text:
\section*{Introduction}

It is well known that $3$-cohomology in an algebraic category like the
categories of (discrete) groups, Lie algebras or associative algebras is
related to crossed modules of groups, Lie algebras or associative algebras
respectively. A {\it crossed module of groups} is, roughly speaking, a
homomorphism of groups $\mu:M\to N$ together with an action by automorphisms
of $N$ on $M$ which is compatible in some sense with $\mu$. Passing to
kernel and cokernel of $\mu$, one gets a $4$-term exact sequence of groups
$$1\to V={\rm ker}\,\mu\to M\to N\to G=\mathrm coker\,\mu\to 1,$$
such that $V$ is an abelian group and a $G$-module. The general algebraic
picture associates to such a crossed module a cohomology class in $H^3(G,V)$
which is the obstruction to come from an extension of $G$ by $M$.
Related notions which take into account topology exist for Lie groupoids
\cite{Mack1}, \cite{Andr} and for topological Lie algebras \cite{Neeb},
\cite{Wage}.

In \cite{Neeb}, Karl-Hermann Neeb defines for a crossed module of topological
Lie algebras which is split as a sequence of topological vector spaces a
cohomology class $[\omega_{\rm top\,\,alg}]$. He shows that
$[\omega_{\rm top\,\,alg}]$ has a specific meaning in the following context:
let $K$ be some Lie group and $P$ be a $K$-principal bundle on some manifold
$M$. Suppose given furthermore a central extension
$$1\to Z\to \hat{K}\to K\to 1$$
of $K$. The question is now whether there exists a $\hat{K}$-principal
bundle $\hat{P}$ on $M$ such that $\hat{P}/Z\cong P$. Neeb uses the
ingredients of the problem to associate a crossed module of topological Lie
algebras to it such that its obstruction class $[\omega_{\rm top\,\,alg}]$
is a $3$-de Rham cohomology class on $M$ which is an obstruction to the
existence of $\hat{P}$.

The origin of this paper is the question of Neeb at the end of section IV
of \cite{Neeb} whether $[\omega_{\rm top\,\,alg}]$ is the full obstruction
to the existence of $\hat{P}$, which $3$-de Rham forms arise as obstructions,
and on the relation of $[\omega_{\rm top\,\,alg}]$ to gerbes.

We answer his questions in the framework of crossed modules of Lie algebroids
and groupoids and show that $[\omega_{\rm top\,\,alg}]$ can be identified
with the obstruction class of a certain crossed module of Lie algebroids
associated to the above problem (theorem $1$), and, up to torsion, even to
the obstruction class of a certain crossed module of Lie groupoids
associated to the above problem (theorem $2$, main theorem of this paper),
which is known \cite{Mack1} to be the full obstruction to the existence of
$\hat{P}$.

In section $5$, we show that it follows from Serre's identification of the
Brauer group $Br(M)$ of $M$ (cf \cite{Grot2}) that Neeb's obstruction class
is zero for finite dimensional structure group. In section $7$, we deduce from
the observation that gerbes and crossed modules of Lie groupoids are classified
by the same kind of cohomology classes a direct relation between these two
kinds of objects.

In the appendix on Deligne cohomology, we present the
necessary material for the proof of theorem $2$.

{\bf Acknowledgements:} We are indepted to Karl-Hermann Neeb for spotting and
correcting an error. Both authors thank the Rencontres
Math\'ematiques de Glanon where some of the work presented here was done.

\section{The Atiyah sequence}

In this section, we recall the Atiyah sequence associated to a $K$-principal
bundle on a fixed base manifold $M$, and we explain the main object of this
article.

Let $M$ be a (finite dimensional, connected, second countable, smooth)
manifold with finite dimensional de Rham cohomology,
and let $K$ be a (not necessarily finite dimensional) Lie group
with Lie algebra ${\mathfrak k}$. We usually take infinite dimensional
Lie groups to be modeled on locally convex spaces.
Furthermore, let $\pi:P\to M$ be a $K$-principal bundle on $M$. As for finite
dimensional structure groups, $P$ can be represented by a smooth \v{C}ech
$1$-cocycle, cf \cite{Wock}. Connections on $P$ can be constructed by
patching on the finite dimensional manifold $M$. $K$ acts on
$P$ from the right, and this action induces an action on $TP$. The induced
map between tangent bundles $T\pi:TP\to TM$ factors to a map
$\pi_*:TP/K\to TM$. The kernel of $\pi_*$ can be identified (cf \cite{Mack}
p. 92 or \cite{Neeb} IV) with $(P\times{\mathfrak k})/K$, where $K$ acts on
the product by the diagonal action, using the adjoint action on the second
factor. We denote this bundle by ${\rm Ad}(P)$. Therefore, we get the
{\it Atiyah sequence} of vector bundles
$$0\to {\rm Ad}(P)\to TP/K\to TM \to 0.$$

The main question we address in this paper is the following:
given a central extension
$$1\to Z\to \hat{K}\to K\to 1$$
of $K$ by $Z$, is there a $\hat{K}$-principal bundle $\hat{P}$ on $M$ such
that $\hat{P}/Z\cong P$ ? More precisely, one wants to construct computable
obstructions for the existence of such a $\hat{P}$. This is the point of view
expressed in Neeb's paper \cite{Neeb} section VI.

\section{Crossed modules of topological Lie algebras}

In \cite{Neeb}, Neeb associates to a given principal bundle $P$ on $M$
and a central extension
$$1\to Z\to \hat{K}\to K\to 1$$
of its structure group $K$, a differential $3$-form $\omega_{\rm top\,\,alg}$
defining a class in $H^3_{\rm dR}(M,{\mathfrak z}) =H^3_{\rm dR}(M)
\otimes {\mathfrak z} $ which is an obstruction to
the existence of a principal $\hat{K}$-bundle $\hat{P}$ such that
$\hat{P}/Z\cong P$. Let us recall its construction:

\begin{defi}
A crossed module of Lie algebras is a homomorphism of
Lie algebras $\mu:\mathfrak{m}\to\mathfrak{n}$ together with an action $\eta$
of $\mathfrak{n}$ on $\mathfrak{m}$ by derivations such that

(a) $\mu(\eta(n)\cdot m)\,=\,[n,\mu(m)]$ for all $n\in\mathfrak{n}$
and all $m\in\mathfrak{m}$,

(b) $\eta(\mu(m))\cdot m'\,=\,[m,m']$ for all $m,m'\in\mathfrak{m}$.
\end{defi}

In the framework of {\it topological Lie algebras}, one requires all maps to
be continuous and topologically split, cf \cite{Neeb}.

To an Atiyah sequence
$$0\to {\rm Ad}(P)\to TP/K\to TM\to 0,$$
and a central extension of the structure group
$$1\to Z\to \hat{K}\to K\to 1,$$
one associates a crossed module of topological Lie algebras. For this,
denoting ${\mathfrak z}$, ${\mathfrak k}$ and $\hat{\mathfrak k}$ the Lie
algebras of $Z$, $K$ and $\hat{K}$ respectively, we first associate to the
extension the sequence of vector bundles
\begin{equation}  \label{extension}
0\to M\times{\mathfrak z}\to\widehat{{\rm Ad}}(P)\to{\rm Ad}(P)\to 0,
\end{equation}
where $\widehat{{\rm Ad}}(P)$ is the bundle $(P\times\hat{\mathfrak k})/K$.
Observe that the adjoint action of $\hat{K}$ on $\hat{\mathfrak k}$ factors
to an action of $K$ on $\hat{\mathfrak k}$. The bundle $\widehat{{\rm Ad}}(P)$,
which is isomorphic to ${\rm Ad}(\hat{P})$ in case $\hat{P}$ exists, is
constructed from the ingredients of the problem and thus exists even if
$\hat{P}$ does not exist.

We now pass to the spaces of global sections of the above vector bundles.
For the Atiyah sequence, we get
${\mathfrak a}{\mathfrak u}{\mathfrak t}(P)=\Gamma(TP/K)={\mathcal V}(P)^K$
the Lie algebra of $K$-invariant vector fields on $P$,
${\mathcal V}(M)=\Gamma(TM)$ the Lie algebra of vector fields on $M$, and
${\mathfrak g}{\mathfrak a}{\mathfrak u}(P)={\mathfrak n}=\Gamma({\rm Ad}(P))$
the {\it gauge Lie algebra}, i.e. the Lie algebra of vertical $K$-invariant
vector fields. For the sequence (\ref{extension}), we get Lie algebras
${\mathfrak c}=\Gamma(M\times{\mathfrak z})$,
${\mathfrak n}=\Gamma({\rm Ad}(P))$ and
$\hat{\mathfrak n}=\Gamma(\widehat{\rm Ad}(P))$.
All these are given the topology of uniform convergence on compact sets of the
function and all of its derivatives, and become in this way locally convex
topological Lie algebras.

One gets a crossed module of topological Lie algebras
$\mu:\hat{\mathfrak n}\to{\mathfrak a}{\mathfrak u}{\mathfrak t}(P)$
by projecting onto ${\mathfrak n}$ and including then ${\mathfrak n}$ into
${\mathfrak a}{\mathfrak u}{\mathfrak t}(P)$, or, passing to the kernel and
the cokernel of $\mu$, we get a four term exact sequence
$$0\to{\mathfrak c}\stackrel{i}{\to}\hat{\mathfrak n}\stackrel{\mu}{\to}
{\mathfrak a}{\mathfrak u}{\mathfrak t}(P)\stackrel{\pi}{\to}{\mathcal V}(M)
\to 0.$$
The action $\eta$ of ${\mathfrak a}{\mathfrak u}{\mathfrak t}(P)$ on
$\hat{\mathfrak n}$ is induced by the derivation action of ${\mathcal V}(P)$
on $\hat{\mathfrak n}\subset{\mathcal C}^{\infty}(P,\hat{\mathfrak k})$.

The differential $3$-form $\omega_{\rm top\,\,alg}$ in
$\Omega^3(M,{\mathfrak z})$ is constructed as a cocycle associated to
this crossed module (cf lemma VI.2 in \cite{Neeb}). Namely, take a
principal connection $1$-form $\theta\in\Omega^1(P,{\mathfrak k})^K$.
It serves two purposes: first, it defines a section $\sigma$ of $\pi$,
second, it gives a projection from
${\mathfrak a}{\mathfrak u}{\mathfrak t}(P)$ to ${\mathfrak n}$.
The curvature of $\theta$ is the $2$-form
$$R_{\theta}=d\theta\,+\,\frac{1}{2}[\theta,\theta]\in\Omega^2(P,
{\mathfrak k})^K.$$
Now, regarding $R_{\theta}$ as an ${\rm Ad}(P)$-valued $2$-form, it can be
lifted to an $\widehat{\rm Ad}(P)$-valued $2$-form $\Omega$. The section
$\sigma$, and therefore the connection $\theta$, permit to define an
{\it outer action}
$$S=\eta\circ\sigma:{\mathcal V}(M)\to{\mathfrak d}{\mathfrak e}
{\mathfrak r}(\hat{\mathfrak n}),$$
meaning that the linear map $S$ is a homomorphism when projected to
${\mathfrak o}{\mathfrak u}{\mathfrak t}(\hat{\mathfrak n})$ and thus gives,
in particular, an action of ${\mathcal V}(M)$ on ${\mathfrak c}$.
The class $\omega_{\rm top\,\,alg}$ is then just $d_S\Omega$, but interpreted
as an element of $\Omega^3(M,{\mathfrak z})$. Here $d_S$ means that one takes
the Chevalley-Eilenberg coboundary with values in $\hat{\mathfrak n}$ as
if $ S$ were an action:
\begin{eqnarray*}
d_S\Omega(x_1,x_2,x_3)&=&S(x_1)\cdot\Omega(\sigma x_2,\sigma x_3)+
S(x_2)\cdot\Omega(\sigma x_3,\sigma x_1)+S(x_3)\cdot\Omega(\sigma x_1,
\sigma x_2)  \\
&-&\Omega(\sigma [x_1,x_2],\sigma x_3)-
\Omega(\sigma [x_2,x_3],\sigma x_1)-\Omega(\sigma [x_3,x_1],\sigma x_2)
\end{eqnarray*}
for $x_1,x_2,x_3\in{\mathcal V}(M)$. Geometrically speaking, $d_S\Omega$ is
the covariant derivative $d_S\Omega=d\Omega+S\wedge\Omega$, cf \cite{Neeb}
section VI.

\begin{rem}
We consider that we have fixed a decomposition as vector spaces
$\hat{\mathfrak k}={\mathfrak k}\oplus{\mathfrak z}$, and that with respect
to this decomposition, the above lift sends $x\in{\mathfrak k}$ to
$(x,0)\in{\mathfrak k}\oplus{\mathfrak z}$.
Observe that as $d_S\Omega(x_1,x_2,x_3)$ has values in ${\mathfrak z}$,
$R_{\theta}$ has
values in ${\mathfrak k}$ and each term
$S(x_i)\cdot\Omega(\sigma x_j,\sigma x_k)$ has a component in
${\mathfrak z}$ and one in ${\mathfrak k}$, the sum in the second line
of the equation for $d_S\Omega(x_1,x_2,x_3)$ together with the components with
values in ${\mathfrak k}$ from the first line must give zero. $d_S\Omega$
is thus just the component with values in ${\mathfrak z}$ of the cyclic sum
of the terms of the form $S(x_i)\cdot\Omega(\sigma x_j,\sigma x_k)$.
\end{rem}

\section{Crossed modules of Lie algebroids}

We recall in this section the main definitions on Lie algebroids, their
cohomology and crossed modules, cf \cite{Andr}, \cite{Mack}.

\begin{defi}\label{def:algebroids}
Let $M$ be a fixed manifold. A Lie algebroid $A$ over $M$ is a vector bundle
$q:A\to M$ together with a morphism of vector bundles $a:A\to TM$ from $M$ to
the tangent bundle $TM$ of $A$, called the anchor, and an $\mathbb{R}$-linear
skewsymmetric bracket $[,]:\Gamma A\times\Gamma A\to\Gamma A$ on the space of
smooth sections $\Gamma A$ of $A$ which satisfies the Jacobi identity, and
such that

\begin{itemize}
\item[(1)] $[X,fY]\,=\,f[X,Y]+a(X)(f)Y$
\item[(2)] $a([X,Y])\,=\,[a(X),a(Y)]$ (this second point being in fact
a consequence of the first one).
\end{itemize}
A Lie algebroid $A$ is called transitive if $a$ is fibrewise surjective. In
this case, $a$ is of constant rank and $TM$ is a quotient bundle of
$A$.
Given two Lie algebroids $A$ and $A' $  over the same manifold $M$,
a
morphism $\phi:A\to A'$ of vector bundles
such that, first, $q' \circ \phi =q $, second
 $\phi$  induces a Lie algebra homomorphism on the spaces of
sections, and such that $a'\circ\phi\,=\,a$ is said to be a Lie
algebroid homomorphism over the identity of $M$.
\end{defi}

We may say simply {\it Lie algebroid morphism},
when it is obvious that it is over the identity of $M$.

Note that if the anchor map is trivial at each point of $M$,
a Lie algebroid is precisely what is called a {\it Lie algebra bundle},
i.e. a vector bundle, endowed fiberwise with a Lie algebra structure,
which is required to satisfy the following assumption of regularity:
for two smooths sections of the vector bundle, taking the bracket
pointwise yields an other smooth section.

The main example of a Lie algebroid and Lie algebroid
morphisms which we have in mind is the Atiyah
algebroid and its associated sequence of a principal $K$-bundle $P$ on a
manifold $M$, cf \cite{Mack} p. 97. It is a transitive Lie algebroid.

\begin{rem}
Regarding a Lie algebra as a Lie algebroid over the point defines a fully
faithful functor from the category of Lie algebras to the category of Lie
algebroids. Sending a Lie algebroid to its space of global sections defines
a functor $\Gamma$ from the category of Lie algebroids to the category of
(topological) Lie algebras which is an inverse to the previous one when
restricted to Lie algebroids over the point.
\end{rem}

\begin{defi}
Let $A$ be a Lie algebroid on $M$, and $E$ be a vector bundle on $M$. A
representation of $A$ on $E$ is a morphism of Lie algebroids
$\rho:A\to{\mathcal D}(E)$ where ${\mathcal D}(E)$ is the Lie algebroid of
first order differential operators $D:\Gamma E\to \Gamma E$ such that there
is a vector fields $X=a_{\mathcal D}(D)$ with
$D(fs)\,=\,fD(s)+a_{\mathcal D}(D)(f)s$.

We shall then say that $E$ is an $A$-module. A morphism of representations
$\phi:E\to E'$ where $E$, $E'$ are $A$-modules via $\rho$ and $\rho'$, is a
morphism of vector bundles satisfying $\phi(\rho(X)(s))=\rho'(X)(\phi(s))$
for all $s\in\Gamma E$, $X\in\Gamma A$.
\end{defi}

For instance, the trivial bundle $M \times {\mathbb R}^k  \to M$ is an
$A$-module, when the Lie algebroid morphism $\rho$ of the definition
is choosen to be the anchor map (and vector fields are considered as
derivations of the space of $k $-tuples of smooth functions).

Let us now recall the complex which computes the cohomology of a Lie
algebroid with values in some representation:

\begin{defi}
Let $A$ be a Lie algebroid and $E$ be an $A$-module, both being vector bundles
on $M$. The standard complex $C^*(A,E)$ is the complex of vector bundles
$C^p(A,E)\,=\,{\rm Alt}^p(A,E)$ together with the usual differential
$d:\Gamma C^p(A,E)\to\Gamma C^{p+1}(A,E)$. Cocycles, coboundaries and
cohomology spaces are defined in the usual way.
\end{defi}

The tangent bundle $TM \to M$ is a Lie algebroid: the anchor map is
the identity and the bracket is the bracket of vector fields.
With respect to the trivial representation  $M \times {\mathbb R}^k  \to M$,
the space of cochains $C^p(A,E)$ is precisely the space of $k$-tuples
of $p$-forms.
It is a direct verification that the algebroid differential becomes,
under this identification, the de Rham differential.
In conclusion,

\begin{lem}\label{lem:deRham-algebroid}
The algebroid cohomology of the tangent bundle $TM \to M $
taking values in the
trivial module $M \times {\mathbb R}^k  \to M $  is  the de Rham
cohomology tensored by ${\mathbb R}^k $, in short
$$ H^*(A) = \bigoplus_{i \in  \mathbb
  N} H_{\rm dR}^i(M) \otimes  {\mathbb R}^k .$$
\end{lem}

This lemma stays true for values in a vector bundle with infinite dimensional
fiber, seen as a trivial $TM \to M $-module.

The following definition is taken from \cite{Andr}:

\begin{defi}
A crossed module of Lie algebroids over the manifold $M$ is a quadruple
$(L,\mu,A,\rho)$ where $L\to M$ is a Lie algebra bundle, $A\to M$ is a
transitive Lie algebroid, $\mu:L\to A$ is a morphism of Lie algebroids and
$\rho:A\to {\mathcal D}(L)$ is a representation of $A$ in $L$ such that
\begin{itemize}
\item[(a)] $\rho(X)([U,V])\,=\,[\rho(X)(U),V]+[U,\rho(X)(V)]$ for all
  $X\in\Gamma A$ and for all $U,V\in\Gamma L$ and
\item[(b)] $\rho(\mu(V))(W)\,=\,[V,W]$ for all $V,W\in\Gamma L$ and
\item[(c)] $\mu(\rho(X)(V))\,=\,[X,\mu(V)]$ for all $X\in\Gamma A$ and
$V\in\Gamma L$.
\end{itemize}
\end{defi}

Observe that this notion is actually the notion of a crossed module of
{\it transitive} Lie algebroids.

According to \cite{Andr}, Section 2, the map $\mu $ is of constant rank.

Let us recall \cite{Mack} p. 272-273 how to associate a $3$-cocycle
$$\omega_{\rm alg} \in  C^3(\mathrm coker(\mu),{\rm ker}(\mu))$$
to a crossed module of Lie algebroids. Every crossed
module of Lie algebroids $(L,\mu,A,\rho)$ induces (using $\rho$) a
{\it coupling}, i.e. a morphism of Lie algebroids
$\mathrm coker(\mu)\,\to\,{\rm Out}{\mathcal D}(L)$. A lift to
$\nabla:\mathrm coker(\mu)\,\to\,{\mathcal D}(L)$ is called a
{\it Lie derivation law} covering the coupling. $\nabla$ can be taken formally
to be an action of $\mathrm coker\mu$ on $L$, although it is not an action in
general. The curvature of $\nabla$ takes values in the inner derivations of
$L$, and can thus be lifted to an alternating vector bundle map
$\Lambda:(\mathrm coker\,\,\mu)\oplus(\mathrm coker\,\,\mu)\to L$. The cocycle is now
obtained as
$$\gamma(X,Y,Z):=\sum_{\rm cycl.}\left(\nabla_X(\Lambda(Y,Z))-\Lambda([X,Y],Z)
\right),$$
i.e. as the formal Chevalley-Eilenberg coboundary operator applied to
the cochain $\Lambda$ using the ``action'' of $\mathrm coker\,\,\mu$ on $L$.
It is clear that $\nabla$ can be given by a section of the quotient map
$A\to\mathrm coker\,\,\mu$ and the action $\rho$.

Now let us look at the special case which is of interest in this paper: let
$P$ be a $K$-principal bundle on a manifold $M$, and
$$1\to Z\to \hat{K}\to K\to 1$$
be a central extension of the structure group. To these data, we attach the
crossed module of Lie algebroids $\mu:\widehat{{\rm Ad}}(P)\to TP/K$. Here,
$\widehat{{\rm Ad}}(P)$ denotes the bundle of Lie algebras
$(P\times\hat{\mathfrak k})/K$, $TP/K$ is the transitive Atiyah Lie algebroid,
and a representation of $TP/K$ on $\widehat{{\rm Ad}}(P)$ by first order
differential operators is given by the action of $K$-invariant vector fields
on $P$ as derivations on functions from $P$ to $\hat{\mathfrak k}$.
This action makes $\mu$ equivariant and restricts to the action of
${{\rm Ad}}(P)$ on $\widehat{{\rm Ad}}(P)$ given by the central
extension.
In this case, $\mathrm coker(\mu) $ is the tangent bundle $TM \to M$
and $ {\rm ker}(\mu)$ is the trivial bundle $M \times {\mathfrak z} \to M
$,
which amounts to the fact that  $\omega_{\rm alg} $ is an element of
$\Omega^3(M,{\mathfrak z}) $, and, by Lemma \ref{lem:deRham-algebroid},
 the class
$ [\omega_{\rm alg}]$ belongs to $H^3_{\rm dR}(M,{\mathfrak z})=
 H^3_{\rm dR}(M)
 \otimes {\mathfrak z} $.

\begin{theo}
Given a $K$-principal bundle $P$ on a manifold $M$ and a central extension
$$1\to Z\to \hat{K}\to K\to 1$$
of the structure group, the $3$-cohomology class $[\omega_{\rm top\,\,alg}]$
of the crossed module of topological Lie algebras
$$0\to{\mathfrak c}\stackrel{i}{\to}\hat{\mathfrak n}\stackrel{\mu}{\to}
{\mathfrak a}{\mathfrak u}{\mathfrak t}(P)\stackrel{\pi}{\to}{\mathcal V}(M)
\to 0$$
coincides with the class $[\omega_{\rm alg}]$ associated to the crossed
module of Lie algebroids
$$0\to M\times{\mathfrak z}\to\widehat{{\rm Ad}}(P)\to TP/K\to TM\to 0.$$
\end{theo}

\pr

Choosing a principal connection $1$-form $\theta$ gives a horizontal lifting
for the Lie algebroids, as for the Lie algebras of global sections. $\theta$
gives rise on the one hand to a coupling and a $\nabla$ which corresponds on
the other hand to the outer action $S=\eta\circ\sigma$ where $\sigma$ is
also defined by $\theta$. The curvature $R_{\theta}$ of $\theta$ can be
regarded as an ${\rm Ad}(P)$-valued $2$-form. It is then lifted to an
$\widehat{\rm Ad}(P)$-valued $2$-form $\Omega$ on $M$ which corresponds to
$\Lambda$ in the discussion preceeding the theorem.
The next step is to compute the above mentioned formal Chevalley-Eilenberg
coboundary of $\Omega$ with values in $\widehat{\rm Ad}(P)$
denoted by $d_S$ in the previous section. Up to identification, we get thus
that for this special crossed module of Lie algebroids, $\omega_{\rm alg}$
has the same expression as $\omega_{\rm top\,\,alg}$.

\fin

By the above proposition, we have at our disposal all results on Lie
algebroids in part $2$, ch. $7$ of \cite{Mack}; applying the global section
functor, they give results on the class of $\omega_{\rm top\,\,alg}$. For
example, Neeb shows in \cite{Neeb} prop. VI.3 that if an extension
$\hat{P}$, i.e. a principal $\hat{K}$-bundle $\hat{P}$ such that
$\hat{P}/Z\cong P$, exists, the class of $\omega_{\rm top\,\,alg}$ is
trivial. But on the other hand, cor. 7.3.9 on p. 281 in \cite{Mack} shows
that in case $[\omega_{\rm alg}]=0$, there exists an extension of Lie
algebroids
$$0\to \widehat{\rm Ad}(P)\to R\to TM\to 0.$$
This is in some sense the converse of Neeb's prop. VI.3. However, we do not
know how $R$ is related to the existence of a $\hat{K}$-principal
bundle $\hat{P}$, but we will see that the question of the existence
of $\hat{P}$ can be solved in terms of Lie groupoids associated to the
Lie algebroids studied here.

\section{Crossed modules of Lie groupoids}

\begin{defi}
A Lie groupoid consists of two manifolds $\Gamma$ and $M$, together with two
surjective submersions $s,t:\Gamma\to M$, called the source and the target
map, and a smooth object inclusion $M\to \Gamma$ such that for composable
elements (i.e. $g,h\in \Gamma$ such that $t(g)=s(h)$), there is a smooth
group law having the images of the object inclusion as its unit elements.

A morphism of Lie groupoids ${\cal F}:\Gamma\to \Xi$ where $\Gamma$ is on $M$
and $\Xi$ is on $N$, is a pair of smooth maps
$({\cal F}:\Gamma\to \Xi,f:M\to N)$ intertwining sources and targets such
that ${\cal F}$ is a homomorphism of groups for composable elements.
\end{defi}

A Lie groupoid where the source and target maps are equal is what is
called a {\it Lie group bundle}, i.e. a bundle $ \Gamma$ over $M$
where all the fibers admit a
Lie group structure which is smooth in the sense
that taking the pointwise product of two smooth local
sections, or the pointwise inverse of a local smooth section,
yields a smooth local section.

We do not limit ourself to the case of manifolds of finite dimension, but
$M$ and $H^*_{\rm dR}(M)$ will always be supposed finite dimensional;
as remarked before, Lemma \ref{lem:deRham-algebroid} can be easily adapted
to (this version of) the infinite dimensional framework. If $\Gamma$ is
infinite dimensional, one can use the notion of submersion with the help
of an implicit function theorem with parameter like in \cite{Gloe}, as it is
shown explicitly in \cite{NeebWage}.

\begin{defi}
Let $\Gamma$ be a Lie groupoid on $M$, and let $\pi:F\to M$ be a bundle of
Lie groups on $M$. A representation of $\Gamma$ on $F$ is a smooth map
$\rho:\Gamma\times_M F\to F$ (where the fiber product is taken with respect
to the source map $s$), such that
\begin{itemize}
\item $\pi\circ\rho= t\circ{\rm proj}_1$, with the target map $t$,
\item $\rho$ is an action for composable elements, and
\item $\rho(x):F_{s(x)}\to F_{t(x)}$ is a Lie group isomorphism for all
$x\in\Gamma$.
\end{itemize}
\end{defi}

\begin{defi}
A crossed module of Lie groupoids ${\cal F}:F\to\Gamma$ is a Lie
groupoid $\Gamma$ on $M$, a Lie group bundle $F$ on $M$, a morphism of Lie
groupoids ${\cal F}:F\to\Gamma$ (with $f$ the identity), and a
representation $\rho$ of $\Gamma$ on $F$ such that
\begin{itemize}
\item ${\cal F}(\rho(x,y))=x{\cal F}(y)x^{-1}$ for all
$(x,y)\in\Gamma\times_MF$,
\item $\rho({\cal F}(x),y)=xyx^{-1}$ for all $x,y\in F$ avec $\pi(x)=\pi(y)$,
and
\item the image of ${\cal F}$ is a closed regularly embedded submanifold of
$\Gamma$.
\end{itemize}
\end{defi}

We will only work here with crossed modules of Lie groupoids
${\cal F}:F\to\Gamma$ such that $\mathrm coker\,\,{\cal F}=M\times M$  (the pair
groupoid)  and
${\rm ker}({\cal F}) \simeq M \times Z$ for some Abelian Lie group $Z$.
As a crossed module of transitive Lie algebroids is in a sense a
crossed module of $TM$ by a trivial ${\mathfrak z}$-bundle, the crossed modules
of Lie groupoids we discuss here are crossed modules of $M\times M$ by a
trivial Lie group bundle $Z\times M$.
In other words, we restrict ourself to the case of crossed modules over
{\it transitive} Lie groupoids.

An important point is that, in our case, $\Gamma $ and $F$ may be infinite
dimensional, while $\mathrm coker \, \, {\cal F}$ is finite dimensional.

Suppose given a $K$-principal bundle $P$ and a central extension $\hat{K}$ of
the structure group $K$. Note that $K$ acts by conjugation, not only on
$K$ itself, but also on $\hat{K}$.
Denote by $P_K(\hat{K})={\mathcal C}^{\infty}(P,\hat{K})^K$ the space of
$K$-equivariant smooth maps from $P_m$ to $\hat{K}$, where $P_m$ is the
fiber over an arbitrary $m \in M$.
$P_K(\hat{K})$ is naturally a group bundle over $M$, and there
is a natural groupoid homomorphism
${\cal F}: P_K(\hat{K}) \to (P \times P)/K $ mapping
$\phi\in{\mathcal C}^{\infty}(P_m,\hat{K})^K $ to
$\overline{(p\cdot\nu(\phi(p)),p)}$
where  $p$ is an arbitrary element of $P_m$, $\nu$ is the map from
$\hat{K}$ to $P$, and the bar means the class in $(P\times P)\,/\,K$.
It is easy to check that the groupoid homomorphism
${\cal F}$ is a crossed module. Conversely, we have:

\begin{prop}  \label{prop1}
Let ${\cal F}:F\to\Gamma$ be a crossed modules of Lie groupoids
such that $\mathrm coker\,\,{\cal F}=M\times M$ and such
that the kernel of ${\cal F}$ is trivial: ${\rm ker}\,{\cal F}=Z\times M$.

Fix a point $m \in M$, and define the Lie group $K$ to be the
quotient of $\hat{K}:=F_m$ by $Z$. Then there exists a $K$-principal bundle
$P$ such that ${\cal F}:F\to\Gamma$ is isomorphic to the crossed module
$$0\to M\times Z\to P_K(\hat{K})\to (P\times P)/K\to M\times M\to 0.$$
\end{prop}

\pr
We fix a point $m \in M$, and define the Lie group $K$ to be the
quotient of $\hat{K}:=F_m$ by $Z$.
Denote by $s$ and $t$ the source and target maps of $\Gamma$.
The space $t^{-1}(m)$ is a submanifold of $\Gamma$
(because the target map is a submersion) on which $K$
acts freely by right multiplication.
Now, the source map $s:t^{-1}(m)\to M$ is a surjective submersion
onto $M$ as $\mathrm coker({\cal F})=M\times M$, and the fibers are precisely given
the right $K$-action. Hence $ t^{-1 } (m) \to M$ is  a principal
$K$-bundle that we denote by $P$.

It is easy to check that the groupoid $\Gamma$ is isomorphic to the
Atiyah groupoid $(P \times P)/K $. The isomorphism $\Psi$ is as follows.
Any element of $\Gamma $ can be written in the form $\gamma = \gamma_1 \cdot
\gamma_2 $
with $\gamma_1,\gamma_2^{-1} \in t^{-1}(m) $. Define
$\Psi(\gamma)  $ to be $  \overline{(\gamma_1,\gamma_2^{-1})} \in
(P \times P)/K$.

Let us also define a map
$\Phi:F\to P_K(\hat{K})={\mathcal C}^{\infty}(P,\hat{K})^K$. To an $f\in F$,
we associate the map $(x\mapsto \rho(x,f))$, where $\rho$ is the action of
$\Gamma$ on $F$, given in the data of the crossed module, and let us recall
that the map ${\mathcal G}:P_K(\hat{K})\to (P\times P)\,/\,K$ of the Atiyah
crossed module corresponding to the central extension of the structural
group $K$ of $P$ to $\hat{K}$, is given in our context by
$$\phi\mapsto {\mathcal G}(\phi):=\overline{(p\cdot_{\Gamma}{\mathcal F}
(\phi(p)),p)},$$
where $p\in P$ is an arbitrary point, $\cdot_{\Gamma}$ is the
multiplication in $\Gamma$.

Let us show that the square

\vspace{.5cm}
\hspace{3cm}
\xymatrix{
P_K(\hat{K})  \ar[r]_{\mathcal G}  &   (P\times P)\,/\,K \\
F \ar[u]_{\Phi} \ar[r]_{\mathcal F} & \Gamma \ar[u]_{\Psi}}
\vspace{.5cm}

\noindent is commutative. Indeed, by the axioms of a crossed module, we have a
commutative square

\vspace{.5cm}
\hspace{3cm}
\xymatrix{
F_n  \ar[r]_{{\mathcal F}_n}  \ar[d]_{\rho(p,1)} &  \Gamma_n^n \\
F_m \ar[r]_{{\mathcal F}_m} & \Gamma_m^m \ar[u]_{{\rm Conj}(p^{-1})}}
\vspace{.5cm}

\noindent where $n=t'(p)$ and $m=s'(p)$, $t'$ and $s'$ being the source and
target maps
of $P$ (i.e. the restrictions of those of $\Gamma$ to $P\subset\Gamma$).
In $n$, we have ${\mathcal F}(f)=\gamma_1\gamma_2^{-1}$ with
$\gamma_1,\gamma_2\in\Gamma_n^m$ and $f\in F_n$. Now compute
$$\Psi({\mathcal F}(f))\,=\,\overline{(\gamma_1,\gamma_2)}\,=\,
\overline{({\mathcal F}(f)p,p)}\,=\,\overline{(pp^{-1}{\mathcal F}(f)p,p)}
\,=\,\overline{(p{\mathcal F}(\rho(p,f)),p)}.$$

\fin

There is a standard way to associate to a crossed module of Lie groupoids
a characteristic class $\omega_{\rm grp}$ (cf \cite{Mack1} p. 197 or
\cite{Andr} p. 13):
let us choose a covering
(trivializing the principal bundle $P$ described above) ${\cal U}=\{U_i\}$
on the manifold
$M$, and denote as usual $U_{ij}=U_i\cap U_j$, $U_{ijk}=U_i\cap U_j\cap U_k$.
The principal bundle $P$ is given by transition functions
$g_{ij}:U_{ij}\to K$, which we may as well regard as a \v{C}ech $1$-cocycle.
Lift the functions $g_{ij}$ to functions $\hat{g}_{ij}$ with values in
$\hat{K}$, and denote by $\hat{g}_{ij}\hat{g}_{jk}\hat{g}_{ki}=h_{ijk}$
their default for forming a \v{C}ech $1$-cocycle.
$h_{ijk}$ has values in the sheaf $\underline{Z}$ of differentiable
$Z$-valued functions, as the $g_{ij}$ do form a cocycle.
The \v{C}ech $2$-cocycle $h_{ijk}$ with values in $\underline{Z}$
is by definition $\omega_{\rm grp}$. Its class will also be regarded as
a class in $H^3(M,\pi_1(Z))$, provided that $Z$ is a connected regular
abelian Lie group,
according to the isomorphism
$$H^2(M,\underline{Z})\,\cong\,H^3(M,\pi_1(Z)),$$
which stems from the fact that for a connected regular abelian Lie group
$$1\to \pi_1(Z)\to{\mathfrak z}\to Z\to 1$$
is an exact sequence of groups, where the abelian group ${\mathfrak z}$ is
the Lie algebra of $Z$, and that the sheaf $\underline{\mathfrak z}$ is fine.

The last step is to consider the image of $H^3(M,\pi_1(Z))$ in the de
Rham cohomology group $H^3_{\rm dR}(M)\otimes {\mathfrak z}$ obtained by
composing the map  $H^3(M,\pi_1(Z)) \to H^3(M,{\mathbb R}) \otimes
{\mathfrak z}$ coming from  the inclusion $\pi_1(Z) $ into
${\mathfrak z} $ with the \v{C}ech-de Rham isomorphism
$H^3(M,{\mathbb R}) \, \simeq \, H^3_{\rm dR}(M) $.

The following theorem is the main result of our paper:

\begin{theo}  \label{mainthm}
Given a $K$-principal bundle $P$ on a manifold $M$ and a central extension
$$1\to Z\to \hat{K}\to K\to 1$$
of the structure group, the $3$-cohomology class $[\omega_{\rm top\,\,alg}]$
of the crossed module of topological Lie algebras
$$0\to{\mathfrak c}\stackrel{i}{\to}\hat{\mathfrak n}\stackrel{\mu}{\to}
{\mathfrak a}{\mathfrak u}{\mathfrak t}(P)\stackrel{\pi}{\to}{\mathcal V}(M)
\to 0$$
defines the same de Rham cohomology class as the $3$-cocycle
$\omega_{\rm grp}$ associated to the crossed module of Lie groupoids
$$0\to M\times Z\to P_K(\hat{K})\to (P\times P)/K\to M\times M\to 0.$$
\end{theo}

\pr We use the previously established notations. Let us choose a
connection on the principal bundle $P \to M $ given, in local
trivializing coordinates $\{U_i\}_{i \in I}$,
by  a family $\theta_i$ of ${\mathfrak k} $-valued $1$-forms on $U_i$.
As usual, the relation
$\theta_j={\rm Ad}_{g_{ji}}\theta_i+g_{ji}^{-1}dg_{ji}$ expresses how to pass
from $\theta_i$ to $\theta_j$ by gauge transformation. Lifting to
$\hat{\mathfrak k}$, we get
$\hat{\theta}_j-{\rm Ad}_{\hat{g}_{ji}}\hat{\theta}_i=\hat{g}_{ij}^{-1}
d\hat{g}_{ij}-\alpha_{ij}$ with $\alpha_{ij}\in{\mathfrak z}$
(remember that we fixed a vector space decomposition
$\hat{\mathfrak k}={\mathfrak k}\oplus{\mathfrak z}$).
We get furthermore for the curvature as usual
$R_i=d\theta_i+\frac{1}{2}[\theta_i,\theta_i]$,
$R_i={\rm Ad}_{g_{ij}}R_j$ and
$dR_i=[\theta_i,R_i]$, and for the lifted curvature
$\hat{R}_i=d\hat{\theta}_i+\frac{1}{2}[\hat{\theta}_i,\hat{\theta}_i]$,
$\hat{R}_i={\rm Ad}_{\hat{g}_{ij}}\hat{R}_j+d\alpha_{ij}$.

The \v{C}ech $2$-cocycle $h_{ijk}$ represents by definition the class
$\omega_{\rm grp}$. In order to establish the link with the class
$\omega_{\rm top\,\,alg}$, we translate \v{C}ech cocycles into differential
forms via the \v{C}ech-de Rham bicomplex: a straight forward computation,
using that $h_{ijk}$ and $\alpha_{ij}$ have values in $Z$, gives

\begin{equation}   \label{****}
\alpha_{ij}-\alpha_{kj}+\alpha_{ki}\,=\,h_{ijk}^{-1}dh_{ijk}.
\end{equation}

Now choose a form
$\omega_i\in\Omega^2(U_i,\hat{\mathfrak k})$ which has the right
transformation
behaviour $\omega_j={\rm Ad}_{\hat{g}_{ij}}\omega_i$ and which has
$R_i$ as its
component in ${\mathfrak k}$ and a form $\omega^{\mathfrak z}$ as its
component in ${\mathfrak z}$. Observe that the ${\mathfrak z}$-component
$\omega^{\mathfrak z}$ is globally defined. It is then the form
$\beta_i:=\hat{R}_i-\omega_i\in\Omega^2(U_i,{\mathfrak z})$ which satisfies
$\beta_i-\beta_j=d\alpha_{ij}$.

The relation (\ref{****}) means that $(h,\alpha)$ forms a \v{C}ech $2$-cocycle
with values in the complex of sheaves
$$d\log:\Omega^0_M(Z)\to\Omega^1_M({\mathfrak z}).$$

Then $d\beta_i$, which is just the component in ${\mathfrak z}$ of
$d\hat{R}_i=[\hat{\theta}_i,\hat{R}_i]$ up to an exact form,
is exactly the expression of a
representative of $\omega_{\rm top\,\,alg}$ (cf remark $1$).

The general procedure means here that $(h,\alpha,\beta)$ forms a \v{C}ech
$2$-cocycle with values in the complex of sheaves

\begin{equation}  \label{*****}
\Omega^0_M(Z)\stackrel{d\log}{\to}\Omega^1_M({\mathfrak z})\stackrel{d}{\to}
\Omega^1_M({\mathfrak z}).
\end{equation}

The rest of the proof of the theorem is inspired by the proof of proposition
$4.2.7$ in \cite{Bryl} p. $174$:
By corollary $2$ in the appendix, $(h,\alpha,\beta)$
defines a cohomology class in the smooth Deligne cohomology group
$H^3(M,\pi_1(Z)(3)^{\infty}_D)$. Furthermore, the class is sent to
the \v{C}ech-connecting homomorphism of $g_{ij}$, i.e. $h_{ijk}$, under the
map in cohomology from $H^3(M,\pi_1(Z)(3)^{\infty}_D)$ to
$H^3(M,\pi_1(Z))\cong H^2(M,\underline{Z})$, induced by the canonical
projection from the complex of sheaves $\pi_1(Z)(3)^{\infty}_D$ to its first
term (which is the constant sheaf $\pi_1(Z)$), see proposition $2$ in the
appendix. On the other hand, the Deligne cohomology class maps to
$[d\beta]$, and thus to $[\omega_{\rm top\,\,alg}]$, under the map of
complexes of sheaves given by prolongation of the complex one step further,
see proposition $3$ of the appendix. It is therefore clear that the image of
$[h_{ijk}]$ or $[\omega_{\rm grp}]$ under the map
$$H^2(M,\underline{Z})\cong H^3(M,\pi_1(Z))\to H^3(M,{\mathfrak z})$$
is the class $[d\beta]$ or $[\omega_{\rm top\,\,alg}]$.\fin

\begin{rem}
In conclusion, the Lie groupoid class $\omega_{\rm grp}$ and the Lie
algebroid class $\omega_{\rm alg}$ coincide up
to torsion (i.e. have the same image in $H^3(M,{\mathfrak z})$)
in the above context.
\end{rem}

\begin{cor}
Let ${\mathcal F}:F\to\Gamma$ be a crossed module of Lie groupoids such that
$\mathrm coker\,{\mathcal F} = M\times M$ and such that the Lie group bundle
${\rm ker}\,{\mathcal F}=:Z\times M$ is trivial. Set as in proposition \ref{prop1}
$K:=F_x/Z$, and $P$ for the principal $K$-bundle $t^{-1}(m)\to M$.

Then the $3$-cohomology class $[\omega_{\rm top\,\,alg}]$
of the crossed module of topological Lie algebras
$$0\to{\mathfrak c}\stackrel{i}{\to}\hat{\mathfrak n}\stackrel{\mu}{\to}
{\mathfrak a}{\mathfrak u}{\mathfrak t}(P)\stackrel{\pi}{\to}{\mathcal V}(M)
\to 0$$
defines the same de Rham cohomology class as the $3$-cocycle
$\omega_{\rm grp}$ associated to the crossed module of Lie groupoids
$$0\to M\times Z\to P_K(\hat{K})\to (P\times P)/K\to M\times M\to 0.$$
\end{cor}

\pr This follows immediately from theorem \ref{mainthm} and proposition
\ref{prop1}. \fin

\section{Finite dimensional structure group}

In this section, we look at the special case of a finite dimensional
structure group $K$. Let us start with $K=PU_n$, the projective unitary group
of an $n$-dimensional complex vector space. $K$-principal bundles on $M$
which are non-trivial in the sense that they cannot be lifted to a principal
$U_n$-bundle define elements of the {\it Brauer group} $Br(M)$, cf
\cite{Grot2}. A theorem of Serre determines $Br(M)$:

\begin{theo}[Serre]
On the manifold $M$, $Br(M)$ can be identified with the torsion subgroup of
the sheaf cohomology group $H^2(M,\underline{\mathbb{C}^*})\cong H^3(M,\Z)$.
\end{theo}

The identification is given by the obstruction class $[\omega_{\rm grp}]$
which measures the obstruction for a given $PU_n$-principal bundle $P$ to be
lifted to a $U_n$-principal bundle. By Serre's theorem together with
theorem $2$, we arrive thus at the following conclusion:

\begin{cor}
Neeb's class $[\omega_{\rm top\,\,alg}]$ associated to the problem of lifting
a given $PU_n$-principal bundle $P$ to a $U_n$-bundle is always zero.
\end{cor}

We can go a step further. Murray shows in \cite{Mur} \S 13 that the class
$[d\beta_i]=[\omega_{\rm top\,\,alg}]$ (in the notation of the proof of
theorem $2$) is trivial in case the structure group is finite dimensional and
structure group and base space are $1$-connected (this condition is not
written in \S 13 of \cite{Mur}, but as he refers to \cite{GLSW}, the condition
seems inevitable).

\begin{cor}
Let $K$ be a finite dimensional $1$-connected group, $P$ a $K$-principal
bundle with $1$-connected base, and $\hat{K}$ a central extension of $K$ by
$Z$.

Then Neeb's class $[\omega_{\rm top\,\,alg}]$ associated to the problem of
lifting $P$ to a $\hat{K}$-bundle $\hat{P}$ such that $\hat{P}/Z\cong P$ is
zero.
\end{cor}

We believe that the study of the triviality of Neeb's class in the general case
is an interesting subject.

On the other hand, in some cases, the significance of the (torsion) class
$[\omega_{\rm grp}]$ is well known; see examples in \cite{Mack1} pp. 206-207.
To cite just one
example, for $K=SO(n)$ and a principal $K$-bundle $P$ on $M$, $n\geq 3$,
the obstruction class in $H^2(M,\Z_2)$ for the existence of a $Spin(n)$-bundle
lifting $P$ is the second Stiefel-Whitney class and gives a $2$-torsion element
in $H^3(M,\Z)$ under the Bockstein map.

\section{Existence of principal bundles}

We will now use the previously established results to answer the question
concerning the existence of a principal $\hat{K}$-bundle $\hat{P}$ such that
$\hat{P}\,/\,Z\cong P$. We use for this the general theory of fiber spaces
set up by Grothendieck \cite{Grot1}, and we will get back the classification
of principal bundles via cohomology classes of Lie groupoids due to
Mackenzie \cite{Mack1}.

Indeed, the short exact sequence of groups
$$1\to Z\to \hat{K}\to K\to 1$$
induces (regardless of the finiteness of the dimension of $K$) an exact
sequence
$$H^1(M,\underline{\hat{K}})\to H^1(M,\underline{K})\to H^2(M,\underline{Z}).$$
The obstruction class $[\omega_{\rm grp}]$ is nothing else than the image of
the class of a given $K$-principal bundle $P$ under the connecting
homomorphism, i.e. the map on the right, and the exactness of the sequence
means that there exists a $\hat{K}$-principal bundle $\hat{P}$ extending $P$
if and only if $[\omega_{\rm grp}]=0$:

\vspace{.5cm}
\hspace{2cm}
\xymatrix{
H^1(M,\underline{\hat{K}})  \ar[r]  & H^1(M,\underline{K}) \ar[r] &
H^2(M,\underline{Z}) \\
[\hat{g}_{ij}]  \ar@{|-->}[r] & [g_{ij}] \ar@{|->}[r] &
[h_{ijk}=\omega_{\rm grp}]}
\vspace{.5cm}

This reasoning gives back theorem $3.4^{\scriptstyle \prime}$ in \cite{Mack1}
which shows the existence of a principal bundle $\hat{P}$ if and only if
$[\omega_{\rm grp}]=0$. By Grothendieck's theory, we get of course a
\v{C}ech cocycle $\hat{g}_{ij}$ of transition functions which is only
continuous, but by \cite{Wock}, we may choose a smooth representative of
the same equivalence class.

Up to torsion, we get thus an equivalence between the existence of a principal
$\hat{K}$-bundle $\hat{P}$ such that $\hat{P}/Z\cong P$ and the vanishing of
Neeb's class $[\omega_{\rm top\,\,alg}]$.

\begin{rem}
Actually, in the context of lifting the structure group to a central extension,
we prescribe not only the outer
action of the Lie groupoid (or principal bundle) whose existence we study,
but also the Lie algebroid (i.e. $P_K(\hat{\mathfrak k})$). This is the reason
why the \v{C}ech cohomology class $\omega_{\rm grp}$ takes values in a sheaf of
germs of constant maps, cf \cite{Mack1} p. $203$.
\end{rem}

\section{Gerbes}

In this section, we reinterprete our results in terms of gerbes and note
that crossed modules of Lie groupoids and gerbes are related to similar
cohomological problems. We then explore a more direct link between these
two objects.

The theory of gerbes and stacks is rather heavy from the point of
view of definitions, so we content ourselves with intuitive explanations for
the objects of study in this section, in order not to blow up this paper.

{\it Differentiable stacks} are more general objects than differentiable
manifolds; their notion grew out of Grothendieck's attempt to characterize the
functor of points of an algebraic variety by abstract conditions. A stack is
in particular a sheaf of categories on a site, but here, our stacks will always
be sheaves on the manifold $M$. The guiding example for us here is the stack
given by the local liftings of the structure group of a $K$-principal bundle
$P$ from $K$ to a $Z$ central extension $\hat{K}$ of $K$, cf \cite{Bryl}
pp. 171-172. On an open set $U$ of $M$, the associated category $C_U$ consists
of $\hat{K}$-principal bundles $\hat{P}$ on $U$ together with an isomorphism
of $K$-principal bundles $f:\hat{P}/Z\cong P$. Morphisms in $C_U$ are bundle
morphisms which commute with the isomorphisms $f$. Note that up to refinement
of $U$, the category $C_U$ is never empty (for non-empty $U$), and any two
objects of $C_U$ are isomorphic. These are the basic properties of a
sheaf of groupoids which is a {\it gerbe}.

The functor of points associates to each Lie groupoid a (pre)stack. Up to
Morita equivalence, this gives a fully faithful functor from Lie groupoids to
differentiable stacks. In this sense a Morita equivalence class of Lie
groupoids defines a differentiable stack.

The manifold $M$, seen as a Lie groupoid, is Morita equivalent to the groupoid
associated to an open covering $\coprod_{i\in I}U_i$ of $M$. Now given a class
$\omega\in H^3(M,\Z)$, there is a central extension of Lie groupoids
(see \cite{TuXu} p. $863$)

\vspace{.5cm}
\hspace{3cm}
\xymatrix{
R_{\omega}  \ar[r]  &
\coprod_{i,j\in I}U_{ij} \dar[r] &
\coprod_{i\in I}U_i}
\vspace{.5cm}

On the other hand, the central extension $1\to \mathbb{C}^*\to U({\cal H})\to
PU({\cal H})\to 1$ of the projective unitary group $PU({\cal H})$ of an
infinite dimensional
separable Hilbert space ${\cal H}$ gives rise to an extension of Lie groupoids

\vspace{.5cm}
\hspace{3cm}
\xymatrix{
U({\cal H})  \ar[r]  &
PU({\cal H}) \dar[r] &
{\rm pt}}
\vspace{.5cm}

\noindent over the point ${\rm pt}$. Now it is shown in {\it loc. cit.}
that there exists
a morphism of stacks from the first extension of Lie groupoids to the second,
and that this implies the existence of a $PU({\cal H})$-principal bundle $P$
on $M$ whose obstruction class is $\omega$.

All this can be summarized by the morphism $\Phi$
$$\Phi:H^1(M,\underline{K})\times H^2(K,\underline{S^1})\to
H^2(M,\underline{S^1})$$
to be found in \cite{TuXu} p. 860. Choosing the particular $S^1$-central
extension of $PU{\cal H}$ given by $U{\cal H}$ (for an infinite dimensional
separable Hilbert space ${\cal H}$) gives an isomorphism
$$\Phi':H^1(M,{\underline PU{\cal H}})\to H^2(M,\underline{S^1}),$$
see \cite{TuXu} p. 861. We see thus that $PU{\cal H}$-principal bundles (LHS)
correspond in a one-to-one manner to integral $3$-cohomology of $M$ (RHS)
via the obstruction mechanism.
Note that the proofs in this section of \cite{TuXu}
do certainly not extend to an infinite dimensional center $Z$, and that it
is not even clear how they extend to a non-compact group $Z$.

In the framework of \cite{TuXu}, the space $H^2(M,\underline{S^1})$ is
interpreted as the space of equivalence classes of gerbes; the natural gerbe
associated to our initial problem of lifting the structure group of a principal
bundle to a central extension is the one described in the beginning. The
above mentioned central extension of Lie groupoids also gives rise to a gerbe.
The {\it band} of a gerbe is the sheaf of groups given by the outer action
(see \cite{LSX} p.15 def. 3.5). In the above central extension, the band
happens to be trivial.

On the
other hand, we started our study from equivalence classes of crossed modules
of Lie groupoids. We are therefore led to search a direct link, i.e. one which
does not pass by cohomology classes, between gerbes and crossed modules of
Lie groupoids.

\begin{theo}
There is a one-to-one correspondence between gerbes with abelian,
trivialized band on a connected manifold $M$, and crossed modules of Lie
groupoids with trivial kernel and cokernel $M\times M$, which induces
the above one-to-one correspondence between cohomology classes when passing to
equivalence classes.
\end{theo}

\pr
We will first describe a map $\phi$ associating to a crossed module
$(F,\delta,\Gamma)$ of Lie groupoids
$$1\to M\times Z\to F\to \Gamma\to M\times M\to 1$$
a gerbe. $\phi$ will respect cohomology classes in
$H^2(M,\underline{Z})$.
Recall that $M$ is connected. Call $H$ the fiber $F_{x_0}$ of the Lie group
bundle $F$ over $x_0\in M$. As described in \cite{Andr} p. 13, one may
associate to $(F,\delta,\Gamma)$ a cocycle of transition functions
$s_{ij}:U_{ij}\to\delta(H)$. Denote by
$\hat{s}_{ij}:U_{ij}\to H$ a lift of $s_{ij}$ with values in $H$.
We now define a gerbe as a
sheaf of categories on $M$ by taking for $C_U$ (for an open set $U$ of $M$)
the category of $H$-principal bundles, given for example by a \v{C}ech cocycle
$\hat{s}_{ij}:U_{ij}\to H$, together with an isomorphism of
$\delta(H)$-principal bundles $(\delta\circ\hat{s}_{ij})\cong (s_{ij})$. By
the general theory, it is clear that this defines a gerbe
$C_{(F,\delta,\Gamma)}$ on $M$. We define thus $\phi$ by
$$\phi(F,\delta,\Gamma)\,=\,C_{(F,\delta,\Gamma)}.$$
It is also clear from \cite{Bryl} p. 172 that $\phi$ respects cohomology
classes.

Let us now define a map $\psi$ in the reverse direction: a gerbe
${\mathfrak G}$ (with trivialized abelian band) on $M$ comes together with an
identification of the sheaf of (abelian) groups of automorphisms of the
objects of the local categories $C_U$; let this sheaf be $\underline{Z}$
and the abelian group be $Z$.
Choosing a local section $s:U\to{\mathfrak G}$ (whose existence is due to
the local existence of objects in $C_U$ - this is one of the
axioms of a gerbe),
this identification implies that $U\times_{\mathfrak G}U\to U$ is a locally
trivial principal bundle $\hat{P}_U$, cf \cite{Hein} rem. 5.2.
The local isomorphy of any
two objects in $C_U$ (which is the other axiom of a gerbe) implies that the
fiber of $U\times_{\mathfrak G}U\to U$ is a group. Denote it by $\hat{K}$.
Now the locally defined $\hat{P}_U/Z=:P_U$ form a globally defined
$K$-principal bundle $P$ on $M$, because the ${\rm Aut}(C_U)=Z$-valued
\v{C}ech cochain defined by the local sections becomes a cocycle for $P$.
This defines as in the beginning of section $4$ a crossed module of
Lie groupoids $(F,\delta,\Gamma)$ of $M\times M$ by $\underline{Z}$. We set
$$\psi({\mathfrak G})\,=\,(F,\delta,\Gamma).$$

$\phi$ and $\psi$ are mutually inverse bijections, descending to bijections
of equivalence classes. In this sense, gerbes and crossed modules of Lie
groupoids are the same objects.\fin

\begin{rem}
Note that this construction does not involve the construction of a principal
bundle $P$ for a given integral cohomology class in \cite{TuXu}. Indeed,
passing by their construction, one could show a relation between gerbes and
crossed modules only for $K=S^1$.
\end{rem}

\begin{appendix}

\section{Smooth Deligne cohomology}

In this appendix, we present the definitions and results from smooth
Deligne cohomology which we need in section $4$. Our main reference is
\cite{Bryl} Ch. 1.5, while our definition differs slightly from his.

Deligne cohomology in our sense is the hypercohomology of truncated
complexes of sheaves. The complex we consider in section $4$, denoted by
$\pi_1(Z)(3)^{\infty}_D$, is
$$\pi_1(Z)\stackrel{\rm incl}{\to}\Omega^0_M({\mathfrak z})
\stackrel{d}{\to}\Omega^1_M({\mathfrak z})\stackrel{d}{\to}
\Omega^2_M({\mathfrak z}).$$
The notation $\pi_1(Z)(3)^{\infty}_D$ indicates that the $0$th term is
the simple (or constant) sheaf $\pi_1(Z)$, and that the complex is
truncated at the $3$rd place. It will be related to the complex
$$\Omega^0_M(Z)\stackrel{d\log}{\to}\Omega^1_M({\mathfrak z})\stackrel{d}{\to}
\Omega^2_M({\mathfrak z}).$$
Let us explain the ingredients: $Z$ is here some (possibly infinite
dimensional) Lie group, ${\mathfrak z}$ is
its Lie algebra. The only property from the setting of infinite dimensional
Lie groups we might choose is that the composition of smooth maps must be
smooth. The main geometric property of $Z$ and ${\mathfrak z}$ we use is that
${\mathfrak z}$ is a universal covering space for $Z$, in particular
$$1\to\pi_1(Z)\to{\mathfrak z}\stackrel{\pi}{\to} Z\to 1$$
is an exact sequence of groups. In the above complex, $\Omega^0_M(Z)$ denotes
the sheaf of smooth maps on $M$ with values in $Z$, and
$\Omega^i_M({\mathfrak z})$ for $i=1,2$ have a similar meaning. $d\log$ is the
{\it logarithmic derivative}, i.e. $d\log f=f^{-1}df$ for $f\in \Omega^0_M(Z)$.
$d\log f$ for $x\in M$ is thus the composition
$$L_{f^{-1}(x)}\circ d_xf:T_xM\to T_{f(x)}Z\to T_eZ={\mathfrak z}.$$

\begin{prop}
The following square is commutative:

\hspace{3cm}
\xymatrix{
\Omega^0_M({\mathfrak z}) \ar[r]^{d} \ar[d]^{\pi^*} &
\Omega^1_M({\mathfrak z}) \ar[d]^{\rm id} \\
\Omega^0_M(Z) \ar[r]^{d\log} & \Omega^1_M({\mathfrak z})
}\vspace{.5cm}
\end{prop}

\pr It is enough to have $d(\pi\circ f)=(\pi\circ f)\cdot df$ for each
$f\in \Omega^0_M({\mathfrak z})$ (where $\cdot$ means once again the left
translation of $df$ in $Z$).

Observe that for $a\in{\mathfrak z}$, the composition
$$L_{\pi(a)^{-1}}\circ d\pi(a):{\mathfrak z}\to T_{\pi(a)}Z\to
T_eZ={\mathfrak z}$$
is the identity, as $\pi:{\mathfrak z}\to Z$ is the universal covering.
This gives $d\pi(a)=L_{\pi(a)}$ for all $a\in{\mathfrak z}$.

Therefore, $d(\pi\circ f)=(d\pi\circ f)\cdot df=(\pi\circ f)\cdot df$
as claimed.\fin

\begin{cor}
We have a quasi-isomorphism of complexes of sheaves:

\hspace{1cm}
\xymatrix{
\pi_1(Z)  \ar[r]^{{\rm incl}} \ar[d]^{0}  &
\Omega^0_M({\mathfrak z}) \ar[r]^{d} \ar[d]^{\pi^*} &
\Omega^1_M({\mathfrak z}) \ar[r]^{d} \ar[d]^{\rm id}  &
\Omega^2_M({\mathfrak z}) \ar[d]^{\rm id} \\
0 \ar[r]^{0} &
\Omega^0_M(Z) \ar[r]^{d\log} &
\Omega^1_M({\mathfrak z}) \ar[r]^{d} &
\Omega^2_M({\mathfrak z})  }\vspace{.5cm}
\end{cor}

\begin{prop}
Let $M$ be a smooth paracompact manifold of dimension $n$, and let
$$0\to{\cal F}^0\to{\cal F}^1\to\ldots\to{\cal F}^n\to 0$$
be a complex of sheaves. Denote by
$H^q(M,{\cal F}^0(p)^{\infty}_D)$ the $q$-th Deligne cohomology group, i.e. the
$q$th hypercohomology group of the truncated sheaf complex
$\{{\cal F}^i\}_{0\leq i\leq p-1}$. Then the projection of
$\{{\cal F}^i\}_{0\leq i\leq p-1}$ onto ${\cal F}^0$ induces an epimorphism
$$H^q(M,{\cal F}^0(p)^{\infty}_D)\,\to\,H^q(M,\underline{\cal F}^0)$$
for each $q\geq p$.
\end{prop}

\pr This proposition is analoguous to a part of theorem 1.5.3 in \cite{Bryl}
p. 48 and is shown in the same way.\fin

\begin{prop}
In the same setting as the previous proposition, the one-step-extension of
the truncated complex
$\{{\cal F}^i\}_{0\leq i\leq p-1}$

\hspace{1cm}
\xymatrix{
{\cal F}^0  \ar[r] \ar[d]^{0}  &
{\cal F}^1 \ar[r] \ar[d]^0 &
\ldots \ar[r] \ar[d]^0  &
{\cal F}^{p-1} \ar[d] \\
0 \ar[r]^0 &
0 \ar[r]^0 &
\ldots \ar[r] &
{\cal F}^p  }\vspace{.5cm}

induces a map
$$H^p(M,{\cal F}^0(p)^{\infty}_D)\,\to\,{\cal F}^p(M).$$
\end{prop}

\end{appendix}

redpajama_set_name: RedPajamaArXiv


text:
Whats the weather like where you are right now
Page 39 of 39 First ... 29373839
Thread: Whats the weather like where you are right now
Deep inversion layer over Sofia, Bulgaria on Jan 8, 2018
Levels of particulate matter in Sofia today reached close to 1000 µg/m³
If everyone would just breathe deeply, those particles would be filtered out of the air in no time. Meatpie, would you consider wearing a mask when you go outside? Do you see people in Sofia wearing masks, the way they do in Asian cities?
Haven't seen anyone wearing a mask people here are ignorant about the true dangers of air pollution. Most people don't know what PM10 PM2.5 mean. Earth is facing a climate change that and environmental degradation that will lead to social upheaval affecting every country on Earth.
Brexit and the election of Donald Trump are direct consequences of two great nations trying to protect themselves from influx of migrants from nations who have undergone societal collapse. This processes will intensify as more and more parts of the world become unstable.
My country has invested millions of euros into building a border fence with Turkey.
Societal upheaval will include desperate fights over fresh water. I had to look at my world map, and I'm still confused about the wall. Are you saying Bulgaria is building a fence between itself and Turkey? Wouldn't the Turks just go into Greece and then from Greece into Bulgaria?
The Greeks built a fence as well so what remains of the European Union is sealed off.
barryds
Rain tonight
Last edited by barryds; 21-01-18 at 05:39 PM.
taynbowen
Forum Inhabitant
Sheboygan, WI
It is Friday, February 9th, 2018 in Sheboygan, wisconsin. It's sunny, but, cold. It's 18 degrees Farenheit, which is -8 degrees Celsius. There's snow on the ground and it's windy. I really miss fall. That's all for today's weather report.
22. Mama. Forensic pathology. Wisconsin to Idaho. True crime, all the time.
Back to normal (greydark and drizzly, around 10C), after an incredible week or so.
Until last week, the Scottish all-time February record high temperature was 17.9C, the Welsh 18.6C , and the English/UK overall 19.7C (67.4F).
The Scottish record fell last Thursday (18.3 in Aberdeenshire). Wales broke its and the UK record on Monday (20.3C) - both of which lasted 24 hours before Wales hit 20.8C, and Kew Gardens in London managed 21.2C (70.2F) - and it was 19C in the Lake District, and in Northumberland only 10 miles from the Scottish border. Yesterday didn't quite break any of those records but it was still amazingly warm and sunny for what's still officially still winter.
I have to say my personal reaction has been 'bring it on', but actually i's scary - just look at those huge red temperature anomalies over much of the Arctic in the pic below.
https://www.euronews.com/2019/02/26/...climate-expert
I'm aware of global warming and the gradual melting of the polar ice, but why in the world would it be so RED HOT north of the Bering Strait?
Utahraptor
rain and shine. rains every 8~9 days
Sun and 12°C.
Warm and sunny, low 20s C - no complaints at all!
And blessed peace and quiet on the Brexit front, since Chief Nurse Theresa sent all the Asylum staff home to calm down since they were much madder than the patients. On a quiet night you can just hear the Monster Raving Loony Brexit werewolves howling at the Moon - it was the Pink Moon on Thursday so that will have driven them apoplectic so with a bit of luck there will be fewer of them when the home leave is over tomorrow. If I can't get to sleep I try to distinguish the howls:
Jacob Rees-Mogg (the MP for the 18th Century) has a howl like the creak of an antique and slightly distressed staircase in a stately home - specifically the tread which the family and its equally stately and distressed retainers know to avoid, since it is designed to precipitate the unwary guest into a pit of vipers beneath the mansion;
Boris Johnson (the MP for Boris Johnson) howls complete piffle (which is actually his middle name!);
Christopher Chope (the MP for Up Skirting, formerly known as Christchurch) has a howl like an upper-class yahoo who has just been told that date-rape is now a crime rather than an accomplishment.
It was the Queen's 93rd birthday yesterday, so of course they fired a 42-gun salute today (don't ask - this is the Asylum after all!). I assume John Bercow (the Speaker of the House of Commons) gave her an extra-special prezzie, since he will be petitioning her for the unprecedented secondment of WaspSwatter Royal to the Houses of Parliament to deal with Andrea Loathsome who is nesting under the Government front bench and keeps trying to sting him. As for the rest of the raving bunch of werewolves and undead, if they get beyond all bearing he can deal with them himself by the traditional Speaker's ultimate sanction - to have them dragged to the dispatch box (so that's how it got its name) and impaled on Black Rod.
Oh, and Theresa May is on a walking holiday in Wales so everyone is wondering what seriously bad idea she will come back with this time (last time it was a general election to wipe out the Labour Party, which succeeded in wiping out her own majority and making Corbyn look electable). But we are assured by a senior Cabinet minister (unnamed) that there is very little risk, as it's years since she came up any idea at all.
But the Moon is rising, and I must go ...
« Primordial soup | HAPPY BIRTHDAY, MeatPie !! »
Even scientists freaking about our SEVERE weather NOW
By ArrowMan in forum General Discussion
The weather is getting warmer, and socks start to reek
By Meatpie in forum Sexual Deviance
whats down his pants?
By Darkside in forum Crime and Death Video
i do not whats hapening!
By kitanaprincess in forum General Discussion
Wow.. suck a nice weather in april
By 0pasan in forum General Discussion
2010, art, bitch, boy, burn, cock, cold, cute, cutedeadguys, dead, drowned, feet, fit, forum, gore, hard, hot, ice, major, men, play, post, russia, war, years

redpajama_set_name: RedPajamaCommonCrawl


The Council of Europe – Commission on Democracy Through Law ("the Venice Commission") and the Organization for Security and Cooperation in Europe – Office for Democratic Institutions and Human Rights (OSCE/ODIHR) received a request from the Parliament of Montenegro to conduct a review of a working version of the Law on Amendments and Supplements to the Law on Election of Councilors and Members of Parliament (hereinafter the draft Law), prepared by a multi-party working group. This request was submitted on 23 March 2010. The joint assessment below takes into consideration previous findings and recommendations made by the OSCE/ODIHR in its final reports from election observation missions in Montenegro as well as progress made by authorities and political parties in Montenegro in their endeavors to improve the legal framework for elections and meet Council of Europe and OSCE commitments and other international standards.


text:
Doctor Who Series 8, Episode 6
Doctor Who S8E6 review: "The Caretaker"
The Doctor takes a job at Coal Hill School in order to stop a clicky-clacky robot who wants to shoot people full of laser beams. But that's really beside the point in an episode that's all about dialogue, character, and the relationships that the Doctor, Clara, and Danny Pink have. We break down the latest episode of "Doctor Who," "The Caretaker," and also ponder Matt Smith lookalikes, the Doctor as an officer and a Time Lord, and Courtney Woods, Disruptive Influence.
Want only The Incomparable Network's Doctor Who-themed episodes? Subscribe to this feed: http://feeds.theincomparable.com/doctorwho
Doctor Who • September 27, 2014 • 34 minutes • Jason Snell with Dan Moren and Glenn Fleishman
Subscribe to just the Doctor Who episodes of TeeVee
Subscribe to the entire TeeVee podcast
Tweet to @teeveenet


An awesome new opportunity has is currently available for a Senior Software Engineer to join a growing Biotech company, based in Cambridge. This company have created an incredibly unique approach to drug discovery, using complex biological systems and now require a talented Software Engineer to join their passionate team.
The successful Senior Software Developer will have a proven background in C#/.net development, as well as demonstrable experience in Database design and development! This is the perfect opportunity for someone who wants to work on an extremely innovative project as well as being given the opportunity to work with the latest tools and technologies.
- Knowledge of .NET (C#) and WPF.
- Good understanding of SQL Server/SQL Server Compact.
- A number of years of hands on experience in designing/implementing Databases.
On offer is a competitive salary & package (Up to £48,000 DOE), as well as the opportunity to work with a company who are making a real world difference in the field of Drug Discovery. You can expect the opportunity to grow professionally and personally among a group of people, passionate about the work they're involved with.
If you're interested or would like some more information then please don't hesitate to get in touch with Jack Lacy via jlacy@pararecruit.com or call 0121,616,3467 for a confidential call.
Software, Developer, Engineer, Programmer, Software Engineer, Software Developer, Senior Software Engineer, Senior Software Developer, C#, C# .net, .NET, C# developer, SQL, SQL Server, Databases, Design, Development, Database Design, Database Development, Implementation, Cambridge, Biotech, Life Science, Drug, Drug discovery, Scientific, Science, Jobs, Careers, Job in IT, Jobs in Software, Jobs in Science, Life Science Jobs, Software Developer Jobs, Careers in Software.


text:
Human Dignity Curriculum
About WYA
History of WYA
Africa Events
Asia Pacific Events
Europe Events
Latin America Events
North America Events
Chapters and National Committees
WYA Awards
UN Statements
Book & Film List
Emerging Leaders of the Arab Region
International Solidarity Forum
Emerging Leaders Conference
Youth Voices at the UN Program
Manhattan International Film Festival
International Arts Forum
WYAAP, DLSU EconOrg partner for 14th YEC NCR series
World Youth Alliance Asia Pacific (WYAAP) and De La Salle University (DLSU) Economics Organization (EconOrg) partnered for the 14th Young Economists' Convention (YEC) National Capital Region (NCR) series last December 2, 2014 at the Escaler Hall of Ateneo de Manila University, Philippines.
Themed Soaring Powerhouse: Enhancing the Possibilities of the Philippine Financial System, the NCR convention gathered over two hundred student leaders and engaged them through different talks about the ASEAN and Philippine Financial Systems. A research proposal competition was also held.
Established in 2001, the YEC is a product of the collective effort of De La Salle University – Angelo King Institute for Economic and Business Studies (DLSU – AKI), Anthony Aguirre Memorial Foundation, USAID, and the Konrad Adenauer Foundation (KAF), in cooperation with the Junior Philippine Economics Society (JPES) and the De La Salle University – Economics Organization (DLSU – EconOrg).
The YEC is dedicated to enriching the Economics education of students in and outside the Philippines through conventions and conferences. It is an avenue for idea exchange through interaction with fellow students, and the wide range of professionals from both public and private sectors—government officials and representatives, businessmen, members of the academe and civil society.
The YEC presents relevant issues to stimulate critical thinking among the youth, and enables them the practical use of economic principles and theories as the future leaders and policymakers of the Philippines.
If you would like to know more about the conference series, please visit the links below:
Official website: www.theyoungeconomists.org
Facebook: www.facebook.com/theYECdlsu/
Twitter: @theYECdlsu
Instagram: @14thYEC
For inquiries, you may also email the organizing committee at youngeconomistsconvention@gmail.com.
Copyright ©1999 World Youth Alliance
The World Youth Alliance is a 501(c)(3) nonprofit organization.
Follow WYA
Check out regional social media accounts
World Youth Alliance Europe is co-funded by the Erasmus+ Programme of the European Union.
World Youth Alliance and the WYA Logo are registered trademarks of WYA Foundation, Inc. No claim is made to the exclusive right to use "Alliance" apart from the mark as shown.


We Are Thank You For Being Here.
The main concern which people nowadays feel is with the confidentiality of the information which is imparted to an investigation firm by the client of the subject. Our company has certain ethics for handling the case.
Handing all the information carefully is the main aspect which is done by us.
Venus detective has a special team which distributes the case to the assigning team which looks after the confined information.
Maintaining the information of the clients a secret is the main thing that our company holds.
In our system, even our team doesn't know who the client is. Their job is just to find the information of the subject and has nothing to do with the information of the client.
In our company even our left hand doesn't know what the right hand is doing.
The job of the team is just to gather the information and share it only with the client.
Our company even doesn't save the data of the subject after being submitted to the client.
Once the report is delivered to the client it is removed from our system too in order to maintain the confidentiality.
To save the client from any illegal matter, the things are handled with care.
All the employees of our company follow all the policies in which the client is safeguard and maintain his secrecy is the main criteria and policy of our company. We have a team of well trained and expert people who knows how to safeguard all the confidential information.
© Copyright © 2017 Avows Consulting & Corporate Solutions Pvt. Ltd, All rights reserved.


The Annual Church Meeting will be held on Thursday 14th April in Church at 7.30pm. Coffee will be available from 7.15pm. All papers (Annual Report etc.) will be available for collection from the Church, or to download from the website, from Thursday 7th April.
Completed nomination forms for all the above positions should be sent to Phillipa Cook by Monday 4th April. A list of all Nominations will be posted on the Notice Board in the Church porch by 7th. April.


In the early part of the growing season, I visited the Rhone region's sweetheart Chapoutier, in L'Hermite when the vines were just coming up from their sleep and tenderness was most important to kick start what would eventually become a hearty, thriving, fruit bearing, bountiful crop.
Chapoutier has many different branded vintages and wine blends offered. Too many to taste. However, prior to a beautiful luncheon at the Chapoutier restaurant, I was taken on a tour of the estate and the high hilly slopes to see the vines up close and learn the intricacies of growing and harvesting vines on such steep mountainsides. Much of the cultivation and harvest is old school because of the terrain.
The Chapoutier wines boast such rich flavours. Deep succulent textured reds, with warm finshes, nutty and herbed nuances. A family operation with a rich history of lots of hard work and tanacity over many years that produces some of the Rhone's best and beautiful wines.


The Third Meeting of States Parties (3MSP) took place in Oslo, Norway, from 11 to 14 September 2012 and gathered some 800 delegates from 60 States Parties, 30 signatory States, 31 observer States and 15 international organizations (see list below).The 3MSP was opened by Dr. Adnan Mansour, Minister of Foreign Affairs and Emigrants of the Republic of Lebanon. During the first plenary meeting, Ambassador Steffen Kongstad, Permanent Representative of Norway to the United Nations and other international organizations in Geneva, was elected by the Meeting as President of the 3MSP.
A number of on-site side events sponsored by States Parties, UN entities, CMC and other non-governmental organisations (NGOs) were organized during the 3MSP. The programme can be found here.
A recording of the opening session can be found here.


Carillion plc run the libraries on behalf of local authorities in Croydon, Ealing, Harrow and Hounslow. We believe that books are just the beginning of what we can do, so in June 2015, we began our exciting new partnership with LEGO® Education, the first of its kind in UK libraries!
This has meant far more than just introducing LEGO bricks into the library. For the last eight months, our staff have been delivering workshops using innovative resources from the LEGO Education range. Research from the University of Cambridge, Towards a playful Pedagogy in Literacy, [published here as LEGO - a system for learning] found that by incorporating model-building into discussions around books, children can take control over their own learning.
This also allows them to be imaginative, create ideas and share them with others, which in turn will inspire them to write creatively. Literacy and creativity are, after all, at the heart of what libraries are about, and by introducing a hands-on element to the reading and writing process, we can create something very powerful and engaging. When asked about the workshops, children who visited said that they enjoyed using their imagination to build a story!
We invested in the full range of LEGO Education products for all the libraries in each of our boroughs and received training from LEGO Education so that we could deliver the pedagogy effectively. These include: StoryStarter, a kit that allows children to design their own storyboards; MoreToMaths, a simple mathematical problem-solving resource that challenges pupils to create structures based on numbers; and WeDo, a resource that combines engineering with computing to get young children involved with programming and real-life science. The LEGO Education workshops combine coding with literacy, numeracy and the STEM curriculum. We also run a weekly MINDSTORMS robotics club in 11 libraries across the boroughs, which have been really popular, especially in Croydon where there has been more interest in the area from Tech City companies.
Thanks to the LEGO Education sessions, we've seen a noticeable increase in the number of people visiting our libraries. The launch events in the Summer of 2015 attracted 290 children and the response was fantastic. The buzz at each event was amazing: all of the children were absorbed for two hours enjoying the activity along with their parents. 83 per cent of visitors told us that the LEGO Education Taster Sessions were excellent, and the remaining 17 per cent said they were very good. Everyone who attended said that they would recommend LEGO Education sessions to a friend!
The library staff have been trained by LEGO Education to deliver the sessions which is part of the LEGO Education Innovation Studio Service Level Agreement. There is a small charge for children to attend each session.
Although the full range of resources were acquired as an Innovation Studio, the investment is scalable and not all the resources need be invested in at once.
Looking forward, we will be developing our partnership with LEGO Education and we are planning to use our libraries as venues for the Junior First LEGO league in May. The Junior First LEGO League is organised by The Institution of Engineering and Technology and encourages teams to solve a challenge using LEGO before attending the event to share individual creations and different solutions. Our library staff have also helped to organise and host a LEGO convention in April for schools in Croydon and neighbouring boroughs. We're keen to develop links with schools and explore how to work together so that these resources reach as many children as possible!
LEGO® Education offers playful learning experiences and teaching solutions based on the LEGO® system of bricks, curriculum-relevant material, and physical and digital resources to early years, primary and after school clubs. In partnership with educators for more than 35 years, we support teaching in an inspiring, engaging, and effective way. Our educational solutions, which range from humanities to science, enable every student to succeed by encouraging them to become active, collaborative learners, build skills for future challenges, and establish a positive mind-set toward learning.
For further information about LEGO please contact Elizabeth Moore: Elizabeth.Moore @ mangomarketing.com or Jasmin De Vivo: Jasmin.DeVivo @ mangomarketing.com, and for more about Carillion's work please contact Fiona: Fiona.Tarn @ carillionservices.co.uk.


Motorcycles crash best of, Motorcycles crash compilation 2017, driving Motorcycles 2017 #646. Driving in russia compilation 2017 as Russian drivers drive cars. Collection of road accidents taken on DVR – Car crashes compilation – A montage of some of the funniest motorcycle crashes in the last few years.


Edited by Gary Taylor, John Jowett, Terri Bourus and Gabriel Egan.
Chapter. Subjects: Social and Political Philosophy. 3833 words.
October 2014; p ublished online May 2015 .
Chapter. Subjects: Social and Political Philosophy. 575 words.


text:
/**
 * Created by adam on 11/22/13.
 */

var geoApp = angular.module('geoApp', ['geoServices', 'taskService', 'gTasks', 'oauth', 'util']);

geoApp.constant('audienceList',
    {
        'myself' : { key: 'myself',  label: 'Myself' },
        'anybody': { key: 'anybody', label: 'Anybody' },
        'blah'   : { key: 'blah',    label: 'Blah label' }
    }
);

geoApp.controller('mapTestCtrl', ['$scope', 'gMap', 'audienceList', 'taskService', 'gTasks', 'oauth', 'util', '$log', function ($scope, map, al, taskService, gTasks, oauth, util, $log) {
    var clientId = '559380446604-rvvui7svj0s98hontlo974fkrtnnms99.apps.googleusercontent.com',
        scope = ['https://www.googleapis.com/auth/tasks', 'https://www.googleapis.com/auth/tasks.readonly'];

    oauth.init(clientId, scope);

    $scope.audienceList = al;
    $scope.message = null;
    $scope.messages = taskService.listTasks;

    map.init($scope);

    taskService.loadTasks(function(tl) {
        tl.forEach(function(t){
            map.addNewMarker(t.location, t.title);
         });
    });

    $scope.change = util.debounce(function() {
            taskService.updateTask($scope.message);
        }, 500);

    oauth.checkAccess().then(function(a) {
        $scope.hideAuthorizationPanel = a;
        $log.debug("v: " + a);
    });

    $scope.upClicked = function() {
        map.moveActiveMarker(0.001, 0);
    };

    $scope.authorize = function() {
        oauth.allowAccess().then(function(r) {
            $scope.hideAuthorizationPanel = r;
        }, function(error) {
            // bummer
        });
        return false;
    }

    $scope.doNotAuthorize = function() {
        $scope.hideAuthorizationPanel = true;
        oauth.denyAccess();
    }
}]);

geoApp.directive('infoWindow', function() {
    return {
        // template: "
<textarea ng-model='message.msg'></textarea>
Audience: <select ng-model='message.audience'><option label='Myself' value='myself'></option><option label='Anybody' value='everybody'></option></select>
"
        templateUrl: 'index.html'
    }
});

redpajama_set_name: RedPajamaGithub


text:
<?xml version="1.0" encoding="utf-8"?>

<menu xmlns:android="http://schemas.android.com/apk/res/android"
      xmlns:custom="http://schemas.android.com/apk/res-auto">

    <item
        android:id="@+id/refresh_item"
        android:title="@string/refresh_label"
        android:menuCategory="container"
        custom:showAsAction="ifRoom"
        android:icon="?attr/navigation_refresh"/>

    <item
        android:id="@+id/queue_sort"
        android:title="@string/sort">

        <menu>
            <item
                android:id="@+id/queue_sort_alpha"
                android:title="@string/alpha">

                <menu>
                    <item
                        android:id="@+id/queue_sort_alpha_asc"
                        android:title="@string/ascending"/>
                    <item
                        android:id="@+id/queue_sort_alpha_desc"
                        android:title="@string/descending"/>
                </menu>
            </item>

            <item
                android:id="@+id/queue_sort_date"
                android:title="@string/date">

                <menu>
                    <item
                        android:id="@+id/queue_sort_date_asc"
                        android:title="@string/ascending"/>
                    <item
                        android:id="@+id/queue_sort_date_desc"
                        android:title="@string/descending"/>
                </menu>
            </item>

            <item
                android:id="@+id/queue_sort_duration"
                android:title="@string/duration">

                <menu>
                    <item
                        android:id="@+id/queue_sort_duration_asc"
                        android:title="@string/ascending"/>
                    <item
                        android:id="@+id/queue_sort_duration_desc"
                        android:title="@string/descending"/>
                </menu>
            </item>

        </menu>
    </item>

</menu>


text:
Welcome to TravMed.com, home of Dr. Rose's Destination Guide and Health Guide.
Summer 2019: Store Offline
Home › Destination Guide › Tuvalu
Choose a Destination by Continent
translation missing: en.layout.general.menu_title
Africa +
Algeria Angola Botswana Benin Burkina Faso Burundi Cameroon Cape Verde Central African Republic Chad Comoros Islands Congo Democratic Republic of Congo Djibouti Egypt Eritrea Ethiopia Gabon Gambia Ghana Guinea Guinea-Bissau Ivory Coast Kenya Lesotho Liberia Libya Madagascar Madeira (Portugal) Malawi Mali Mauritania Mauritius Mayotte Morocco Mozambique Namibia Niger Nigeria Réunion Rwanda São Tomé and Príncipe Senegal Seychelles Sierra Leone Somalia South African Sudan Swaziland Tanzania Togo Tunisia Uganda Western Sahara Zambia Zimbabwe
Asia +
Afghanistan Armenia Azerbijan Bahrain Bangladesh Bhutan Brunei Darussalam Cambodia China Cyprus Georgia Hong Kong (China) India Indonesia Iran Iraq Israel Japan Jordan Kazakhstan Kuwait Kyrgyzstan Laos Lebanon Malaysia Maldives Mongolia Myanmar (Burma) Nepal North Korea Oman Pakistan Philippines Qatar Russia Saudi Arabia Singapore South Korea Sri Lanka Syria Taiwan Tajikistan Thailand Turkey Turkmenistan United Arab Emirates Uzbekistan Vietnam Yemen
Australia & Oceania -
American Samoa Australia Christmas Island (Australia) Cook Islands (New Zealand) Fiji French Polynesia Guam (USA) Kiribati Marshall Islands Micronesia Nauru New Caledonia New Zealand Northern Mariana Islands (USA) Palau Papua New Guinea Samoa (Western) Solomon Islands Tokelau Tonga Tuvalu Vanuatu Wallis and Futuna (France)
Europe +
Albania Armenia Austria Azores (Portugal) Belarus Belgium Bosnia and Herzegovina Bulgaria Croatia Cyprus Czech Republic Denmark England Estonia Finland France Georgia Germany Greece Hungary Iceland Ireland Italy Latvia Lithuania Luxembourg Macedonia Malta Moldova Mongolia Netherlands Norway Poland Portugal Romania Russia Serbia Slovenia Slovakia Spain Sweden Switzerland Turkey Ukraine United Kingdom
North America +
Anguilla Antigua & Barbuda Bahamas Barbados Belize Canada Cayman Islands Costa Rica Cuba Dominica Dominican Republic El Salvador Greenland Grenada Guatemala Guadeloupe Haiti Honduras Jamaica Martinique Mexico Nicaragua Panama Puerto Rico Sint Maarten (Netherlands Antilles) St. Barthelemy St. Kitts & Nevis St. Lucia St. Martin St. Vincent & the Grenadines Trinidad & Tobago Virgin Islands (UK) Virgin Islands (US) United States Turks & Caicos
South America +
Argentina Bolivia Bonaire Brazil Chile Colombia Equador French Guiana Guyana Paraguay Peru Suriname Uruguay Venezuela
Capital: Funafuti
Time Zone: +12 hours. No day light savings time in 2008.
Tel. Country Code: 688
USADirect Tel.: 0
Electrical Standards: Electrical current is 220/50 (volts/hz). South Pacific Style Adaptor Plug. Grounding Adaptor Plug E.
NaTHNaC
Travel Health Services
Foreign Commonweatlh Office
Neither the United States, Canada, Australia, nor the United Kingdom has an embassy in Tuvalu.
• The U.S. Embassy in Fiji provides assistance for U.S. citizens in Tuvalu.
31 Loftus Street
Tel: [679] 331-4466
Fax: [679] 330-2267
Website:hhttp://suva.usembassy.gov
• High Commission of Canada
125 The Terrace
Tel: [64] (4) 473-9577
Fax: [64] (4) 471-2082
E-Mail: wlgtn@international.gc.ca
Website: http://www.international.gc.ca/newzealand
• British High Commission
Tel: [679] 322 9100
Fax: [679] 322 9132
Email: ukpassportsuva@fco.gov.uk
A passport, onward/return ticket, and proof of sufficient funds for the stay in Tuvalu are required. Visitor permits valid for up to three months are issued upon arrival. For further information about entry requirements, travelers may wish to contact the Tuvalu Permanent Mission to the United Nations in New York, Suite 400 B, 800 2nd Avenue, NY 10017, tel: (212) 490-0534, fax: (212) 808-4975. This is particularly true for those persons planning to enter by sea.
HIV Test: Not required.
Required Vaccinations: None required.
Tuvalu is a South Pacific island nation consisting of four reef islands and five atolls. A self-governing member of the British Commonwealth, Tuvalu has a parliamentary system of government.
Vaccinations: Recommended and Routine
Hepatitis A: Recommended for all travelers >1 year of age not previously immunized against hepatitis A.
Hepatitis B: Recommended for all non-immune travelers at potential risk for acquiring this infection. Hepatitis B is transmitted via infected blood or bodily fluids. Travelers may be exposed by needle sharing and unprotected sex; from acupuncture, tattooing or body piercing; when receiving non-sterile medical or dental injections, or unscreened blood transfusions; by direct contact with open skin sores on an infected person. Recommended for long-term travelers, expatriates, and any traveler requesting protection against hepatitis B infection.
Influenza: Vaccination recommended for all travelers >6 months of age who have not received a flu shot in the previous 12 months.
Routine Immunizations: Routine
Immunizations against tetanus-diphtheria, measles, mumps, rubella (MMR vaccine) and varicella (chickenpox) should be updated, if necessary, before departure. MMR protection is especially important for any female of childbearing age who may become pregnant.
• The new Tdap vaccine, ADACEL, which also boosts immunity against pertussis (whooping cough) should be considered when a tetanus-diphtheria booster is indicated.
Typhoid: Recommended for all travelers with the exception of short-stay visitors who restrict their meals to hotels or resorts.
Hospitals / Doctors
Tuvalu's only hospital is on Funafuti and the outer islands have trained nurses only. Medical facilities are generally adequate for routine medical treatment.
All travelers should be up-to-date on their immunizations and are advised to carry a medical kit as well as antibiotics to treat travelers diarrhea or other infections. Travelers who are taking regular medications should carry them properly labeled and in sufficient quantity to last for the duration of their trip; they should not expect to obtain prescription or over-the-counter drugs in local stores or pharmacies in this country - the equivalent drugs may not be available or may be of unreliable quality.
• Travelers are advised to obtain comprehensive travel insurance with specific overseas coverage. Policies should cover: ground and air ambulance transport, including evacuation to home country; payment of hospital bills; 24-hour telephone assistance. For more serious or complicated problems, medical evacuation to Fiji, Hawaii, or Australia may be required.
• Princess Margaret Hospital
Tel: 752
A 36-bed facility opened in 1978.
Internet: http://www.panoramio.com/photo/4502998
Destination Health Info for Travelers
AIDS/HIV: A small number of AIDS cases and HIV infections have been reported but little data are available for Tuvalu. Seamen are at particular risk. (http://www.unaids.org.fj/_resources/main/files/Country-Profiles/Tuvalu_profile.pdf) The over-all prevalence of adults living with HIV in Oceania is 0.4%.
• Transmission of HIV can be prevented by avoiding: sexual contact with a high-risk partner; injecting drug use with shared needles; non-sterile medical injections; unscreened blood transfusions.
• The threat of HIV/AIDS should not be a primary concern for the traveler. However, there may be a concern for a subset of travelers who may be exposed to HIV, the virus that causes AIDS, through contact with the body fluids or blood of another person. Although travel has contributed in a general way to the global spread of AIDS, fear of traveling because of this disease is not warranted.
Accidents & Medical Insurance: Accidents and injuries are the leading cause of death among travelers under the age of 55 and are most often caused by motor vehicle and motorcycle crashes; drownings, aircraft crashes, homicides, and burns are lesser causes.
• Heart attacks cause most fatalities in older travelers.
• Infections cause only 1% of fatalities in overseas travelers, but, overall, infections are the most common cause of travel-related illness.
• MEDICAL INSURANCE: Travelers are advised to obtain, prior to departure, supplemental travel health insurance with specific overseas coverage. The policy should provide for direct payment to the overseas hospital and/or physician at the time of service and include a medical evacuation benefit. The policy should also provide 24-hour hotline access to a multilingual assistance center that can help arrange and monitor delivery of medical care and determine if medevac or air ambulance services are required.
Dengue Fever: Sporadic cases and outbreaks are reported. Dengue fever is a mosquito-transmitted, flu-like viral illness occurring throughout the Pacific Islands. Symptoms consist of sudden onset of fever, headache, muscle aches, and a rash. A syndrome of hemorrhagic shock can occur in severe cases.
• Dengue is transmitted via the bite of an infected Aedes aegypti mosquito. Aedes mosquitoes feed predominantly during daylight hours. All travelers are at risk and should take measures to prevent daytime mosquito bites. Insect-bite prevention measures include applying a DEET-containing repellent to exposed skin and applying permethrin spray or solution to clothing and gear.
• There is no vaccination or medication to prevent or treat dengue.
A dengue fever map is at:http://www.nathnac.org/ds/c_pages/documents/dengue_map.pdf
Filariasis: Sporadic cases and outbreaks are reported. Travelers should take protective measures against mosquito bites.
Hepatitis: All travelers not previously immunized against hepatitis A should be vaccinated against this disease. Travelers who are non-immune to hepatitis A (i.e. have never had the disease and have not been vaccinated) should take particular care to avoid potentially contaminated food and water. Travelers who will have access to safe food and water are at lower risk. Those at higher risk include travelers visiting friends and relatives, long-term travelers, and those visiting areas of poor sanitation.
• Hepatitis E is may be endemic but levels are unclear. Sporadic cases may be underdiagnosed or underreported. Transmission of the hepatitis E virus (HEV) occurs primarily through drinking water contaminated by sewage and also through raw or uncooked shellfish. Farm animals, such as swine, may serve as a viral reservoirs. (HEV is one of the few viruses which has been shown to be transmitted directly from animals through food.) In developing countries, prevention of hepatitis E relies primarily on the provision of clean water supplies and overall improved sanitation and hygiene. There is no vaccine.
• Hepatitis B is hyperendemic. The overall hepatitis B (HBsAg) carrier rate in the general population is estimated at >8%. Hepatitis B is transmitted via infected blood or bodily fluids. Travelers may be exposed by needle sharing and unprotected sex; from non-sterile medical or dental injections, and acupuncture; from unscreened blood transfusions; by direct contact with open skin lesions of an infected person. The average traveler is at low risk for acquiring this infection. Vaccination against hepatitis B is recommended for: persons having casual/unprotected sex with new partners; sexual tourists; injecting drug users; long-term visitors; expatriates, and anybody wanting increased protection against the hepatitis B virus.
• Hepatitis C is endemic in the general population. Most hepatitis C virus (HCV) is spread either through intravenous drug use or, in lesser-developed countries, through blood contamination during medical procedures. Over 200 million people around the world are infected with hepatitis C - an overall incidence of around 3.3% of the population of the world. Statistically, as many people are infected with HCV as are with HIV, the virus that causes AIDS.
Malaria: There is no risk of malaria in Tuvalu.
Marine Hazards: Stingrays, poisonous fish, sea anemones, corals and jellyfish are hazards to swimmers.
• Ciguatera poisoning occurs and can result from eating coral reef fish such as grouper, snapper, sea bass, jack, and barracuda. The ciguatoxin is not destroyed by cooking.
Rabies: Tuvalu is considered rabies-free, but surveillance is limited. Stray and free-roaming dogs may be encountered. All unprovoked animal bites should be medically evaluated for possible post-exposure treatment.
Travelers' Diarrhea: Outside of hotels and resorts, we recommend that you boil, filter or purify all drinking water or drink only bottled water or other bottled beverages and do not use ice cubes. Avoid unpasteurized dairy products. Do not eat raw or undercooked food, especially meat, fish, raw vegetables. Peel all fruits.
• Good hand hygiene reduces the incidence of travelers' diarrhea by 30%.
• A quinolone antibiotic, or azithromycin, combined with loperamide (Imodium), is recommended for the treatment of diarrhea. Diarrhea not responding to antibiotic treatment may be due to a parasitic disease such as giardiasis, amebiasis, or cryptosporidiosis.
• Seek qualified medical care if you have bloody diarrhea and fever, severe abdominal pain, uncontrolled vomiting, or dehydration.
Typhoid Fever: Typhoid fever is the most serious of the Salmonella infections. Typhoid vaccine is recommended by the CDC for all people (except short-stay visitors and cruise ship passengers) traveling to or working in Oceania, especially if visiting smaller cities, villages, or rural areas and staying with friends or relatives where exposure might occur through food or water. Current vaccines against Salmonella typhi are only 50-80% protective and do not protect against Salmonella paratyphi, the cause of paratyphoid fever. (Paratyphoid fever bears similarities with typhoid fever, but the course is generally more benign.) Travelers should continue to practice strict food, water and personal hygiene precautions, even if vaccinated.
Please note: We an an Amazon Affiliate site.
© 2019 Travel Medicine, Inc.
Proud member of ISTM: The International Society of Travel Medicine.


text:
Video shows Brazos Co. deputy saving 20-day-old baby boy who stopped breathing
Watch the chilling video of a deputy helping a family after their 20-day-old baby stopped breathing.
BRYAN, Texas (KTRK) -- A Brazos County Sheriff's Office deputy is being deemed a hero and was honored with a special award after he saved the life of a 20-day-old baby boy who stopped breathing.
Deputy Adam Dodson was awarded the Life Saving Medal.
The life-changing incident happened on Sept. 18. Dodson was dispatched to a mobile home where CPR was being performed on the baby.
READ ALSO: 10-month-old falls 8-feet down into heating vent
When he arrived, he said the baby wasn't conscious and wasn't breathing.
"With the baby's parents looking on in desperation, [Dodson] began to perform life-saving measures, clearing the child's airway and performing the Heimlich maneuver for infants," said the sheriff's office in a statement.
READ ALSO: Body cam footage shows Atlanta police officer rescue 7-month-old baby locked in hot car
Dodson said after what seemed to be the longest minute of his life, he noticed the baby became responsive and started to breathe on his own.
The baby was sent to the hospital. Authorities said one of the responding medics later reached out to Dodson and acknowledged his quick life-saving actions by saying, "Good job on that call, you saved that baby's life."
"Adam's calm and decisive action, unquestionably saved this baby's life, and he, without a doubt, deserves to be awarded the Life Saving Medal," said the sheriff's office.
SOCIETY BRYAN BABY RESCUED TEXAS NEWS RESCUE CAUGHT ON VIDEO GOOD SAMARITAN U.S. & WORLD GOOD NEWS FEEL GOOD