]{\perp} & \mathsf{sCat}.\ar[shift left=2]{l}{\mathfrak{N}}
\end{tikzcd}
\]
The functor $\mathfrak{N}:\mathsf{sCat} \rightarrow \mathsf{sSet}$ is the homotopy coherent nerve, or the simplicial nerve, as in \cite[1.1.5.5]{htt}.
After a fibrant replacement, the functor $\mathfrak{N}$ sends  $\mathsf{L}^H(\M, \W)$ to the equivalence class of $\N(\M)[\W^{-1}]$,
as seen in \cite[1.3.1]{hin}.
\end{rem}

\begin{rem}\label{rem: DK loc on cof and fib are same}
As noted in \cite[1.3.4.16]{lurie1}, \cite[1.3.4]{hin}, and \cite[8.4]{dwyer-kan}, if the model category $\M$ admits \emph{functorial} fibrant and cofibrant replacement, in the sense of \cite[1.1.1. 1.1.3]{hovey}, then the following $\infty$-categories are equivalent:
\[
\N(\M_c)[\W^{-1}] \simeq \N(\M)[\W^{-1}] \simeq \N(\M_f)[\W^{-1}],
\]
where $\M_c\subseteq \M$ is the full subcategory of cofibrant objects, and $\M_f\subseteq \M$ is the full subcategory of fibrant objects.
\end{rem}

\subsection{Symmetric Monoidal Dwyer-Kan Localization}\label{DKloc: sym DK}

We now construct the symmetric mono\-idal structure on the Dwyer-Kan localization of a symmetric mono\-idal model category $\M$. This is a recollection of Appendix A in \cite{tch} and Section 4.1.7 on monoidal model categories in \cite{lurie1}.

\begin{defi}[{\cite[4.1.7.4]{lurie1}, \cite[A.4, A.5]{tch}}]\label{def: dk sym mon loc}
Let $\Cinf^\o$ be a symmetric mono\-idal $\infty$-category. Let $\mathcal{W}\subseteq \mathsf{Hom}_{\mathsf{sSet}}(\Delta^1, \Cinf)$ be a class of edges in $\Cinf$ that is stable under homotopy, composition and contains all equivalences.
Suppose further that $\o: \Cinf \times \Cinf \rightarrow \Cinf$ preserves the class $\mathcal{W}$ separately in each variable.
The \emph{symmetric monoidal Dywer-Kan localization of $\Cinf^\o$ with respect to $\mathcal{W}$} is a symmetric monoidal $\infty$-category, denoted $\Cinf[\mathcal{W}^{-1}]^\o$, together with a symmetric monoidal functor $i:\Cinf^\o\rightarrow\Cinf[\mathcal{W}^{-1}]^\o$ which is characterized by the following universal property.
\begin{enumerate}
\item[(U)]For any other symmetric monoidal $\infty$-category $\mathcal{D}^\o$, the functor $i$ induces an equivalence of $\infty$-categories:
\[
\mathcal{Fun}_\o(\Cinf[\mathcal{W}^{-1}]^\o, \mathcal{D}^\o) \stackrel{\simeq}\longrightarrow \mathcal{Fun}_\o^\mathcal{W}(\Cinf^\o, \mathcal{D}^\o),
\]
where $\mathcal{Fun}_\o^\mathcal{W}(\Cinf^\o, \mathcal{D}^\o)$ is the full subcategory of symmetric monoidal functors $\Cinf^\o\rightarrow \mathcal{D}^\o$ that sends $\mathcal{W}$ to equivalences.
\end{enumerate}
\end{defi}

As noticed in \cite[A.5]{tch}, the underlying $\infty$-category of the symmetric monoidal category $\Cinf[\mathcal{W}^{-1}]^\o$ is precisely the Dwyer-Kan localization of $\Cinf$ with respect to $\mathcal{W}$ in the sense of Definition \ref{def: dk loc C[W-1] of any inf cat}, i.e.:
\[
\Big(\Cinf[\mathcal{W}^{-1}]^\o\Big)_{\langle 1 \rangle} \simeq \Cinf [\mathcal{W}^{-1}].
\]

\begin{rem}\label{rem: W edges are over id}
Let $\Cinf^\o$ and $\mathcal{W}$ be as in Definition \ref{def: dk sym mon loc}. Given the symmetric monoidal structure $\Cinf^\o\rightarrow \N(\Fin)$, products of $n$ edges in $\mathcal{W}$ in $\Cinf$ correspond precisely, under the equivalence:
\[
\Cinf^{\times n} \simeq \Cinf^\o_{\langle n \rangle},
\]
to morphisms lying over $\id_{\langle n \rangle}$ in $\N(\Fin)$. This defines a class of edges $\mathcal{W}^\o\subseteq \mathsf{Hom}_{\mathsf{sSet}}(\Delta^1, \Cinf^\o)$. Then the Dwyer-Kan localization of $\Cinf^\o$ with respect to $\mathcal{W}^\o$, in the sense of Definition \ref{def: dk loc C[W-1] of any inf cat}, denoted $\Cinf^\o\left[{\left(\mathcal{W}^\o\right)}^{-1}\right]$, is equivalent to $\Cinf[\mathcal{W}^{-1}]^\o$ defined above.
\end{rem}

We would like to study the case where the underlying $\infty$-category of $\Cinf^\o$ is the Dwyer-Kan localization $\N(\M)[\W^{-1}]$ of a model category $\M$.
We first recall the induced symmetric monoidal structure on the nerve of a symmetric monoidal category.

\begin{defi}[{\cite[2.0.0.1]{lurie1}}]\label{def: C otimes of a sym mon cat from lurie}
Let $(\C, \o, \I)$ be a symmetric monoidal category. Define a new category $\C^\o$, called \emph{the operator category of $\C$}, as follows.
\begin{itemize}
\item Objects are sequences $(C_1, \ldots, C_n)$ where each $C_i$ is an object in $\C$, for all $1\leq i \leq n$, for some $n \geq 1$. We allow the case $n=0$ and thus the empty set $\emptyset$  as a sequence.

\item A morphism $(C_1, \ldots, C_n) \rightarrow (C'_1, \ldots, C'_m)$ in $\C^\o$ is a pair $(\alpha, \{ f_j \})$, where $\alpha$ is a map of finite sets $\alpha: \langle n \rangle \rightarrow \langle m \rangle $ and $\{f_j \}$ is a collection of $m$-morphisms in $\C$:
\[
f_j:\bigotimes_{i\in\alpha^{-1}(j)} C_i \longrightarrow C'_j,
\]
for all $1 \leq j \leq m$. If $\alpha^{-1}(j)=\emptyset$, then $f_j$ is a morphism $\I\rightarrow C_j'$.
\item The composition of morphisms in $\C^\o$ is defined using the compositions in $\Fin$ and $\C$ together with the associativity of the symmetric monoidal structure of $\C$.
\item The identity morphism on an object $(C_1, \ldots, C_n)$ is given by the identities in $\Fin$ and $\C$: $(\id_{\langle n \rangle}, \{\id_{C_j}\})$.
\end{itemize}
\end{defi}

We obtain a functor:
\[
\C^\o \longrightarrow \Fin,
\]
that sends $(C_1, \ldots C_n)$ to $\langle n \rangle$. The induced functor $\N(\C^\o) \rightarrow \N(\Fin)$ in $\infty$-categories is coCartesian and defines a symmetric monoidal structure.

\begin{prop}[{\cite[2.1.2.21]{lurie1}}]\label{prop: N(C) is a sym mon cat}
Given $(\C, \o, \I)$ a symmetric monoidal category, let $\C^\o$ be the operator category of $\C$. Then the nerve $\N(\C^\o)$ is a symmetric monoidal $\infty$-category whose underlying $\infty$-category is $\N(\C)$.
\end{prop}

\begin{rem}\label{rem: explicit coCart lift for C^o}
Let us provide the coCartesian lifts for $\N(\C^\o)\rightarrow \N(\Fin)$. Given $(C_1, \ldots, C_n)$ in $\C^\o$, and $\alpha: \langle n \rangle  \rightarrow \langle m \rangle$ a map in $\Fin$, the associated coCartesian lift is induced by defining $C'_j$ as follows:
\[
C'_j:= \bigotimes_{i\in \alpha^{-1}(j)} C_i,
\]
for each $1\leq j \leq m$. Define $C_j'=\I$ whenever $\alpha^{-1}(j)=\emptyset$. This determines a morphism $(C_1, \ldots, C_n) \rightarrow (C'_1, \ldots, C'_m)$ in the operator category $\C^\o$ as desired.
\end{rem}

If the symmetric monoidal category $(\C, \o, \I)$ happens to  be endowed with a model structure, the bifunctor $\o: \C\times \C \rightarrow \C$ need not preserve weak equivalences in either variable. We need to restrict to \emph{(symmetric) monoidal model categories} as in \cite[4.2.6]{hovey}.
In any  monoidal model category $(\M, \o, \I)$, the tensor $\o:\M\times \M \rightarrow \M$ preserves weak equivalences in each variable, if we restrict to cofibrant objects $\M_c\subseteq \M$. Moreover. the tensor product of cofibrant objects is again cofibrant. In model categories, this allows us to define a \emph{derived tensor product} for the homotopy category $\mathsf{Ho}(\M)=\M[\W^{-1}]$, see \cite[4.3.2]{hovey}.
In higher category, the transition between the tensor product and the derived tensor product is exactly through the Dwyer-Kan localization of a symmetric monoidal $\infty$-category as in Definition \ref{def: dk sym mon loc}.
If we suppose in addition that $\I$ is cofibrant, then, as in Definition \ref{def: C otimes of a sym mon cat from lurie}, we can define $\M_c^\o\subseteq \M^\o$ from the full subcategory of cofibrant objects $\M_c\subseteq \M$, since $(\M_c, \o, \I)$ is symmetric monoidal.

\begin{prop}[{\cite[4.1.7.6]{lurie1}, \cite[A.7]{tch}}]\label{prop: compatibility of DK in sym mon}
Let $(\M, \o, \I)$ be a symmetric monoidal model category. Suppose that $\I$ is cofibrant.
Then the Dwyer-Kan localization $\N(\M_c)[{\W}^{-1}]$ of $\M$ can be given the structure of symmetric monoidal $\infty$-category via the symmetric monoidal Dwyer-Kan localization of $\N(\M_c^\o)$ in the sense of \textup{Definition \ref{def: dk sym mon loc}},
\[
\begin{tikzcd}
\N(\M_c^\o) \ar{r} & \N(\M_c)[{\W}^{-1}]^\o,
\end{tikzcd}
\]
where ${\W}$ is the class of weak equivalences restricted to cofibrant objects in $\M$.
\end{prop}

\begin{rem}\label{rem:Dk SYM loc on cof or not is the same }
We warn the reader of the following technicality.
The inclusion of cofibrant objects $\M_c\subseteq \M$ induces a \emph{lax} symmetric monoidal functor $\N(\M_c^\o)\rightarrow\N(\M^\o)$.
From Remark \ref{rem: DK loc on cof and fib are same}, Proposition \ref{prop: compatibility of DK in sym mon} implies that we can also construct a symmetric monoidal $\infty$-category  $\N(\M)[\W^{-1}]^\o$ whose fiber over $\langle 1 \rangle$ is precisely $\N(\M)[\mathsf{W}^{-1}]$.
However, cofibrant replacement induces a functor $\N(\M^\o)\rightarrow \N(\M)[{\W}^{-1}]^\o$ that is only lax symmetric monoidal and does not share the same properties of universality as in Definition \ref{def: dk sym mon loc}. We invite the interested reader to look at \cite[A.7]{tch} for more details.
\end{rem}

\subsection{Weak Monoidal Quillen Equivalence}\label{DKloc: QE}
Given model categories $\C$ and $\D$, denote $\W_\C$ and $\W_\D$ their respective class of weak equivalences. Suppose we are given a Quillen adjunction:
 \[\begin{tikzcd}
L:\C \ar[shift left=2]{r} & \ar[shift left=2]{l}[swap]{\perp} \D:R.
\end{tikzcd}\]
Then as the left adjoint functor $L$ preserves weak equivalences between cofibrant objects and the right adjoint functor $R$ preserves weak equivalences between fibrant objects, we get, by \cite[1.5.1]{hin}, a pair of adjoint functors in $\infty$-categories between the Dwyer-Kan localizations of $\C$ and $\D$:
\[
\begin{tikzcd}
\mathbb{L}:\N(\C)[\W^{-1}_\C] \ar[shift left=2]{r} & \ar[shift left=2]{l}[swap]{\perp} \N(\D)[\W^{-1}_\D]:\mathbb{R},
\end{tikzcd}
\]
where $\mathbb{L}$ and $\mathbb{R}$ represent the derived functors of $L$ and $R$.
If $\C$ and $\D$ are symmetric monoidal model categories, we investigate when the derived functors are symmetric monoidal functors of $\infty$-categories.

\begin{defi}[{\cite[3.6]{monmodSS}}]\label{def: weak mon qui pair}
Let $(\C, \o, \bI)$ and $(\D, \sm, \bJ)$ be symmetric monoidal model categories.
A \emph{weak monoidal Quillen pair} consists of a Quillen adjunction:
\[\begin{tikzcd}
L:(\C, \o, \bI) \ar[shift left=2]{r} & \ar[shift left=2]{l}[swap]{\perp} (\D, \sm, \bJ):R,
\end{tikzcd}
\]
where $L$ is lax comonoidal such that the following two conditions hold.
\begin{enumerate}[label=(\roman*)]
\item\label{enum: weak monoidal} For all cofibrant objects $X$ and $Y$ in $\C$, the comonoidal map:
\[
\begin{tikzcd}
L(X\o Y) \ar{r} & L(X)\Smash L(Y),
\end{tikzcd}
\]
is a weak equivalence in $\D$.
\item\label{enum; weak monoidal 2} For some (hence any) cofibrant replacement $\lambda:c\bI \stackrel{\sim}\longrightarrow \bI$ in $\C$, the composite map:
\[
\begin{tikzcd}
L(c\mathbb{I}) \ar{r}{L(\lambda)} & L(\I) \ar{r} & \bJ,
\end{tikzcd}
\]
is a weak equivalence in $\D$, where the unlabeled map is the natural comono\-idal structure of $L$.
\end{enumerate}
A weak monoidal Quillen pair is a \emph{weak monoidal Quillen equivalence} if the underlying Quillen pair is a Quillen equivalence.
\end{defi}

\begin{thm}\label{thm: Weak monoidal eq imply strong in infinity}
Let $(\C, \o, \bI)$ and $(\D, \sm, \bJ)$ be symmetric monoidal model categories with cofibrant units.
Let $\W_\C$ and $\W_\D$ be the classes of weak equivalence in $\C$ and $\D$ respectively.
Let: \[\begin{tikzcd}
L:(\C, \o, \bI) \ar[shift left=2]{r} & \ar[shift left=2]{l}[swap]{\perp} (\D, \sm, \bJ):R,
\end{tikzcd}\] be a weak monoidal Quillen pair.
Then the derived functor of  $L:\C \r \D$ induces a symmetric monoidal functor between the Dwyer-Kan localizations:
\[
\begin{tikzcd}
\mathbb{L}:\N (\C_c) \left[\W_\C^{-1}\right] \ar{r} & \N(\D_c) \left[\W_\D^{-1}\right],
\end{tikzcd}
\]
where $\C_c\subseteq \C$ and $\D_c\subseteq \D$ are the full subcategories of cofibrant objects. It is the left adjoint of the derived functor of $R:\D\rightarrow \C$.
If $L$ and  $R$ form a weak monoidal Quillen equivalence, then $\mathbb{L}$ is a symmetric monoidal equivalence of $\infty$-categories.
\end{thm}

\begin{proof}
Let $\C_c^\o$ and $\D_c^\o$ be the operator categories (Definition \ref{def: C otimes of a sym mon cat from lurie}).
Denote the symmetric monoidal Dwyer-Kan localizations (Definition \ref{def: dk sym mon loc}) by:
\[
\begin{tikzcd}
i_\C:\N(\C_c^\o) \ar{r} & \N(\C_c)[{\W}_\C^{-1}]^\o, & i_\D:\N(\D_c^\o) \ar{r} & \N(\D_c)[{\W}_\D^{-1}]^\o,
\end{tikzcd}
\]
and denote their coCartesian fibrations by:
\[
\begin{tikzcd}
p: \N(\C_c)[{\W}_\C^{-1}]^\o \ar{r} & \N(\Fin), & q:  \N(\D_c)[{\W}_\D^{-1}]^\o \ar{r} & \N(\Fin).
\end{tikzcd}
\]
The functor $L:\C\rightarrow \D$, as a left Quillen functor, defines $\N(\C_c)\rightarrow \N(\D_c)$, and hence a functor $L^\o: \N(\C_c^\o) \rightarrow \N(\D_c^\o)$ that is compatible with the coCartesian structures:
\[
\begin{tikzcd}
 \N(\C_c^\o) \ar{r}{L^\o} \ar{dr} & \N(\D_c^\o)\ar{d} \ar{r}{i_\D}&  \N(\D_c)[{\W}_\D^{-1}]^\o\ar{dl}{q}\\
 & \N(\Fin).&
\end{tikzcd}
\]
We show that the composite:
\[
\begin{tikzcd}
\N(\C_c^\o) \ar{r}{L^\o} & \N(\D_c^\o) \ar{r}{i_\D} & \N(\D_c)[{\W}_\D^{-1}]^\o
\end{tikzcd}
\]
is a symmetric monoidal functor that sends ${\W}_\C$ to equivalences, i.e., belongs to the $\infty$-category: \[\mathcal{Fun}^{{\W}_\C}_\o(\N(\C_c^\o),  \N(\D_c)[{\W}_\D^{-1}]^\o),\] as in Definition \ref{def: dk sym mon loc}. The latter is clear as $L$ is a left Quillen functor.
We are left to show that the composite sends $p$-coCartesian lifts to $q$-coCartesian lifts.
Let $(C_1, \ldots, C_n)$ be an object of $\C_c^\o$, and let $\alpha: \langle n \rangle \rightarrow \langle m \rangle$ be a morphism of finite sets.
A $p$-lift of $(\alpha, (C_1, \ldots, C_n))$ is given in Remark \ref{rem: explicit coCart lift for C^o} by a certain sequence $(C'_1, \ldots, C'_m)$ in $\C_c^\o$, i.e., the induced map $(C_1, \ldots, C_n)\rightarrow (C'_1, \ldots, C'_m)$ is sent to $\alpha$ via the coCartesian functor $p$.
Since $L$ is weak monoidal functor, from \ref{enum: weak monoidal} of Definition \ref{def: weak mon qui pair}, we get that:
\[
\begin{tikzcd}
\displaystyle\bigwedge_{i\in \alpha^{-1}(j)} L(C_i) & \displaystyle\ar{l}[swap]{\sim} L \left( \bigotimes_{i\in \alpha^{-1}(j)} C_i \right)= L(C'_j),
\end{tikzcd}
\]
is a weak equivalence in $\D$, for all $1\leq j \leq m$.
In the case $\alpha^{-1}(j)=\emptyset$, we apply \ref{enum; weak monoidal 2} of Definition \ref{def: weak mon qui pair} to obtain a weak equivalence:
\[
\begin{tikzcd}
\mathbb{J} & L(\bI)=L(C_j). \ar{l}[swap]{\sim}
\end{tikzcd}
\]
Applying the localization $i_D$ and Remark \ref{rem: W edges are over id}, we get that $(L(C'_1), \ldots, L(C'_m))$ defines the desired $q$-coCartesian lift.

By the universal property (U) of the symmetric monoidal Dwyer-Kan localization in Definition \ref{def: dk sym mon loc}, the composite functor $i_\D\circ L^\o$ represents a symmetric monoidal $\infty$-functor:
\[
\begin{tikzcd}
\mathbb{L}^\o:\N(\C_c)[{\W}_\C^{-1}]^\o\ar{r} & \N(\D_c)[{\W}_\D^{-1}]^\o.
\end{tikzcd}
\]
Fiberwise over $\N(\Fin)$, the functor $\mathbb{L}^\o$ is precisely the product of the derived left adjoint functor $\mathbb{L}:\N(\C_c)[\W_\C^{-1}] \rightarrow \N(\D_c)[\W_\D^{-1}]$.
In particular, if $L$ is a Quillen equivalence, then $\mathbb{L}$ is an equivalence of $\infty$-categories, and hence $\mathbb{L}^\o$ is an equivalence of symmetric monoidal $\infty$-categories.
\end{proof}

\begin{rem}\label{rem: weak monoidal eq imply eq of coalgebras}
In \cite[3.12]{monmodSS}, Schwede and Shipley show that given a weak monoidal Quillen equivalence: \[\begin{tikzcd}
L:(\C, \o, \bI) \ar[shift left=2]{r} & \ar[shift left=2]{l}[swap]{\perp} (\D, \sm, \bJ):R,
\end{tikzcd}\] with cofibrant units, then the right adjoint $R$ induces Quillen equivalences between the category of monoids $\mathsf{Mon}(\D)$ and $\mathsf{Mon}(\C)$.
Our Theorem \ref{thm: Weak monoidal eq imply strong in infinity} strengthen the results when we work with $\infty$-categories.
In particular, given any $\infty$-operad $\Oinf^\o$,
we get an equivalence of $\infty$-categories:
\[\mathcal{Alg}_\Oinf \left( \N(\C_c)[\W_\C^{-1}] \right)\simeq \mathcal{Alg}_\Oinf \left( \N(\D_c)[\W_\D^{-1}] \right),\]
which has been challenging to prove in the case of $\Oinf=\mathbb{E}_\infty$ in the past, see for instance \cite{richtershipley} and \cite[1.3, 1.4]{mandell}.
We also obtain an equivalence on the $\infty$-categories of coalgebras (see definition in \cite[2.1]{coalgenr}):
\[
\mathcal{CoAlg}_\Oinf \left( \N(\C_c)[\W_\C^{-1}] \right)\simeq \mathcal{CoAlg}_\Oinf \left( \N(\D_c)[\W_\D^{-1}] \right).
\]
Such a result on coalgebras has been shown to be untrue in some model categories, see for instance \cite{sore3} and Section \ref{subsection: rigi failure in connective} below.
\end{rem}

\section{The Rigidification Problem}\label{sec: rigidification setting}

In this section, we want to compare homotopy coherent coalgebras  with their strict analogues. On one hand, given a nice enough symmetric monoidal model category $\M$, we can obtain its Dwyer-Kan localization which is a symmetric monoidal $\infty$-category. We can then define the $\infty$-category of $\ai$ or $\ei$-coalgebras as in \cite[2.1]{coalgenr}. On the other hand, we can consider coalgebras (also called comonoids) in the monoidal category $\M$ in the classical sense and then take their Dwyer-Kan localization as in Definition \ref{def: dk loc C[W-1] of any inf cat}. We are interested to know if the $\infty$-categories are equivalent.

There are classical rigidification results that compare $\mathbb{A}_\infty$-algebras with their strict associative analogue, see \cite[4.1.8.4]{lurie1}. There is also a comparison between the $\mathbb{E}_\infty$-case with the commutative case in \cite[4.5.4.7]{lurie1}. However, there is no reason to expect that these results dualize in general. In particular, if $\mathbb{A}_\infty$-algebras can be rigidified to strict associative algebras in a model category $\M$, there is no reason to expect that $\mathbb{A}_\infty$-coalgebras correspond to strict coassociative coalgebras in $\M$ as we shall see in Corollary \ref{cor: failure of rigi} and Theorem \ref{thm: perouxshipley result} below.

\subsection{Rigidification Properties}
Let $\C$ be a (symmetric) monoidal category and denote $\C^\o$ its operator category, as in Definition \ref{def: C otimes of a sym mon cat from lurie}.
Let $p:\C^\o\rightarrow \Delta\op$ be its associated Grothendieck opfibration (see \cite[4.5]{groth}) that determines the monoidal structure of $\C$, and induces the coCartesian fibration $\N(\C^\o)\rightarrow \N(\Delta\op)$.
There is a correspondence between monoids in $\C$ and sections of $p$ that send convex morphisms to $p$-coCartesian arrows (see \cite[4.21]{groth}).
In particular, we obtain the following equivalence of $\infty$-categories:
\[
\begin{tikzcd}
\N(\mathsf{Mon}(\C)) \ar{r} & \mathcal{Alg}_\ai(\N(\C)).
\end{tikzcd}
\]
By using opposite categories, we obtain therefore an equivalence:
\[
\begin{tikzcd}
\N(\comon(\C)) \ar{r} & \coalginf_\ai(\N(\C)).
\end{tikzcd}
\]
Let $\M$ be a symmetric monoidal model category with cofibrant unit.
Consider $\M_c\subseteq \M$ the full subcategory of cofibrant objects. Apply the above identification to $\C=\M_c$ to obtain the following equivalence in $\infty$-categories:
\[
\begin{tikzcd}
\N\Big( \comon(\M_c) \Big) \ar{r} & \coalginf_\ai( \N(\M_c))
\end{tikzcd}
\]
Let $\W$ be the class of weak equivalences in $\M$.
By Proposition \ref{prop: compatibility of DK in sym mon} there is a symmetric monoidal functor $\N(\M_c^\o)\rightarrow \N(\M_c) \left[\W^{-1}\right]^\o$, which thus provides a map of $\infty$-categories:
\[
\begin{tikzcd}
\coalginf_\ai( \N(\M_c)) \ar{r} & \coalginf_\ai\left(\N(\M_c) \left[\W^{-1}\right]\right),
\end{tikzcd}
\]
and therefore we obtain a functor of $\infty$-categories:
\[
\begin{tikzcd}
\alpha:\N\Big(\comon(\M_c)\Big) \ar{r} & \coalginf_\ai\left(\N(\M_c) \left[\W^{-1}\right]\right).
\end{tikzcd}
\]
Denote $\W_\comon$ the class of morphisms in $\comon(\M_c)$ that are weak equivalences as underlying morphisms in $\M$. Notice that the above functor $\alpha$ sends $\W_\comon$ to equivalences.
By the universal property of Dwyer-Kan localizations as in Definition \ref{def: dk loc C[W-1] of any inf cat}, we obtain the following natural functor of $\infty$-categories:
\[
\begin{tikzcd}
\alpha:\N\Big(\comon(\M_c)\Big) \left[ \W_\comon^{-1} \right] \ar{r} & \coalginf_\ai\left(\N(\M_c) \left[\W^{-1}\right]\right).
\end{tikzcd}
\]
Similarly, for the cocommutative case we obtain the natural functor of $\infty$-categories:
\[
\begin{tikzcd}
\beta:\N\Big(\ccomon(\M_c)\Big) \left[ \W_\ccomon^{-1} \right] \ar{r} & \coalginf_\ei\left(\N(\M_c) \left[ \W^{-1} \right]\right).
\end{tikzcd}
\]
\begin{defi}\label{defi: coassoc and cocom rigidification}
Let $\M$ be a symmetric monoidal model category with cofibrant unit. Let $\alpha$ and $\beta$ be the functors described above.
If $\alpha$ is an equivalence of $\infty$-categories, we say that the model category $\M$ (or its Dwyer-Kan localization) \emph{ satisfies the coassociative rigidification}.
If $\beta$ is an equivalence of $\infty$-categories, we say that $\M$ (or its Dwyer-Kan localization) \emph{satisfies the cocommutative rigidification}.
\end{defi}

\begin{rem}
In general there is no reason to expect that if a model category $\M$ respects the associative rigidification then it also respects the coassociative rigidification. We see examples below in Corollaries \ref{cor: failure of rigi} and \ref{cor: non rigi  for coassociative}. Since coalgebras in $\M$ are algebras in $\M\op$ this might be surprising to the reader.
However, a key requirement for the associative rigidification is for $\M$ to be presentable, see \cite[4.1.8.4, 4.5.4.7]{lurie1}. But if $\M$ is presentable, then $\M\op$ is \emph{not} presentable, unless $\M$ is a complete lattice, see \cite[1.64]{Adamek-Rosicky}.
\end{rem}

\begin{rem}\label{rem: not easy to do rigidification in full generality}
If we inspect the dual case of algebras \cite[4.1.8.4, 4.5.4.7]{lurie1}, we see that we should have considered the $\infty$-category $\N(\comon(\M)) \left[ \W_\comon^{-1} \right]$ and not the $\infty$-category $\N(\comon(\M_c))\left[ \W_\comon^{-1} \right]$. There are several issues with that.
\begin{itemize}

\item In general, these $\infty$-categories are not equivalent unless for instance $\M$ admits a functorial lax comonoidal cofibrant replacement. This means there is a functor $Q:\M\rightarrow \M_c$ such that there is a natural map: \[Q(X\otimes Y)\rightarrow Q(X)\otimes Q(Y),\] for any $X$ and $Y$ in $\M$. The main issue is that in general the functor $\N(\M_c^\o)\rightarrow \N(\M^\o)$ is only lax symmetric monoidal, see Remark \ref{rem:Dk SYM loc on cof or not is the same }.
Of course, if all objects in $\M$ are cofibrant, no such issues appear.

\item There is no guarantee to have a model structure on $\comon(\M)$ whose weak equivalences are $\W_\comon$, even when using the left-induced methods from \cite{left2}.
Even though we do not need a model category to define $\N(\comon(\M))\left[ \W_\comon^{-1} \right]$, this would help to determine if there was some compa\-tibility with $\M$.
For instance, if we suppose $\M$ is combinatorial monoidal model category and there exists a model structure on the category of coalgebra so that the forgetful-cofree adjunction:
\[
\begin{tikzcd}
U:\comon(\M) \ar[shift left=2]{r} & \ar[shift left=2]{l}[swap]{\perp} \M:\cofree,
\end{tikzcd}
\]
is a Quillen adjunction, then there exists a functorial cofibrant replacement $\comon(\M)\rightarrow \comon(\M_c)$ that induces an equivalence of $\infty$-categories:
\[
\N(\comon(\M_c))\left[ \W_\comon^{-1} \right] \simeq \N(\comon(\M))\left[ \W_\comon^{-1} \right].
\]
\item In the cases where $\comon(\M)$ does admit a model structure, it is in general left-induced by a model category that is not a monoidal model category. For instance, in chain complexes the left induced-model structure stem from the injective model structure (which is not monoidal) instead of the projective model structure, see \cite{left2}. See also Section \ref{sec: mod structure for coalg in sp} below.
\end{itemize}
All the above also applies to the cocommutative case.
\end{rem}

\subsection{The Cartesian Case}
We provide here a simple case of model categories satisfying the coassociative and cocommutative rigidification in the sense of Definition \ref{defi: coassoc and cocom rigidification}.
Let $(\M, \times, *)$ be a symmetric monoidal model category with respect to its Cartesian monoidal structure.  Let $\W$ be the class of weak equivalences in $\M$. Suppose the terminal object $*$ is cofibrant.

\begin{prop}\label{prop: works for cartesian}
Let $(\M, \times, *)$ be as above. Then, $\M$ satisfies the coassociative and cocommutative rigidification, i.e. the following natural maps are equivalences of $\infty$-categories:
\[
\begin{tikzcd}
\N(\comon(\M_c)) \left[ \W_\comon^{-1} \right] \ar{r}{\simeq} & \coalginf_\ai\left(\N(\M_c) \left[ \W^{-1} \right]\right),
\end{tikzcd}
\]
\[
\begin{tikzcd}
\N(\ccomon(\M_c)) \left[ \W_\ccomon^{-1} \right] \ar{r}{\simeq} & \coalginf_\ei\left(\N(\M_c) \left[ \W^{-1} \right]\right),
\end{tikzcd}
\]
and all four of the $\infty$-categories above are equivalent to the Dwyer-Kan localization $\N(\M) \left[ \W^{-1} \right]$.
\end{prop}

\begin{proof}
For any Cartesian monoidal $\infty$-category $\Cinf$, we have the equivalence:
\[
\coalginf_\ai (\Cinf) \simeq \coalginf_\ei(\Cinf) \simeq \Cinf,
\]
see \cite[2.4.3.10]{lurie1}.
For any Cartesian monoidal (ordinary) category $\C$, we have the isomorphism of categories:
\[
\comon(\C) \cong \ccomon(\C) \cong \C.
\]
We obtain that the natural maps fit in the diagram:
\[
\begin{tikzcd}
\N(\comon(\M_c)) \left[ \W_\comon^{-1} \right] \ar{r}\ar{d}[swap]{\simeq} & \coalginf_\ai\left(\N(\M_c) \left[ \W^{-1} \right]\right)\ar{d}{\simeq}\\
\N(\M_c) \left[ \W^{-1} \right] \ar[equals]{r} & \N(\M_c) \left[ \W^{-1} \right]\\
\N(\ccomon(\M_c)) \left[ \W_\ccomon^{-1} \right] \ar{u}{\simeq}\ar{r} & \coalginf_\ei\left(\N(\M_c) \left[ \W^{-1} \right]\right)\ar{u}[swap]{\simeq}.
\end{tikzcd}
\]
Thus we obtain our desired equivalences.
\end{proof}

\section{Dold-Kan Correspondence For Coalgebras}
\label{section: DK loc for coalgebras}

We now apply Theorem \ref{thm: Weak monoidal eq imply strong in infinity} to a weak monoidal Quillen equivalence induced by the Dold-Kan correspondence as in \cite{monmodSS}, all missing details can be found there.
From now on, let $R$ be a commutative discrete ring.
Let $\mathsf{sMod}_R$ denote the category of simplicial $R$-modules, and let $\mathsf{Ch}^{\geq 0}_R$ denote the category of non-negatively graded chain complexes.
The \emph{Dold-Kan correspondence} says that the \emph{normalization functor}:
\begin{equation}\label{eq: Dold-Kan equivalence}
\begin{tikzcd}
\mathsf{N}:\mathsf{sMod}_R \ar{r}{\cong} &  \mathsf{Ch}^{\geq 0}_R,
\end{tikzcd}
\end{equation}
is an equivalence of categories.
Its inverse functor is denoted $\Gamma: \mathsf{Ch}^{\geq 0}_R \rightarrow \mathsf{sMod}_R$.

We show in Corollary \ref{cor: dold kan colag} that if we derive the Dold-Kan correspondence, then we obtain a correspondence between the coalgebraic objects. Moreover, Theorem \ref{thm: Weak monoidal eq imply strong in infinity} clarifies the equivalence of the stable Dold-Kan correspondence, see Theorem \ref{cor: stable dold-kan eq}. Comparing this to a result of \cite{sore3}, we obtain that $\ai$-coalgebras of connective modules over a discrete commutative ring cannot be rigidified in the Dold-Kan context, see Section \ref{cor: failure of rigi}.

\subsection{The Derived Dold-Kan Equivalence}\label{DKloc: derived DK eq}
We endow each category with a model structure as follows.
For $\mathsf{sMod}_R$, the weak equivalences and fibrations are the underlying weak equivalences and fibrations in simplicial sets, i.e., they are weak homotopy equivalences and Kan fibrations.
For $\mathsf{Ch}^{\geq 0}_R$, we use the usual projective model structure. The weak equivalences are the quasi-isomorphisms, and the fibrations are the positive levelwise epimorphisms.
Both categories can be endowed with their usual symmetric monoidal structure induced by the tensor product of $R$-modules. However, the Dold-Kan equivalence (\ref{eq: Dold-Kan equivalence}) does \emph{not} preserve the monoidal structure.
Nonetheless, with respect to the above choice of model structures, the categories $\mathsf{sMod}_R$ and $\mathsf{Ch}^{\geq 0}_R$ are both symmetric monoidal model categories with cofibrant units.
The isomorphism of categories from (\ref{eq: Dold-Kan equivalence}) can be regarded now as either of two Quillen equivalences, depending on the choice of left and right adjoints:
\begin{equation}\label{eq: dold-kan sym mon}
\begin{tikzcd}[column sep=large]
\mathsf{Ch}^{\geq 0}_R \ar[shift left=2]{r}{\Gamma} & \ar[shift left=2]{l}{\mathsf{N}}[swap]{\perp} \mathsf{sMod}_R,
\end{tikzcd}
\end{equation}
and:
\begin{equation}\label{eq: dold-kan lax mon}
\begin{tikzcd}[column sep=large]
\mathsf{sMod}_R \ar[shift left=2]{r}{\mathsf{N}} & \ar[shift left=2]{l}{\Gamma}[swap]{\perp}\mathsf{Ch}^{\geq 0}_R .
\end{tikzcd}
\end{equation}
If we choose the normalization functor $\mathsf{N}$ to be the right adjoint as in (\ref{eq: dold-kan sym mon}), then it can be considered as lax symmetric monoidal via the shuffle map.
If we choose $\mathsf{N}$ to be the left adjoint as in (\ref{eq: dold-kan lax mon}), then the Alexander-Whitney formula gives a lax comonoidal structure which is \emph{not} symmetric.

Nevertheless, this shows that both Quillen equivalences form a weak monoidal Quillen equivalence with cofibrant units, which is symmetric in the case where $\mathsf{N}$ is a right adjoint (\ref{eq: dold-kan sym mon}).
We can therefore apply Theorem \ref{thm: Weak monoidal eq imply strong in infinity} to obtain the following.

\begin{cor}[The Derived Dold-Kan Correspondence]\label{cor: derived dold-kan eq}
Let $R$ be a commutative discrete ring.
Then the Dwyer-Kan localizations of $\mathsf{sMod}_R$ and $\mathsf{Ch}^{\geq 0}_R$ are equivalent as symmetric monoidal $\infty$-categories:
\[
 \N\left(\mathsf{sMod}_R\right) \left[\W_\Delta^{-1}\right] \simeq \N\left(\mathsf{Ch}^{\geq 0}_R\right)\left[\W_\mathsf{dg}^{-1} \right],
\]
via the left Quillen derived functor $\Gamma: \mathsf{Ch}^{\geq 0}_R\rightarrow \mathsf{sMod}_R$ from the Quillen equivalence of \textup{(\ref{eq: dold-kan sym mon})},
where $\W_\Delta$ is the class of weak homotopy equivalences between simplicial $R$-modules, and $\W_\mathsf{dg}$ is the class of quasi-isomorphisms between non-negatively graded chain complexes over $R$.
\end{cor}

In particular, applying Remark \ref{rem: weak monoidal eq imply eq of coalgebras}, we get the following result.

\begin{cor}[Dold-Kan Correspondence For Coalgebras]\label{cor: dold kan colag}
For any $\infty$-operad $\Oinf$, the normalization functor induces an equivalence of $\infty$-categories:
\[
\begin{tikzcd}
\mathcal{CoAlg}_\Oinf\left( \N\left(\mathsf{sMod}_R\right) \left[\W_\Delta^{-1}\right]\right) \ar{r}{\simeq}&  \mathcal{CoAlg}_\Oinf\left(\N\left(\mathsf{Ch}^{\geq 0}_R\right)\left[\W_\mathsf{dg}^{-1} \right]\right).
\end{tikzcd}
\]
\end{cor}

\subsection{Rigidification Failure of Connective Coalgebras}\label{subsection: rigi failure in connective}
The above result bypasses a difficulty on the level of model categories and strict coalgebras.
If we choose the second adjunction (\ref{eq: dold-kan lax mon}) as a weak Quillen monoidal equivalence, then the normalization functor, being lax comonoidal, lifts to coalgebras: \[\mathsf{N}:\coalg_R \left(\mathsf{sMod}_R\right)\rightarrow \coalg_R ( \mathsf{Ch}^{\geq 0}_R ),\]
but its inverse $\Gamma$, being only lax monoidal, does not lift to coalgebras.
Nevertheless, a right adjoint exists on the level of $R$-coalgebras, either by presentability, or using dual methods as in section 3.3 of \cite{monmodSS}. We shall denote it by $\Gamma_\coalg$.
Then, using left-induced methods of \cite{left2}, we obtain model structures and a Quillen adjunction:
\[
\begin{tikzcd}[column sep=large]
\coalg_R \left(\mathsf{sMod}_R\right) \ar[shift left =2]{r}{\mathsf{N}}[swap]{\perp} & \ar[shift left=2]{l}{\Gamma_\coalg} \coalg_R \left( \mathsf{Ch}^{\geq 0}_R \right).
\end{tikzcd}
\]
The weak equivalences are the underlying weak equivalences and every object is cofibrant, in both model categories.
However, it was shown in \cite[4.16]{sore3} that the above Quillen pair is \emph{not} a Quillen equivalence, at least when $R$ is a field. It was shown that for a particular choice of fibrant object $C$ in $\coalg_R ( \mathsf{Ch}^{\geq 0}_R)$, the counit $\mathsf{N}\left( \Gamma_\coalg \left( C \right) \right) \longrightarrow C$ is not a weak equivalence (i.e. not a quasi-isomorphism).

Therefore, the normalization functor does not induce an equivalence of $\infty$-categories on the Dwyer-Kan localizations:
\[
\begin{tikzcd}[column sep= small]
\mathsf{N}:\N\Big(\comon(\mathsf{sMod_R})\Big) \left[ \W_{\Delta, \mathsf{CoAlg}}^{-1} \right] \ar{r}{\not\simeq} & \N\Big(\comon(\mathsf{\ch}_R^{\geq 0})\Big) \left[ \W_{\mathsf{dg}, \mathsf{CoAlg}}^{-1} \right].
\end{tikzcd}
\]
Here $\W_{\Delta, \mathsf{CoAlg}}\subseteq \W_\Delta$ and $\W_{\mathsf{dg}, \mathsf{CoAlg}}\subseteq \W_\mathsf{dg}$ denote the subclasses of their respective weak equivalences between coalgebra objects.
However, Corollary \ref{cor: dold kan colag} shows that the normalization functor induces an equivalence between the $\mathbb{A}_\infty$-coalgebras:
\[
\begin{tikzcd}
\mathsf{N}:\mathcal{CoAlg}_{\mathbb{A}_\infty}\left( \N\left(\mathsf{sMod}_R\right) \left[\W_\Delta^{-1}\right]\right) \ar{r}{\simeq} &  \mathcal{CoAlg}_{\mathbb{A}_\infty}\left(\N\left(\mathsf{Ch}^{\geq 0}_R\right)\left[\W_\mathsf{dg}^{-1} \right]\right).
\end{tikzcd}
\]
By \cite[7.1.3.10]{lurie1}, the $\infty$-categories $\N(\mathsf{sMod}_R) [\W_\Delta^{-1}]$ and  $\N(\mathsf{Ch}^{\geq 0}_R)[\W_\mathsf{dg}^{-1}]$ represent the symmetric monoidal $\infty$-category $\mathcal{Mod}_{HR}^{\geq 0}$ of $HR$-modules in connective spectra. From our above discussion, we obtain the following result which says that we cannot rigidify coassociative coalgebras in one of the model categories $\ch_R^{\geq 0}$ or $\mathsf{sMod}_R$.

\begin{cor}\label{cor: failure of rigi}
Let $R$ be a commutative ring.
Consider $\mathsf{sMod}_R$ and $\mathsf{Ch}^{\geq 0}_R$ with their standard monoidal model structures.
Then one of these two categories, which we denote $\M$, does not satisfy the coassociative rigidification, i.e.:
\[
\begin{tikzcd}
\N(\coalg(\M_c)) \left[\W^{-1}\right] \ar{r}{\not\simeq} & \coalginf_{\mathbb{A}_\infty} \left(\mathcal{Mod}_{HR}^{\geq 0}\right),
\end{tikzcd}
\]
where $\W$ is the class of maps of coalgebras which are weak equivalences in $\M$.
\end{cor}

\subsection{The Stable Dold-Kan Correspondence}

Our approach also gives a new proof of the \emph{stable Dold-Kan correspondence}. This well-known result was formalized with $\infty$-categories in {\cite[7.1.2.13]{lurie1}} as follows. Let $R$ be a commutative discrete ring. Then the $\infty$-category of $HR$-modules in spectra $\mathcal{Mod}_{HR}$ is equivalent to $\infty$-category of derived $R$-modules $\mathcal{D}(R)$ as symmetric monoidal $\infty$-categories: $\mathcal{Mod}_{HR} \simeq \mathcal{D}(R)$.
However, the equivalence was not described explicitly in \cite{lurie1}.
In \cite[2.10]{hzalgshipley}, Shipley provided an explicit zig-zag of (weak monoidal) Quillen equivalences between the standard model category $\mathsf{Mod}_{HR}$  of $HR$-modules in symmetric spectra and the projective model category of chain complexes over $R$:
\[
\begin{tikzcd}
\mathsf{Mod}_{HR}  \ar[shift left =2]{r}{}[swap]{\perp} & \ar[shift left=2]{l} \mathsf{Sp}^\Sigma\left(\mathsf{sMod}_R\right) \ar[shift left=2]{d} & \\
& \mathsf{Sp}^\Sigma\left(\mathsf{Ch}^{\geq 0}_R\right)\ar[shift left=2]{u}[swap]{\dashv} \ar[shift left =2]{r}{}[swap]{\perp} &  \ar[shift left=2]{l}\mathsf{Ch}_R.
\end{tikzcd}
\]
Notice that the Dwyer-Kan localizations of $\mathsf{Mod}_{HR}$ and $\mathsf{Ch}_R$ are precisely the $\infty$-categories $\mathcal{Mod}_{HR}$ and $\mathcal{D}(R)$ respectively.
If we derive and combine the Quillen functors above, we obtain a functor of $\infty$-categories $\Theta:\mathcal{Mod}_{HR} \rightarrow \mathcal{D}(R)$.
Recall that both $\mathsf{sMod}_R$ and $\mathsf{Ch}^{\geq 0}_R$ are left proper cellular symmetric monoidal model categories.
Recall from \cite[B.3]{parsp} that there is a compatibility with stabilization and the Dwyer-Kan localization of a left proper cellular simplicial symmetric monoidal model category. Combining  the above with Corollary \ref{cor: derived dold-kan eq} and Theorem \ref{thm: Weak monoidal eq imply strong in infinity} yields the following.

\begin{cor}[The Stable Dold-Kan Correspondence]\label{cor: stable dold-kan eq}
Let $R$ be a commutative discrete ring. Then the $\infty$-category of $HR$-modules is equivalent to the $\infty$-category of derived  $R$-modules as symmetric monoidal $\infty$-categories via the functor:\[
\Theta:\mathcal{Mod}_{HR}\stackrel{\simeq}\longrightarrow \mathcal{D}(R).\]
Moreover, the functor induces an equivalence for any $\infty$-operad $\Oinf$:
\[
\coalginf_\Oinf\left(\mathcal{Mod}_{HR}\right)\stackrel{\simeq}\longrightarrow \coalginf_\Oinf\left(\mathcal{D}(R)\right).
\]
\end{cor}

\section{Coalgebras in Spectra}\label{sec: non rigi for spectra}
Based on the main result of \cite{perouxshipley}, we prove here, in Corollary \ref{cor: non rigi  for coassociative}, that the monoidal model categories of symmetric spectra (see \cite{SS}), orthogonal spectra (see \cite{MMSS} \cite{MM}), $\Gamma$-spaces (see \cite{Segal} \cite{BousfieldFrie}), and $\mathscr{W}$-spaces (see \cite{Anderson}), do not respect the coassociative rigidification, in the sense of Definition \ref{defi: coassoc and cocom rigidification}.
In other words, the strictly coassociative counital coalgebras in these monoidal categories of spectra do \emph{not} have the correct homotopy type.

We work with the symmetric monoidal model category of symmetric spectra, denoted $\SpS$ (see \cite{SS}), and claim that similar results can be obtained with the other categories mentioned above, following \cite{perouxshipley}.
Notice that we have the equivalence of $\infty$-categories:
\[
\N(\SpS)[\mathsf{W}^{-1}] \simeq \mathcal{Sp},
\]
where $\W$ is the class of stable equivalences of symmetric spectra, and $\mathcal{Sp}$ is the $\infty$-category of spectra as in \cite[1.4.3.1]{lurie1}.

\subsection{Model Structures for Coalgebras}\label{sec: mod structure for coalg in sp}
Although not necessary to show the non-rigidification, as seen in Remark \ref{rem: not easy to do rigidification in full generality}, we provide a model category for coalgebras and cocommutative coalgebras in symmetric spectra.
We shall use the left-induced methods from \cite{left2}.
In \cite[Section 5]{SS} there is a simplicial, combinatorial model structure on $\SpS$ with all objects cofibrant called the \emph{(absolute) injective stable model structure}, see also \cite[Remark III.4.13]{schwede-book}.

\begin{prop}[{\cite[5.0.1, 5.0.2]{left2}}]
For any $\bS$-algebra $A$ in $\SpS$, there exists an injective model structure on $\Mod_A(\SpS)$ left-induced from the injective stable model structure on $\SpS$:
\[
\begin{tikzcd}[column sep=large]
\Mod_A(\SpS) \ar[shift left=2]{r}{U} & \ar[shift left=2]{l}{\Hom_{\SpS}(A,-)}[swap]{\perp} \SpS,
\end{tikzcd}
\]
with cofibrations the monomorphisms and weak equivalences the stable equivalences. This model structure on $\Mod_A(\SpS)$ is simplicial and combinatorial.
\end{prop}

Let $A$ be a commutative ring spectrum (i.e. a commutative $\bS$-algebra).
The symmetric monoidal category $(\Mod_A(\SpS), \sm_A, A)$ is presentable and the smash product $\sm_A$ preserves colimits in both variables.
Thus we can apply \cite[2.7]{Porst} and we obtain the (forgetful-cofree)-adjunction between $A$-coalgebras and $A$-modules in $\SpS$:
\[
\begin{tikzcd}[column sep=large]
\coalg_A(\SpS) \ar[shift left=2]{r}{U} & \ar[shift left=2]{l}{\cofree}[swap]{\perp} \Mod_A(\SpS).
\end{tikzcd}
\]

\begin{prop} \label{prop: model cat for coalgebras in spectra }
Let $A$ be any commutative $\mathbb{S}$-algebra in symmetric spectra $\SpS$.
There exists a combinatorial model structure on $A$-coalgebras $\coalg_A(\SpS)$ left-induced by the (forgetful-cofree) adjunction from the injective stable model structure on $\Mod_A(\SpS)$.
In particular, the weak equivalences in  $\coalg_A(\SpS)$ are the underlying stable equivalences, and the cofibrations are the underlying monomorphisms.
\end{prop}

\begin{proof}
We mimic the proof of \cite[Theorem 5.0.3]{left2}. We apply \cite[2.2.1]{left2}. Tensoring with a simplicial set lifts to $A$-coalgebras. Indeed, let $K$ be a simplicial set and $(C, \Delta_C, \varepsilon_C)$ be an $A$-coalgebra. Then the free $\bS$-module $\si_+K$ is endowed with a unique (cocommutative) $\bS$-coalgebra structure $(\si_+K, \Delta_K, \varepsilon_K)$, see \cite[Lemma 2.4]{perouxshipley}, where the comultiplication $\Delta_K$ is induced by the diagonal $K_+\r K_+\sm K_+$ and the counit $\varepsilon_K$ is induced by the non-trivial map $K_+\r S^0$.
Then the tensor $K\o C:=\Sigma^\infty_+ K\sm_\bS C$ is an $A$-coalgebra with comultiplication:
\begin{eqnarray*}
\si_+ K\sm_\bS C & \stackrel{\Delta_K\sm \Delta_C}\longrightarrow & (\si_+ K\sm_\bS \si_+ K) \sm_\bS (C\sm_A C) \\
& & \cong (\si_+K\sm_\bS C)\sm_A (\si_+ K\sm_\bS C),
\end{eqnarray*}

and counit:
\[
\begin{tikzcd}
\si_+K\sm_\bS C \ar{r}{\varepsilon_K\sm \varepsilon_C} & \bS \sm_\bS A\cong A.
\end{tikzcd}
\]
There is a good cylinder object in $\sset$ given by the factorization:
\[
\begin{tikzcd}
\displaystyle S^0\coprod S^0 \ar[rightarrowtail]{r} & \Delta[1]_+=I\ar{r}{\sim} & S^0.
\end{tikzcd}
\]
Since $\Mod_A(\SpS)$ is simplicial, all objects are cofibrant, and that the smash product of an $A$-coalgebra with this factorization in $\sset$ lifts to $\coalg_A(\SpS)$, this defines a good cylinder object in $\coalg_A(\SpS)$ for any $A$-coalgebra $C$:
\[
\begin{tikzcd}
\displaystyle C\coprod C \ar[rightarrowtail]{r} & C\otimes I \ar{r}{\sim} & C,
\end{tikzcd}
\]
as $C\otimes S^0\cong C$, and colimits in $\coalg_A(\SpS)$ are computed in $\Mod_A(\SpS)$. This is a combinatorial model structure by \cite[2.23]{left1} and \cite[3.3.4]{left2}.
\end{proof}

In the proof above, we see that the tensor $K\o C$ remains a cocommutative $A$-coalgebra if $C$ is cocommutative, since the coalgebra $\si_+ K$ is always cocommutative. Thus,
we can extend the result to cocommutative $A$-coalgebras.

\begin{prop} \label{prop: model cat for COCOMMUTATIVE coalgebras in spectra}
Let $A$ be any commutative $\mathbb{S}$-algebra in symmetric spectra $\SpS$.
There exists a combinatorial model structure on the category of cocommutative $A$-coalgebras $\ccoalg_A(\SpS)$ left-induced by the (forgetful-cofree) adjunction from the injective stable model structure on $\Mod_A(\SpS)$.
In particular, the weak equivalences in  $\ccoalg_A(\SpS)$ are the underlying stable equivalences, and the cofibrations are the underlying monomorphisms.
\end{prop}

\subsection{The Failure of Rigidification}
We show here the failure of rigidification. Let $A$ and $B$ be commutative $\bS$-coalgebras. A map $A\rightarrow B$ is defined to be a \emph{positive flat cofibration of commutative $\bS$-algebras} if it is a cofibration in the model category of commutative $\bS$-algebras defined in \cite[3.2]{convenient} (or the positive flat stable model structure defined in \cite[III.6.1]{schwede-book}). Any $\mathbb{E}_\infty$-ring spectrum is equivalent to a positive flat cofibrant commutative $\bS$-algebra in $\SpS$.
As noted in \cite[2.4]{perouxshipley}, every comonoid in $(\mathsf{sSet}_*, \Smash, S^0)$ is of the form $Y_+$ and the comultiplication is given by the diagonal $Y_+\rightarrow (Y\times Y)_+\cong Y_+ \Smash Y_+$.

\begin{thm}[{\cite[3.4, 3.6]{perouxshipley}}]\label{thm: perouxshipley result}
Let $A$ be a positive flat cofibrant commutative $\mathbb{S}$-algebra in $\SpS$. Then, given any counital coassociative $A$-coalgebra $C$ in $\SpS$, the comultiplication is cocommutative and induced by the following epimorphism of $A$-coalgebras:
\[
A\Smash C_0 \longrightarrow C,
\]
where $A \Smash C_0$ is given an $A$-coalgebra structure via the diagonal on the pointed space $C_0\rightarrow C_0\Smash C_0$.
\end{thm}

Let $A$ be any commutative $\bS$-algebra. Let $\coalg_A(\SpS_c)$ denote the comonoid in the cofibrant objects of $A$-modules in $\SpS$ endowed with the absolute projective stable model structure (as in \cite[IV.6.1]{schwede-book}).  There is a natural map of $\infty$-categories:
\[
\begin{tikzcd}
 \alpha:\N(\mathsf{CoAlg}_A(\SpS_c)) \left[ \W^{-1}  \right] \ar{r} & \mathcal{CoAlg}_{\mathbb{A}_\infty}(\mathcal{Mod}_A(\mathcal{Sp})),
\end{tikzcd}
\]
where $\W$ is the class of stable equivalences between $A$-coalgebras.

\begin{cor}\label{cor: non rigi  for coassociative}
Let $A$ be a positive flat cofibrant commutative $\bS$-algebra in $\SpS$. Then the $\infty$-category of $A$-modules $\mathcal{Mod}_A(\mathcal{Sp})$ does not satisfy the coassociative rigidification.
In parti\-cular, for $A=\bS$ we have:
\[
\begin{tikzcd}
\N(\coalg_\bS(\SpS_c)) \left[ \W^{-1}  \right] \ar{r}{\not\simeq} & \coalginf_{\mathbb{A}_\infty} (\mathcal{Sp})
\end{tikzcd}
\]
\end{cor}

\begin{proof}
Let $(C, \Delta, \varepsilon)$ be an $A$-coalgebra in $\SpS$ that is cofibrant as an $A$-module in the (absolute) projective stable model structure.
Suppose the functor:
\[
\begin{tikzcd}
 \alpha:\N(\mathsf{CoAlg}_A(\SpS_c)) \left[ \W^{-1}  \right] \ar{r} & \mathcal{CoAlg}_{\mathbb{A}_\infty}(\mathcal{Mod}_A(\mathcal{Sp})),
\end{tikzcd}
\]
is an equivalence of $\infty$-categories.
By Theorem \ref{thm: perouxshipley result}, we see that $\alpha(C)$ is automatically an $\ei$-coalgebra.
But there exist $\ai$-coalgebras in $\mathcal{Sp}$ that are not $\ei$-coalgebras.
Indeed, take any compact topological group that is not Abelian (say $O(2)$), then $A\wedge O(2)_+$ is an $\ai$-algebra in $\mathcal{Mod}_A(\mathcal{Sp})$ that is not commutative and is a compact spectrum. By Spanier-Whitehead duality, we obtain an $\ai$-coalgebra that is not $\ei$ in spectra.
\end{proof}

\begin{rem}
Let $A$ be a positive flat cofibrant commutative $\bS$-algebra in $\SpS$. Let $G$ be a compact Abelian Lie group (say a torus). Then the Spanier-Whitehead dual of the commutative ring spectrum $A\wedge G_+$ is an $\ei$-coalgebra in spectra with comultiplication that is not the diagonal.
Therefore the $\infty$-category of $A$-modules $\mathcal{Mod}_A(\mathcal{Sp})$ does not satisfy the cocommutative rigidification.
In parti\-cular, for $A=\bS$ we have:
\[
\begin{tikzcd}
\N(\ccoalg_\bS(\SpS_c)) \left[ \W^{-1}  \right] \ar{r}{\not \simeq} & \coalginf_{\ei} (\mathcal{Sp}).
\end{tikzcd}
\]
\end{rem}

\renewcommand{\bibname}{References}
\bibliographystyle{amsalpha}
\bibliography{biblio}
\end{document}


TITLE: A quiz question based on finding sum of series
QUESTION [1 upvotes]: This question was asked in my  analysis quiz and I had  no clue in the exam how it could be solved.So, I am asking here .

Find the sum of series $\sum_{n=0}^{\infty} \frac{n^2} {2^{n} } $ .

Unfortunately, I have no idea on how to approach this problem.

REPLY [3 votes]: We start with $$\sum_{n=0}^\infty \frac{x^n}{2^n} = \frac{2}{2-x}.$$
Taking the derivative, we see that $$\sum_{n=1}^\infty \frac{nx^{n-1}}{2^n} = \frac{2}{(2-x)^2}.$$
Multiplying by $x$ and taking the derivative once more, we see that $$\sum_{n=1}^\infty \frac{n^2 x^{n-1}}{2^n} = \frac{4+2x}{(2-x)^3}.$$
Lastly, plugging in $x=1$ gives that $$\sum_{n=1}^\infty \frac{n^2}{2^n} = 6.$$


TITLE: Subadditivity of Lebesgue inner measure
QUESTION [2 upvotes]: Im in a trouble. I want to prove this proposition:
Let $A,B \subseteq \Re^n $ such that $d(A,B)>0$.
Then, $ m_*(A\cup B) \le m_*(A) + m_*(B)  $
.
Where  $m_*$ is the Lebesgue inner measure.
I don't know if this lemma is true. But it would help me if it were.
Thank you! $<2+1$

REPLY [2 votes]: Your statement is true.
Recall that
$$m_{*}(A) = \sup_{U\subset A, U \textrm{meas}} m(U)= \sup_{K\subset A, K \textrm{compact}}m(K) $$
Let $d = d(A,B)$.  Consider $A_1 = \{x \ | d(x,A) \le \frac{d}{3}\}$, $B_1= \{x \ | d(x,B) \le \frac{d}{3}\}$. We have $A_1$, $B_1$ closed and disjoint, $A\subset A_1$, $B\subset B_1$. This measurable separation of $A$, $B$ is all we need.
Take $U\subset A\cup B$, $U$ measurable. We have $U\cap A_1 \subset A$, $U\cap B_1 \subset B$ and $U = (U\cap A_1) \cup U\cap B_1 $ a partition of $U$.
We have
$$m(U) = m(U\cap A_1) + m(U \cap B_1) \le m_{*}(A) + m_{*}(B)$$
Now take the supremum over all $U$ measurable, $U\subset A\cup B$.
Note that in general for $A$, $B$ disjoint we have
$$m_{*}(A)+m_{*}(B) \le m_*(A\cup B)$$
Therefore, if $A$, $B$ are moreover separated, we have equality.


TITLE: Let $a_0+a_1x+\dots+a_nx^n$ be a non zero polynomial with integer coefficients. If $p(√2+√3+√6)=0$, the smallest possible value of n is?
QUESTION [4 upvotes]: Question

Let $a_0+a_1x+\dots+a_nx^n$ be a non zero polynomial with integer coefficients. If $p(√2+√3+√6)=0$, the smallest possible value of n is?

Honestly I have no idea how to begin to solve this question.
And I'm guessing that the $a$'s are in arithmetic progression. Any help is appreciated.Thanks :)
Options for the answers are: (A) 8;   (B) 6;  (C) 4; (D) 2.

REPLY [2 votes]: Let's delve into the algebra behind @piquito's answer to see where the decreased degree comes from when one radical is the product of the other two.
Start with $x=\sqrt{a}+\sqrt{b}+\sqrt{c}$, with $a, b, c$ all positive and rational but none of them a rational square.  A polynomial having this as a root must contain the factor
$x-\sqrt{a}-\sqrt{b}-\sqrt{c}$.
We incorporate this factor into the following fourth-degree product in which the signs on the radicals form an orthgonal array:
$P(x)=$
$(x-\sqrt{a}-\sqrt{b}-\sqrt{c})·$
$(x+\sqrt{a}+\sqrt{b}-\sqrt{c})·$
$(x+\sqrt{a}-\sqrt{b}+\sqrt{c})·$
$(x-\sqrt{a}+\sqrt{b}+\sqrt{c})$.
With such an orthogonal array of signs we might expect that when we multiply the factors out all the  radicals will cancel out because a positive radical term is balanced by a corresponding negative one.  In fact such a prediction is almost correct:
$P(x)=x^4-2(a+b+c)x^2-8\sqrt{abc}x+2(ab+ac+bc)-(a^2+b^2+c^2)$.
There is one surviving radical term symmetric in $a, b, c$.  To clear it we multiply $P(x)$ as defined above by $P(-x)$; we have to incorporate the sign of $x$ itself into the orthogonal array which is thereby doubled in size.  That product is the usual eighth-degree minimal polynomial.
But suppose that in the above development $c=r^2ab$ where $r$ is any positive rational number (it does not have to be $1$).  Then $\sqrt{abc}=rab$ and now our fourth-degree polynomial has only rational coefficients:
$P(x)=x^4-2(a+b+c)x^2-8(rab)x+2(ab+ac+bc)-(a^2+b^2+c^2)$.
Thereby the minimal polynomial degree is reduced from eight to four.


TITLE: Prove that $\lim_{y \to x-}f(y) = a$ if for every sequence $(x_n)_n$ in $\mathbb{R}$ that increases towards x, the sequence $f(x_n)$ converges to $a$
QUESTION [2 upvotes]: Let $f: \mathbb{R} \to \mathbb{R}$ be a function.
Prove that: $$\forall (x_n)_n \subseteq \mathbb{R}: (x_n \to x \land
 \forall n: x_n \leq x_{n+1} \implies f(x_n) \to a)$$ implies that
   $\lim_{y \to x-}f(y) = a$

My attempt:
Suppose $\lim_{y\to x-}f(y) \neq a$. This means,
$$\exists \epsilon >0: \forall \delta >0: \exists y \in \mathbb{R}: 0 < x-y < \delta \land |f(y) - a | \geq \epsilon$$
Choosing this $\epsilon > 0$ and letting $\delta := 1/n$, we can find a sequence $(y_n)_n$ in the reals such that $0 < x-y_n< 1/n$ and $|f(y_n)-a| \geq \epsilon$
This implies that $y_n \to x$ and $f(y_n) \not\to a$ and it is also clear that $(y_n)_n$ is increasing. The result follows by contraposition.
Is this correct?

REPLY [0 votes]: Attempt to fix it , correct me if wrong:
You have:
1)$0<x-y_n \lt 1/n$  and
2)$|f(y_n)-a| \ge \epsilon$.
1) Implies $\lim_{ n \rightarrow \infty} y_n =a.$
You need to choose an increasing subsequence such that
$y_{n_k} \le y_{n_{k+1}} $.
Inductively:
0) Start: $x-y_0\lt 1/n_0.$
1) Choose $n_1$ such that :
$1/n_1 \lt (x-y_0).$
Step:
2) Choose $n_{k+1}$ such that :
$1/n_{k+1} \lt (x-y_{n_k})$.
Now apply your argument for the increasing subsequence $y_{n_{k}}.$


TITLE: Uniform Convergence of $f_n$ with sequence $\{a_k \}.$
QUESTION [1 upvotes]: Let $\{a_k\}_{k=1}^\infty \subset \mathbb R$ be a sequence, and suppose there exists constant $M>0$ s.t. $|a_k|\leqq \dfrac{M}{k^3}$ for all $k.$
Then, prove that the series of function $\{f_n : \mathbb R \to \mathbb R\}_{n=1}^\infty$ where $f_n(x):=\displaystyle\sum_{k=1}^n a_k \sin (kx)$ uniformly converges to a $C^1$ function $g$ defined on $\mathbb R.$

My attempt is below.
Let $h_n(x):=f'_n (x)= \displaystyle\sum_{k=1}^n k a_k \cos kx,$ and $h(x):=\displaystyle\sum_{k=1}^\infty k a_k \cos kx.$
$\left( h(x) \mathrm{ \ exists \ since\ }
\left|\displaystyle\sum_{k=1}^n k a_k \cos kx\right|\leqq \displaystyle\sum_{k=1}^n \dfrac{M}{k^2} \to \displaystyle\sum_{k=1}^\infty \dfrac{M}{k^2}<\infty.\right)$
Then, (maybe) $h_n$ converges to $h$ uniformly.
Noting that $f_n(x)=f_n(x)-f_n(0)=\displaystyle\int_0^x f'_n(t) dt =\int_0^x h_n(t) dt$, I get
\begin{align}
\left| f_n(x)-\int_0^x h(t) dt \right|
&=\left| \int_0^x (h_n(t)- h(t)) dt \right| \\
&\leqq \left|\int_0^x \left|h_n(t)- h(t) \right|dt \right|.
\end{align}
Since (maybe) $h_n \to h$ uniformly, I expect that $f_n(x)\to \displaystyle\int_0^x h(t) dt$ uniformly. And $g(x):=\displaystyle\int_0^x h(t) dt$ seems to work.
The problems are these.
(i) Does $h_n$ actually converge to $h$ uniformly ?
(ii) How can I firmly show $f_n(x)\to \displaystyle\int_0^x h(t) dt$ uniformly ?

REPLY [2 votes]: $ \sum\limits_{k=1}^{\infty} a_k \sin (kx)$ converges uniformly by M-Test since $|a_k \sin kx)| \leq \frac M {k^{3}}$ and $\sum \frac M {k^{3}}<\infty$.
$ \sum\limits_{k=1}^{\infty} ka_k \cos (kx)$ converges uniformly by M-Test since $|ka_k \cos (kx)| \leq \frac M {k^{2}}$ and $\sum \frac M {k^{2}}<\infty$.
Lemma
If $f_n \to f$ uniformly and $f_n' \to g$ uniformly then $f$ is differentiable and $f'=g$.
Proof: $f_n(x)=f_n(0)+\int_0^{x} f_n'(t)dt$. Taking limit on both sides we get $f(x)-f(0)=\int_0^{x} g(t) dt$. Also, $g$ is continuous (by uniformity of convergence). Hence, $f$ is differentiable and $f'=g$.
Can you finish? [Take $ f_n(x)=\sum\limits_{k=1}^{n} a_k \sin (kx)$].


TITLE: Finding the constant k from density function
QUESTION [0 upvotes]: Wondering how to find the value of the constant k for this. I know you have to integrate, so I integrated both functions, however I'm not quite sure on the steps afterwards. Any help would be greatly appreciated!

REPLY [1 votes]: We know that the integral over the density function must be equal to $1$.
That is:
$$\int_0^1 kx^2~ dx+\int_1^2 k(2-x)~ dx=1$$
Computing the two integrals gives:$$\frac{k}{2}+\frac{k}{3}=1 \iff \frac{5k}{6}=1$$ and so we have $$k=\frac{6}{5}$$