text:
\section{Outline of the paper}

The first part \cite{1eme} of this series was an expository paper
about the geometric intuition underlying the notion of T-homotopy. The
purpose of this second paper is to prove that the class of old
T-homotopy equivalences introduced in \cite{diCW} and in \cite{model2}
is actually not big enough. Indeed, the only kind of old T-homotopy
equivalence consists of the deformations which locally act like in
Figure~\ref{ex1}. So it becomes impossible with this old definition to
identify the directed segment of Figure~\ref{ex1} with the full
$3$-cube of Figure~\ref{3cube} by a zig-zag sequence of weak
S-homotopy and of T-homotopy equivalences preserving the initial state
and the final state of the $3$-cube since every point of the $3$-cube
is related to three distinct edges. This contradicts the fact that
concurrent execution paths cannot be distinguished by observation.
The end of the paper proposes a new definition of T-homotopy
equivalence following the paradigm of invariance by refinement of
observation. It will be checked that the preceding drawback is then
overcome.

This second part gives only a motivation for the new definition of
T-homotopy. Further developments and applications are given in
\cite{3eme}, \cite{4eme} and \cite{hocont}. The left-properness of the
model category structure of \cite{model3} is also established in this
paper. The latter result is used several times in the next papers of
this series (e.g., \cite{3eme} Theorem~11.2, \cite{4eme} Theorem~9.2).

\begin{figure}
\[
\xymatrix{\widehat{0} \ar@{->}[rrrr]^-{U} &&&& \widehat{1} \\
\widehat{0} \ar@{->}[rr]^-{U'} && A \ar@{->}[rr]^-{U''} && \widehat{1}}
\]
\caption{The simplest example of T-homotopy equivalence}
\label{ex1}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=5cm]{3cube.eps}
\end{center}
\caption{The full $3$-cube}
\label{3cube}
\end{figure}

Section~\ref{gldef} collects some facts about globular complexes and
their relationship with the category of flows. Indeed, it is not known
how to establish the limitations of the old form of T-homotopy
equivalence without using globular complexes together with a
compactness argument.  Section~\ref{rappelT} recalls the notion of old
T-homotopy equivalence of flows which is a kind of morphism between
flows coming from globular complexes (the class of flows ${\cell(\dtop)}$).
Section~\ref{factI} presents elementary facts about relative
$I^{gl}_+$-cell complexes which will be used later in the paper.
Section~\ref{sectionleftproperflow} proves that the model category of
flows is left proper. This technical fact is used in the proof of the
main theorem of the paper, and it was not established in
\cite{model3}. Section~\ref{main1} proves the first main theorem of
the paper:
\begin{thmN} (Theorem~\ref{pasiso}) Let $n\geq 3$. There does not
  exist any zig-zag sequence of S-homotopy equivalences and of old
  T-homotopy equivalences between the flow associated with the
  $n$-cube and the flow associated with the directed segment.
\end{thmN} Finally Section~\ref{main2} proposes a new definition of
T-homotopy equivalence and the second main theorem of the paper is
proved:
\begin{thmN} (Theorem~\ref{OKnew}) Every T-homotopy in the old sense
  is the composite of an S-homotopy equivalence with a T-homotopy
  equivalence in the new sense~\footnote{Since a T-homotopy in the old
    sense is a T-homotopy in the new sense only up to S-homotopy, the
    terminology ``generalized T-homotopy'' used in Section~\ref{main2}
    may not be the best one. However, this terminology is used in the
    other papers of this series so we keep it to avoid any
    confusion.}.
\end{thmN}

\section{Prerequisites and notations}

The initial object (resp. the terminal object) of a category $\mathcal{C}$, if
it exists, is denoted by $\varnothing$ (resp. $\mathbf{1}$).

Let $\mathcal{C}$ be a cocomplete category.  If $K$ is a set of morphisms of
$\mathcal{C}$, then the class of morphisms of $\mathcal{C}$ that satisfy the RLP
(\textit{right lifting property}) with respect to any morphism of $K$
is denoted by $\inj(K)$ and the class of morphisms of $\mathcal{C}$ that are
transfinite compositions of pushouts of elements of $K$ is denoted by
$\cell(K)$. Denote by $\cof(K)$ the class of morphisms of $\mathcal{C}$ that
satisfy the LLP (\textit{left lifting property}) with respect to the
morphisms of $\inj(K)$.  It is a purely categorical fact that
$\cell(K)\subset \cof(K)$. Moreover, every morphism of $\cof(K)$ is a
retract of a morphism of $\cell(K)$ as soon as the domains of $K$ are
small relative to $\cell(K)$ (\cite{MR99h:55031} Corollary~2.1.15). An
element of $\cell(K)$ is called a \textit{relative $K$-cell complex}.
If $X$ is an object of $\mathcal{C}$, and if the canonical morphism
$\varnothing\longrightarrow X$ is a relative $K$-cell complex, then
the object $X$ is called a \textit{$K$-cell complex}.

Let $\mathcal{C}$ be a cocomplete category with a distinguished set of
morphisms $I$. Then let $\cell(\mathcal{C},I)$ be the full subcategory of $\mathcal{C}$
consisting of the object $X$ of $\mathcal{C}$ such that the canonical morphism
$\varnothing\longrightarrow X$ is an object of $\cell(I)$. In other
terms, $\cell(\mathcal{C},I)=(\varnothing\!\downarrow \! \mathcal{C}) \cap \cell(I)$.

It is obviously impossible to read this paper without a strong
familiarity with \textit{model categories}. Possible references for
model categories are \cite{MR99h:55031}, \cite{ref_model2} and
\cite{MR1361887}.  The original reference is \cite{MR36:6480} but
Quillen's axiomatization is not used in this paper. The axiomatization
from Hovey's book is preferred.  If $\mathcal{M}$ is a
\textit{cofibrantly generated} model category with set of generating
cofibrations $I$, let $\cell(\mathcal{M}) := \cell(\mathcal{M},I)$:
this is the full subcategory of \textit{cell complexes} of the model
category $\mathcal{M}$. A cofibrantly generated model structure
$\mathcal{M}$ comes with a \textit{cofibrant replacement functor}
$Q:\mathcal{M} \longrightarrow \cell(\mathcal{M})$. In all usual model
categories which are cellular (\cite{ref_model2} Definition~12.1.1),
all the cofibrations are monomorphisms.  Then for every monomorphism
$f$ of such a model category $\mathcal{M}$, the morphism $Q(f)$ is a
cofibration, and even an inclusion of subcomplexes (\cite{ref_model2}
Definition~10.6.7) because the cofibrant replacement functor $Q$ is
obtained by the small object argument, starting from the identity of
the initial object. This is still true in the model category of flows
reminded in Section~\ref{remflow} since the class of cofibrations
which are monomorphisms is closed under pushout and transfinite
composition.

A \textit{partially ordered set} $(P,\leq)$ (or \textit{poset}) is a
set equipped with a reflexive antisymmetric and transitive binary
relation $\leq$. A poset is \textit{locally finite} if for any
$(x,y)\in P\p P$, the set $[x,y]=\{z\in P,x\leq z\leq y\}$ is finite.
A poset $(P,\leq)$ is \textit{bounded} if there exist $\widehat{0}\in
P$ and $\widehat{1}\in P$ such that $P = [\widehat{0},\widehat{1}]$
and such that $\widehat{0} \neq \widehat{1}$. For a bounded poset $P$,
let $\widehat{0}=\min P$ (the bottom element) and $\widehat{1}=\max P$
(the top element). In a poset $P$, the interval $]\alpha,-]$ (the
sub-poset of elements of $P$ strictly bigger than $\alpha$) can also
be denoted by $P_{>\alpha}$.

A poset $P$, and in particular an ordinal, can be viewed as a small
category denoted in the same way: the objects are the elements of $P$
and there exists a morphism from $x$ to $y$ if and only if $x\leq y$.
If $\lambda$ is an ordinal, a \textit{$\lambda$-sequence} in a
cocomplete category $\mathcal{C}$ is a colimit-preserving functor $X$ from
$\lambda$ to $\mathcal{C}$. We denote by $X_\lambda$ the colimit $\varinjlim X$
and the morphism $X_0\longrightarrow X_\lambda$ is called the
\textit{transfinite composition} of the morphisms
$X_\mu\longrightarrow X_{\mu+1}$.

A model category is \textit{left proper} if the pushout of a weak
equivalence along a cofibration is a weak equivalence. The model
categories $\top$ and ${\brm{Flow}}$ (see below) are both left proper (cf.
Theorem~\ref{leftproperflow} for ${\brm{Flow}}$).

In this paper, the notation $\xymatrix@1{\ar@{^{(}->}[r]&}$ means
\textit{cofibration}, the notation $\xymatrix@1{\ar@{->>}[r]&}$ means
\textit{fibration}, the notation $\simeq$ means \textit{weak equivalence},
and the notation $\cong$ means \textit{isomorphism}.

\section{Reminder about the category of flows}
\label{remflow}

The category $\top$ of \textit{compactly generated topological spaces}
(i.e. of weak Hausdorff $k$-spaces) is complete, cocomplete and
cartesian closed (more details for this kind of topological spaces in
\cite{MR90k:54001,MR2000h:55002}, the appendix of \cite{Ref_wH} and
also the preliminaries of \cite{model3}). For the sequel, any
topological space will be supposed to be compactly generated. A
\textit{compact space} is always Hausdorff.

The category $\top$ is equipped with the unique model structure having
the \textit{weak homotopy equivalences} as weak equivalences and
having the \textit{Serre fibrations}~\footnote{that is a continuous
map having the RLP with respect to the inclusion $\mathbf{D}^n\p
0\subset \mathbf{D}^n\p [0,1]$ for any $n\geq 0$ where $\mathbf{D}^n$
is the $n$-dimensional disk.} as fibrations.

The time flow of a higher dimensional automaton is encoded in an
object called a \textit{flow} \cite{model3}. A flow $X$ contains a set
$X^0$ called the \textit{$0$-skeleton} whose elements correspond to
the \textit{states} (or \textit{constant execution paths}) of the
higher dimensional automaton. For each pair of states
$(\alpha,\beta)\in X^0\p X^0$, there is a topological space
$\P_{\alpha,\beta}X$ whose elements correspond to the
\textit{(non-constant) execution paths} of the higher dimensional
automaton \textit{beginning at} $\alpha$ and \textit{ending at}
$\beta$. For $x\in \P_{\alpha,\beta}X$, let $\alpha=s(x)$ and
$\beta=t(x)$. For each triple $(\alpha,\beta,\gamma)\in X^0\p X^0\p
X^0$, there exists a continuous map $*:\P_{\alpha,\beta}X\p
\P_{\beta,\gamma}X\longrightarrow \P_{\alpha,\gamma}X$ called the
\textit{composition law} which is supposed to be associative in an
obvious sense. The topological space $\P
X=\bigsqcup_{(\alpha,\beta)\in X^0\p X^0}\P_{\alpha,\beta}X$ is called
the \textit{path space} of $X$. The category of flows is denoted by
${\brm{Flow}}$. A point $\alpha$ of $X^0$ such that there are no non-constant
execution paths ending at $\alpha$ (resp. starting from $\alpha$) is
called an \textit{initial state} (resp. a \textit{final state}). A
morphism of flows $f$ from $X$ to $Y$ consists of a set map $f^0:X^0
\longrightarrow Y^0$ and a continuous map $\P f: \P X \longrightarrow
\P Y$ preserving the structure. A flow is therefore ``almost'' a small
category enriched in $\top$. A flow $X$ is \textit{loopless} if for
every $\alpha\in X^0$, the space $\P_{\alpha,\alpha}X$ is empty.

Here are four fundamental examples of flows:
\begin{enumerate}
\item Let $S$ be a set. The flow associated with $S$, still denoted by
  $S$, has $S$ as set of states and the empty space as path space.
  This construction induces a functor ${\brm{Set}} \rightarrow {\brm{Flow}}$ from
  the category of sets to that of flows. The flow associated with a
  set is loopless.
\item Let $(P,\leq)$ be a poset. The flow associated with $(P,\leq)$,
  and still denoted by $P$ is defined as follows: the set of states of
  $P$ is the underlying set of $P$; the space of morphisms from
  $\alpha$ to $\beta$ is empty if $\alpha\geq \beta$ and equals to
  $\{(\alpha,\beta)\}$ if $\alpha<\beta$ and the composition law is
  defined by $(\alpha,\beta)*(\beta,\gamma) = (\alpha,\gamma)$. This
  construction induces a functor ${\brm{PoSet}} \rightarrow {\brm{Flow}}$ from the
  category of posets together with the strictly increasing maps to the
  category of flows. The flow associated with a poset is loopless.
\item The flow ${\rm{Glob}}(Z)$ defined by \begin{eqnarray*}
  && {\rm{Glob}}(Z)^0=\{\widehat{0},\widehat{1}\} \\
  && \P {\rm{Glob}}(Z)=Z \hbox{ with }s(z)=\widehat{0}\hbox{ and }t(z)=\widehat{1}\hbox{ for all }z\in Z
\end{eqnarray*} and a trivial composition law (cf.
  Figure~\ref{exglob}). It is called the \textit{globe} of $Z$.
\item The \textit{directed segment} $\vec{I}$ is by definition
  ${\rm{Glob}}(\{0\}) \cong \{\widehat{0} < \widehat{1}\}$.
\end{enumerate}

The category ${\brm{Flow}}$ is equipped with the unique model structure
such that \cite{model3}:
\begin{itemize}
\item The weak equivalences are the \textit{weak S-homotopy equivalences},
i.e. the morphisms of flows $f:X\longrightarrow Y$ such that
$f^0:X^0\longrightarrow Y^0$ is a bijection and such that $\P f:\P
X\longrightarrow \P Y$ is a weak homotopy equivalence.
\item The fibrations are the morphisms of flows
$f:X\longrightarrow Y$ such that $\P f:\P X\longrightarrow \P Y$ is a
Serre fibration.
\end{itemize}
This model structure is cofibrantly generated. The set of generating
cofibrations is the set $I^{gl}_+=I^{gl}\cup
\{R:\{0,1\}\longrightarrow \{0\},C:\varnothing\longrightarrow \{0\}\}$
with
\[I^{gl}=\{{\rm{Glob}}(\mathbf{S}^{n-1})\subset {\rm{Glob}}(\mathbf{D}^{n}), n\geq
0\}\] where $\mathbf{D}^{n}$ is the $n$-dimensional disk and
$\mathbf{S}^{n-1}$ the $(n-1)$-dimensional sphere. The set of
generating trivial cofibrations is
\[J^{gl}=\{{\rm{Glob}}(\mathbf{D}^{n}\p\{0\})\subset
{\rm{Glob}}(\mathbf{D}^{n}\p [0,1]), n\geq 0\}.\]

\begin{figure}
\begin{center}
\includegraphics[width=7cm]{exglob.eps}
\end{center}
\caption{Symbolic representation of
${\rm{Glob}}(Z)$ for some topological space $Z$} \label{exglob}
\end{figure}

If $X$ is an object of $\cell({\brm{Flow}})$, then a presentation of the
morphism $\varnothing \longrightarrow X$ as a transfinite composition
of pushouts of morphisms of $I^{gl}_+$ is called a \textit{globular
decomposition} of $X$.

\section{Globular complex}
\label{gldef}

The reference is \cite{model2}. A \textit{globular complex} is a
topological space together with a structure describing the sequential
process of attaching \textit{globular cells}. A general globular
complex may require an arbitrary long transfinite construction. We
restrict our attention in this paper to globular complexes whose
globular cells are morphisms of the form
${\rm{Glob}}^{top}(\mathbf{S}^{n-1}) \longrightarrow
{\rm{Glob}}^{top}(\mathbf{D}^{n})$ (cf. Definition~\ref{globe_top}).

\begin{defn}
A {\rm multipointed topological space} $(X,X^0)$ is a pair of
topological spaces such that $X^0$ is a discrete subspace of $X$.  A
morphism of multipointed topological spaces $f:(X,X^0)\longrightarrow
(Y,Y^0)$ is a continuous map $f:X\longrightarrow Y$ such that
$f(X^0)\subset Y^0$. The corresponding category is denoted by
$\top^m$. The set $X^0$ is called the {\rm $0$-skeleton} of $(X,X^0)$.
The space $X$ is called the {\rm underlying topological space} of
$(X,X^0)$.
\end{defn}

The category of multipointed spaces is cocomplete.

\begin{defn} \label{globe_top}
Let $Z$ be a topological space. The {\rm globe of $Z$}, which is
denoted by ${\rm{Glob}}^{top}(Z)$, is the multipointed space
\[(|{\rm{Glob}}^{top}(Z)|,\{\widehat{0},\widehat{1}\})\] where the topological space
$|{\rm{Glob}}^{top}(Z)|$ is the quotient of
$\{\widehat{0},\widehat{1}\}\sqcup (Z\p[0,1])$ by the relations
$(z,0)=(z',0)=\widehat{0}$ and $(z,1)=(z',1)=\widehat{1}$ for any
$z,z'\in Z$. In particular, ${\rm{Glob}}^{top}(\varnothing)$ is the
multipointed space
$(\{\widehat{0},\widehat{1}\},\{\widehat{0},\widehat{1}\})$.
\end{defn}

\begin{nota}
  If $Z$ is a singleton, then the globe of $Z$ is denoted by
  $\vec{I}^{top}$.
\end{nota}

\begin{defn}
Let $I^{gl,top}:=\{{\rm{Glob}}^{top}(\mathbf{S}^{n-1})\longrightarrow
{\rm{Glob}}^{top}(\mathbf{D}^{n}),n\geq 0\}$.
A {\rm relative globular precomplex} is a relative $I^{gl,top}$-cell
complex in the category of multipointed topological spaces.
\end{defn}

\begin{defn}
A {\rm globular precomplex} is a $\lambda$-sequence of multipointed
topological spaces $X:\lambda\longrightarrow \top^m$ such that $X$ is
a relative globular precomplex and such that $X_0=(X^0,X^0)$ with
$X^0$ a discrete space. This $\lambda$-sequence is characterized by a
presentation ordinal $\lambda$, and for any $\beta<\lambda$, an
integer $n_\beta\geq 0$ and an attaching map
$\phi_\beta:{\rm{Glob}}^{top}(\mathbf{S}^{n_\beta-1}) \longrightarrow
X_\beta$. The family $(n_\beta,\phi_\beta)_{\beta<\lambda}$ is called
the {\rm globular decomposition} of $X$.
\end{defn}

Let $X$ be a globular precomplex. The $0$-skeleton of $\varinjlim X$ is
equal to $X^0$.

\begin{defn} A morphism of globular precomplexes $f:X\longrightarrow Y$ is a
morphism of multipointed spaces still denoted by $f$ from $\varinjlim X$
to $\varinjlim Y$. \end{defn}

\begin{nota}
If $X$ is a globular precomplex, then the underlying topological space
of the multipointed space $\varinjlim X$ is denoted by $|X|$ and the
$0$-skeleton of the multipointed space $\varinjlim X$ is denoted by
$X^0$.
\end{nota}

\begin{defn}
Let $X$ be a globular precomplex. The space $|X|$ is called the {\rm
underlying topological space} of $X$. The set $X^0$ is called the {\rm
$0$-skeleton} of $X$.
\end{defn}

\begin{defn}
Let $X$ be a globular precomplex. A morphism of globular precomplexes
$\gamma:\vec{I}^{top}\longrightarrow X$ is a {\rm non-constant execution
path} of $X$ if there exists $t_0=0<t_1<\dots<t_{n}=1$ such that:
\begin{enumerate}
\item  $\gamma(t_i)\in X^0$ for any $0 \leq i \leq n$,
\item $\gamma(]t_i,t_{i+1}[)\subset {\rm{Glob}}^{top}(\mathbf{D}^{n_{\beta_i}}
\backslash \mathbf{S}^{n_{\beta_i}-1})$ for some $(n_{\beta_i},\phi_{\beta_i})$
of the globular decomposition of $X$,
\item  for $0\leq i<n$, there exists $z^i_\gamma\in \mathbf{D}^{n_{\beta_i}}\backslash
\mathbf{S}^{n_{\beta_i}-1}$ and a strictly increasing continuous map
$\psi^i_\gamma:[t_i,t_{i+1}]\longrightarrow [0,1]$ such that
$\psi^i_\gamma(t_i)=0$ and $\psi^i_\gamma(t_{i+1})=1$ and for any
$t\in [t_i,t_{i+1}]$, $\gamma(t)=(z^i_\gamma,\psi^i_\gamma(t))$.
\end{enumerate}
In particular, the restriction $\gamma\!\restriction_{]t_i,t_{i+1}[}$
of $\gamma$ to $]t_i,t_{i+1}[$ is one-to-one. The set of non-constant
execution paths of $X$ is denoted by ${\P}^{top}(X)$.
\end{defn}

\begin{defn}
A morphism of globular precomplexes $f:X\longrightarrow Y$ is {\rm
non-decreasing} if the canonical set map
$\top([0,1],|X|)\longrightarrow \top([0,1],|Y|)$ induced by
composition by $f$ yields a set map ${\P}^{top}(X)\longrightarrow
{\P}^{top}(Y)$. In other terms, one has the commutative diagram of
sets
\[\xymatrix{
{\P}^{top}(X)\fr{}\fd{\subset}& {\P}^{top}(Y)\fd{\subset}\\
\top([0,1],|X|) \fr{} &\top([0,1],|Y|).}
\]
\end{defn}

\begin{defn}
A {\rm globular complex } (resp. a {\rm relative globular complex})
$X$ is a globular precomplex (resp. a relative globular precomplex)
such that the attaching maps $\phi_\beta$ are non-decreasing. A
morphism of globular complexes is a morphism of globular precomplexes
which is non-decreasing. The category of globular complexes together
with the morphisms of globular complexes as defined above is denoted
by ${\brm{glTop}}$. The set ${\brm{glTop}}(X,Y)$ of morphisms of globular complexes
from $X$ to $Y$ equipped with the Kelleyfication of the compact-open
topology is denoted by ${\brm{glTOP}}(X,Y)$.
\end{defn}

\begin{defn}
Let $X$ be a globular complex. A point $\alpha$ of $X^0$ such that
there are no non-constant execution paths ending to $\alpha$
(resp. starting from $\alpha$) is called {\rm initial state} (resp.
{\rm final state}). More generally, a point of $X^0$ will be sometime
called {\rm a state} as well.
\end{defn}

\begin{thm} (\cite{model2} Theorem~III.3.1)
There exists a unique functor $cat:{\brm{glTop}}\longrightarrow{\brm{Flow}}$ such
that
\begin{enumerate}
\item if $X=X^0$ is a discrete globular complex, then $cat(X)$ is
the achronal flow $X^0$ (``achronal'' meaning with an empty path space),
\item if $Z=\mathbf{S}^{n-1}$ or $Z=\mathbf{D}^{n}$ for some integer $n\geq 0$,
then $cat({\rm{Glob}}^{top}(Z))={\rm{Glob}}(Z)$,
\item for any globular complex $X$ with globular decomposition
$(n_\beta,\phi_\beta)_{\beta<\lambda}$, for any limit
ordinal $\beta\leq\lambda$, the canonical morphism of flows
\[\varinjlim_{\alpha<\beta} cat(X_\alpha)\longrightarrow
cat(X_\beta)\] is an isomorphism of flows,
\item for any globular complex $X$ with globular decomposition
$(n_\beta,\phi_\beta)_{\beta<\lambda}$, for any
$\beta<\lambda$, one has the pushout of flows
\[\xymatrix{{\rm{Glob}}(\mathbf{S}^{n_\beta-1})\fr{cat(\phi_\beta)}\fd{}& cat(X_\beta)\fd{}\\
{\rm{Glob}}(\mathbf{D}^{n_\beta})\fr{} & cat(X_{\beta+1}).\cocartesien}\]
\end{enumerate}
\eth

The following theorem is important for the sequel:

\begin{thm} \label{remonte} The functor $cat$ induces a functor, still
denoted by $cat$ from ${\brm{glTop}}$ to ${\cell(\dtop)}\subset {\brm{Flow}}$ since its image
is contained in ${\cell(\dtop)}$. For any flow $X$ of ${\cell(\dtop)}$, there exists a
globular complex $Y$ such that $cat(U)=X$, which is constructed by
using the globular decomposition of $X$.  \eth

\begin{proof}
The construction of $U$ is made in the proof of \cite{model2}
Theorem~V.4.1.
\end{proof}

\section{T-homotopy equivalence}
\label{rappelT}

The old notion of T-homotopy equivalence for globular complexes was
given in \cite{diCW}. A notion of T-homotopy equivalence of flows was
given in \cite{model2} and it was proved in the same paper that these
two notions are equivalent.

We first recall the definition of the branching and merging space
functors, and then the definition of a T-homotopy equivalence of
flows, exactly as given in \cite{model2} (Definition~\ref{Tdi} below),
and finally a characterization of T-homotopy of flows using globular
complexes (Theorem~\ref{TTT} below).

Roughly speaking, the branching space of a flow is the space of germs
of non-constant execution paths beginning in the same way.

\begin{prop} \label{universalpm} (\cite{exbranch} Proposition~3.1)
Let $X$ be a flow. There exists a topological space $\P^-X$ unique up
to homeomorphism and a continuous map $h^-:\P X\longrightarrow \P^- X$
satisfying the following universal property:
\begin{enumerate}
\item For any $x$ and $y$ in $\P X$ such that $t(x)=s(y)$, the equality
$h^-(x)=h^-(x*y)$ holds.
\item Let $\phi:\P X\longrightarrow Y$ be a
continuous map such that for any $x$ and $y$ of $\P X$ such that
$t(x)=s(y)$, the equality $\phi(x)=\phi(x*y)$ holds. Then there exists a
unique continuous map $\overline{\phi}:\P^-X\longrightarrow Y$ such that
$\phi=\overline{\phi}\circ h^-$.
\end{enumerate}
Moreover, one has the homeomorphism
\[\P^-X\cong \bigsqcup_{\alpha\in X^0} \P^-_\alpha X\]
where $\P^-_\alpha X:=h^-\left( \bigsqcup_{\beta\in
X^0} \P^-_{\alpha,\beta} X\right)$. The mapping $X\mapsto \P^-X$
yields a functor $\P^-$ from ${\brm{Flow}}$ to $\top$.
\end{prop}

\begin{defn}
Let $X$ be a flow. The topological space $\P^-X$ is called the {\rm
branching space} of the flow $X$. The functor $\P^-$ is called the
{\rm branching space functor}. \end{defn}

\begin{prop} \label{universalmp}(\cite{exbranch} Proposition~A.1)
Let $X$ be a flow. There exists a topological space $\P^+X$ unique up
to homeomorphism and a continuous map $h^+:\P X\longrightarrow \P^+ X$
satisfying the following universal property:
\begin{enumerate}
\item For any $x$ and $y$ in $\P X$ such that $t(x)=s(y)$, the equality
$h^+(y)=h^+(x*y)$ holds.
\item Let $\phi:\P X\longrightarrow Y$ be a
continuous map such that for any $x$ and $y$ of $\P X$ such that
$t(x)=s(y)$, the equality $\phi(y)=\phi(x*y)$ holds. Then there exists a
unique continuous map $\overline{\phi}:\P^+X\longrightarrow Y$ such that
$\phi=\overline{\phi}\circ h^+$.
\end{enumerate}
Moreover, one has the homeomorphism
\[\P^+X\cong \bigsqcup_{\alpha\in X^0} \P^+_\alpha X\]
where $\P^+_\alpha X:=h^+\left( \bigsqcup_{\beta\in
X^0} \P^+_{\alpha,\beta} X\right)$. The mapping $X\mapsto \P^+X$
yields a functor $\P^+$ from ${\brm{Flow}}$ to $\top$.
\end{prop}

Roughly speaking, the merging space of a flow is the space of germs of
non-constant execution paths ending in the same way.

\begin{defn} \label{defplus}
Let $X$ be a flow. The topological space $\P^+X$ is called the {\rm
merging space} of the flow $X$. The functor $\P^+$ is called the {\rm
merging space functor}. \end{defn}

\begin{defn} \cite{model2}
Let $X$ be a flow. Let $A$ and $B$ be two subsets of $X^0$. One says
that $A$ is {\rm surrounded} by $B$ (in $X$) if for any $\alpha\in A$,
either $\alpha \in B$ or there exists execution paths $\gamma_1$ and
$\gamma_2$ of $\P X$ such that $s(\gamma_1)\in B$,
$t(\gamma_1)=s(\gamma_2)=\alpha$ and $t(\gamma_2)\in B$. We denote
this situation by $A\lll B$.  \end{defn}

\begin{defn} \cite{model2}
Let $X$ be a flow. Let $A$ be a subset of $X^0$. Then the
\textit{restriction} $X\!\restriction_A$ of $X$ over $A$ is the unique
flow such that $(X\!\restriction_A)^0=A$, such that
$\P_{\alpha,\beta}(X\!\restriction_A)=\P_{\alpha,\beta}X$ for any
$(\alpha,\beta)\in A\p A$ and such that the inclusions $A\subset X^0$
and $\P (X\!\restriction_A)\subset \P X$ induces a morphism of flows
$X\!\restriction_A\longrightarrow X$.
\end{defn}

\begin{defn} \label{Tdi}\cite{model2}
Let $X$ and $Y$ be two objects of ${\cell(\dtop)}$.  A morphism of flows
$f:X\longrightarrow Y$ is a {\rm T-homotopy equivalence} if and only
if the following conditions are satisfied:
\begin{enumerate}
\item The morphism of flows $f:X\longrightarrow
Y\!\restriction_{f(X^0)}$ is an isomorphism of flows. In particular,
the set map $f^0:X^0\longrightarrow Y^0$ is one-to-one.
\item For $\alpha\in Y^0\backslash f(X^0)$, the topological spaces
$\P^-_\alpha Y$ and $\P^+_\alpha Y$ are singletons.
\item $Y^0\lll f(X^0)$.
\end{enumerate}
\end{defn}

We recall the following important theorem for the sequel:

\begin{thm} (\cite{model2} Theorem~VI.3.5) \label{TTT} Let $X$ and $Y$ be two
objects of ${\cell(\dtop)}$. Let $U$ and $V$ be two globular complexes with
$cat(U)=X$ and $cat(V)=Y$ ($U$ and $V$ always exist by
Theorem~\ref{remonte}). Then a morphism of flows $f:X\longrightarrow Y$
is a T-homotopy equivalence if and only if there exists a morphism of
globular complexes $g:U\longrightarrow V$ such that $cat(g)=f$ and
such that the continuous map $|g|:|U|\longrightarrow |V|$ between the
underlying topological spaces is an homeomorphism.  \eth

This characterization was actually the first definition of a
T-homotopy equivalence proposed in \cite{diCW} (see Definition~4.10
p66).

\section{Some facts about relative $I^{gl}_+$-cell complexes}
\label{factI}

Recall that $I^{gl}_+=I^{gl}\cup
\{R:\{0,1\}\longrightarrow \{0\},C:\varnothing\longrightarrow \{0\}\}$
with
\[I^{gl}=\{{\rm{Glob}}(\mathbf{S}^{n-1})\subset {\rm{Glob}}(\mathbf{D}^{n}), n\geq
0\}.\] Let $I_g=I^{gl}\cup\{C\}$. Since for any $n\geq 0$, the
inclusion $\mathbf{S}^{n-1}\subset \mathbf{D}^n$ is a closed inclusion
of topological spaces, so an effective monomorphism of the category
$\top$ of compactly generated topological spaces, every morphism of
$I_g$, and therefore every morphism of $\cell(I_g)$, is an effective
monomorphism of flows as well (cf. also \cite{model3} Theorem~10.6).

\begin{prop} \label{removingR}
If $f:X\longrightarrow Y$ is a relative $I^{gl}_+$-cell complex and if
$f$ induces a one-to-one set map from $X^0$ to $Y^0$, then
$f:X\longrightarrow Y$ is a relative $I_g$-cell subcomplex.
\end{prop}

\begin{proof} A pushout of $R$ appearing in the presentation of $f$ cannot
identify two elements of $X^0$ since, by hypothesis, $f^0:X^0
\rightarrow Y^0$ is one-to-one. So either such a pushout is trivial,
or it identifies two elements added by a pushout of $C$.
\end{proof}

\begin{prop} \label{factorR}
If $f:X\longrightarrow Y$ is a relative $I^{gl}_+$-cell complex, then
$f$ factors as a composite $g\circ h\circ k$ where $k:X\longrightarrow
Z$ is a morphism of $\cell(\{R\})$, where $h:Z \longrightarrow T$ is a
morphism of $\cell(\{C\})$, and where $g:T\longrightarrow Y$ is a
relative $I^{gl}$-cell complex.
\end{prop}

\begin{proof}
One can use the small object argument with $\{R\}$ by \cite{model3}
Proposition~11.8. Therefore the morphism $f:X\longrightarrow Y$
factors as a composite $g\circ h$ where $h:X\longrightarrow Z$ is a
morphism of $\cell(\{R\})$ and where the morphism $Z\longrightarrow Y$
is a morphism of $\inj(\{R\})$. One deduces that the set map
$Z^0\longrightarrow Y^0$ is one-to-one. One has the pushout diagram of
flows
\[
\xymatrix{
X\fd{f} \fr{k} & Z \fd{g} \\
Y  \ar@{=}[r] & Y. \cocartesien}
\]
Therefore the morphism $Z\longrightarrow Y$ is a relative
$I^{gl}_+$-cell complex. Proposition~\ref{removingR} implies that the
morphism $Z\longrightarrow Y$ is a relative $I_g$-cell complex. The
morphism $Z\longrightarrow Y$ factors as a composite
$h:Z\longrightarrow Z \sqcup (Y^0\backslash Z^0)$ and the inclusion
$g:Z \sqcup (Y^0\backslash Z^0)\longrightarrow Y$.
\end{proof}

\begin{prop}\label{tjrcof}
Let $X=X^0$ be a set viewed as a flow (i.e. with an empty path space).
Let $Y$ be an object of ${\cell(\dtop)}$. Then any morphism from $X$ to $Y$ is a
cofibration.
\end{prop}

\begin{proof}
Let $f:X\longrightarrow Y$ be a morphism of flows. Then $f$ factors as
a composite $X=X^0\longrightarrow Y^0\longrightarrow Y$.  Any set map
$X^0\longrightarrow Y^0$ is a transfinite composition of pushouts of
$C$ and $R$. So any set morphism $X^0\longrightarrow Y^0$ is a
cofibration of flows. And for any flow $Y$, the canonical morphism of
flows $Y^0\longrightarrow Y$ is a cofibration since it is a relative
$I_g$-cell complex. Hence the result.
\end{proof}

\section{Left properness of the weak S-homotopy model structure of ${\brm{Flow}}$}
\label{sectionleftproperflow}

\begin{prop}\label{pushexplicit}\label{model3} (\cite{model3} Proposition~15.1)
Let $f:U\longrightarrow V$ be a continuous map.  Consider the pushout
diagram of flows
\[
\xymatrix{
{\rm{Glob}}(U)\fr{}\fd{{\rm{Glob}}(f)} & X \fd{g}\\
{\rm{Glob}}(V)\fr{} & Y. \cocartesien}
\]
Then the continuous map $\P g:\P X\longrightarrow \P Y$ is a
transfinite composition of pushouts of continuous maps of the form a
finite product $\id\p\dots\p f\p\dots \p\id$ where the symbol $\id$
denotes identity maps.
\end{prop}

\begin{prop} \label{versleft}
Let $f:U\longrightarrow V$ be a Serre cofibration. Then the pushout of
a weak homotopy equivalence along a map of the form a finite product
$\id_{X_1}\p\dots\p f\p\dots \p\id_{X_p}$ with $p\geq 0$ is still a
weak homotopy equivalence.
\end{prop}

If the topological spaces $X_i$ for $1\leq i\leq p$ are cofibrant,
then the continuous map $\id_{X_1}\p\dots\p f\p\dots \p\dots\p
\id_{X_p}$ is a cofibration since the model category of compactly
generated topological spaces is monoidal with the categorical product
as monoidal structure.  So in this case, the result follows from the
left properness of this model category (\cite{ref_model2}
Theorem~13.1.10). In the general case, $\id_{X_1} \p \dots \p f\p\dots
\p \dots \p \id_{X_p}$ is not a cofibration anymore. But any
cofibration $f$ for the Quillen model structure of $\top$ is a
cofibration for the Str{\o}m model structure of $\top$
\cite{MR35:2284} \cite{MR39:4846} \cite{ruse} \cite{strom2}. In the
latter model structure, any space is cofibrant. Therefore the
continuous map $\id_{X_1} \p \dots \p f \p \dots \p \dots \p
\id_{X_p}$ is a cofibration of the Str{\o}m model structure of $\top$,
that is a NDR pair. So the continuous map $\id_{X_1} \p \dots \p f \p
\dots \p \dots \p \id_{X_p}$ is a closed $T_1$-inclusion anyway. This
fact will be used below.

\begin{proof}
We already know that the pushout of a weak homotopy equivalence along
a cofibration is a weak homotopy equivalence. The proof of this
proposition is actually an adaptation of the proof of the left
properness of the model category of compactly generated topological
spaces. Any cofibration is a retract of a transfinite composition of
pushouts of inclusions of the form $\mathbf{S}^{n-1}\subset
\mathbf{D}^{n}$ for $n\geq 0$. Since the category of compactly generated
topological spaces is cartesian closed, the binary product preserves
colimits. Thus we are reduced to considering a diagram of topological
spaces like
\[
\xymatrix{
X_1\p \dots \p \mathbf{S}^{n-1}\p\dots \p X_p\fd{} \fr{} & U \fd{} \fr{s}& X \fd{} \\
 X_1\p \dots \p \mathbf{D}^{n} \p\dots\p X_p\fr{} & \widehat{U} \cocartesien \fr{\widehat{s}} & \widehat{X} \cocartesien}
\]
where $s$ is a weak homotopy equivalence and we have to prove that
$\widehat{s}$ is a weak homotopy equivalence as well. By
\cite{MR36:6480} and \cite{MR48:6288}, it suffices to prove that
$\widehat{s}$ induces a bijection between the path-connected
components of $\widehat{U}$ and $\widehat{X}$, a bijection between the
fundamental groupoids $\pi(\widehat{U})$ and $\pi(\widehat{X})$, and
that for any local coefficient system of abelian groups $A$ of
$\widehat{X}$, one has the isomorphism
$\widehat{s}^*:H^*(\widehat{X},A)\cong H^*(\widehat{U},\widehat{s}^*A)$.

For $n=0$, one has $\mathbf{S}^{n-1}=\varnothing$ and
$\mathbf{D}^{n}=\{0\}$.  So $X_1\p \dots \p \mathbf{S}^{n-1}\p\dots \p
X_p = \varnothing$ and $X_1\p \dots \p \mathbf{D}^{n} \p \dots \p X_p
= X_1\p \dots \p X_p$. So $\widehat{U}\cong U\sqcup (X_1\p \dots \p
X_p)$ and $\widehat{X}\cong X\sqcup (X_1\p \dots \p X_p)$.  Therefore
the mapping $t$ is the disjoint sum $s\sqcup \id_{X_1\p \dots \p
  X_p}$. So it is a weak homotopy equivalence.

Let $n\geq 1$. The assertion concerning the path-connected components
is clear. Let $\mathbf{T}^n = \{x\in \mathbb{R}^n,0<|x|\leq 1\}$.
Consider the diagram of topological spaces
\[
\xymatrix{
X_1\p \dots \p \mathbf{S}^{n-1}\p\dots \p X_p\fd{} \fr{} & U \fd{} \fr{s}& X \fd{} \\
 X_1\p \dots \p \mathbf{T}^{n} \p \dots \p X_p\fr{} & \widetilde{U} \cocartesien \fr{\widetilde{s}} & \widetilde{X}. \cocartesien}
\]
Since the pair $(\mathbf{T}^{n},\mathbf{S}^{n-1})$ is a deformation
retract, the three pairs $(X_1\p \dots \p \mathbf{T}^{n} \p \dots \p
X_p,X_1\p \dots
\p \mathbf{S}^{n-1}\p\dots \p X_p)$, $(\widetilde{U},U)$ and
$(\widetilde{X},X)$ are deformation retracts as well. So the
continuous maps $U\longrightarrow \widetilde{U}$ and $X\longrightarrow
\widetilde{X}$ are both homotopy equivalences.
The Seifert-Van-Kampen theorem for the fundamental groupoid
(cf. \cite{MR48:6288} again) then yields the diagram of groupoids
\[
\xymatrix{
\pi(X_1\p \dots \p \mathbf{T}^{n}\p\dots \p X_p)\fd{} \fr{} & \pi(\widetilde{U}) \fd{} \fr{\pi(\widetilde{s})}& \pi(\widetilde{X}) \fd{} \\
 \pi(X_1\p \dots \p \mathbf{D}^{n} \p X_p)\fr{} & \pi(\widehat{U})
 \cocartesien \fr{\pi(\widehat{s})} & \pi(\widehat{X}). \cocartesien}
\]
Since $\pi(\widetilde{s})$ is an isomorphism of groupoids, then so is $\pi(\widehat{s})$.

Let $\mathbf{B}^n=\{x\in
\mathbb{R}^n,0\leq |x|< 1\}$.  Then $(\mathbf{B}^n,\widetilde{U})$ is an
excisive pair of $\widehat{U}$ and $(\mathbf{B}^n,\widetilde{X})$ is an
excisive pair of $\widehat{X}$. The Mayer-Vietoris long exact sequence then yields
the commutative diagram of groups
\[
\xymatrix{
\dots\fr{}& H^p(\widehat{X},A)\fr{} \fd{} & H^p({X},A)\oplus H^p(\mathbf{B}^n,A)\fr{}\fd{\cong} &
H^p(\mathbf{B}^n\backslash\{0\},A)\fr{}\fd{\cong}&\dots\\
\dots\fr{}& H^p(\widehat{U},\widehat{s}^*A)\fr{} & H^p({U},s^*A)\oplus H^p(\mathbf{B}^n,s^*A)\fr{} &
H^p(\mathbf{B}^n\backslash\{0\},s^*A)\fr{}&\dots}
\]
A five-lemma argument completes the proof.
\end{proof}

\begin{prop} \label{petitplus} Let $\lambda$ be an ordinal. Let $M:\lambda
\longrightarrow \top$ and $N:\lambda \longrightarrow \top$ be two
$\lambda$-sequences of topological spaces. Let $s:M\longrightarrow N$
be a morphism of $\lambda$-sequences which is also an objectwise weak
homotopy equivalence. Finally, let us suppose that for all
$\mu<\lambda$, the continuous maps $M_\mu \longrightarrow M_{\mu+1}$
and $N_\mu \longrightarrow N_{\mu+1}$ are of the form a finite product
$\id_{X_1}\p\dots\p f\p\dots \p\id_{X_p}$ with $p\geq 0$ and with $f$
a Serre cofibration.  Then the continuous map $\varinjlim s:\varinjlim M
\longrightarrow \varinjlim N$ is a weak homotopy equivalence.  \end{prop}

If for all $\mu<\lambda$, the continuous maps $M_\mu \longrightarrow
M_{\mu+1}$ and $N_\mu \longrightarrow N_{\mu+1}$ are cofibrations,
then Proposition~\ref{petitplus} above is a consequence of
\cite{ref_model2} Proposition~17.9.3 and of the fact that the model
category $\top$ is left proper. With the same additional hypotheses,
Proposition~\ref{petitplus} above is also a consequence of
\cite{MR2045835} Theorem~A.7. Indeed, the latter states that an
homotopy colimit can be calculated either in the usual Quillen model
structure of $\top$, or in the Str{\o}m model structure of $\top$
\cite{ruse} \cite{strom2}.

\begin{proof} The principle of the proof is standard.  If the ordinal $\lambda$
is not a limit ordinal, then this is a consequence of
Proposition~\ref{versleft}. Assume now that $\lambda$ is a limit
ordinal. Then $\lambda \geq \aleph_0$.

Let $u:\mathbf{S}^n\longrightarrow \varinjlim N$ be a continuous map.
Then $u$ factors as a composite $\mathbf{S}^n\longrightarrow N_\mu
\longrightarrow \varinjlim N$ since the $n$-dimensional sphere
$\mathbf{S}^n$ is compact and since any compact space is
$\aleph_0$-small relative to closed $T_1$-inclusions
(\cite{MR99h:55031} Proposition~2.4.2). By hypothesis, there exists a
continuous map $\mathbf{S}^n \longrightarrow M_\mu$ such that the
composite $\mathbf{S}^n \longrightarrow M_\mu \longrightarrow N_\mu$
is homotopic to $\mathbf{S}^n\longrightarrow N_\mu$. Hence the
surjectivity of the set map $\pi_n(\varinjlim M,*) \longrightarrow
\pi_n(\varinjlim N,*)$ (where $\pi_n$ denotes the $n$-th homotopy group)
for $n\geq 0$ and for any base point $*$.

Let $u,v:\mathbf{S}^n \longrightarrow \varinjlim M$ be two continuous
maps such that there exists an homotopy $H:\mathbf{S}^n\p [0,1]
\longrightarrow \varinjlim N$ between $\varinjlim s \circ f$ and $\varinjlim s
\circ g$. Since the space $\mathbf{S}^n\p [0,1]$ is compact, the homotopy
$H$ factors as a composite $\mathbf{S}^n\p [0,1] \longrightarrow
N_{\mu_0} \longrightarrow \varinjlim N$ for some $\mu_0<\lambda$. And
again since the space $\mathbf{S}^n$ is compact, the map $f$
(resp. $g$) factors as a composite $\mathbf{S}^n \longrightarrow
M_{\mu_1} \longrightarrow \varinjlim M$ (resp. $\mathbf{S}^n
\longrightarrow M_{\mu_2}
\longrightarrow \varinjlim M$) with $\mu_1<\lambda$ (resp. $\mu_2<\lambda$).
Then $\mu_4=\max(\mu_0,\mu_1,\mu_2)<\lambda$ since $\lambda$ is a
limit ordinal. And the map $H:\mathbf{S}^n\p [0,1] \longrightarrow
N_{\mu_4}$ is an homotopy between $f:\mathbf{S}^n \longrightarrow
M_{\mu_4}$ and $g:\mathbf{S}^n \longrightarrow M_{\mu_4}$. So the set
map $\pi_n(\varinjlim M,*) \longrightarrow \pi_n(\varinjlim N,*)$ for $n\geq
0$ and for any base point $*$ is one-to-one.
\end{proof}

\begin{thm} \label{leftproperflow}
The model category ${\brm{Flow}}$ is left proper.
\eth

\begin{proof} Consider the pushout diagram of ${\brm{Flow}}$
\[
\xymatrix{
U \fr{s}\fd{i} & X \fd{} \\ V \fr{t} & \cocartesien Y}
\]
where $i$ is a cofibration of ${\brm{Flow}}$ and $s$ a weak S-homotopy
equivalence. We have to check that $t$ is a weak S-homotopy
equivalence as well. The morphism $i$ is a retract of a
$I^{gl}_+$-cell complex $j:U\longrightarrow W$. If one considers
the pushout diagram of ${\brm{Flow}}$
\[\xymatrix{
U \fr{s}\fd{j} & X \fd{} \\
W \fr{u} & \cocartesien Y}
\]
then $t$ must be a retract of $u$. Therefore it suffices to prove that
$u$ is a weak S-homotopy equivalence. So one can suppose that one has
a diagram of flows of the form
\[
\xymatrix{
A\fr{\phi} \fd{k} & U \fr{s}\fd{i} & X \fd{} \\
B \fr{} & \cocartesien V \fr{t} & \cocartesien Y}
\]
where $k\in \cell(I^{gl}_+)$. By Proposition~\ref{factorR}, the
morphism $k:A\longrightarrow B$ factors as a composite $A
\longrightarrow A' \longrightarrow A'' \longrightarrow B$ where the morphism
$A \longrightarrow A'$ is an element of $\cell(\{R\})$, where the
morphism $A' \longrightarrow A''$ is an element of $\cell(\{C\})$, and
where the morphism $A'' \longrightarrow B$ is a morphism of
$\cell(I^{gl})$. So we have to treat the cases $k\in \cell(\{R\})$,
$k\in \cell(\{C\})$ and $k\in \cell(I^{gl})$.

The case $k\in \cell(I^{gl})$ is a consequence of
Proposition~\ref{pushexplicit}, Proposition~\ref{versleft} and
Proposition~\ref{petitplus}. The case $k\in \cell(\{C\})$ is trivial.

Let $k\in \cell(\{R\})$. Let $(\alpha,\beta)\in U^0\p U^0$. Then
$\P_{i(\alpha),i(\beta)}V$ (resp.
$\P_{i(\alpha),i(\beta)}Y$) is a coproduct of terms of the form
$\P_{\alpha,u_0}U\p \P_{v_0,u_1}U\p \dots\p \P_{v_p,\beta}U$
(resp. $\P_{\alpha,u_0}X\p \P_{v_0,u_1}X\p \dots\p
\P_{v_p,\beta}X$) such that $(u_i,v_i)$ is a pair of distinct elements of
$U^0=X^0$ identified by $k$. So $t$ is a weak S-homotopy equivalence
since a binary product of weak homotopy equivalences is a weak
homotopy equivalence.
\end{proof}

\section{T-homotopy equivalence and $I^{gl}_+$-cell complex}
\label{main1}

The first step to understand the reason why Definition~\ref{Tdi} is
badly-behaved is the following theorem which gives a description of
the T-homotopy equivalences $f:X\longrightarrow Y$ such that the
$0$-skeleton of $Y$ contains exactly one more state than the
$0$-skeleton of $X$.

\begin{thm} \label{re1}
Let $X$ and $Y$ be two objects of ${\cell(\dtop)}$. Let $f:X\longrightarrow Y$ be
a T-homotopy equivalence. Assume that $Y^0=X^0\sqcup \{\alpha\}$. Then
the canonical morphism $\varnothing\longrightarrow X$ factors as a
composite $\varnothing\longrightarrow u_f(X)\longrightarrow
v_f(X)\longrightarrow X$ such that
\begin{enumerate}
\item one has the diagram
\[
\xymatrix{
& \varnothing \fd{}&\\
\{\widehat{0},\widehat{1}\}={\rm{Glob}}(\mathbf{S}^{-1}) \fr{} \fd{} & u_f(X) \fd{}&&\\
\vec{I}={\rm{Glob}}(\mathbf{D}^0) \fr{}\fd{\phi} & \cocartesien v_f(X)  \fd{} \ar@{->}[rr] && X \fd{}\\
\vec{I}*\vec{I} \fr{} & \cocartesien \widehat{v}_f(X) \ar@{->}[rr] && \cocartesien Y
}
\]
\item the morphisms $\varnothing\longrightarrow u_f(X)$ and
$v_f(X)\longrightarrow X$ are relative $I_g$-cell complexes.
\end{enumerate}
\eth

By Proposition~\ref{tjrcof}, the morphism
$\{\widehat{0},\widehat{1}\}={\rm{Glob}}(\mathbf{S}^{-1})\longrightarrow
u_f(X)$ is a cofibration.  Therefore the morphism $\vec{I}\longrightarrow
v_f(X)$ is a cofibration as well. The morphism $u_f(X)\longrightarrow
\widehat{v}_f(X)$ is a relative $I_g$-cell complex as well since it is
a pushout of the inclusion $\{\widehat{0},\widehat{1}\}\subset
\vec{I}*\vec{I}$ sending $\widehat{0}$ to the initial state of $\vec{I}*\vec{I}$ and $\widehat{1}$ to the
final state of $\vec{I}*\vec{I}$.

\begin{proof} By Proposition~\ref{removingR}, and since $Y$ is an object of
${\cell(\dtop)}$, the canonical morphism of flows $Y^0\longrightarrow Y$ is a
relative $I_g$-cell complex. So there exists an ordinal $\lambda$ and
a $\lambda$-sequence $\mu \mapsto Y_\mu :\lambda\longrightarrow {\brm{Flow}}$
(so also denoted by $Y$) such that $Y=\varinjlim_{\mu<\lambda} Y_\mu$ and
such that for any ordinal $\mu<\lambda$, the morphism
$Y_\mu\longrightarrow Y_{\mu+1}$ is a pushout of the form
\[
\xymatrix{
{\rm{Glob}}(\mathbf{S}^{n_\mu-1})\fd{} \fr{\phi_\mu}& Y_\mu\fd{}\\
{\rm{Glob}}(\mathbf{D}^{n_\mu}) \ar@{->}[r]_-{\psi_\mu}& Y_{\mu+1}\cocartesien
}
\]
of the inclusion of flows ${\rm{Glob}}(\mathbf{S}^{n_\mu})\longrightarrow
{\rm{Glob}}(\mathbf{D}^{n_\mu+1})$ for some $n_\mu\geq 0$.

For any ordinal $\mu$, the morphism of flows $Y_\mu\longrightarrow
Y_{\mu+1}$ induces an isomorphism between the $0$-skeletons $Y_\mu^0$
and $Y_{\mu+1}^0$. If $n_\mu\geq 1$ for some $\mu$, then for any
$\beta,\gamma\in Y^0$, the topological space $\P_{\beta,\gamma}Y_\mu$
is non empty if and only if the topological space
$\P_{\beta,\gamma}Y_{\mu+1}$ is non empty. Consider the set of
ordinals
\[\left\{\mu<\lambda; \bigsqcup_{\beta\in X^0}\P_{\beta,\alpha}Y_\mu\neq \varnothing\right\}\]
It is non-empty since $f$ is a T-homotopy equivalence. Take its smallest element
$\mu_0$. Consider the set of ordinals
\[\left\{\mu<\lambda; \bigsqcup_{\beta\in X^0}\P_{\alpha,\beta}Y_\mu\neq \varnothing\right\}\]
Take its smallest element $\mu_1$. Let us suppose for instance that
$\mu_0<\mu_1$.

The ordinal $\mu_0$ cannot be a limit ordinal. Otherwise for any
$\mu<\mu_0$, the isomorphisms of flows $Y_\mu=Z_\mu\sqcup \{\alpha\}$
and $Y_{\mu_0}\cong \varinjlim_{\mu<\mu_0}\left( Z_\mu\sqcup
\{\alpha\}\right)\cong\left(
\varinjlim_{\mu<\mu_0}Z_\mu\right) \sqcup \{\alpha\}$ would hold:
contradiction. Therefore $\mu_0=\mu_2+1$ and $n_{\mu_2}=0$. There
does not exist other ordinal $\mu$ such that
$\phi_\mu(\widehat{1})=\alpha$ otherwise $\P_\alpha^+Y$ could not be a
singleton anymore.

For a slightly different reason, the ordinal $\mu_1$ cannot be a limit
ordinal either. Otherwise if $\mu_1$ was a limit ordinal, then the
isomorphism of flows $Y_{\mu_1}\cong \varinjlim_{\mu<\mu_1} Y_\mu$ would
hold. The path space of a colimit of flows is in general not the
colimit of the path spaces. But any element of $\P Y_{\mu_1}$ is a
composite $\gamma_1*\dots*\gamma_p$ where the $\gamma_i$ for $1\leq
i\leq p$ belong to $\varinjlim_{\mu<\mu_1}\P Y_\mu$. By hypothesis, there
exists an execution path $\gamma_1*\dots*\gamma_p\in
\P_{\alpha,\beta}Y_{\mu_1}$ for some $\beta\in X^0$.  So
$s(\gamma_1)=\alpha$, which contradicts the definition of
$\mu_1$. Therefore $\mu_1=\mu_3+1$ and necessarily $n_{\mu_3}=0$.
There does not exist any other ordinal $\mu$ such that
$\phi_\mu(\widehat{0})=\alpha$ otherwise $\P_\alpha^-Y$ could not be a
singleton anymore.

Therefore one has the following situation: $Y_{\mu_2}$ is a flow of
the form $Z_{\mu_2}\sqcup \{\alpha\}$. The passage from $Y_{\mu_2}$ to
$Y_{\mu_2+1}$ is as follows:
\[
\xymatrix{
{\rm{Glob}}(\mathbf{S}^{-1}) \fd{} \fr{\phi_{\mu_2}} & Y_{\mu_2} \fd{}\\
{\rm{Glob}}(\mathbf{D}^0) \fr{} & \cocartesien Y_{\mu_2+1}}
\]
where $\phi_{\mu_2}(\widehat{0})\in X^0$ and
$\phi_{\mu_2}(\widehat{1})=\alpha$. The morphism of flows
$Y_{\mu_2+1}\longrightarrow Y_{\mu_3}$ is a transfinite composition of
pushouts of the inclusion of flows
${\rm{Glob}}(\mathbf{S}^{n})\longrightarrow {\rm{Glob}}(\mathbf{D}^{n+1})$ where
$\phi_\mu(\widehat{0})$ and $\phi_\mu(\widehat{1})$ are never equal to
$\alpha$. The passage from $Y_{\mu_3}$ to $Y_{\mu_3+1}$ is as follows:
\[
\xymatrix{
{\rm{Glob}}(\mathbf{S}^{-1}) \fd{} \fr{\phi_{\mu_3}} & Y_{\mu_3} \fd{}\\
{\rm{Glob}}(\mathbf{D}^0) \fr{} & \cocartesien Y_{\mu_3+1}}
\]
where $\phi_{\mu_3}(\widehat{0})=\alpha$ and
$\phi_{\mu_3}(\widehat{1})\in X^0$.  The morphism of flows
$Y_{\mu_3+1}\longrightarrow Y_{\lambda}$ is a transfinite composition
of pushouts of the inclusion of flows
${\rm{Glob}}(\mathbf{S}^{n})\longrightarrow {\rm{Glob}}(\mathbf{D}^{n+1})$ where
$\phi_\mu(\widehat{0})$ and $\phi_\mu(\widehat{1})$ are never equal to
$\alpha$.  Hence the result.
\end{proof}

We are now ready to give a characterization of the old T-homotopy
equivalences:

\begin{thm}
\label{re3}
Let $X$ and $Y$ be two objects of ${\cell(\dtop)}$.  Then a morphism of flows
$f:X\longrightarrow Y$ is a T-homotopy equivalence if and only if
there exists a commutative diagram of flows of the form (with $\vec{I}^{*
(n+1)}:=\vec{I}^{* n}*\vec{I}$ and $\vec{I}^{* 1}:=\vec{I}$ for $n\geq 1$):
\[
\xymatrix{
& \varnothing \fd{}&\\
\bigsqcup_{i\in I}\{0,1\}=\bigsqcup_{i\in I}{\rm{Glob}}(\mathbf{S}^{-1}) \fr{} \fd{} & u_f(X) \fd{}&&\\
\bigsqcup_{i\in I}\vec{I}=\bigsqcup_{i\in I}{\rm{Glob}}(\mathbf{D}^0) \fr{}\fd{\bigsqcup_{i\in I} r_i} & \cocartesien v_f(X)  \fd{} \ar@{->}[rr] && X \fd{f}\\
\bigsqcup_{i\in I}\vec{I}^{* n_i} \fr{} & \cocartesien \widehat{v}_f(X) \ar@{->}[rr] && \cocartesien Y
}
\]
where for any $i\in I$, $n_i$ is an integer with $n_i\geq 1$ and such
that $r_i:\vec{I}\longrightarrow \vec{I}^{* n_i}$ is the unique morphism of
flows preserving the initial and final states and where
the morphisms $\varnothing\longrightarrow u_f(X)$ and
$v_f(X)\longrightarrow X$ are relative $I_g$-cell complexes.
\eth

The pushout above tells us that the copy of $\vec{I}$ corresponding to the
indexing $i\in I$ is divided in the concatenation of $n_i$ copies of
$\vec{I}$. This intuitively corresponds to a refinement of observation.

\begin{proof}
By Theorem~\ref{remonte}, there exists a globular complex $U$
(resp. $V$) such that $cat(U)=X$ (resp. $cat(V)=Y$). If a morphism of
flows $f:X\longrightarrow Y$ is a T-homotopy equivalence, then by
Theorem~\ref{TTT}, there exists a morphism of globular complexes
$g:U\longrightarrow V$ such that $cat(g)=f$ and such that the
continuous map $|g|:|U|\longrightarrow |V|$ between the underlying
topological spaces is an homeomorphism. So for any pair of points
$(\alpha,\beta)$ of $X^0\p X^0$, and any morphism $\vec{I}\longrightarrow
X$ appearing in the globular decomposition of $X$, the set of
subdivision of this segment in $Y$ is finite since $Y^0$ is discrete
and since the segment $[0,1]$ is compact. The result is then
established by repeatedly applying Theorem~\ref{re1}.

Now suppose that a morphism of flows $f:X=cat(U)\longrightarrow
Y=cat(V)$ can be written as a pushout of the form of the statement of
the theorem. Then start from a globular decomposition of $U$ which is
compatible with the composite $\varnothing\longrightarrow
u_f(X)\longrightarrow v_f(X)\longrightarrow X$. Then let us divide
each segment of $[0,1]$ corresponding to the copy of $\vec{I}$ indexed by
$i\in I$ in $n_i$ pieces. Then one obtains a globular decomposition of
$V$ and the identity of $U$ gives rise to a morphism of globular
complexes $g:U\longrightarrow V$ which induces an homeomorphism
between the underlying topological spaces and such that
$cat(g)=f$. Hence the result.
\end{proof}

\begin{defn} Let $n\geq 1$. The {\rm full $n$-cube} $\vec{C}_n$ is by
definition the flow $Q(\{\widehat{0}<\widehat{1}\}^n)$, where $Q$ is
the cofibrant replacement functor. \end{defn}

The flow $\vec{C}_3$ is represented in Figure~\ref{3cube}.

\begin{lem} \label{ordrestate} If a flow $X$ is loopless, then the
  transitive closure of the set \[\{(\alpha,\beta)\in X^0\p X^0\hbox{
    such that }\P_{\alpha,\beta}X\neq\varnothing\}\] induces a partial
  ordering on $X^0$.
\end{lem}

\begin{proof} If $(\alpha,\beta)$ and $(\beta,\alpha)$ with $\alpha\neq \beta$
belong to the transitive closure, then there exists a finite sequence
$(x_1,\dots,x_\ell)$ of elements of $X^0$ with $x_1=\alpha$,
$x_\ell=\alpha$, $\ell>1$ and with $\P_{x_m,x_{m+1}}X$ non-empty for
each $m$. Consequently, the space $\P_{\alpha,\alpha}X$ is non-empty
because of the existence of the composition law of $X$: contradiction.
\end{proof}

\begin{thm} \label{pasiso}
Let $n\geq 3$. There does not exist any zig-zag sequence
\[
\xymatrix{
\vec{C}_n=X_0 \fr{f_0} & X_1 & X_2 \fl{f_1}\fr{f_2} & \dots &  X_{2n}=\vec{I} \fl{f_{2n-1}}
}
\]
where every $X_i$ is an object of ${\cell(\dtop)}$ and where every $f_i$ is
either a S-homotopy equivalence or a T-homotopy equivalence. \eth

\begin{proof} By an immediate induction, one sees that each flow $X_i$ is loopless,
with a finite $0$-skeleton. Moreover by construction, each poset
$(X_i^0,\leq)$ is bounded, i.e. with one bottom element $\widehat{0}$
and one top element $\widehat{1}$. So the zig-zag sequence above gives
rise to a zig-zag sequence of posets:
\[
\xymatrix{
\vec{C}_n^0=X_0^0=\{\widehat{0}<\widehat{1}\}^n \fr{} & X_1^0 &
X_2^0 \fl{}\fr{} & \dots & X_{2n}^0=\vec{I}^0=\{\widehat{0}<\widehat{1}\}
\fl{} }
\]
where $\{\widehat{0}<\widehat{1}\}^n$ is the product
$\{\widehat{0}<\widehat{1}\}\p \dots \p \{\widehat{0}<\widehat{1}\}$
($n$ times) in the category of posets.  Each morphism of posets is an
isomorphism if the corresponding morphism of flows is a S-homotopy
equivalence because a S-homotopy equivalence induces a bijection
between the $0$-skeletons. Otherwise one can suppose by
Theorem~\ref{re1} that the morphism of posets $P_1\longrightarrow P_2$
can be described as follows: take a segment $[x,y]$ of $P_1$ such that
$]x,y[=\varnothing$; add a vertex $z\in ]x,y[$; then let
$P_2=P_1\cup\{z\}$ with the partial ordering $x<z<y$.  In such a
situation, $\min(]z,-[)$ exists and is equal to $y$, and $\max(]-,z[)$
exists and is equal to $x$. So by an immediate induction, there must
exist $x,y,z\in \{0<1\}^n$ with $x<z<y$ and such that $\min(]z,-[)=y$
and $\max(]-,z[)=x$. This situation is impossible in the poset
$\{\widehat{0}<\widehat{1}\}^n$ for $n\geq 3$.
\end{proof}

\section{Generalized T-homotopy equivalence}
\label{main2}

As explained in the introduction, it is not satisfactory not to be
able to identify $\vec{C}_3$, and more generally $\vec{C}_n$ for
$n\geq 3$, with $\vec{I}$. The following definitions are going to be
important for the sequel of the paper, and also for the whole series.

\begin{defn}~\footnote{The statement of the definition is slightly different,
but equivalent to the statement given in other parts of this series.}
A {\rm full directed ball} is a flow $\vec{D}$ such that:
\begin{itemize}
\item $\vec{D}$ is loopless (so by Lemma~\ref{ordrestate}, the set
  $\vec{D}^0$ is equipped with a partial ordering $\leq$)
\item $(\vec{D}^0,\leq)$ is finite bounded
\item for all $(\alpha,\beta)\in \vec{D}^0\p \vec{D}^0$, the
  topological space $\P_{\alpha,\beta}\vec{D}$ is weakly contractible
  if $\alpha<\beta$, and empty otherwise by definition of $\leq$.
\end{itemize}
\end{defn}

Let $\vec{D}$ be a full directed ball. Then by Lemma~\ref{ordrestate},
the set $\vec{D}^0$ can be viewed as a finite bounded poset.
Conversely, if $P$ is a finite bounded poset, let us consider the
\textit{flow} $F(P)$ \textit{associated with} $P$: it is of course
defined as the unique flow $F(P)$ such that $F(P)^0=P$ and
$\P_{\alpha,\beta}F(P)=\{u\}$ if $\alpha<\beta$ and
$\P_{\alpha,\beta}F(P)=\varnothing$ otherwise. Then $F(P)$ is a full
directed ball and for any full directed ball $\vec{D}$, the two flows
$\vec{D}$ and $F(\vec{D}^0)$ are weakly S-homotopy equivalent.

Let $\vec{E}$ be another full directed ball. Let $f: \vec{D}
\longrightarrow \vec{E}$ be a morphism of flows preserving the initial
and final states. Then $f$ induces a morphism of posets from
$\vec{D}^0$ to $\vec{E}^0$ such that $f(\min
\vec{D}^0) = \min \vec{E}^0$ and $f(\max \vec{D}^0) = \max \vec{E}^0$.
Hence the following definition:

\begin{defn} \label{definitiondeT}
Let $\mathcal{T}$ be the class of morphisms of posets
$f:P_1\longrightarrow P_2$ such that:
\begin{enumerate}
\item The posets $P_1$ and $P_2$ are finite and bounded.
\item The morphism of posets $f:P_1 \longrightarrow P_2$ is one-to-one;
in particular, if $x$ and $y$ are two elements of $P_1$ with $x<y$,
then $f(x)<f(y)$.
\item One has $f(\min P_1)=\min P_2$ and  $f(\max P_1)=\max P_2$.
\end{enumerate}
Then a {\rm generalized T-homotopy equivalence} is a morphism of
$\cof(\{Q(F(f)),f\in\mathcal{T}\})$ where $Q$ is the cofibrant
replacement functor of the model category ${\brm{Flow}}$.
\end{defn}

It is of course possible to identity $\vec{C}_n$ ($n\geq 1$) with
$\vec{I}$ by the following zig-zag sequence of S-homotopy and generalized
T-homotopy equivalences
\[
\xymatrix{ \vec{I} & \fl{\simeq} Q(\vec{I}) \ar@{->}[rr]^-{Q(F(g_n))} &&
  Q(\{\widehat{0} < \widehat{1}\}^n), }
\]
where $g_n:\{\widehat{0} < \widehat{1}\} \longrightarrow \{\widehat{0} <
\widehat{1}\}^n \in \mathcal{T}$.

The relationship between the new definition of T-homotopy equivalence
and the old definition is as follows:

\begin{thm}  \label{OKnew}
Let $X$ and $Y$ be two objects of ${\cell(\dtop)}$.  Let $f:X\longrightarrow Y$
be a T-homotopy equivalence. Then $f$ can be written as a composite
$X\longrightarrow Z\longrightarrow Y$ where $g:X\longrightarrow Z$ is
a generalized T-homotopy equivalence and where $h:Z\longrightarrow Y$
is a weak S-homotopy equivalence.
\eth

\begin{proof} By Theorem~\ref{re3}, there exists a pushout
diagram of flows of the form (with $\vec{I}^{* (n+1)}:=\vec{I}^{* n}*\vec{I}$ and
$\vec{I}^{* 1}:=\vec{I}$ for $n\geq 1$):
\[
\xymatrix{
\bigsqcup_{k\in K} \vec{I} \ar@{^{(}->}[r]^{}\fd{\bigsqcup_{k\in K} r_k} & X \fd{}\\
\bigsqcup_{k\in K} \vec{I}^{* n_k} \fr{} & Y \cocartesien}
\]
where for any $k\in K$, $n_k$ is an integer with $n_k\geq 1$ and such
that $r_k:\vec{I}\longrightarrow \vec{I}^{* n_k}$ is the unique morphism of
flows preserving the initial and final states. Notice that each
$\vec{I}^{* n_k}$ is a full directed ball. Thus one obtains the following
commutative diagram:
\[
\xymatrix{
\bigsqcup_{k\in K} Q(\vec{I}) \ar@{->>}[r]^{\simeq}\fd{} & \bigsqcup_{k\in K} \vec{I}
\ar@{^{(}->}[r]\fd{} & X \ar@{^{(}->}[d]\\
\bigsqcup_{k\in K} Q(\vec{I}^{* n_k}) \ar@{=}[r] & \bigsqcup_{k\in K} Q(\vec{I}^{* n_k}) \ar@{^{(}->}[r]\fd{\simeq} & \cocartesien Z \fd{\simeq} \\
&\bigsqcup_{k\in K} \vec{I}^{* n_k} \ar@{^{(}->}[r] & Y. \cocartesien}
\]
Now here are some justifications for this diagram. First of all, a
morphism of flows $f:M\longrightarrow N$ is a fibration of flows if
and only if the continuous map $\P f :\P M\longrightarrow \P N$ is a
Serre fibration of topological spaces. Since any coproduct of Serre
fibration is a Serre fibration, the morphism of flows $\bigsqcup_{i\in
I} Q(\vec{I})\longrightarrow \bigsqcup_{k\in K} \vec{I} $ is a trivial
fibration of flows. Thus the underlying set map $\bigsqcup_{k\in K}
Q(\vec{I})\longrightarrow \bigsqcup_{k\in K} \vec{I} $ is surjective. So the
commutative square
\[
\xymatrix{
\bigsqcup_{k\in K} Q(\vec{I})\ar@{->}[d]\fr{} & \fd{} X\\
\bigsqcup_{k\in K} Q(\vec{I}^{* n_k})\fr{} & \cocartesien Z}
\]
is cocartesian and the morphism of flows $X\longrightarrow Z$ is then
a generalized T-homotopy equivalence. It is clear that the morphism
$\bigsqcup_{k\in K} Q(\vec{I}^{* n_k})\longrightarrow
\bigsqcup_{k\in K} \vec{I}^{* n_k}$ is a weak S-homotopy equivalence. The latter
morphism is even a fibration of flows, but that does not matter
here. So the morphism $Z\longrightarrow Y$ is the pushout of a weak
S-homotopy equivalence along the cofibration $\bigsqcup_{k\in K}
Q(\vec{I}^{* n_k})\longrightarrow Z$.  Since the model category ${\brm{Flow}}$ is
left proper by Theorem~\ref{leftproperflow}, the proof is complete.
\end{proof}

\section{Conclusion}

This new definition of T-homotopy equivalence contains the old one up
to S-homotopy equivalence.  The drawback of the old definition
presented in \cite{model2} is overcome. It is proved in \cite{3eme}
that this new notion of T-homotopy equivalence does preserve the
branching and merging homology theories. And it is proved in
\cite{4eme} that the underlying homotopy type of a flow is also
preserved by this new definition of T-homotopy equivalence. Finally,
the paper \cite{hocont} proposes an application of this new notion of
dihomotopy: that is a Whitehead theorem for the full dihomotopy
relation.

redpajama_set_name: RedPajamaArXiv


Friday's all-day snowstorm left fifteen inches of new snow. Saturday's "flurries" left an additional three inches. Last night's "chance of snow" left two inches.
It continues to rain almost every day. Our meadow garden is WET… standing water wet… and the cucumbers and squash are producing like crazy.


Calvin Klein Minimal 35mm K3M226G6 is an amazing and attractive Gents watch. Case is made out of PVD rose plating and the Silver dial gives the watch that unique look. In regards to the water resistance, the watch has got a resistancy up to 30 metres. It means it can be worn in scenarios where it is likely to be splashed but not immersed in water. It can be worn while washing your hands and will be fine in rain. The watch is shipped with an original box and a guarantee from the manufacturer.


The shares of Laredo Petroleum, Inc. (NYSE:LPI) has been pegged with a rating of Hold by Societe Generale in its latest research note that was published on February 15th, 2019. Societe Generale wasn't the only research firm that published a report of Laredo Petroleum, Inc., with other equities research analysts also giving their opinion on the stock. MKM Partners advised investors in its research note published on February 15th, 2019, to Neutral the LPI stock. The stock had earned Neutral rating from Piper Jaffray when it published its report on January 9th, 2019. The stock was given Underweight rating by CapitalOne in its report released on December 20th, 2018. Credit Suisse was of a view that LPI is Underperform in its latest report on December 19th, 2018. MKM Partners thinks that LPI is worth Buy rating. This was contained in the firm's report on December 6th, 2018 in which the stock's price target was also moved to $8.
Amongst the analysts that rated the stock, 0 have recommended investors to sell it, 10 believe it has the potential for further growth, thus rating it as Hold while 1 advised investors to purchase the stock. The consensus currently stands at a Hold while its average price target is $4.27. The price of the stock the last time has raised by 12.14% from its Week high price while it is raised higher than its 52-Week low price. A look at the stock's other technical shows that its 14-day RSI now stands at 47.78.
The shares of the company dipped by -1.87% during the trading session on Monday, reaching a low of $3.14 while ending the day at $3.14. During the trading session, a total of 3.64 million shares were traded which represents a 33.35% incline from the average session volume which is 5.46M shares. LPI had ended its last session trading at 3.20. Laredo Petroleum, Inc. currently has a market cap of $723.05M, while its P/E ratio stands at 2.25, while its P/E earnings growth sits at 0.65, with a beta of 1.48. Laredo Petroleum, Inc. debt-to-equity ratio currently stands at 0.84, while its quick ratio hovers at . LPI 52-week low price stands at $2.80 while its 52-week high price is $11.18.
The company in its last quarterly report recorded $0.16 earnings per share which is below the $0.24 predicted by most analysts. The Laredo Petroleum, Inc. generated $215.29 million in revenue during the last quarter, which is slightly lower than the $233.71 million predicted by analysts. In the second quarter last year, the firm recorded $0.27 earnings per share. Compared to the same quarter last year, the firm's revenue was down by -68.75%. Laredo Petroleum, Inc. has the potential to record 1.39 EPS for the current fiscal year, according to equities analysts.
Investment analysts at Jefferies published a research note on April 8th, 2019 where it informed investors and clients that New York Community Bancorp, Inc. (NYSE:NYCB) is now rated as Hold. Their price target on the stock stands at $12. MKM Partners also rated NYCB as Downgrade on February 15th, 2019, with its price target of $70 suggesting that NYCB could down by -1.36% from its current share price. Even though the stock has been trading at $12.00/share, analysts expect it to surge higher by -0.75% to reach $11.75/share. It started the day trading at $12.01 and traded between $11.86 and $11.91 throughout the trading session.
A look at its technical shows that NYCB's 50-day SMA is 12.00 while its 200-day SMA stands at 10.77. The stock has a high of $13.07 for the year while the low is $8.61. The company's P/E ratio currently sits at 15.10, while the P/B ratio is 0.94. At the moment, only of New York Community Bancorp, Inc. shares were sold short. The company's average trading volume currently stands at 5.59M shares, which means that the short-interest ratio is just 7.77 days. Over the past seven days, the company moved, with its shift of 0.59%. Looking further, the stock has raised 17.11% over the past 90 days while it gained 17.69% over the last six months.
The change in the stock's fortunes has led to several institutional investors altering their holdings of the stock. BlackRock Fund Advisors bought more NYCB shares, increasing its portfolio by +2.75% during the last quarter. This move now sees BlackRock Fund Advisors purchasing 1,362,737 shares in the last quarter, thus it now holds 50,827,564 shares of NYCB, with a total valuation of $588,074,915. The Vanguard Group Inc meanwhile bought more NYCB shares in the recently filed quarter, changing its stake to $518,926,220 worth of shares. Barrow Hanley Mewhinney Strau followed the path by increasing its NYCB portfolio by +7.74% in the quarter. This means that Barrow Hanley Mewhinney Strau bought 2,960,395 shares in the last quarter and now controls 41,199,922 shares of the NYCB stock, with the valuation hitting $476,683,098.
Similarly, Goldman Sachs Co LLC decreased its New York Community Bancorp, Inc. shares by -26.06% during the recently filed quarter. After selling -5,215,372 shares in the last quarter, the firm now controls 14,799,285 shares of New York Community Bancorp, Inc. which are valued at $171,227,727. In the same vein, Alecta Pension Insurance Mutual decreased its New York Community Bancorp, Inc. shares by during the most recent reported quarter. The firm sold 0 shares during the quarter which decreased its stakes to 10,400,000 shares and is now valued at $120,328,000. Following these latest developments, around 1.90% of New York Community Bancorp, Inc. stocks are owned by institutional investors and hedge funds.
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text:
ROD THOMPSON / RTHOMPSON@STARBULLETIN.COM
The Hawaiian Acres subdivision home of shooting victims John and Doraline McDonald was visible yesterday from the home of neighbor Paul Cyr through a thicket of uluhe ferns and other bushes. Cyr told police he heard two series of fast shots from the house, which is about 80 feet from his house.
Big Isle shooting
suspect dead
Police find the man dead
of a gunshot wound, an
apparent suicide, in a
home south of Hilo
By Rod Thompson
rthompson@starbulletin.com
HILO >> Big Island shooting suspect Nat Kalae Eckman apparently committed suicide yesterday, a day after allegedly shooting a man to death, critically wounding the man's wife and abducting the couple's 3-year-old girl.
Nat Kalae Eckman: He also had abducted a 3-year-old girl
Eckman, 33, was found dead of a gunshot wound in a Hawaiian Acres subdivision home south of Hilo, a few miles from the Hawaiian Acres house where he allegedly shot the victims Sunday evening, according to police.
The girl had been dropped off, unharmed, at another house, a police source said.
The victims were identified as John C. McDonald, 62, found dead at the scene, and his wife, Doraline, 57, shot in the face and in critical condition at the Queen's Medical Center.
The motive for the shooting seemed to be the abduction of the couple's 3-year-old granddaughter, whom they had adopted as their daughter, said neighbor Paul Cyr.
Eckman is the girl's father, according to Cyr. However, police Capt. Samuel Thomas declined to comment on the girl, and neither of two written police statements mentioned her.
The location of the girl's mother was not clear.
Cyr heard the girl crying during the shootings, then heard the intruder say something like, "I've got you," he said.
Eckman was found after police received a report of a possible suicide victim, a statement said. Police confirmed that the man was Eckman but stopped short of confirming that his death was a suicide until an autopsy is conducted today.
Cyr said he has been living in his home since 1993, about the same time that the McDonalds moved in next door. John McDonald was a disabled mechanic who had lost sight in one eye in an industrial accident, Cyr said.
"You couldn't ask for a better neighbor," he said. "He always had a smile on his face, a happy-go-lucky guy."
A few years ago the McDonalds' minor teenage daughter had the baby girl, Cyr said. The daughter was unable to take care of the child, so her parents adopted the girl, he said.
Cyr had never heard the McDonalds mention any trouble with the girl's father.
Cyr said he was watching television Sunday night when he heard shots from the McDonald home, about 80 feet from his, partially visible through some bushes in forested, sparsely populated Hawaiian Acres.
Police put the time of the shooting at about 8:15 p.m.
Cyr said he heard, "pop, pop, pop, pop, pop," as many as five shots in quick succession.
"It sounded like a kid playing with a cap gun, but it was real fast," he said. He heard the girl crying. Then he heard four more shots, "pop, pop, pop, pop," he said.
He heard a man say something like, "Get out of here," or, "I've got you."
"Right after that the kid stopped crying and it was real quiet." Cyr did not hear a car or any sound of someone leaving.
A bit later he heard a soft voice, he said.
Capt. Thomas said the wounded Doraline McDonald telephoned for help.
About 100 feet up the road, another neighbor, who asked that his name not be used, also heard the shots and called police.
The neighbor said his dog started barking 20 or 30 minutes before the shooting, which was unusual because his dog rarely barks.
Unlike Cyr, he said the shooting began with just two separate shots. "Then I heard someone yelling 'Help!' screaming," he said.
"We have a lot of drugs out here. I thought it was related to that. You hear gunshots all the time," he said. "But I never heard anyone yelling 'Help!' before."
He called 911.
"Probably while I was talking to them, I heard three or four more shots," he said. "It was a small caliber. It was more of a pop than a bang."
"John (McDonald) was a nice guy. He walked the granddaughter out here in a little red wagon," the neighbor said.
"He always said he'd come back and have a beer, but he never did."
An autopsy was scheduled today for McDonald, police said.

redpajama_set_name: RedPajamaCommonCrawl


HiLine has served customers in a number of industries ranging from commercial nuclear power to agriculture. Though HiLine's customers are diverse in terms of industry, they have in common a need for high-quality equipment that's been designed, built and tested by people they trust. Because HiLine's engineers and craftspeople posess wide-ranging experience and expertise, most projects can be completed in-house. This increases confidentiality, facilitates quality and reduces project risk for our customers. When necessary, we will team with trusted partners for specialized and/or larger projects.


April 25-26, 2015: The inaugural Jackalope Arts fair kicked off its two-day event by bringing hundreds of patrons and vendors to Central Park in Old Town Pasadena. The traveling, semi-annual fair takes place in Old Pasadena, CA and Phoenix, AZ in the spring and fall, Denver, CO in the summer and winter. For more information about events or being a vendor, visit their website.


text:
World War I: A History
Title World War I: A History
Authors Strachan, Hew
Publisher Oxford University Press
City Oxford
The First World War has shaped the history of the twentieth century. In this extensively illustrated book, an international team of experts explores the war in all its different aspects. From its causes to its consequences, from the strategy of the politicians to the tactics of the generals, the course of the war is charted and its political and human consequences assessed. Chapters on economic mobilization, the impact on women, the role of propaganda, and the rise of socialism establish the wider social context of fighting which took place at sea and in the air and which ranged on land from the Flanders trenches to the Balkan mountains and the deserts of the Middle East. The legacy of 'the war to end wars'--in poetry and prose, in collective memory and political culture--is with us still, eighty years after that first Armistice Day. This remarkable book helps us understand that legacy.
Anthology/Edited Volume
Textbook/Survey
War & Warfare
War & Violence
War & International Politics
War & Nation/Nation-building
Military & Military Systems
War Mobilization
War Memories & Gender
Postwar Societies & Gender
First World War (1914-19)
-North America
Transatlantic Connections
Britain/British Empire
Austria/Habsburg Empire
Turkey/Ottoman Empire


Nick and Benji bring you all the latest from the Big Finish audio world. Guest star: Jacob Dudman (the Tenth and Eleventh Doctor Chronicles). Drama tease: Jago and Litefoot Forever (Doctor Who spin-off).
Latest Releases - with reviews and clips.
Big Finish News - with a Star Cops featurette and so much more, including an introduction to Pathfinder Legends.
Listeners' Emails - including a brand new competition. Join in by emailing podcast@bigfinish.com.
Guest Star - rising star Jacob (Jake) Dudman chats to Nick Briggs about what's generally tingling his molecules, and demonstrates a rather nifty Matt Smith voice.
Randomoid Selectortron - another surprise from the Big Finish archives.
Drama Tease - the first fifteen minutes of our much anticipated celebratory release, Jago and Litefoot Forever.
This podcast is released on Sunday 27th May 2018. You can download or stream it now simply by pressing one of those handy little buttons below this text. Go on, you know it'll brighten up your day.


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Report: Houston Losing Tech Jobs, But Outlook Positive
The numbers may not account for some jobs in the energy industry.
| Posted on April 1, 2019, 10:09 AM
Houston has been working on improving its startup ecosystem and is developing an innovation hub, called "Ion," on the site of the former Midtown Sears building.
A new report by tech trade association CompTIA says Houston is losing tech jobs, even as the city has been working hard to improve its standing as a tech innovation hub.
From 2017 to 2018, Greater Houston lost about 2,300 tech jobs — more than any other metro area analyzed.
The vast majority of cities gained jobs in the tech industry, including the other major Texas metros.
But these numbers don't tell the whole story, said Gaby Rowe, president and CEO of the startup support organization Station Houston.
Many oil and gas companies are no longer labeling some jobs as "tech" jobs, she said. They are now simply part of operations.
"It moves someone, say, to the lubricant side that is actually working on tech, but it no longer calls them a tech position and instead classifies them as upstream," Rowe said. "Then they fall off on those numbers."
A better indicator is the number of tech job postings in Houston, she said, which went up by 140 percent last year, "and that sounds a lot more like what we're seeing."
In total, there were more than 62,000 tech job postings last year.
Rowe said another good sign is the increase in median wages for these jobs in Houston, which pay more here than in the nation as a whole. The median wage for a tech job in Houston is $86,700, compared to $81,900 nationally, according to CompTIA.
Courtesy of Rice University
Tags employment houston economy innovation jobs technology
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Business Reporter
Florian Martin is currently the News 88.7 business reporter. Florian's stories can frequently be heard on other public radio stations throughout Texas and on NPR nationwide. Some of them have earned him awards from Texas AP Broadcasters, the Houston Press Club, National Association of Real Estate Editors, and Public Radio...
16-Year-Old Houston Entrepreneur Aspires To Be The Queen Of Sauces
Houston Astros Fire Manager, GM After MLB Investigation
Wind Energy On Track To Surpass Coal Power In Texas


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redpajama_set_name: RedPajamaGithub


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Where is Danny Keough today? His low-key life
Ron Galella/Getty Images
The world was shocked to learn about Lisa Marie Presley's sudden death on 12th January 2023, days after she'd attended the Golden Globes ceremony with her mother, Priscilla Presley. Reports state that a housekeeper found Lisa Marie unresponsive in her room.
Lisa's ex-husband, Danny Keough, arrived home shortly afterward and performed CPR until paramedics arrived. They restarted Lisa's heart and rushed her to a local hospital, where doctors placed her on life support. Reports claim that Lisa's family asked doctors not to revive her after her heart stopped a second time as she was pronounced brain dead on admission.
Danny Keough lives in Los Angeles and works as an actor
Danny Keough is an actor who lives in Los Angeles. He likely has another gig, as his acting projects have been few and far between, according to IMDb. Keough's last appearance came in 2019's The Lodge alongside his daughter, Riley Keough.
Keough and Lisa Marie Presley married in 1988. "I am thrilled for Lisa Marie," Lisa told People after the low-key Church of Scientology ceremony. "Danny is a great guy, and I couldn't be happier for the two of them."
Rumors claimed that Lisa forced Danny, who'd met Lisa in rehab, to sign away his claim to Marie's fortune. The alleged prenup would prove beneficial six years later – the couple divorced in 1994 but remained close friends. Lisa told People:
"We are like brother and sister, so it is not weird at all. You don't have to put your crap on your kids even if you are nor together. You can still be civilized. I knew at the age that I had the kids with him that Danny was someone I could be connected with for the rest of my life."
Danny and Lisa started living together again following the suicide death of their son, Benjamin Keough. Lisa wrote on Facebook that Danny was 'completely lost'. A source told The Sun:
"In her [Lisa's] final years, she was racked with grief over Ben, he was her baby, and he relied on her heavily. Her world completely fell apart when he died. She was comforted having her ex Danny around again. They weren't romantic but they were close."
Lisa Marie Presley's ethnicity: The late singer-songwriter's surprising heritage
Where is Walter Noel now? His private life detailed


Who doesn't love cheeky Peter Rabbit, possibly Beatrix Potter's most famous creation? Peter's back with more 'tails' of fun, mischief and friendship in a new DVD collection "Peter Rabbit – The Tale Of The Great Rabbit And Squirrel Adventure" which is released by Abbey Media and on sale from Monday 24th October 2016.
I have one copy to give away. Entry is via the Rafflecopter widget. UK entries only and terms and conditions apply (please see my Competitions Page). The giveaway will end at 11:59 pm on Friday 11th November.
You can find this DVD at all major retailers and it would make a great Christmas stocking filler.
"Beck Valley Book Tour: The Fabulous @50 Re-experience"
"Top Tips For Avoiding Colds & Flu From Dr Dawn Harper & ColdZyme"


So....you are in the right place. This IS electric lion. You are not seeing things. The above image is of a newborn. It is true.
Don't worry, it's weird for me too.
I got a call from my buddy Jon today to ask me to come grab some shots of him and Becca with their now-one-day-old child person. This is a service I don't mind providing for them at all. Our relationship goes back a few years when they asked me to shoot their wedding have no CLUE if i shot them or not. They just liked my work and wanted me to do whatever I wanted at their wedding. That is pretty much the dream scenario when it comes to me + weddings. I shoot them here and there and actually do get a lot out of them*, but I don't love them. So to be hired basically to just shoot..and then it happens to be a wedding...well thats alright.
So I get this call, head over to their hospital place later in the day, and I sit here now really satisfied. It is no secret that I also don't love kids (nor slobbery dogs nor bad drivers nor gimmicky photography). The very fact I am posting a tiny-child stems from just how in love with this kid they were. They weren't OHMYGODTHISISTHESMARTESTEVERCHILD ITWASBORNWITHANIVYLEAGUEDEGREE SERIOUSLYITSRIGHTHEREANDDIDWEMENTIONHOWFUCKINGCUTEITIS OHMYGOOOOODDDDDD. They were just happy. You could see it all over them. It was internal, which is where I like it. It seemed real. They weren't having a joygasm over telling everyone about their kid. They just...were happy.
So that's why you saw a baby on electric lion. Jon, kissing Stella, with Becca in the background**.
*"get a lot out of them" means here the following: i use them as a means of learning more technical skills, dealing with pressure, etc, but mostly dealing with high stress situations in a manner that FEELS relaxed. that and executing my craft in a situation that will not be repeated. you get one shot and you've gotta be ready. its good training for photography in general.
**i debated what to do with beccas eyes. they look a little racoony but its in a hospital room with less-than-optimal light. if i blow them out it doesnt make any sense, if i dodge them a little she looks sort of like an alien. hopefully this note is moot because you, dear reader, didn't even notice. though of course now you may go look again and be bugged by it too.
oh and post script, I thought I'd mention that, yes, Stella has a cleft lip/palette. in case you saw it and wondered.


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Separate swim periods for women pit New York Times against city's Orthodox
Halbfinger replaces Ian Fisher, who held the post for less than a year. (Jonathan Torgovnik/Getty Images)
(JTA) — An only-in-New-York story about a public swimming pool that offered women-only swim periods for the area's Orthodox community turned into a full-blown media firestorm when The New York Times weighed in on the subject.
The pool, located in the heavily Orthodox Williamsburg neighborhood of Brooklyn, had been offering women-only hours since the 1990s to accommodate those whose religious sensitivities forbid women and men from swimming together.
Last week, the Parks Department canceled the women-only swim periods after an anonymous complaint was made to the city's Commission on Human Rights, only to reverse itself following objections by Assemblyman Dov Hikind, an Orthodox politician representing the nearby Borough Park and Midwood neighborhoods.
The reversal led to a strongly worded editorial in the Times on Wednesday asserting that setting aside a special time for a religious group at a public facility violated "the laws of New York City and the Constitution, and commonly held principles of fairness and equal access."
"The city's human rights law is quite clear that public accommodations like a swimming pool cannot exclude people based on sex," the editorial argued, adding that the current practice has a "a strong odor of religious intrusion into a secular space."
The editorial drew a swift backlash from parts of the Jewish community, who accused the paper of unfairly rejecting a reasonable religious accommodation and of applying a double standard to Orthodox Jews.
Seth Lipsky, the founding editor of the New York Sun and a former editor of the Forward, wrote a heated missive in the New York Post titled "Let My People Swim — and Damn the New York Times."
In a letter to the Times, Rabbi Avi Shafran, the director of public affairs for Agudath Israel of America, a haredi Orthodox umbrella organization, called the women-only hours a "reasonable accommodation," according to the Jewish Daily Forward.
"Rescinding the special sex-segregated hours would be the equivalent of a sign saying, 'No people with traditional values allowed,'" he wrote. "The classical concept of modesty that is embraced by many citizens may have its roots in religious systems. But reasonable accommodation of the needs of such New Yorkers is not an endorsement of any religion."
A group in New York's Rockland County associated with that area's Hasidic Satmar community, the Orthodox Jewish Public Affairs Council, also released a statement saying"The hypocrisy and inflammatory language in the New York Times is astounding in many ways." It pointed to a February story in which the Times reported favorably on accommodations made at a public housing project in Toronto that offered a women's-only swim period for Muslim women.
"If it is great when the wishes of women in the Muslim community are accommodated, why is it a problem when the same is granted to women in the Orthodox Jewish community?" the council asked.
In Tablet magazine, Yair Rosenberg pointed to examples in St. Paul, Minnesota, San Diego and Seattle in which accommodations made for Muslim women to swim without men were applauded in some cases and sparked controversy in others. But he questioned why the Times editorial failed to mention these precedents and focused exclusively on Orthodox Jews.
"It is exceedingly odd that the national paper of record only excoriated the practice of sex-segregated swimming when it became aware of religious Jews engaging in it, and even then, omitted the identical practices of religious Muslims," he wrote.
Hikind, meanwhile, called the city's decision to continue the separate swimming hours "a major victory for human rights."
"It is a major victory for the people, and the community can rest much easier this Shabbos knowing that men and women can continue to swim separately," he wrote in a statement.
The Commission on Human Rights and the Parks Department continue to review the pool policies, a spokesperson told Gothamist.


text:
Capstone – projekt prowadzony przez rząd Stanów Zjednoczonych, mający na celu opracowanie standardów kryptografii do celów publicznych i rządowych. Projekt ten był kierowany przez NIST oraz NSA. Projekt został zapoczątkowany w 1993 roku.

Inicjatywa obejmowała cztery podstawowe algorytmy: algorytm szyfrowania danych o nazwie Skipjack wraz z chipem Clipper; algorytm podpisu cyfrowego DSA; funkcję skrótu SHA-1 oraz protokół wymiany klucza. Pierwszą impelementacją Capstone była karta Fortezza PCMCIA.

Wszystkie części Capstone były ukierunkowane na 80-bitowy poziom bezpieczeństwa. Klucze symetryczne miały długość 80 bitów.

Przypisy

Kryptologia

redpajama_set_name: RedPajamaWikipedia