text:
News > Washington
3 young men convicted of 2015 killing in Tacoma
UPDATED: Fri., Feb. 16, 2018
TACOMA – Three young men have been convicted of fatally shooting an 18-year-old in 2015.
The News Tribune reported Thursday that jurors found one of the men guilty this week for the killing of Elijah Crawford, while the other two pleaded guilty.
Prosecutors say the men's intended target was Crawford's friend, who had been feuding with 25-year-old Alberto Colt-Sarmiento.
The shooting happened while those involved were gathered for a planned fistfight between the two.
The gunman, 19-year-old Juan Javier Zuniga-Gonzalez, pleaded guilty last year to first-degree murder, second-degree murder and two counts of first-degree attempted murder.
Colt-Sarmiento was convicted of first-degree manslaughter, second-degree murder, two counts of first-degree assault and unlawful gun possession.
Trino Valentino Martinez, who gave Zuniga-Gonzalez the gun, was sentenced to more than 21 years in prison.
Top stories in Washington
New state Rep. Rob Chase hires former Spokane County GOP chairwoman who resigned after defending white nationalist
Body cam footage captures Seattle officers directing homeless person to Cal Anderson Park for services …
Reports: Seattle police violated policies at summer protests …
Trident Seafoods reports 4 COVID-19 cases at plant in Alaska …
State Ecology Department rejects permit for Kalama methanol plant on Columbia River …


Get the Pricing Right - Never ever go any higher than market value if you are selling your home yourself. This is because buyers will not get in contact with you at all if they think that you are trying to get more than your home is worth from them. You need to do a little market analysis to see what similar homes are going for in order to set a realistic price and make sure that you incorporate all costs into a breakdown that is available for buyers.
First Impressions Count - If you are selling your home then you need to make sure that it is 100% ready, including all repairs necessary. This will set an excellent first impression and will draw in more offers.
Get Legal Documents Ready - Always make sure that all legal contracts and documents are ready for potential buyers, from the mortgage payoff, loan application and property profile right through to the property survey, exclusion list and personal property confirmation. That way you will have everything you need to hand.
Marketing is Key - Marketing is one of the most important elements of selling your home. You should take full advantage of all local resources, from newspapers to bulletin boards. The Internet is also key because you can reach more buyers there and ensure that your home is listed in house hunting searches that agents and brokers perform on a regular basis.
Be Professional - When you are showing your home, you should be a professional rather than an individual with an investment in a property. An agent will remain objective and merge into the background and you should do the same.
Set Pre-Qualifications - Do not waste your time on buyers that cannot go through the entire process from start to finish. If they cannot afford your home then screen them out.
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Understand Your Motivations - Why are you selling? This is an essential question that you need to ask yourself in order to plan how you are to sell your home. Although you need not tell anyone, the reason will definitely help you to determine exactly what terms and price you are prepared to accept.


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Q: trying to capture intervening sequence in perl regex So I am working with a Perl regex to do a basic restriction digest on DNA sequences with two enzymes.  I have a larger program that works, but it seems that I should be able to accomplish this in a few lines with a regex.  With the input DNA sequence I need to capture the enzyme cut sites, plus all the DNA in between consecutive sites.
Input: CATCCCCCGTCAAAAACACTTGAAAAACAT
Enzyme 1 cut site: CA
Enzyme 2 cut site: TG
Expected Output:
CATCCCCCGTCA

CAAAAACA

CACTTG

TGAAAAACA

CAT

I am using the following script:
my $input = "CATCCCTCGTCAAAAACACTTGAAAAACAT";

my $enzyme1="CA";
my $enzyme2="TG";

@cuts = $input =~ /(CA|TG[.]*CA|TG)/ig;

foreach $cut (@cuts){
    print "$cut\n";
}

However when I get the following:
CA

CA

CA

TG

CA

Why is my regex not capturing the both cut sites plus the intervening sequence?  I am using the parentheses for capture, but it must not be correct.
Thanks for the help in advance.

A: You can use a Positive Lookahead assertion for capturing to get overlapping matches:
my @cuts = ( $input =~ /(?=((?:CA|TG).*?(?:CA|TG|$)))/ig );

Working Demo

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text:
\section{Introduction}

Let $\mathsf{M}$ be a model category and $\mathsf{W}$ its morphism class of weak equivalences.
Recall that the homotopy category $\mathsf{Ho}(\mathsf{M})$, associated to $\mathsf{M}$, is an ordinary category obtained by inverting all weak equivalences, and can also be denoted $\mathsf{M}[ \mathsf{W}^{-1}]$, see \cite[1.2.1, 1.2.10]{hovey}. However, the higher homotopy information is lost in $\mathsf{Ho}(\mathsf{M})$. Dwyer and Kan, in \cite{dwyer-kan}, suggested instead a simplicial category $\mathsf{L}^H(\mathsf{M}, \mathsf{W})$ sometimes called \emph{the hammock localization} of $\mathsf{M}$, 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 ${\mathbb{A}_\infty}$-ring spectrum is homotopic to a strictly unital and associative ring spectrum, in some monoidal model category representing spectra, say symmetric spectra.
Similarly, any ${\mathbb{E}_\infty}$-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 $\mathsf{M}$ are precisely the ${\mathbb{A}_\infty}$-algebras and ${\mathbb{E}_\infty}$-algebras of $\mathsf{M}$, see  \cite[4.1.8.4, 4.5.4.7]{lurie1}.

In this paper, we answer the following question: can ${\mathbb{A}_\infty}$-coalgebras and ${\mathbb{E}_\infty}$-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 ${\mathbb{A}_\infty}$-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 ${\mathbb{A}_\infty}$ and ${\mathbb{E}_\infty}$-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 ${\mathbb{A}_\infty}$-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 ${\mathbb{A}_\infty}$ and ${\mathbb{E}_\infty}$ 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 ${\mathbb{A}_\infty}$-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 $\mathcal{C}$ be an $\infty$-category and fix a collection  $\mathcal{W}\subseteq \mathsf{Hom}_{\mathsf{sSet}}(\Delta^1, \mathcal{C})$ of morphisms in $\mathcal{C}$. The \emph{Dwyer-Kan localization of $\mathcal{C}$ with respect to the collection $\mathcal{W}$} is an $\infty$-category, denoted $\mathcal{C} [\mathcal{W}^{-1}]$, together with a functor $f:\mathcal{C}\rightarrow \mathcal{C} [\mathcal{W}^{-1}]$ 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:
\[
\begin{tikzcd}
\mathcal{Fun}( \mathcal{C} [\mathcal{W}^{-1}] , \mathcal{D}) \ar{r}{\simeq} & \mathcal{Fun}^\mathcal{W}(\mathcal{C}, \mathcal{D}),
\end{tikzcd}
\]
where $\mathcal{Fun}^\mathcal{W}(\mathcal{C}, \mathcal{D})$ is the full subcategory of functors $\mathcal{C} \rightarrow \mathcal{D}$ that sends morphisms in $\mathcal{W}$ to equivalences in $\mathcal{D}$.
\end{enumerate}
\end{defi}

The Dywer-Kan localization $\mathcal{C}[\mathcal{W}^{-1}]$ always exists, for any choice of $\mathcal{C}$ 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 $\mathcal{C}$ equals $\mathscr{N}(\mathsf{M})$, the nerve of a model category $\mathsf{M}$.

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

Notice that the homotopy category of $\mathscr{N}(\mathsf{M})[{\mathsf{W}}^{-1}]$ is precisely the homotopy category $\mathsf{Ho}(\mathsf{M})$ associated to $\mathsf{M}$ 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 $\mathsf{L}^H(\mathsf{M}, \mathsf{W})$, 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 $\mathsf{L}^H(\mathsf{M}, \mathsf{W})$ is a model for the Dwyer-Kan localization $\mathscr{N}(\mathsf{M})[\mathsf{W}^{-1}]$.
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  $\mathsf{L}^H(\mathsf{M}, \mathsf{W})$ to the equivalence class of $\mathscr{N}(\mathsf{M})[\mathsf{W}^{-1}]$,
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 $\mathsf{M}$ 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:
\[
\mathscr{N}(\mathsf{M}_c)[\mathsf{W}^{-1}] \simeq \mathscr{N}(\mathsf{M})[\mathsf{W}^{-1}] \simeq \mathscr{N}(\mathsf{M}_f)[\mathsf{W}^{-1}],
\]
where $\mathsf{M}_c\subseteq \mathsf{M}$ is the full subcategory of cofibrant objects, and $\mathsf{M}_f\subseteq \mathsf{M}$ 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 $\mathsf{M}$. 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 $\mathcal{C}^\o$ be a symmetric mono\-idal $\infty$-category. Let $\mathcal{W}\subseteq \mathsf{Hom}_{\mathsf{sSet}}(\Delta^1, \mathcal{C})$ be a class of edges in $\mathcal{C}$ that is stable under homotopy, composition and contains all equivalences.
Suppose further that $\o: \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ preserves the class $\mathcal{W}$ separately in each variable.
The \emph{symmetric monoidal Dywer-Kan localization of $\mathcal{C}^\o$ with respect to $\mathcal{W}$} is a symmetric monoidal $\infty$-category, denoted $\mathcal{C}[\mathcal{W}^{-1}]^\o$, together with a symmetric monoidal functor $i:\mathcal{C}^\o\rightarrow\mathcal{C}[\mathcal{W}^{-1}]^\o$ 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:
\[
\mathcal{Fun}_\o(\mathcal{C}[\mathcal{W}^{-1}]^\o, \mathcal{D}^\o) \stackrel{\simeq}\longrightarrow \mathcal{Fun}_\o^\mathcal{W}(\mathcal{C}^\o, \mathcal{D}^\o),
\]
where $\mathcal{Fun}_\o^\mathcal{W}(\mathcal{C}^\o, \mathcal{D}^\o)$ is the full subcategory of symmetric monoidal functors $\mathcal{C}^\o\rightarrow \mathcal{D}^\o$ 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 $\mathcal{C}[\mathcal{W}^{-1}]^\o$ is precisely the Dwyer-Kan localization of $\mathcal{C}$ with respect to $\mathcal{W}$ in the sense of Definition \ref{def: dk loc C[W-1] of any inf cat}, i.e.:
\[
\Big(\mathcal{C}[\mathcal{W}^{-1}]^\o\Big)_{\langle 1 \rangle} \simeq \mathcal{C} [\mathcal{W}^{-1}].
\]

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

We would like to study the case where the underlying $\infty$-category of $\mathcal{C}^\o$ is the Dwyer-Kan localization $\mathscr{N}(\mathsf{M})[\mathsf{W}^{-1}]$ of a model category $\mathsf{M}$.
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 $(\mathsf{C}, \o, \mathbb I)$ be a symmetric monoidal category. Define a new category $\mathsf{C}^\o$, called \emph{the operator category of $\mathsf{C}$}, as follows.
\begin{itemize}
\item Objects are sequences $(C_1, \ldots, C_n)$ where each $C_i$ is an object in $\mathsf{C}$, 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 $\mathsf{C}^\o$ 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 $\mathsf{C}$:
\[
f_j:\bigotimes_{i\in\alpha^{-1}(j)} C_i \longrightarrow C'_j,
\]
for all $1 \leq j \leq m$. If $\alpha^{-1}(j)=\emptyset$, then $f_j$ is a morphism $\mathbb I\rightarrow C_j'$.
\item The composition of morphisms in $\mathsf{C}^\o$ is defined using the compositions in $\mathsf{Fin}_*$ and $\mathsf{C}$ together with the associativity of the symmetric monoidal structure of $\mathsf{C}$.
\item The identity morphism on an object $(C_1, \ldots, C_n)$ is given by the identities in $\mathsf{Fin}_*$ and $\mathsf{C}$: $(\mathsf{id}_{\langle n \rangle}, \{\mathsf{id}_{C_j}\})$.
\end{itemize}
\end{defi}

We obtain a functor:
\[
\mathsf{C}^\o \longrightarrow \mathsf{Fin}_*,
\]
that sends $(C_1, \ldots C_n)$ to $\langle n \rangle$. The induced functor $\mathscr{N}(\mathsf{C}^\o) \rightarrow \mathscr{N}(\mathsf{Fin}_*)$ 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 $(\mathsf{C}, \o, \mathbb I)$ a symmetric monoidal category, let $\mathsf{C}^\o$ be the operator category of $\mathsf{C}$. Then the nerve $\mathscr{N}(\mathsf{C}^\o)$ is a symmetric monoidal $\infty$-category whose underlying $\infty$-category is $\mathscr{N}(\mathsf{C})$.
\end{prop}

\begin{rem}\label{rem: explicit coCart lift for C^o}
Let us provide the coCartesian lifts for $\mathscr{N}(\mathsf{C}^\o)\rightarrow \mathscr{N}(\mathsf{Fin}_*)$. Given $(C_1, \ldots, C_n)$ in $\mathsf{C}^\o$, and $\alpha: \langle n \rangle  \rightarrow \langle m \rangle$ a map in $\mathsf{Fin}_*$, 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 $C_j'=\mathbb I$ whenever $\alpha^{-1}(j)=\emptyset$. This determines a morphism $(C_1, \ldots, C_n) \rightarrow (C'_1, \ldots, C'_m)$ in the operator category $\mathsf{C}^\o$ as desired.
\end{rem}

If the symmetric monoidal category $(\mathsf{C}, \o, \mathbb I)$ happens to  be endowed with a model structure, the bifunctor $\o: \mathsf{C}\times \mathsf{C} \rightarrow \mathsf{C}$ 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 $(\mathsf{M}, \o, \mathbb I)$, the tensor $\o:\mathsf{M}\times \mathsf{M} \rightarrow \mathsf{M}$ preserves weak equivalences in each variable, if we restrict to cofibrant objects $\mathsf{M}_c\subseteq \mathsf{M}$. 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 $\mathsf{Ho}(\mathsf{M})=\mathsf{M}[\mathsf{W}^{-1}]$, 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 $\mathbb I$ is cofibrant, then, as in Definition \ref{def: C otimes of a sym mon cat from lurie}, we can define $\mathsf{M}_c^\o\subseteq \mathsf{M}^\o$ from the full subcategory of cofibrant objects $\mathsf{M}_c\subseteq \mathsf{M}$, since $(\mathsf{M}_c, \o, \mathbb I)$ is symmetric monoidal.

\begin{prop}[{\cite[4.1.7.6]{lurie1}, \cite[A.7]{tch}}]\label{prop: compatibility of DK in sym mon}
Let $(\mathsf{M}, \o, \mathbb I)$ be a symmetric monoidal model category. Suppose that $\mathbb I$ is cofibrant.
Then the Dwyer-Kan localization $\mathscr{N}(\mathsf{M}_c)[{\mathsf{W}}^{-1}]$ of $\mathsf{M}$ can be given the structure of symmetric monoidal $\infty$-category via the symmetric monoidal Dwyer-Kan localization of $\mathscr{N}(\mathsf{M}_c^\o)$ in the sense of \textup{Definition \ref{def: dk sym mon loc}},
\[
\begin{tikzcd}
\mathscr{N}(\mathsf{M}_c^\o) \ar{r} & \mathscr{N}(\mathsf{M}_c)[{\mathsf{W}}^{-1}]^\o,
\end{tikzcd}
\]
where ${\mathsf{W}}$ is the class of weak equivalences restricted to cofibrant objects in $\mathsf{M}$.
\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 $\mathsf{M}_c\subseteq \mathsf{M}$ induces a \emph{lax} symmetric monoidal functor $\mathscr{N}(\mathsf{M}_c^\o)\rightarrow\mathscr{N}(\mathsf{M}^\o)$.
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  $\mathscr{N}(\mathsf{M})[\mathsf{W}^{-1}]^\o$ whose fiber over $\langle 1 \rangle$ is precisely $\mathscr{N}(\mathsf{M})[\mathsf{W}^{-1}]$.
However, cofibrant replacement induces a functor $\mathscr{N}(\mathsf{M}^\o)\rightarrow \mathscr{N}(\mathsf{M})[{\mathsf{W}}^{-1}]^\o$ 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 $\mathsf{C}$ and $\mathsf{D}$, denote $\mathsf{W}_\mathsf{C}$ and $\mathsf{W}_\mathsf{D}$ their respective class of weak equivalences. Suppose we are given a Quillen adjunction:
 \[\begin{tikzcd}
L:\mathsf{C} \ar[shift left=2]{r} & \ar[shift left=2]{l}[swap]{\perp} \mathsf{D}:R.
\end{tikzcd}\]
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 $\mathsf{C}$ and $\mathsf{D}$:
\[
\begin{tikzcd}
\mathbb{L}:\mathscr{N}(\mathsf{C})[\mathsf{W}^{-1}_\mathsf{C}] \ar[shift left=2]{r} & \ar[shift left=2]{l}[swap]{\perp} \mathscr{N}(\mathsf{D})[\mathsf{W}^{-1}_\mathsf{D}]:\mathbb{R},
\end{tikzcd}
\]
where $\mathbb{L}$ and $\mathbb{R}$ represent the derived functors of $L$ and $R$.
If $\mathsf{C}$ and $\mathsf{D}$ 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 $(\mathsf{C}, \o, \mathbb I)$ and $(\mathsf{D}, \wedge, \mathbb{J})$ be symmetric monoidal model categories.
A \emph{weak monoidal Quillen pair} consists of a Quillen adjunction:
\[\begin{tikzcd}
L:(\mathsf{C}, \o, \mathbb I) \ar[shift left=2]{r} & \ar[shift left=2]{l}[swap]{\perp} (\mathsf{D}, \wedge, \mathbb{J}):R,
\end{tikzcd}
\]
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 $\mathsf{C}$, the comonoidal map:
\[
\begin{tikzcd}
L(X\o Y) \ar{r} & L(X)\wedge L(Y),
\end{tikzcd}
\]
is a weak equivalence in $\mathsf{D}$.
\item\label{enum; weak monoidal 2} For some (hence any) cofibrant replacement $\lambda:c\mathbb I \stackrel{\sim}\longrightarrow \mathbb I$ in $\mathsf{C}$, the composite map:
\[
\begin{tikzcd}
L(c\mathbb{I}) \ar{r}{L(\lambda)} & L(\mathbb I) \ar{r} & \mathbb{J},
\end{tikzcd}
\]
is a weak equivalence in $\mathsf{D}$, 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 $(\mathsf{C}, \o, \mathbb I)$ and $(\mathsf{D}, \wedge, \mathbb{J})$ be symmetric monoidal model categories with cofibrant units.
Let $\mathsf{W}_\mathsf{C}$ and $\mathsf{W}_\mathsf{D}$ be the classes of weak equivalence in $\mathsf{C}$ and $\mathsf{D}$ respectively.
Let: \[\begin{tikzcd}
L:(\mathsf{C}, \o, \mathbb I) \ar[shift left=2]{r} & \ar[shift left=2]{l}[swap]{\perp} (\mathsf{D}, \wedge, \mathbb{J}):R,
\end{tikzcd}\] be a weak monoidal Quillen pair.
Then the derived functor of  $L:\mathsf{C} \r \mathsf{D}$ induces a symmetric monoidal functor between the Dwyer-Kan localizations:
\[
\begin{tikzcd}
\mathbb{L}:\mathscr{N} (\mathsf{C}_c) \left[\mathsf{W}_\mathsf{C}^{-1}\right] \ar{r} & \mathscr{N}(\mathsf{D}_c) \left[\mathsf{W}_\mathsf{D}^{-1}\right],
\end{tikzcd}
\]
where $\mathsf{C}_c\subseteq \mathsf{C}$ and $\mathsf{D}_c\subseteq \mathsf{D}$ are the full subcategories of cofibrant objects. It is the left adjoint of the derived functor of $R:\mathsf{D}\rightarrow \mathsf{C}$.
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 $\mathsf{C}_c^\o$ and $\mathsf{D}_c^\o$ 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:
\[
\begin{tikzcd}
i_\mathsf{C}:\mathscr{N}(\mathsf{C}_c^\o) \ar{r} & \mathscr{N}(\mathsf{C}_c)[{\mathsf{W}}_\mathsf{C}^{-1}]^\o, & i_\mathsf{D}:\mathscr{N}(\mathsf{D}_c^\o) \ar{r} & \mathscr{N}(\mathsf{D}_c)[{\mathsf{W}}_\mathsf{D}^{-1}]^\o,
\end{tikzcd}
\]
and denote their coCartesian fibrations by:
\[
\begin{tikzcd}
p: \mathscr{N}(\mathsf{C}_c)[{\mathsf{W}}_\mathsf{C}^{-1}]^\o \ar{r} & \mathscr{N}(\mathsf{Fin}_*), & q:  \mathscr{N}(\mathsf{D}_c)[{\mathsf{W}}_\mathsf{D}^{-1}]^\o \ar{r} & \mathscr{N}(\mathsf{Fin}_*).
\end{tikzcd}
\]
The functor $L:\mathsf{C}\rightarrow \mathsf{D}$, as a left Quillen functor, defines $\mathscr{N}(\mathsf{C}_c)\rightarrow \mathscr{N}(\mathsf{D}_c)$, and hence a functor $L^\o: \mathscr{N}(\mathsf{C}_c^\o) \rightarrow \mathscr{N}(\mathsf{D}_c^\o)$ that is compatible with the coCartesian structures:
\[
\begin{tikzcd}
 \mathscr{N}(\mathsf{C}_c^\o) \ar{r}{L^\o} \ar{dr} & \mathscr{N}(\mathsf{D}_c^\o)\ar{d} \ar{r}{i_\mathsf{D}}&  \mathscr{N}(\mathsf{D}_c)[{\mathsf{W}}_\mathsf{D}^{-1}]^\o\ar{dl}{q}\\
 & \mathscr{N}(\mathsf{Fin}_*).&
\end{tikzcd}
\]
We show that the composite:
\[
\begin{tikzcd}
\mathscr{N}(\mathsf{C}_c^\o) \ar{r}{L^\o} & \mathscr{N}(\mathsf{D}_c^\o) \ar{r}{i_\mathsf{D}} & \mathscr{N}(\mathsf{D}_c)[{\mathsf{W}}_\mathsf{D}^{-1}]^\o
\end{tikzcd}
\]
is a symmetric monoidal functor that sends ${\mathsf{W}}_\mathsf{C}$ to equivalences, i.e., belongs to the $\infty$-category: \[\mathcal{Fun}^{{\mathsf{W}}_\mathsf{C}}_\o(\mathscr{N}(\mathsf{C}_c^\o),  \mathscr{N}(\mathsf{D}_c)[{\mathsf{W}}_\mathsf{D}^{-1}]^\o),\] 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 $\mathsf{C}_c^\o$, 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 $\mathsf{C}_c^\o$, 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 $\mathsf{D}$, 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:
\[
\begin{tikzcd}
\mathbb{J} & L(\mathbb I)=L(C_j). \ar{l}[swap]{\sim}
\end{tikzcd}
\]
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 $i_\mathsf{D}\circ L^\o$ represents a symmetric monoidal $\infty$-functor:
\[
\begin{tikzcd}
\mathbb{L}^\o:\mathscr{N}(\mathsf{C}_c)[{\mathsf{W}}_\mathsf{C}^{-1}]^\o\ar{r} & \mathscr{N}(\mathsf{D}_c)[{\mathsf{W}}_\mathsf{D}^{-1}]^\o.
\end{tikzcd}
\]
Fiberwise over $\mathscr{N}(\mathsf{Fin}_*)$, the functor $\mathbb{L}^\o$ is precisely the product of the derived left adjoint functor $\mathbb{L}:\mathscr{N}(\mathsf{C}_c)[\mathsf{W}_\mathsf{C}^{-1}] \rightarrow \mathscr{N}(\mathsf{D}_c)[\mathsf{W}_\mathsf{D}^{-1}]$.
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: \[\begin{tikzcd}
L:(\mathsf{C}, \o, \mathbb I) \ar[shift left=2]{r} & \ar[shift left=2]{l}[swap]{\perp} (\mathsf{D}, \wedge, \mathbb{J}):R,
\end{tikzcd}\] with cofibrant units, then the right adjoint $R$ induces Quillen equivalences between the category of monoids $\mathsf{Mon}(\mathsf{D})$ and $\mathsf{Mon}(\mathsf{C})$.
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 $\mathcal{O}^\o$,
we get an equivalence of $\infty$-categories:
\[\mathcal{Alg}_\mathcal{O} \left( \mathscr{N}(\mathsf{C}_c)[\mathsf{W}_\mathsf{C}^{-1}] \right)\simeq \mathcal{Alg}_\mathcal{O} \left( \mathscr{N}(\mathsf{D}_c)[\mathsf{W}_\mathsf{D}^{-1}] \right),\]
which has been challenging to prove in the case of $\mathcal{O}=\mathbb{E}_\infty$ 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}):
\[
\mathcal{CoAlg}_\mathcal{O} \left( \mathscr{N}(\mathsf{C}_c)[\mathsf{W}_\mathsf{C}^{-1}] \right)\simeq \mathcal{CoAlg}_\mathcal{O} \left( \mathscr{N}(\mathsf{D}_c)[\mathsf{W}_\mathsf{D}^{-1}] \right).
\]
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 $\mathsf{M}$, we can obtain its Dwyer-Kan localization which is a symmetric monoidal $\infty$-category. We can then define the $\infty$-category of ${\mathbb{A}_\infty}$ or ${\mathbb{E}_\infty}$-coalgebras as in \cite[2.1]{coalgenr}. On the other hand, we can consider coalgebras (also called comonoids) in the monoidal category $\mathsf{M}$ 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 $\mathsf{M}$, there is no reason to expect that $\mathbb{A}_\infty$-coalgebras correspond to strict coassociative coalgebras in $\mathsf{M}$ as we shall see in Corollary \ref{cor: failure of rigi} and Theorem \ref{thm: perouxshipley result} below.

\subsection{Rigidification Properties}
Let $\mathsf{C}$ be a (symmetric) monoidal category and denote $\mathsf{C}^\o$ its operator category, as in Definition \ref{def: C otimes of a sym mon cat from lurie}.
Let $p:\mathsf{C}^\o\rightarrow \Delta^\mathsf{op}$ be its associated Grothendieck opfibration (see \cite[4.5]{groth}) that determines the monoidal structure of $\mathsf{C}$, and induces the coCartesian fibration $\mathscr{N}(\mathsf{C}^\o)\rightarrow \mathscr{N}(\Delta^\mathsf{op})$.
There is a correspondence between monoids in $\mathsf{C}$ 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:
\[
\begin{tikzcd}
\mathscr{N}(\mathsf{Mon}(\mathsf{C})) \ar{r} & \mathcal{Alg}_{\mathbb{A}_\infty}(\mathscr{N}(\mathsf{C})).
\end{tikzcd}
\]
By using opposite categories, we obtain therefore an equivalence:
\[
\begin{tikzcd}
\mathscr{N}(\mathsf{CoAlg}(\mathsf{C})) \ar{r} & \mathcal{CoAlg}_{\mathbb{A}_\infty}(\mathscr{N}(\mathsf{C})).
\end{tikzcd}
\]
Let $\mathsf{M}$ be a symmetric monoidal model category with cofibrant unit.
Consider $\mathsf{M}_c\subseteq \mathsf{M}$ the full subcategory of cofibrant objects. Apply the above identification to $\mathsf{C}=\mathsf{M}_c$ to obtain the following equivalence in $\infty$-categories:
\[
\begin{tikzcd}
\mathscr{N}\Big( \mathsf{CoAlg}(\mathsf{M}_c) \Big) \ar{r} & \mathcal{CoAlg}_{\mathbb{A}_\infty}( \mathscr{N}(\mathsf{M}_c))
\end{tikzcd}
\]
Let $\mathsf{W}$ be the class of weak equivalences in $\mathsf{M}$.
By Proposition \ref{prop: compatibility of DK in sym mon} there is a symmetric monoidal functor $\mathscr{N}(\mathsf{M}_c^\o)\rightarrow \mathscr{N}(\mathsf{M}_c) \left[\mathsf{W}^{-1}\right]^\o$, which thus provides a map of $\infty$-categories:
\[
\begin{tikzcd}
\mathcal{CoAlg}_{\mathbb{A}_\infty}( \mathscr{N}(\mathsf{M}_c)) \ar{r} & \mathcal{CoAlg}_{\mathbb{A}_\infty}\left(\mathscr{N}(\mathsf{M}_c) \left[\mathsf{W}^{-1}\right]\right),
\end{tikzcd}
\]
and therefore we obtain a functor of $\infty$-categories:
\[
\begin{tikzcd}
\alpha:\mathscr{N}\Big(\mathsf{CoAlg}(\mathsf{M}_c)\Big) \ar{r} & \mathcal{CoAlg}_{\mathbb{A}_\infty}\left(\mathscr{N}(\mathsf{M}_c) \left[\mathsf{W}^{-1}\right]\right).
\end{tikzcd}
\]
Denote $\mathsf{W}_\mathsf{CoAlg}$ the class of morphisms in $\mathsf{CoAlg}(\mathsf{M}_c)$ that are weak equivalences as underlying morphisms in $\mathsf{M}$. Notice that the above functor $\alpha$ sends $\mathsf{W}_\mathsf{CoAlg}$ 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:
\[
\begin{tikzcd}
\alpha:\mathscr{N}\Big(\mathsf{CoAlg}(\mathsf{M}_c)\Big) \left[ \mathsf{W}_\mathsf{CoAlg}^{-1} \right] \ar{r} & \mathcal{CoAlg}_{\mathbb{A}_\infty}\left(\mathscr{N}(\mathsf{M}_c) \left[\mathsf{W}^{-1}\right]\right).
\end{tikzcd}
\]
Similarly, for the cocommutative case we obtain the natural functor of $\infty$-categories:
\[
\begin{tikzcd}
\beta:\mathscr{N}\Big(\mathsf{CoCAlg}(\mathsf{M}_c)\Big) \left[ \mathsf{W}_\mathsf{CoCAlg}^{-1} \right] \ar{r} & \mathcal{CoAlg}_{\mathbb{E}_\infty}\left(\mathscr{N}(\mathsf{M}_c) \left[ \mathsf{W}^{-1} \right]\right).
\end{tikzcd}
\]
\begin{defi}\label{defi: coassoc and cocom rigidification}
Let $\mathsf{M}$ 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 $\mathsf{M}$ (or its Dwyer-Kan localization) \emph{ satisfies the coassociative rigidification}.
If $\beta$ is an equivalence of $\infty$-categories, we say that $\mathsf{M}$ (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 $\mathsf{M}$ 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 $\mathsf{M}$ are algebras in $\mathsf{M}^\mathsf{op}$ this might be surprising to the reader.
However, a key requirement for the associative rigidification is for $\mathsf{M}$ to be presentable, see \cite[4.1.8.4, 4.5.4.7]{lurie1}. But if $\mathsf{M}$ is presentable, then $\mathsf{M}^\mathsf{op}$ is \emph{not} presentable, unless $\mathsf{M}$ 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 $\mathscr{N}(\mathsf{CoAlg}(\mathsf{M})) \left[ \mathsf{W}_\mathsf{CoAlg}^{-1} \right]$ and not the $\infty$-category $\mathscr{N}(\mathsf{CoAlg}(\mathsf{M}_c))\left[ \mathsf{W}_\mathsf{CoAlg}^{-1} \right]$. There are several issues with that.
\begin{itemize}

\item In general, these $\infty$-categories are not equivalent unless for instance $\mathsf{M}$ admits a functorial lax comonoidal cofibrant replacement. This means there is a functor $Q:\mathsf{M}\rightarrow \mathsf{M}_c$ such that there is a natural map: \[Q(X\otimes Y)\rightarrow Q(X)\otimes Q(Y),\] for any $X$ and $Y$ in $\mathsf{M}$. The main issue is that in general the functor $\mathscr{N}(\mathsf{M}_c^\o)\rightarrow \mathscr{N}(\mathsf{M}^\o)$ 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 $\mathsf{M}$ are cofibrant, no such issues appear.

\item There is no guarantee to have a model structure on $\mathsf{CoAlg}(\mathsf{M})$ whose weak equivalences are $\mathsf{W}_\mathsf{CoAlg}$, even when using the left-induced methods from \cite{left2}.
Even though we do not need a model category to define $\mathscr{N}(\mathsf{CoAlg}(\mathsf{M}))\left[ \mathsf{W}_\mathsf{CoAlg}^{-1} \right]$, this would help to determine if there was some compa\-tibility with $\mathsf{M}$.
For instance, if we suppose $\mathsf{M}$ is combinatorial monoidal model category and there exists a model structure on the category of coalgebra so that the forgetful-cofree adjunction:
\[
\begin{tikzcd}
U:\mathsf{CoAlg}(\mathsf{M}) \ar[shift left=2]{r} & \ar[shift left=2]{l}[swap]{\perp} \mathsf{M}:\mathsf{T^\vee},
\end{tikzcd}
\]
is a Quillen adjunction, then there exists a functorial cofibrant replacement $\mathsf{CoAlg}(\mathsf{M})\rightarrow \mathsf{CoAlg}(\mathsf{M}_c)$ that induces an equivalence of $\infty$-categories:
\[
\mathscr{N}(\mathsf{CoAlg}(\mathsf{M}_c))\left[ \mathsf{W}_\mathsf{CoAlg}^{-1} \right] \simeq \mathscr{N}(\mathsf{CoAlg}(\mathsf{M}))\left[ \mathsf{W}_\mathsf{CoAlg}^{-1} \right].
\]
\item In the cases where $\mathsf{CoAlg}(\mathsf{M})$ 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 $(\mathsf{M}, \times, *)$ be a symmetric monoidal model category with respect to its Cartesian monoidal structure.  Let $\mathsf{W}$ be the class of weak equivalences in $\mathsf{M}$. Suppose the terminal object $*$ is cofibrant.

\begin{prop}\label{prop: works for cartesian}
Let $(\mathsf{M}, \times, *)$ be as above. Then, $\mathsf{M}$ satisfies the coassociative and cocommutative rigidification, i.e. the following natural maps are equivalences of $\infty$-categories:
\[
\begin{tikzcd}
\mathscr{N}(\mathsf{CoAlg}(\mathsf{M}_c)) \left[ \mathsf{W}_\mathsf{CoAlg}^{-1} \right] \ar{r}{\simeq} & \mathcal{CoAlg}_{\mathbb{A}_\infty}\left(\mathscr{N}(\mathsf{M}_c) \left[ \mathsf{W}^{-1} \right]\right),
\end{tikzcd}
\]
\[
\begin{tikzcd}
\mathscr{N}(\mathsf{CoCAlg}(\mathsf{M}_c)) \left[ \mathsf{W}_\mathsf{CoCAlg}^{-1} \right] \ar{r}{\simeq} & \mathcal{CoAlg}_{\mathbb{E}_\infty}\left(\mathscr{N}(\mathsf{M}_c) \left[ \mathsf{W}^{-1} \right]\right),
\end{tikzcd}
\]
and all four of the $\infty$-categories above are equivalent to the Dwyer-Kan localization $\mathscr{N}(\mathsf{M}) \left[ \mathsf{W}^{-1} \right]$.
\end{prop}

\begin{proof}
For any Cartesian monoidal $\infty$-category $\mathcal{C}$, we have the equivalence:
\[
\mathcal{CoAlg}_{\mathbb{A}_\infty} (\mathcal{C}) \simeq \mathcal{CoAlg}_{\mathbb{E}_\infty}(\mathcal{C}) \simeq \mathcal{C},
\]
see \cite[2.4.3.10]{lurie1}.
For any Cartesian monoidal (ordinary) category $\mathsf{C}$, we have the isomorphism of categories:
\[
\mathsf{CoAlg}(\mathsf{C}) \cong \mathsf{CoCAlg}(\mathsf{C}) \cong \mathsf{C}.
\]
We obtain that the natural maps fit in the diagram:
\[
\begin{tikzcd}
\mathscr{N}(\mathsf{CoAlg}(\mathsf{M}_c)) \left[ \mathsf{W}_\mathsf{CoAlg}^{-1} \right] \ar{r}\ar{d}[swap]{\simeq} & \mathcal{CoAlg}_{\mathbb{A}_\infty}\left(\mathscr{N}(\mathsf{M}_c) \left[ \mathsf{W}^{-1} \right]\right)\ar{d}{\simeq}\\
\mathscr{N}(\mathsf{M}_c) \left[ \mathsf{W}^{-1} \right] \ar[equals]{r} & \mathscr{N}(\mathsf{M}_c) \left[ \mathsf{W}^{-1} \right]\\
\mathscr{N}(\mathsf{CoCAlg}(\mathsf{M}_c)) \left[ \mathsf{W}_\mathsf{CoCAlg}^{-1} \right] \ar{u}{\simeq}\ar{r} & \mathcal{CoAlg}_{\mathbb{E}_\infty}\left(\mathscr{N}(\mathsf{M}_c) \left[ \mathsf{W}^{-1} \right]\right)\ar{u}[swap]{\simeq}.
\end{tikzcd}
\]
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 ${\mathbb{A}_\infty}$-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:
\[
 \mathscr{N}\left(\mathsf{sMod}_R\right) \left[\mathsf{W}_\Delta^{-1}\right] \simeq \mathscr{N}\left(\mathsf{Ch}^{\geq 0}_R\right)\left[\mathsf{W}_\mathsf{dg}^{-1} \right],
\]
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 $\mathsf{W}_\Delta$ is the class of weak homotopy equivalences between simplicial $R$-modules, and $\mathsf{W}_\mathsf{dg}$ 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 $\mathcal{O}$, the normalization functor induces an equivalence of $\infty$-categories:
\[
\begin{tikzcd}
\mathcal{CoAlg}_\mathcal{O}\left( \mathscr{N}\left(\mathsf{sMod}_R\right) \left[\mathsf{W}_\Delta^{-1}\right]\right) \ar{r}{\simeq}&  \mathcal{CoAlg}_\mathcal{O}\left(\mathscr{N}\left(\mathsf{Ch}^{\geq 0}_R\right)\left[\mathsf{W}_\mathsf{dg}^{-1} \right]\right).
\end{tikzcd}
\]
\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: \[\mathsf{N}:\mathsf{CoAlg}_R \left(\mathsf{sMod}_R\right)\rightarrow \mathsf{CoAlg}_R ( \mathsf{Ch}^{\geq 0}_R ),\]
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 $\Gamma_\mathsf{CoAlg}$.
Then, using left-induced methods of \cite{left2}, we obtain model structures and a Quillen adjunction:
\[
\begin{tikzcd}[column sep=large]
\mathsf{CoAlg}_R \left(\mathsf{sMod}_R\right) \ar[shift left =2]{r}{\mathsf{N}}[swap]{\perp} & \ar[shift left=2]{l}{\Gamma_\mathsf{CoAlg}} \mathsf{CoAlg}_R \left( \mathsf{Ch}^{\geq 0}_R \right).
\end{tikzcd}
\]
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 $\mathsf{CoAlg}_R ( \mathsf{Ch}^{\geq 0}_R)$, the counit $\mathsf{N}\left( \Gamma_\mathsf{CoAlg} \left( C \right) \right) \longrightarrow C$ 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:
\[
\begin{tikzcd}[column sep= small]
\mathsf{N}:\mathscr{N}\Big(\mathsf{CoAlg}(\mathsf{sMod_R})\Big) \left[ \mathsf{W}_{\Delta, \mathsf{CoAlg}}^{-1} \right] \ar{r}{\not\simeq} & \mathscr{N}\Big(\mathsf{CoAlg}(\mathsf{\mathsf{Ch}}_R^{\geq 0})\Big) \left[ \mathsf{W}_{\mathsf{dg}, \mathsf{CoAlg}}^{-1} \right].
\end{tikzcd}
\]
Here $\mathsf{W}_{\Delta, \mathsf{CoAlg}}\subseteq \mathsf{W}_\Delta$ and $\mathsf{W}_{\mathsf{dg}, \mathsf{CoAlg}}\subseteq \mathsf{W}_\mathsf{dg}$ 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:
\[
\begin{tikzcd}
\mathsf{N}:\mathcal{CoAlg}_{\mathbb{A}_\infty}\left( \mathscr{N}\left(\mathsf{sMod}_R\right) \left[\mathsf{W}_\Delta^{-1}\right]\right) \ar{r}{\simeq} &  \mathcal{CoAlg}_{\mathbb{A}_\infty}\left(\mathscr{N}\left(\mathsf{Ch}^{\geq 0}_R\right)\left[\mathsf{W}_\mathsf{dg}^{-1} \right]\right).
\end{tikzcd}
\]
By \cite[7.1.3.10]{lurie1}, the $\infty$-categories $\mathscr{N}(\mathsf{sMod}_R) [\mathsf{W}_\Delta^{-1}]$ and  $\mathscr{N}(\mathsf{Ch}^{\geq 0}_R)[\mathsf{W}_\mathsf{dg}^{-1}]$ 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 $\mathsf{Ch}_R^{\geq 0}$ 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 $\mathsf{M}$, does not satisfy the coassociative rigidification, i.e.:
\[
\begin{tikzcd}
\mathscr{N}(\mathsf{CoAlg}(\mathsf{M}_c)) \left[\mathsf{W}^{-1}\right] \ar{r}{\not\simeq} & \mathcal{CoAlg}_{\mathbb{A}_\infty} \left(\mathcal{Mod}_{HR}^{\geq 0}\right),
\end{tikzcd}
\]
where $\mathsf{W}$ is the class of maps of coalgebras which are weak equivalences in $\mathsf{M}$.
\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 $\mathcal{O}$:
\[
\mathcal{CoAlg}_\mathcal{O}\left(\mathcal{Mod}_{HR}\right)\stackrel{\simeq}\longrightarrow \mathcal{CoAlg}_\mathcal{O}\left(\mathcal{D}(R)\right).
\]
\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 $\mathsf{Sp}^{{\Sigma}}$ (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:
\[
\mathscr{N}(\mathsf{Sp}^{{\Sigma}})[\mathsf{W}^{-1}] \simeq \mathcal{Sp},
\]
where $\mathsf{W}$ 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 $\mathsf{Sp}^{{\Sigma}}$ 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 $\mathbb{S}$-algebra $A$ in $\mathsf{Sp}^{{\Sigma}}$, there exists an injective model structure on $\mathsf{Mod}_A(\mathsf{Sp}^{{\Sigma}})$ left-induced from the injective stable model structure on $\mathsf{Sp}^{{\Sigma}}$:
\[
\begin{tikzcd}[column sep=large]
\mathsf{Mod}_A(\mathsf{Sp}^{{\Sigma}}) \ar[shift left=2]{r}{U} & \ar[shift left=2]{l}{\mathsf{Hom}_{\mathsf{Sp}^{{\Sigma}}}(A,-)}[swap]{\perp} \mathsf{Sp}^{{\Sigma}},
\end{tikzcd}
\]
with cofibrations the monomorphisms and weak equivalences the stable equivalences. This model structure on $\mathsf{Mod}_A(\mathsf{Sp}^{{\Sigma}})$ is simplicial and combinatorial.
\end{prop}

Let $A$ be a commutative ring spectrum (i.e. a commutative $\mathbb{S}$-algebra).
The symmetric monoidal category $(\mathsf{Mod}_A(\mathsf{Sp}^{{\Sigma}}), \wedge_A, A)$ is presentable and the smash product $\wedge_A$ 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 $\mathsf{Sp}^{{\Sigma}}$:
\[
\begin{tikzcd}[column sep=large]
\mathsf{CoAlg}_A(\mathsf{Sp}^{{\Sigma}}) \ar[shift left=2]{r}{U} & \ar[shift left=2]{l}{\mathsf{T^\vee}}[swap]{\perp} \mathsf{Mod}_A(\mathsf{Sp}^{{\Sigma}}).
\end{tikzcd}
\]

\begin{prop} \label{prop: model cat for coalgebras in spectra }
Let $A$ be any commutative $\mathbb{S}$-algebra in symmetric spectra $\mathsf{Sp}^{{\Sigma}}$.
There exists a combinatorial model structure on $A$-coalgebras $\mathsf{CoAlg}_A(\mathsf{Sp}^{{\Sigma}})$ left-induced by the (forgetful-cofree) adjunction from the injective stable model structure on $\mathsf{Mod}_A(\mathsf{Sp}^{{\Sigma}})$.
In particular, the weak equivalences in  $\mathsf{CoAlg}_A(\mathsf{Sp}^{{\Sigma}})$ 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 $\mathbb{S}$-module $\Sigma^\infty_+K$ is endowed with a unique (cocommutative) $\mathbb{S}$-coalgebra structure $(\Sigma^\infty_+K, \Delta_K, \varepsilon_K)$, see \cite[Lemma 2.4]{perouxshipley}, where the comultiplication $\Delta_K$ is induced by the diagonal $K_+\r K_+\wedge K_+$ and the counit $\varepsilon_K$ is induced by the non-trivial map $K_+\r S^0$.
Then the tensor $K\o C:=\Sigma^\infty_+ K\wedge_\mathbb{S} C$ is an $A$-coalgebra with comultiplication:
\begin{eqnarray*}
\Sigma^\infty_+ K\wedge_\mathbb{S} C & \stackrel{\Delta_K\wedge \Delta_C}\longrightarrow & (\Sigma^\infty_+ K\wedge_\mathbb{S} \Sigma^\infty_+ K) \wedge_\mathbb{S} (C\wedge_A C) \\
& & \cong (\Sigma^\infty_+K\wedge_\mathbb{S} C)\wedge_A (\Sigma^\infty_+ K\wedge_\mathbb{S} C),
\end{eqnarray*}
and counit:
\[
\begin{tikzcd}
\Sigma^\infty_+K\wedge_\mathbb{S} C \ar{r}{\varepsilon_K\wedge \varepsilon_C} & \mathbb{S} \wedge_\mathbb{S} A\cong A.
\end{tikzcd}
\]
There is a good cylinder object in $\mathsf{sSet}$ 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 $\mathsf{Mod}_A(\mathsf{Sp}^{{\Sigma}})$ is simplicial, all objects are cofibrant, and that the smash product of an $A$-coalgebra with this factorization in $\mathsf{sSet}$ lifts to $\mathsf{CoAlg}_A(\mathsf{Sp}^{{\Sigma}})$, this defines a good cylinder object in $\mathsf{CoAlg}_A(\mathsf{Sp}^{{\Sigma}})$ 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 $\mathsf{CoAlg}_A(\mathsf{Sp}^{{\Sigma}})$ are computed in $\mathsf{Mod}_A(\mathsf{Sp}^{{\Sigma}})$. 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 $\Sigma^\infty_+ K$ 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 $\mathsf{Sp}^{{\Sigma}}$.
There exists a combinatorial model structure on the category of cocommutative $A$-coalgebras $\mathsf{CoCAlg}_A(\mathsf{Sp}^{{\Sigma}})$ left-induced by the (forgetful-cofree) adjunction from the injective stable model structure on $\mathsf{Mod}_A(\mathsf{Sp}^{{\Sigma}})$.
In particular, the weak equivalences in  $\mathsf{CoCAlg}_A(\mathsf{Sp}^{{\Sigma}})$ 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 $\mathbb{S}$-coalgebras. A map $A\rightarrow B$ is defined to be a \emph{positive flat cofibration of commutative $\mathbb{S}$-algebras} if it is a cofibration in the model category of commutative $\mathbb{S}$-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 $\mathbb{S}$-algebra in $\mathsf{Sp}^{{\Sigma}}$.
As noted in \cite[2.4]{perouxshipley}, every comonoid in $(\mathsf{sSet}_*, \wedge, S^0)$ is of the form $Y_+$ and the comultiplication is given by the diagonal $Y_+\rightarrow (Y\times Y)_+\cong Y_+ \wedge Y_+$.

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

Let $A$ be any commutative $\mathbb{S}$-algebra. Let $\mathsf{CoAlg}_A(\mathsf{Sp}^{{\Sigma}}_c)$ denote the comonoid in the cofibrant objects of $A$-modules in $\mathsf{Sp}^{{\Sigma}}$ endowed with the absolute projective stable model structure (as in \cite[IV.6.1]{schwede-book}).  There is a natural map of $\infty$-categories:
\[
\begin{tikzcd}
 \alpha:\mathscr{N}(\mathsf{CoAlg}_A(\mathsf{Sp}^{{\Sigma}}_c)) \left[ \mathsf{W}^{-1}  \right] \ar{r} & \mathcal{CoAlg}_{\mathbb{A}_\infty}(\mathcal{Mod}_A(\mathcal{Sp})),
\end{tikzcd}
\]
where $\mathsf{W}$ 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 $\mathbb{S}$-algebra in $\mathsf{Sp}^{{\Sigma}}$. Then the $\infty$-category of $A$-modules $\mathcal{Mod}_A(\mathcal{Sp})$ does not satisfy the coassociative rigidification.
In parti\-cular, for $A=\mathbb{S}$ we have:
\[
\begin{tikzcd}
\mathscr{N}(\mathsf{CoAlg}_\mathbb{S}(\mathsf{Sp}^{{\Sigma}}_c)) \left[ \mathsf{W}^{-1}  \right] \ar{r}{\not\simeq} & \mathcal{CoAlg}_{\mathbb{A}_\infty} (\mathcal{Sp})
\end{tikzcd}
\]
\end{cor}

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

\begin{rem}
Let $A$ be a positive flat cofibrant commutative $\mathbb{S}$-algebra in $\mathsf{Sp}^{{\Sigma}}$. 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 ${\mathbb{E}_\infty}$-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 $A=\mathbb{S}$ we have:
\[
\begin{tikzcd}
\mathscr{N}(\mathsf{CoCAlg}_\mathbb{S}(\mathsf{Sp}^{{\Sigma}}_c)) \left[ \mathsf{W}^{-1}  \right] \ar{r}{\not \simeq} & \mathcal{CoAlg}_{{\mathbb{E}_\infty}} (\mathcal{Sp}).
\end{tikzcd}
\]
\end{rem}

\renewcommand{\bibname}{References}
\bibliographystyle{amsalpha}

redpajama_set_name: RedPajamaArXiv


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text:
require 'spec_helper_acceptance'

describe 'ceph usecases' do

  # passing it directly as unqoted array is not supported everywhere
  packages = "[ 'python-ceph', 'ceph-common', 'librados2', 'librbd1', 'libcephfs1' ]"

  describe 'I want to try this module, heard of ceph, want to see it in action' do

    it 'should install one monitor and one OSD on /srv/data' do
      pp = <<-EOS
        class { 'ceph::repo': }
        class { 'ceph':
          fsid                       => '82274746-9a2c-426b-8c51-107fb0d890c6',
          mon_host                   => $::ipaddress,
          authentication_type        => 'none',
          osd_pool_default_size      => '1',
          osd_pool_default_min_size  => '1',
        }
        ceph_config {
         'global/osd_journal_size': value => '100';
        }
        ceph::mon { 'a':
          public_addr         => $::ipaddress,
          authentication_type => 'none',
        }
        ceph::osd { '/srv/data': }
      EOS

      apply_manifest(pp, :catch_failures => true)
      apply_manifest(pp, :catch_changes => true)

      shell 'sleep 10' # we need to wait a bit until the OSD is up

      shell 'ceph -s', { :acceptable_exit_codes => [0] } do |r|
        expect(r.stdout).to match(/1 mons at/)
        expect(r.stderr).to be_empty
      end

      shell 'ceph osd tree | grep osd.0', { :acceptable_exit_codes => [0] } do |r|
        expect(r.stdout).to match(/up/)
        expect(r.stderr).to be_empty
      end
    end

    it 'should uninstall one osd' do
      shell 'ceph osd tree | grep osd.0', { :acceptable_exit_codes => [0] } do |r|
        expect(r.stdout).to match(/up/)
        expect(r.stderr).to be_empty
      end

      pp = <<-EOS
        ceph::osd { '/srv/data':
          ensure => absent,
        }
      EOS

      apply_manifest(pp, :catch_failures => true)
      apply_manifest(pp, :catch_changes => true)

      shell 'sleep 10' # we need to wait a bit until the OSD is down

      shell 'ceph osd tree | grep osd.0', { :acceptable_exit_codes => [1] }
    end

    it 'should uninstall one monitor' do
      pp = <<-EOS
        ceph::mon { 'a':
          ensure => absent,
        }
      EOS

      apply_manifest(pp, :catch_failures => true)
      apply_manifest(pp, :catch_changes => true)

      osfamily = fact 'osfamily'
      operatingsystem = fact 'operatingsystem'

      if osfamily == 'Debian' && operatingsystem == 'Ubuntu'
        shell 'status ceph-mon id=a', { :acceptable_exit_codes => [1] } do |r|
          expect(r.stdout).to be_empty
          expect(r.stderr).to match(/Unknown instance: ceph.a/)
        end
      end
      if osfamily == 'RedHat'
        shell 'service ceph status mon.a', { :acceptable_exit_codes => [1] } do |r|
          expect(r.stdout).to match(/mon.a not found/)
          expect(r.stderr).to be_empty
        end
      end
    end

    it 'should purge all packages' do
      pp = <<-EOS
        package { #{packages}:
          ensure => purged
        }
        class { 'ceph::repo':
          ensure  => absent,
        }
        file { [
           '/var/lib/ceph',
           '/srv/data',
          ]:
          ensure => absent,
          recurse => true,
          purge => true,
          force => true,
        }

      EOS

      apply_manifest(pp, :catch_failures => true)
      # can't check for idempotency because of https://tickets.puppetlabs.com/browse/PUP-1198
      #apply_manifest(pp, :catch_changes => true)
      apply_manifest(pp, :catch_failures => true)
    end
  end
end
# Local Variables:
# compile-command: "cd ../..
#   BUNDLE_PATH=/tmp/vendor bundle install
#   BEAKER_set=ubuntu-server-1204-x64 \
#   BUNDLE_PATH=/tmp/vendor \
#   bundle exec rspec spec/acceptance/ceph_usecases_spec.rb
# "
# End:

redpajama_set_name: RedPajamaGithub


Call Us First Home INspection has been providing home buyers/sellers, commercial property buyers/sellers, and realtors in Moore, Valley Brook, Norman, Del City, Midwest City, and the surrounding areas with expert inspections and thorough, fast reports for years. Call us today!
Catch a flick at Moore Warren Theatre.
Head to Oklahoma's only full-time dinner theater, The Yellow Rose, for dinner and a show.
Take your clubs out to Broadmoore Golf Course or Belmar Golf Club.
Get some sun at Buck Thomas Park or watch the kids practice their skateboard skills over at Moore Skatepark.
Hit the outlets at Shops at Moore on Service Rd or visit Showplace Market on Broadway, Fan Outfitters, Maurices, or Daisy Exchange on I-35 Service Rd.
Participate in one of the many events held throughout the year, like the City of Moore Fishing Derby, Dive-In Movie at the Station Aquatic Center on Broadway, Food Truck Fridays at Central Park, or Movie in the Park at Central Park.
On Thursdays and Saturdays, Central Park is also home to the Farmers Market, so you can get some local produce to spice up dinner.
Looking for great places to eat in Moore that don't involve YOU cooking? Try out Fontana Italian Restaurant on 12th (in Malibu Shopping Center), Oliveto Italian Bistro on I-35 Service Rd, Nosh on 19th, Dong A Restaurant on Eastern Ave, Hummus Mediterranean Cafe on 19th, Hideaway Pizza on 19th, Himalayas – Aroma of India on Moore Ave, The Lazy Donkey Mexican Restaurant on Broadway, Sunny Side Up on 19th, Hollie's Flatiron Grill on I-35 Service Rd, or Volcano Sushi Bar & Hibachi on I-35 Service Rd to start.
As experienced buyers and sellers ourselves, we know how important communication, timing, and money are when you're buying or selling a property, and we work to fill the gap left by other inspectors.
We also provide custom pricing and lower priced re-inspections, and we guarantee you'll have the information you need in a written report within 24 hours of your inspection.
So if you live in Moore and you need someone who's thorough, but not nit-picky to inspect your home, new construction home, pool, well, septic tank, sprinkler system, fireplace, air quality, or commercial property, call 405-640-0406 or request an inspection online. We're here to help.
If you live in Midwest City, OK, you can relax because you live in our service area which means we can take care of any commercial or residential inspections you need, when you need them.


text:
Chinese hackers caught trying to steal COVID-19 vaccine research
Agencies are raising the alarm over government-backed hackers from China trying to steal intellectual property amid ongoing disinformation efforts by Beijing to shape the virus outbreak.
A worker is seen inside the Beijing Applied Biological Technologies (XABT) research and development laboratory May 14. [Nicolas Asfouri/AFP]
WASHINGTON -- Chinese-backed hackers are attempting to steal research and intellectual property related to treatments and vaccines for COVID-19, US authorities Wednesday (May 13) warned healthcare and scientific researchers.
A statement from the US Federal Bureau of Investigation (FBI) and the Cybersecurity and Infrastructure Security Agency (CISA) warned organisations researching the disease of "likely targeting and network compromise by the People's Republic of China".
"These actors have been observed attempting to identify and illicitly obtain valuable intellectual property and public health data related to vaccines, treatments, and testing from networks and personnel affiliated with COVID-19-related research," the statement said.
"China's efforts to target these sectors pose a significant threat to our nation's response to COVID-19," it added.
The FBI and CISA urged "all organisations conducting research in these areas to maintain dedicated cybersecurity and insider threat practices to prevent surreptitious review or theft of COVID-19-related material."
"China expresses strong dissatisfaction and firm opposition to such smearing," Foreign Ministry spokesman Zhao Lijian Thursday (May 14) said in response to the allegation.
The warning comes as dozens of companies, institutes and countries around the world are racing to develop vaccines to halt the spread of the coronavirus, which has killed at least 292,000 worldwide and originated in China.
Cyber-attacks, disinformation
The warning adds to a series of alerts and reports accusing government-backed hackers in Iran, North Korea, Russia and China of malicious activity related to the coronavirus pandemic, from pumping out false news to targeting workers and scientists.
Last week, Britain and the United States warned of a rise in cyber-attacks against health professionals by organised criminals "often linked with other state actors."
CISA and Britain's National Cyber Security Centre said they had detected large-scale "password spraying" tactics -- hackers trying to access accounts through commonly used passwords -- aimed at healthcare bodies and medical research organisations.
Meanwhile, the Chinese and Russian regimes in particular have stepped up co-operation to spread false narratives over the coronavirus pandemic.
"Even before the COVID-19 crisis, we assessed a certain level of co-ordination between Russia and the People's Republic of China in the realm of propaganda," said Lea Gabrielle, co-ordinator of the US State Department's Global Engagement Centre, which tracks foreign propaganda.
"But with this pandemic the co-operation has accelerated rapidly," she told reporters May 8, according to AFP.
The Chinese regime has intensified its online campaign to defend its handling of the pandemic and to criticise the United States, according to the centre.
"Beijing is adapting in real time and increasingly using techniques that have long been employed by Moscow," Gabrielle said.
Earlier this month a report from various intelligence agencies said China went so far as to cover up news of the virus by silencing or "disappearing" doctors who spoke out, destroying evidence of COVID-19 in laboratories and refusing to provide samples to scientists working on a vaccine.
"That's not right": Lukashenko is indignant over Russia's conduct "Last night I received information that Russia is selling natural gas to Europe during these difficult times for prices at up to $70 - $65-68, but definitely not for $127, as it does for Belarus," Lukashenko said. Focusing on the fact that, according to his information, Germany is buying natural gas from Russia at a price of up to $70, Lukashenko urged the head of state to study this issue in more detail. "That's not right - I'm not talking about the 75th anniversary [of the Soviet defeat of Nazi Germany]; they're selling natural gas for up to $70, my information (from a variety of sources) says, but definitely not for $127, as they do for Belarus, Belarusian President Alexander Lukashenko stressed.
Of course the Russians [are involved] :))) On this site they're the bad boys :)))
It was more than likely the Russians.
UK 'absolutely confident' Russian intelligence behind COVID-19 vaccine hack
Britain, the United States and Canada have accused a hacking group linked to Russian intelligence of seeking to steal virus research.
China has become 'safe haven for cyber criminals'
The charge comes after two Chinese nationals were indicted for trying to steal COVID-19 vaccine research and intellectual property, among a number of other cyber related crimes.
China's self portrayal as COVID-19 hero contradicts true role in pandemic
Beijing is actively trying to convince the world that its efforts during the global health crisis have been 'heroic'. Facts say otherwise.


Taking this opportunity to send extremely warm Christmas wishes to you and everyone you hold dear.
Look to the person at your left and squeeze them….ow! goddam it why do I always get left with the weirdo?
Jokes (and restraining orders) aside, it's the Christmas season and I, for one am very lucky that I have just a big bunch of lunatics to call my friends. I know it sounds harsh but seriously, these guys are idiots. When I say I'm the sane one you know you need to be worried.
I am aware that I am very lucky in the friend and family department – (the bedding dept not so much) ba dum dum chhh!
To everyone that I've met this year, and those Ive known for far too long, I wish you as much love as Mariah Carey wishes she could still squidge herself into that santa outfit.
Anywhoo – I'm not great at the advice for no other reason than I've been single for 5 years and I live with my parents, but if you have that one person you need to hi 5 right now then let them know. If they are anything like my friends they have no idea what your talking about and will make you leave.
« Why is the Nativity like the Great British Bake Off?


text:
Q: How to enable the drag and drop feature on the node in GetOrgChart library How to enable the drag and drop feature on the node in GetOrgChart plugin?
I have used the free version of GetOrgChart plugin to draw Organization chart. I have a requirement to drag and drop the node present in the OrgChart. What I have to do is to drag a node from the chart and drop it to another node to assign it to that particular node. I haven't got any help and suggestions from the documentation of the GetOrgChart.
Any help is appreciated.
http://www.getorgchart.com/


text:
" AiQ is an integral part of the HAL team working collaboratively to solve challenging project requirements. With their holistic approach and excellent stakeholder management skills, AiQ's involvement as a trusted partner helps ensure a successful outcome, time and time again."
Darren Colderwood – Baggage Development Director, Heathrow Airport Limited
"AiQ Consulting continues to do a great job, working with authenticity and professionalism as always, with the entire Budapest Airport team. Their continued support throughout the transition has supported the airport to grow, ensured stability of BHS operations and enabled to meet new challenges, and we look forward to working with them through the next phase for operational projects."
Huszka Péter – Chief Operating Officer, Budapest Airport
"To support our ambitious expansion plans, we instructed NATS and AiQ Consulting to assess the current and future capacity of the whole Bristol Airport operation, focusing on the experience from our passengers' perspective. The result was an independent report that helped to validate our investment plans and identified some immediate opportunities for us to improve, and ensuring the safe and efficient growth of the operation.
"Thanks to the report we are satisfied that the airport's growth targets are on track and are equipped to work on the areas that really need it. Working with an experienced third party has enabled us to affirm our future investment decisions. The collaboration between NATS and AIQ Consulting provided us with a integrated airside and landside view of our airport, highlighting the strong connections between the different areas of airport operations."
"We are delighted with the work NATS and AiQ have undertaken and look forward to working with them on more projects in the future."
Andrew Goodenough – Development Director, Bristol Airport
" The AiQ team have supported a number of aviation projects I have been involved with over the years. They make it their business to understand operational challenges and project priorities in order to come up with solutions that work for all. It's been great to work with nice people that really care. "
Simon Newland – Project Director at Network Rail
(Previously Delivery Leader at Heathrow Airport)
" AiQ conducted a full operational capacity assessment study of Leeds Bradford International Airport covering Kerbside, Terminal, Airfield and Baggage Operations. This included analysis and recommendations to optimize the current operations and providing an enhanced passenger experience whilst maximizing resource utilisation.
We found the AiQ simulation an accurate predictor and helped ensured we had an action plan in place. The recommendations AiQ provided, enabled us to better plan and manage our peak operations through the implementation of the correct technology and process.
We have been delighted with the work AiQ have undertaken and look forward to engaging with them to look at future capacity planning over the coming months. "
Simon Whitby – Operations Director, Leeds Bradford Airport
Read the case study here
" A very professional and flexible company to work with. AiQ have always strived to go above and beyond expectations and deliver work efficiently, their understanding of planning and operations gives a high level of confidence to myself and the teams I work with."
Dean Wyeth – Infrastructure Manager, Heathrow Airport Limited
"Without the insight and professionalism of the AiQ Consulting team led by Nomaan Ashgar and the capacity models provided, we would have been unable to identify the root causes of issues and how they could be overcome. Clearly defined procedures with SOPs and Process Maps of each process have led to a much more efficient and effective baggage system." Head of Baggage, Budapest Airport
Huszka Péter – Chief Operating Officer,
" AiQ's expertise in capacity and baggage planning assisted Cobalt to ensure a smooth & successful co-location programme for Delta & Virgin at London Heathrow Terminal 3. "
Andy Stewart – Ramp and Baggage Manager, Cobalt Ground Solutions Ltd
" AiQ successfully met the challenge of assessing passenger check-in & baggage make-up demand for multiple airline move scenarios across a multi-terminal airport. "
Chris Owens – UK Head of Operations, LogPlan
" With the largest stakeholder integration exercise ever undertaken by Heathrow Airport, incorporating high levels of passenger and baggage demands, AiQ's capacity, demand and logistics planning expertise enabled Heathrow to successfully fulfil our role as Host Airport for the London 2012 Olympic and Paralympic Games. "
Luke Burton – Head of Baggage Operations, Heathrow Airport Limited
"I have worked with AiQ over many years and have always found them a great company to work with, a company that really knows what their service offer is and always delivers consistently against the brief."
Sally Jenner – Consultant, Alchemmy Management Consultancy
(Previously Business Change Lead at Heathrow Airport)
" AiQ's ability to accurately define resourcing needs for both people and equipment is exceptional. It helped us save money and protected terminal operations during complex construction mitigation. "
Grant Thompson – Managing Director, Aviation Business Continuity
" The practical capacity solutions provided by the AiQ team enabled us to successfully implement new terminal developments and gain agreement from our key stakeholder. "
Simon James – IFS Coordinator, PMO, Development, Heathrow Airport Limited
" Our partnership with Adrian and his team is invaluable. Their expert knowledge in airports and baggage solutions has had a significant influence on the terminal operations at Heathrow. "
David Tooth – Managing Director, Gebler Tooth Architects
> Read our Case Studies