The third cohomology 2-group
Alan S. Cigoli^1 Sandra Mantovani^2Giuseppe Metere^3

In this paper we show that a finite product preserving  opfibration  can be factorized through an opfibration with the same property, but with groupoidal fibres.
If moreover the codomain is additive, one can endow each fibre of the new opfibration  with a canonical symmetric $2$-group structure. We then apply such factorization to the opfibration that sends a crossed extension of a group $C$ to its corresponding $C$-module. The symmetric $2$-group structure so obtained on the fibres, defines the third cohomology $2$-group of $C$, with coefficients in a $C$-module. We show that the usual third and second cohomology groups are recovered as its homotopy invariants. Furthermore, even if all results are presented in the category of groups, their proofs are valid in any strongly protomodular semi-abelian category, once one adopts the corresponding internal notions.

MSC: 20J06, 18E13, 18G45, 18D30.

Keywords: 2-groups, cartesian monoidal opfibration, cohomology, crossed extension, semi-abelian category.

§ INTRODUCTION

For a fixed group $C$, let $\Mod(C)$ denote the category of $C$-modules, whose objects are pairs $(B,\xi)$, with $\xi\colon C\times B\to B$ a group action on the abelian group $B$, and morphisms are  equivariant  group homomorphisms.  $\Mod(C)$ is an abelian category, and it is well known to be equivalent to  the category $\Ab(\Gp/C)$ of abelian groups in $\Gp/C$  (also called  Beck $C$-modules, see [5]).

With any $C$-module $(B,\xi)$, we can associate the groupoid $\OpExt(C,B,\xi)$ of abelian extensions of $C$, with kernel $B$ and induced action $\xi$, where, given an extension of C with abelian kernel $B$

\[
\xymatrix{
0 \ar[r] & B \ar[r]^j & E \ar[r]^p & C \ar[r] & 0\,,
\]

the  action of $C$ on $B$ is given by the conjugation action of $E$, via any pointed set-theoretical section of  $p$.

If we consider the set $\pi_0(\OpExt(C,B,\xi))$ of connected components,  we can extend the previous assignment to a functor
\[H^2(C,-)\colon  \Mod(C) \to \Set\]
which preserves finite products. Actually, likewise any $\Set$-valued functor with  abelian domain which preserves finite products, $H^2(C,-)$ factors through the category $\Ab$ of abelian groups (see [8], where the Baer sum of abelian extensions in a protomodular category is introduced in this way). As a consequence, the set $\pi_0(\OpExt(C,B,\xi))$ can be endowed with an  abelian group  structure, which is isomorphic to the usual  $H^2(C,B,\xi)$ (see [24]).

It is well known since [19] and [21] that also the cohomology functor $H^{3}(C,-)$ can be described in a similar fashion, i.e. using crossed extensions (see <Ref>):

\[
\xymatrix{
0 \ar[r] & B \ar[r]^-j & E_2 \ar[r]^\partial & E_1 \ar[r]^-p & C \ar[r] & 0\,.
\]

In this case the categories $\XExt(C,B,\xi)$ of such extensions are no longer groupoids. However, their connected components define again a $\Set$-valued  functor which preserves finite products. So, it is possible to apply the same argument as for $H^2(C,-)$.

In fact, the abelian group structure on $\pi_0(\OpExt(C,B,\xi))$ is nothing but the shadow of a richer structure of a symmetric 2-group $\cH^2(C,B,\xi)$ which can be defined directly on $\OpExt(C,B,\xi)$  (see [16]).  This result has been anticipated by Bourn in [7], where he singles out some sufficient conditions to lift a monoidal closed structure to the fibres of a given opfibration. The groupoids $\OpExt(C,B,\xi)$ are actually  the fibres of an opfibration $P_C \colon \OpExt(C) \to \Mod(C)$ which preserves finite products. An articulate treatment of the subject can be found in [15], where  cartesian monoidal opfibrations [26] are investigated (see also [29]). It turns out that, when the codomain of a finite product preserving opfibration is additive, then its fibres are symmetric $2$-groups as soon as they are groupoids, and this is precisely what happens in the case of abelian extensions.

In the present paper we deal with the case of  opfibrations as before, but whose fibres are not groupoids. In <Ref> we show that a $2$-group structure can still be defined on the fibres of a new opfibration obtained from the original one via a suitable category of fractions.

The case of crossed extensions is exactly of the kind  described above. In fact, the categories $\XExt(C,B,\xi)$ turn out to be the fibres of an opfibration $\Pi_C\colon \XExt(C)\to\Mod(C)$ which again preserves finite products. The first step is to  investigate the category of fractions $[\BExt](C)$ of $\XExt(C)$ with respect to  $\Pi_C$-vertical maps (which are indeed weak equivalences between the underlying crossed modules, see [1]).  Relying on results in [14] and [13], in <Ref> we show that such fractions can be described using diagrams called butterflies by Noohi (see [27]).  By applying <Ref>, we get that the functor $\Pi_C$ factors through a new cartesian monoidal opfibration $\overline\Pi_C$ with groupoidal fibres. Therefore, for each $C$-module $(B,\xi)$, the corresponding fibre of $\overline\Pi_C$ becomes a symmetric $2$-group which can be considered as the third cohomology $2$-group, and will be denoted by $\cH^{3}(C,B,\xi)$.

The abelian group  $\pi_0(\cH^{3}(C,B,\xi))$ of connected components of the $2$-group  $\cH^{3}(C,B,\xi)$ is isomorphic to the usual $H^{3}(C,B,\xi)$, as observed above. On the other hand, the abelian group  $\pi_1(\cH^{3}(C,B,\xi))$ given by the automorphisms of the monoidal identity  $I_\xi$ is isomorphic to the group $H^{2}(C,B,\xi)\cong \pi_0(\cH^{2}(C,B,\xi))$. This result is obtained as a consequence of <Ref>, where it is proved that the $2$-group $\cH^{2}(C,B,\xi)$ is monoidally equivalent to the $2$-group $Eq(I_\xi)$ of auto-equivalences of $I_\xi$, with tensor product given by butterfly composition.

We remark here that, although all results are presented having as base category the category of groups, all proofs are indeed internal, in the sense that they can be performed in any strongly protomodular semi-abelian category (see [6]), once one adopts the corresponding internal notions.

We use sans-serif capital letters for categories ($\cA, \cB$ etc.).

Composition of arrows (1-cells) $f, g$ is denoted $g\cdot f$, or just $gf$.
Binary products  in a category $\cB$ are denoted as usual, with projections $\pr_1,\pr_2$.

We denote by $\CG$ the $2$-category of $2$-groups; its objects are (not necessarily strict) $2$-groups (also known as categorical groups), morphisms are monoidal functors, and $2$-morphisms are monoidal natural transformations. Their symmetric monoidal version organizes in the $2$-category $\SCG$. There is a $2$-functor

\[
(\pi_0,\pi_1)\colon \CG\to \Mod
\]

where the category of group modules $\Mod$ is considered as a $2$-discrete $2$-category. It associates with any $2$-group $\cG$ the group module $(\pi_0(\cG), \pi_1(\cG))$ of its homotopy invariants, with $\pi_0(\cG)$  the group of connected components of $\cG$ and $\pi_1(\cG)$  the abelian group of the automorphisms of the identity object of $\cG$, endowed with a canonical $\pi_0(\cG)$-module structure (see for instance [3]). Notice that, if the $2$-group is symmetric, the group $\pi_0(\cG)$ is abelian.

Finally, throughout the paper, we will use the additive notation for any group operation.

§ PRELIMINARIES

In this section we recall from [15] some basic results on monoidal opfibrations, and complete the description of the symmetric $2$-group  $\cH^2(C,B,\xi)$ given in the introduction.

Let $\cX, \cB$ be categories with finite products, and $P\colon \cX\to\cB$ be an opfibration. We say that $P$ is cartesian monoidal if it strictly preserves finite products and  the product of two cocartesian morphisms is still cocartesian.

In other words, cartesian monoidal opfibrations are nothing but monoidal opfibrations, with the monoidal structures given by  cartesian products, see  <cit.>.

Let  and  be categories with finite products and $P\colon \cX \to \cB$ be an opfibration with groupoidal fibres. Then $P$ is a cartesian monoidal opfibration if and only if it strictly preserves finite products.

It suffices to notice that each morphism in  factorizes through a cocartesian lifting of its image by $P$, with a $P$-vertical  comparison morphism, hence an isomorphism by assumption. So any morphism, in particular any product of morphisms, is cocartesian.

If the codomain $\cB$ is an additive category, every object of $\cB$ is endowed with a (unique) commutative monoid structure. In this case, the  monoidal opfibration induces on its fibres and on change-of-base functors  canonical symmetric monoidal structures, as shown in <cit.> (see also <cit.>).  Furthermore, we have the following result, see <cit.>.

Let $P\colon \cX\to \cB$ be a cartesian monoidal opfibration. If the category $\cB$ is additive, then the symmetric monoidal structures induced on the  fibres are $2$-groups  if and only if the fibres are groupoids.

Let $C$ be a group, and $\OpExt(C)$ be the category of abelian extensions of the group $C$ and their morphisms. The functor $P\colon \OpExt(C)\to \Mod(C)$, which assigns to each abelian extension the canonical $C$-module structure on the kernel, is an instance of Bourn's direction functor (see [7]). In op.cit., the author proved that the latter is an opfibration that preserves finite products and that cocartesian maps are stable under binary products. In other words, it is cartesian monoidal. As explained in <cit.>, since the codomain of $P$ is additive, thanks to Theorem <ref>, each $P$-fibre $\OpExt(C,B,\xi)$ inherits a symmetric $2$-group structure:

\[
\cH^2(C,B,\xi):=(\OpExt(C,B,\xi),\otimes,E_{\rtimes})
\]

where the tensor product $\otimes$ is obtained by the classical construction used to define Baer sums, before taking the isomorphism classes of extensions. The identity $E_\rtimes$ is the canonical extension associated with the semidirect product  $B\rtimes_{\xi}C$.  Moreover, each $C$-module morphism $f\colon B\to B'$ induces a symmetric monoidal functor

\[
f_*\colon \cH^2(C,B,\xi)\to \cH^2(C,B',\xi')\,.
\]

Concerning the homotopy invariants $\pi_0,\pi_1$, as recalled in the introduction, we have that
\[\pi_0(\cH^2(C,B,\xi))\cong H^2(C,B,\xi)\,.\]
On the other hand, the abelian group  $\pi_1(\cH^2(C,B,\xi))$ consists of the automorphisms of the  extension $E_{\rtimes}$. According to <cit.>, this can be interpreted as the group of 1-cocycles, i.e.
\[\pi_1(\cH^2(C,B,\xi))\cong Z^1(C,B,\xi)\,.\]

§ CROSSED EXTENSIONS

Recall that a crossed module consists of a group homomorphism $\partial \colon E_2\to E_1$, endowed with an action of $E_1$ on $E_2$, satisfying the following conditions:

\begin{align*}
\text{i.} & \quad\partial(g\ast x)=g+\partial x-g & \text{(Pre-crossed module)} \\
\text{ii.} & \quad \partial x_1\ast x_2=x_1+x_2-x_1 & \text{(Peiffer)}
\end{align*}

Crossed modules, and their usual morphisms, organize in a category, which is equivalent to the category of internal groupoids in the category of groups. To fix notations, let us recall that each crossed module $\partial$ corresponds to the internal groupoid

\begin{equation} \label{diag:xmod_gpd}
\xymatrix{
E_2 \rtimes E_1 \ar@<1ex>[r]^-d \ar@<-1ex>[r]_-c & E_1, \ar[l]
\end{equation}

where the semidirect product operation is induced by the given action of $E_1$ on $E_2$ and the morphisms $d$ and $c$ are as follows:

\begin{align*}
d(x,g) & = \partial x+g, \\
c(x,g) & = g.
\end{align*}

Let $C$ be a group. A crossed extension $E$ of $C$ is given by

\[
\xymatrix{
0 \ar[r] & B \ar[r]^{j} & E_2  \ar[r]^{\partial} & E_1 \ar[r]^{p}  & C  \ar[r] & 0
\]

where $\partial$ is endowed with a crossed module structure, with $B$ and $C$  fixed kernel and cokernel, respectively. Morphisms of crossed extensions of $C$ are given by commutative diagrams:

\[
\xymatrix{
0 \ar[r] & B \ar[r]^{j} {\ar[d]_{\beta}} & E_2
{\ar[d]_{f_2}} \ar[r]^{\partial} & E_1 \ar[r]^{p} {\ar[d]_{f_1}} & C {\ar@{=}[d]} \ar[r] & 0 \\
{0} {\ar[r]} &{B'} {\ar[r]^{j'}} & {E_2'} {\ar[r]^{\partial'}} &{E_1'} {\ar[r]^{p'}} & {C} {\ar[r]} & {0}
\]

where  the middle square is a crossed module morphism. We denote by $\XExt(C)$ the category  of crossed extensions of $C$.

In each crossed extension the kernel $B$ is central and it comes equipped with a $C$-module structure $(B,\xi)$, where $\xi$ is the action of $C$ over $B$ induced by  the action of $E_1$ on $E_2$ (see for instance [10]).

One can lift a given crossed extension  along a $C$-module morphism $\beta\colon(B,\xi) \to (B',\xi')$ by means of what is called the push forward along $\beta$ (see [2], or [11] for the semi-abelian version):

\[
\xymatrix{
0 \ar[r] & B \ar[r]^j \ar[d]_{\beta}<EMAIL_ADDRESS>& E_2 \ar[r]^{\partial} {\ar[d]} & E_1 \ar[r]^p {\ar@{=}[d]} & C \ar[r] {\ar@{=}[d]} & 0 \\
0 {\ar[r]} & B' {\ar[r]} & {B'\times^B E_2} {\ar[r]^-{\partial'}} & {E_1}{\ar[r]^p} & {C} {\ar[r]} &{0}
\]

Such liftings are universal, i.e.  the functor
P_C\colon \XExt(C)\to \Mod(C)
is an opfibration (see <cit.>). Actually,  it is also cartesian monoidal. A proof in the context of strongly semi-abelian categories, based on properties of Bourn's 1-dimensional direction functor,  can be found in [28].

Indeed, $P_C$ has not groupoidal fibres, since  morphisms in the fibre over $(B,\xi)$ are of the kind $(1,f_1,f_2,1)$:

\[
\xymatrix{
0 \ar[r] & B\ar[r]^{j} \ar@{=}[d] & E_2 \ar[r]^{\partial} \ar[d]_{f_2} & E_1 \ar[r]^{p} \ar[d]^{f_1} & C \ar@{=}[d] \ar[r] & 0 \\
0 \ar[r] & B \ar[r]^{j'} & E_2' \ar[r]^{\partial'} & E_1' \ar[r]^{p'} & C \ar[r] & 0
\]

which induce weak equivalences of crossed modules (see [1]). Such maps do not have inverses in general, so that it is not possible to endow the fibres of $P_C$ with a 2-group structure. The idea is to turn $P_C$ into an opfibration with groupoidal fibres, but still cartesian monoidal. This will be performed the next section.

§ THE SYMMETRIC $2$-GROUP $\CH^3(C,B,\XI)$

The categorical construction that we need in order to make  $P_C$-vertical morphisms invertible consists in taking the corresponding category of fractions. We rely on results obtained in [1], where such fractions are described by means of suitable diagrams called butterflies (see [27]).

A butterfly between two crossed extensions of $C$ is depicted as a diagram

\begin{equation}\label{diag:butterfly}
\begin{aligned}
\xymatrix{
0 \ar[r] & B \ar[r]^{j} & E_2 \ar[rr]^{\partial} \ar[dr]_{\kappa} & & E_1 \ar[r]^{p} & C \ar[r] & 0 \\
& & & F \ar[dr]_{\gamma} \ar[ur]_{\delta} \\
0 \ar[r] & B' \ar[r]_{j'} & E_2' \ar[rr]_{\partial'} \ar[ur]_{\iota} & & E_1'\ar[r]_{p'} & C \ar[r] & 0
\end{aligned}
\end{equation}

where the following conditions are satisfied:

i. $\delta\cdot\kappa=\partial$,    $\gamma\cdot\iota=\partial'$ and $p\cdot \delta = p'\cdot\gamma$,

ii. $(\kappa,\gamma)$ is a complex, i.e. $\gamma\cdot \kappa = 0$ and $(\delta,\iota)$ is a short exact sequence, i.e. $\delta=\coker \iota$ and $\iota=\ker\delta$,

iii. the action of $F$ on $E_2$, induced by the one of $E_1$ on $E_2$ via $\delta$, makes $\kappa$ a pre-crossed module,

iv. the action of $F$ on $E_2'$, induced by the one of $E_1'$ on $E_2'$ via $\gamma$, makes $\iota$ a pre-crossed module.

We will use the short notation $\widehat{F}\colon E \to E'$ to denote the above butterfly.

With each butterfly $\widehat{F}\colon E\to E'$ as in (<ref>), one can associate a $C$-module morphism $\beta\colon B \to B'$. In order to describe how it is constructed, let us recall from [1] that each butterfly induces a span of crossed module morphisms as in the right-hand side of the following diagram:

\begin{equation} \label{diag:bfly_span}
\begin{aligned}
\xymatrix{
& B \ar@{=}[dl] \ar[r]^j & E_2 \ar[rr]^{\partial} \ar@{-->}[dr]_{\kappa} & & E_1 \\
B \ar[dr]_\beta \ar[r]^(.5){\overline{j}} & E_2 \times E_2' \ar[ur]^{\pr_1} \ar[dr]_(.4){\pr_2} \ar[rr]^{\kappa\sharp\iota} & & F \ar[dr]_{\gamma} \ar[ur]_{\delta} \\
& B' \ar[r]^(.4){j'} & E_2' \ar[rr]^{\partial'} \ar@{-->}[ur]_{\iota} & & E_1'
\end{aligned}
\end{equation}

where the crossed module $\kappa\sharp\iota$ is the cooperator (see [6]) of the arrows $\iota$ and $\kappa$, which exists since the images  $\kappa(E_2)$ and $\iota(E_2')$ commute in $F$ (see [1] for details). More explicitly, $\kappa\sharp\iota(x,x')=\kappa x +\iota x'$. Now, the square $\delta\cdot\kappa\sharp\iota = \partial\cdot\pr_1$ turns out to be a pullback, so that one can choose $B$ as a kernel of $\kappa\sharp\iota$ via a morphism $\overline{j}$ with  $\pr_1\cdot\overline{j}=j$. We call $\beta\colon B \to B'$ the unique morphism such that $j'\cdot\beta=\pr_2\cdot\overline{j}$, whence $\overline{j}(b)=(j(b),j'\beta(b))$.

Actually, $\beta$ is a $C$-module morphism, so it admits an opposite $-\beta$. Notice that

\begin{equation} \label{eq:-beta}
\kappa\cdot j = \iota\cdot j'\cdot (-\beta).
\end{equation}

One can prove this by showing that $\kappa\cdot j + \iota\cdot j'\cdot \beta = 0$. The same argument holds in any semi-abelian category (see <cit.> for  details on symmetrizable morphisms). Consider the commutative diagram

\[
\xymatrix{
& & & & E_2 \\
B \ar@/^2ex/[urrrr]^j \ar@/_2ex/[drrrr]_{j'\beta} \ar[rr]|-{\langle 1,1 \rangle} & & B \times B \ar[r]^{1\times \beta} & B\times B' \ar[r]^{j\times j'} & E_2 \times E_2' \ar[u]_{\pr_1} \ar[d]^{\pr_2} \ar[r]^-{\kappa\sharp\iota} & F \\
& & & & E_2'
\]

The composite $\kappa\sharp\iota\cdot(j\times j')\cdot(1\times\beta)$ is the cooperator of $\kappa\cdot j$ with $\iota\cdot j'\cdot\beta$, so that precomposition with the diagonal $\langle 1,1\rangle\colon B \to B\times B$ yields the sum $\kappa\cdot j + \iota\cdot j'\cdot \beta$ (see Definition 1.3.21 in [6]). This sum is trivial since $(j\times j')\cdot (1\times\beta)\cdot\langle 1,1\rangle = \overline{j}$, the two terms being equal by composition with product projections.

After our discussion above, we are allowed to report more information in the picture of a butterfly. For instance, diagram (<ref>) becomes:

\[
\xymatrix{
0 \ar[r] & B\ar@{-->}[dd]_{\beta} \ar[r]^{j} & E_2 \ar[rr]^{\partial} \ar[dr]_{\kappa} & & E_1 \ar[r]^{p} & C\ar@{=}[dd] \ar[r] & 0 \\
& & & F \ar[dr]_{\gamma} \ar[ur]_{\delta} \\
0 \ar[r] & B' \ar[r]_{j'} & E_2' \ar[rr]_{\partial'} \ar[ur]_{\iota} & & E_1'\ar[r]_{p'} & C \ar[r] & 0
}	\]

where $\beta$ is dashed in order to remind the reader that the  pentagon on the left only commutes up to a $-1$ factor. Notice that, on the other hand, the pentagon on the right does commute, since $p\cdot\delta=p'\cdot\gamma$ by hypotesis.

Butterflies between crossed extensions can be composed, and details can be found in [1]. Here we just describe the construction of the composite butterfly. Let us consider two butterflies $\widehat{F}\colon E \to E'$ and $\widehat{F}'\colon E' \to E''$. The following diagram illustrates the composition  $\widehat{F}'\cdot \widehat{F}$ (the kernels and cokernels of the involved crossed modules are omitted in the diagram, for the sake of clarity)

\begin{equation}\label{diag:butterfly_composition}
\begin{aligned}
\xymatrix{
&& E_2\ar[ddl]_{\langle\kappa,0\rangle} \ar@{-->}[rr]_{\partial} \ar[dr]_{\kappa} & & E_1  \\
&& & F \ar[dr]_{\gamma} \ar[ur]_{\delta} \\
F'\cdot F\ar@/^10ex/[rrrruu]^{\overline{\delta\cdot p_1}}  \ar@/_10ex/[rrrrdd]_{\overline{\gamma'\cdot p_2}}
&\ar[l]_{q}F\times_{E_1'}F' \ar[drr]_{p_2}\ar[urr]^{p_1}  & \ar[l]_(.3){\langle\iota, \kappa' \rangle} E_2' \ar@{-->}[rr]_{\partial'}\ar[dr]^{\kappa'} \ar[ur]_{\iota} & & E_1'\\
&& & F' \ar[dr]_{\gamma'} \ar[ur]_{\delta'} \\
&& E_2'' \ar[uul]^{\langle0,\iota'\rangle} \ar@{-->}[rr]^{\partial''} \ar[ur]_{\iota'} & & E_1''
\end{aligned}
\end{equation}

In diagram above, $F\times_{E_1'}F'$ is the pullback over the pair $(\gamma,\delta')$, $q$ is the cokernel of $\langle\iota, \kappa' \rangle$,   $\overline{\gamma'\cdot p_2}$ and $\overline{\delta\cdot p_1}$ are the unique arrows such that $\overline{\gamma'\cdot p_2}\cdot q =\gamma'\cdot p_2$ and $\overline{\delta\cdot p_1}\cdot q= \delta\cdot p_1$ respectively. The resulting butterfly $\widehat{F'\cdot F}\colon E\to E''$ is given by  the complex  $(q\cdot \langle\kappa,0\rangle,\overline{\gamma'\cdot p_2})$ and the short exact sequence $(q\cdot \langle0,\iota'\rangle, \overline{\delta\cdot p_1})$.

Identity butterflies are depicted as follows:

\begin{equation}\label{diag:id_butterfly}
\begin{aligned}
\xymatrix{
0 \ar[r] & B\ar@{==}[dd] \ar[r]^{j} & E_2 \ar[rr]^{\partial} \ar[dr]_{\ker (d)} & & E_1 \ar[r]^{p} & C \ar[r]\ar@{=}[dd]  & 0 \\
& & & E_2\rtimes E_1 \ar[dr]_{d} \ar[ur]_{c} \\
0 \ar[r] & B \ar[r]_{j} & E_2 \ar[rr]_{\partial} \ar[ur]_{\ker(c)} & & E_1\ar[r]_{p} & C \ar[r] & 0
\end{aligned}
\end{equation}

where $c$ and $d$ are as in (<ref>).

We say that two butterflies $\widehat{F},\widehat{F'}\colon E \to E'$ are isomorphic to each other if there exists an isomorphism $\sigma\colon F \to F'$ such that

\[
\sigma\cdot \iota = \iota',\quad \sigma\cdot\kappa = \kappa', \quad \gamma'\cdot\sigma = \gamma,\quad \delta'\cdot\sigma = \delta.
\]

By taking isomorphism classes of butterflies as morphisms between crossed extensions of $C$, we get a category $[\BExt](C)$. From Proposition 6.4 in [14] and Proposition 3.14 in [13] we obtain the following result.

The category $[\BExt](C)$ is the category of fractions of $\XExt(C)$ with respect to $P_C$-vertical maps, by means of a functor $ Q_C\colon \XExt(C) \to [\BExt](C) $ which is the identity on objects.

A description of $Q_C$ on morphisms can be found in <cit.>. Those butterflies representing a class in the image of $Q_C$ are called representable.
On the other hand, as proved in [16], those butterflies whose isomorphism class is invertible in $[\BExt](C)$ are precisely the ones where also $(\gamma,\kappa)$ is a short exact sequence. These are called flippable butterflies.

Actually, $[\BExt](C)$ is the classifying category of the bicategory of butterflies of crossed extensions of $C$. It was proved in [1] that butterflies provide a bicategory of fractions of crossed modules with respect to weak equivalences, so that the above theorem can be seen as a shadow of this result.

Theorem <ref> is important in our context also because it shows that the category of fractions of $\XExt(C)$  we need, is still a locally small category. Going back to the functor $P_C\colon\XExt(C)\to\Mod(C)$, recall that it sends $P_C$-vertical maps to isomorphisms. Hence, by the universal property of $Q_C$, we get a factorization of $P_C$ given by

\begin{equation} \label{diag:Pc_bar}
\begin{aligned}
\xymatrix{
\XExt(C) \ar[r]_{Q_C} \ar@/^4ex/[rr]^{P_C} & [\BExt](C) \ar[r]_{\overline{P}_C} & \Mod(C),
\end{aligned}
\end{equation}

where $\overline{P}_C$ associates with each isomorphism class of butterflies the $C$-module morphism $\beta$ described in diagram (<ref>) which is invariant under isomorphisms (see Theorem 6.6 of [14] for details).

In the factorization of diagram (<ref>), $\overline{P}_C$ is an opfibration with groupoidal fibres.

The thesis follows by Theorem 4.3 of [13], since, in 2-categorical terms, $Q_C$ is  the coinverter of the identee of $P_C$.

Now we are going to show that $\overline{P}_C$ inherits from $P_C$ the property of being cartesian monoidal. As for the preservation of finite products, we need a preliminary Lemma, which follows from results in Section 3.2 of [22] (see also [17]).

Let $\cA$ be a category with finite products and $\Sigma$ a class of morphisms in $\cA$ which contains identities and such that, if $f$ and $g$ are in $\Sigma$, $f\times g$  is in $\Sigma $ as well. Then the category of fractions $\cA[\Sigma^{-1}]$ has finite products and the localization functor $Q\colon \cA \to \cA[\Sigma^{-1}]$ preserves them.

As an application, we get the following general result.

Let  and  be categories  with finite products, and $P\colon \cX \to \cB$ an opfibration strictly preserving them. Consider the factorization

\[
\xymatrix@!=5ex{
\cX \ar@/^3ex/[rr]^{P} \ar[r]_-{Q} & \cQ \ar[r]_-{\overline{P}} & \cB
\]

of $P$ through  the category of fractions $(Q,\cQ)$, with respect to the class of $P$-vertical morphisms. Then $\overline{P}\colon \cQ \to \cB$ is a cartesian monoidal opfibration with groupoidal fibres.

If moreover $\cB$ is additive, each fibre of $\overline P$ is endowed with a symmetric $2$-group structure, and  change-of-base functors are symmetric monoidal.

Lemma <ref> is applicable, since $P$-vertical morphisms are closed under finite products. Hence, finite products in  serve also as finite products in . So $\overline{P}$ preserves finite products as soon as $P$ does. Then by Theorem 4.3 in [13] and <Ref>, $\overline{P}$ is a cartesian monoidal opfibration with groupoidal fibres. Finally, when $\cB$ is additive, the  result follows from <Ref>.

As observed in <Ref>, $P_C\colon\XExt(C)\to\Mod(C)$ is a cartesian monoidal opfibration, hence we can apply <Ref> to get that ${\overline{P}_C}: [\BExt](C) \to \Mod(C)$ is a cartesian monoidal opfibration with groupoidal fibres. Moreover, the codomain $\Mod(C)$ is an additive category, so by <Ref> each fibre of $\overline{P}_C$ becomes a symmetric 2-group, which can be considered as the third cohomology 2-group of $C$ with coefficients in $(B,\xi)$ and denoted by ${\cH^3(C,B,\xi)}$.

It is easy to see that if one considers connected components, the tensor operation induces precisely the Baer sum, which makes $\pi_0(\cH^3(C,B,\xi))$ a group (isomorphic to $H^3(C,B,\xi)$).

As a matter of fact, for a given crossed extension

\[
\xymatrix{
E\colon & 0 \ar[r]
& B \ar[r]^{j}
& E_2 \ar[r]^{\partial}
& E_1 \ar[r]^{p}
& C \ar[r]
& 0,
\]

any representative of the inverse of the class $[E]$ in $\pi_0(\cH^3(C,B,\xi))$ is a pseudo-inverse of $E$ in $\cH^3(C,B,\xi)$. For example, one can take

\[
\xymatrix{
E^*\colon & 0 \ar[r]
& B \ar[r]^{-j}
& E_2 \ar[r]^{\partial}
& E_1 \ar[r]^{p}
& C \ar[r]
& 0\,.
\]

To the reader's convenience, we provide here a proof of this fact. First, we have to compute the tensor product $E\otimes E^*$  of the two crossed extensions above, then find a flippable butterfly in the fibre over $(B,\xi)$ between $E\otimes E^*$ and the unit object $I_\xi $:

\[
\xymatrix{
0 \ar[r] & B \ar@{=}[r] & B \ar[r]^0 & C \ar@{=}[r] & C \ar[r] & 0
\]

of ${\cH^3(C,B,\xi)}$. This process is summarized in the diagram below:

\[
\xymatrix@C=5ex{
0 \ar[r] & B\times B \ar[d]_{[1,1]} \ar[r]^-{j\times(-j)}<EMAIL_ADDRESS>& E_2 \times E_2 \ar[d]^{f_2} \ar[rr]^-{\partial \times \partial} \ar@/^/@{-->}[ddr]^(.4){\varphi} & & E_1 \times_C E_1 \ar[r]^-{p\cdot \pr_2} \ar@{=}[d] & C \ar[r] \ar@{=}[d] & 0 \\
0 \ar[r] & B \ar[r]^-{j'} \ar@{==}[dd] \ar@/_/@{-->}[drr]_(.4){\ker(c)\cdot(-j)} & \Ker(p\cdot c) \ar[rr]^{\partial'} \ar[dr]|{\ker(p\cdot c)} & & E_1 \times_C E_1 \ar[r]^-{p\cdot \pr_2} & C \ar@{=}[dd] \ar[r] & 0 \\
& & & E_2 \rtimes E_1 \ar[dr]^{p\cdot c} \ar[ur]_{\langle c,d \rangle} \\
0 \ar[r] & B \ar@{=}[r] & B \ar[rr]^{0} \ar[ur]^{\ker(c)\cdot j} & & C \ar@{=}[r] & C \ar[r] & 0
\]

The arrow $\varphi\colon E_2 \times E_2 \to E_2 \rtimes E_1$ is the cooperator of $\ker(d)\colon E_2\to E_2 \rtimes E_1$ with $\ker(c)\colon E_2\to E_2 \rtimes E_1$, where $d$ and $c$ are as in diagram (<ref>). More explicitly, $\varphi(x_1,x_2)=(-x_1+{\partial x_1}\ast x_2,\partial x_1)$. It is easy to check that $\varphi\cdot(j\times(-j))=\ker(c)\cdot(-j)\cdot[1,1]$. Moreover, both the dashed arrows factor uniquely through the kernel of $p\cdot c$, hence we get the upper left commutative diagram.
Now consider the commutative diagram

\[
\xymatrix{
B \ar@{=}[d] \ar[rr]^-{j'} & & \Ker(p\cdot c) \ar@{}[dr]|{(a)} \ar[d]|{\ker(p\cdot c)} \ar[r]^{\overline{\partial}} & \Ker(p\cdot \pr_2) \ar@{}[dr]|{(b)} \ar[d]|{\ker(p\cdot \pr_2)} \ar[r] & 0 \ar[d] \\
B \ar[rr]^-{\ker(c)\cdot (-j)} & & E_2 \rtimes E_1 \ar[r]^-{\langle c,d \rangle} & E_1 \times_C E_1 \ar[r]^-{p\cdot \pr_2} & C.
\]

Both squares $(b)$ and $(a)+(b)$ are pullback, so by cancellation $(a)$ is a pullback. Hence, since $\ker(c)\cdot (-j)$ is a kernel of $\langle c,d \rangle$, $j'$ is a kernel of $\overline{\partial}$. Notice that $\langle c,d \rangle$ is a regular epimorphism, since it is the comparison arrow of the internal groupoid $E_2 \rtimes E_1$ to its support $E_1 \times_C E_1$ (i.e. the kernel pair of the coequalizer $p$ of $d$ and $c$). Let us denote $\partial'=\langle c,d \rangle\cdot \ker(p\cdot c)$. One can check that $\ker(p\cdot \pr_2)\cdot\overline{\partial}\cdot f_2 = \langle c,d \rangle\cdot\varphi = \partial\times\partial$. But $\partial\times\partial = \ker(p\cdot\pr_2)\cdot\overline{\partial\times\partial}$, where $\overline{\partial\times\partial}$ is a cokernel of $j\times(-j)$, so that $\overline{\partial}\cdot f_2 = \overline{\partial\times\partial}$.

The observations above explain that the commutative square  $f_2\cdot (j\times(-j)) = j'\cdot[1,1]$ induces isomorphisms on kernels and cokernels of the horizontal arrows, hence it is a push forward, thanks to <cit.>. As a consequence, by construction $(p\cdot \pr_2,\partial',j')$ is the tensor product $E\otimes E^*$. Furthermore, we obtain an isomorphism, represented by the flippable butterfly $\widehat{E_2 \rtimes E_1}$, between $E\otimes E^*$ and the unit object $I_\xi$.

As recalled in the introduction, associated with any 2-group $\cG$, beside $\pi_0(\cG)$ there is also the abelian group $\pi_1(\cG)$ of automorphisms of its unit object. Let us investigate it in our context.

It was proved in [16] that elements in $\pi_1(\cH^3(C,B,\xi))$ are exactly (isomorphism classes) of  butterflies of the form:

\begin{equation}\label{diag:butt_ext}
\begin{aligned}
\xymatrix{
& B\ar@{=}[r]\ar@{==}[dd]
& B\ar[rr]^{0}\ar[dr]_{-\kappa}
& C\ar@{=}[r]
& C\ar@{=}[dd]\ar[r]
& 0
\\
& & & E \ar[dr]^{\gamma} \ar[ur]_{\gamma}& & &\\
& B\ar@{=}[r]
& B\ar[rr]_{0}\ar[ur]^{\kappa}
& C\ar@{=}[r]
& C\ar[r]
& 0
\end{aligned}
\end{equation}

As observed in [16], both $\pi_1(\cH^3(C,B,\xi))$ and $\pi_0(\cH^2(C,B,\xi))$ are isomorphic to the abelian group $H^2(C,B,\xi)$. More is true:  by defining $\Phi(\kappa,\gamma)$ as the butterfly in (<ref>), one gets a monoidal equivalence

\[\Phi\colon \cH^2(C,B,\xi)\to  \mathsf{Eq}(I_\xi)\,,\]

where $\mathsf{Eq}(I_\xi)$ is the $2$-group with objects the butterflies as in (<ref>), and arrows the isomorphisms of such butterflies, with tensor product given by butterfly composition. Indeed, since $\pi_0(\mathsf{Eq}(I_\xi))$ is isomorphic to $\pi_1(\cH^3(C,B,\xi))$, $\pi_0(\Phi)$ makes the latter isomorphic to $\pi_0(\cH^2(C,B,\xi))$, as recalled above.

Given a $C$-module $B$ with action $\xi$, the assignment $\Phi$ gives rise to a symmetric strict monoidal equivalence between the symmetric $2$-groups $\cH^2(C,B,\xi)$ and $\mathsf{Eq}(I_\xi)$.

Let us recall the construction of the Baer sum in $\cH^2(C,B,\xi)$ of two abelian extensions

\[
E\colon \xymatrix{
\qquad
E'\colon \xymatrix{
\]

of cokernel $C$ and abelian kernel $B$. In order to get their Baer sum $E\oplus E'$ (displayed in the left-most vertical sequence of maps, diagram (<ref>) below)

\begin{equation}\label{diag:Baer_of_ext}
\begin{aligned}
\xymatrix@C=12ex{
B\ar[d]_{\bar \kappa}\ar@{}[dr]|{(a)}
&B\times B\ar[l]_{[1,1]}\ar@{=}[r]\ar[d]^{\kappa\times_C \kappa'}
&B\times B\ar[d]^{\kappa\times \kappa'}
\\
E\oplus E'\ar[d]_{\bar\gamma}
& E\times_{C}E'\ar[l]_p \ar[r]^{\langle p_1,p_2\rangle}\ar[d]_r\ar@{}[dr]|{(b)}
& E\times E'\ar[d]^{\gamma\times \gamma'}
\\
& C\times C
\end{aligned}
\end{equation}

first take the pullback $(b)$ of $\gamma \times \gamma'$ along the diagonal map $\langle1,1\rangle$ of $C$ (i.e., take the pullback of $\gamma$ along $\gamma'$), then take the pushforward $(a)$ of the kernel $\kappa\times_C\kappa'$ along the codiagonal $[1,1]$ of the abelian object $B$. The short exact sequence $(\bar\kappa,\bar\gamma)$ is the Baer sum of $(\kappa,\gamma)$ and $(\kappa',\gamma')$.

On the other hand, if we consider $\Phi(E)$ and $\Phi(E')$ (as in  diagram (<ref>)), and we take their composite as butterflies, we get the following construction:

\begin{equation}\label{diag:Butter_of_ext}
\begin{aligned}
\xymatrix{
&& B\ar[ddl]_{\langle-\kappa,0\rangle} \ar@{->}[rr]_{0} \ar[dr]_{-\kappa} & & C  \\
&& & E \ar[dr]_{\gamma} \ar[ur]_{\gamma} \\
E'\cdot E\ar@/^10ex/[rrrruu]^{\overline{\gamma\cdot p_1}}  \ar@/_10ex/[rrrrdd]_{\overline{\gamma'\cdot p_2}}
&\ar[l]_{q}E\times_{C}E' \ar[drr]_{p_2}\ar[urr]^{p_1}  & \ar[l]_(.3){\langle\kappa, -\kappa' \rangle} B \ar@{->}[rr]_{0}\ar[dr]^{-\kappa'} \ar[ur]_{\kappa} & & C\\
&& & E' \ar[dr]_{\gamma'} \ar[ur]_{\gamma'} \\
&& B \ar[uul]^{\langle0,\kappa'\rangle} \ar@{->}[rr]^{0} \ar[ur]_{\kappa'} & & C
\end{aligned}
\end{equation}

Now, thanks to  <Ref> below,  the composite $q\cdot \langle0,\kappa'\rangle$    in  diagram (<ref>) is actually a pushforward of $\kappa\times_C\kappa'$ along $[1,1]$. As a consequence, the short exact sequence $(\bar\kappa,\bar\gamma)$ of diagram (<ref>) coincides with the short exact sequence $(q\cdot\langle0,\kappa'\rangle,\overline{\gamma \cdot p_1})$ of diagram  4(<ref>).
Therefore, the identity $\Phi(E\oplus E')= \Phi(E')\cdot  \Phi(E)$ holds.

As far as the monoidal units are concerned, it is clear that the image of the unit object of $\cH^2(C,B,\xi)$

\[
E_{\rtimes}\colon \xymatrix{B\ar[r]^-{i_B}&B\rtimes C\ar[r]^-{p_C}&C}
\]

under $\Phi$ is given by the identity butterfly of the crossed extension $I_{\xi}=(1_B,0,1_C)$:

\[
\xymatrix{
0 \ar[r] & B\ar@{==}[dd]  \ar@{=}[r] & B \ar[rr]^{0} \ar[dr]_{-i_B} & &  C\ar@{=}[r] & C \ar[r]\ar@{=}[dd] & 0 \\
& & & B\rtimes C \ar[dr]_{p_C} \ar[ur]_{p_C} \\
0 \ar[r] & B \ar@{=}[r] & B \ar[rr]_{0} \ar[ur]_{i_B} & & C\ar@{=}[r] & C \ar[r] & 0
\]

Finally, by means of the universal properties used in the composition of butterflies, one easily determines a canonical natural isomorphism

\[
\sigma_{E,E'}\colon \Phi(E\oplus E')= \Phi(E')\cdot\Phi(E)\cong\Phi(E)\cdot\Phi(E')=\Phi(E'\oplus E)
\]

which makes $\Phi$ symmetric monoidal.

The square of solid arrows in diagram (<ref>) below presents $q\cdot\langle0,\kappa'\rangle$ as the pushforward of $\kappa\times_C \kappa'$ along $[1,1]$.

\begin{equation}\label{diag:lemma}
\begin{aligned}
\xymatrix@C=10ex{
B\ar[d]_{q\cdot\langle0,\kappa'\rangle }
&B\times B\ar[l]_-{[1,1]}\ar[d]^{\kappa\times_C\kappa'}
\\
E\cdot E'\ar@{-->}[d]_{\overline{\gamma \cdot p_1}}
&E\times_C E'\ar[l]_{q}\ar@{-->}[d]^r
\\
\end{aligned}
\end{equation}

First we prove that the candidate pushforward square is commutative. This can be done by precomposing with the jointly epimorphic pair

\[
\xymatrix@C=10ex{
&B\times B
\]

\[
\]

\[
\]

As a second step we prove that the dashed square  commutes. To this end, let us consider the square $(b)$ of diagram (<ref>). One has:

\[\langle1,1\rangle\cdot r=\langle \gamma\cdot p_1,\gamma'\cdot p_2\rangle=\langle \gamma\cdot p_1,\gamma\cdot p_1\rangle=\langle1,1\rangle\cdot\gamma \cdot p_1=\langle1,1\rangle\cdot\overline{\gamma \cdot p_1}\cdot q
\]

Canceling  the monomorphism $\langle1,1\rangle$ on both sides, one obtains $r= \overline{\gamma \cdot p_1}\cdot q$.

Summarizing, diagram (<ref>) shows that the pair $([1,1],q)$ is a morphism of crossed modules between normal monomorphisms with isomorphic cokernels, hence a pushforward, thanks to <cit.>.

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# On Global Hyperbolicity of Spacetimes:
Some Recent Advances and Open Problems

Felix<EMAIL_ADDRESS>Fakultät für Mathematik, Universität Regensburg,
D-93040 Regensburg  Albert<EMAIL_ADDRESS>Institut für
Theoretische Physik
Universität Leipzig
D-04103 Leipzig  Kyriakos<EMAIL_ADDRESS>Department
of Mathematics, Kuwait University, Safat 13060, Kuwait

###### Abstract

This chapter is an up-to-date account of results on globally hyperbolic
spacetimes, and serves as a multitool. We begin with the exposition of results
from a foundational level, where the main tools are order theory and general
topology, we continue with results of a more geometric nature, and we conclude
with results that are related to current research in theoretical physics. In
each case, we list a number of open questions and finally formulate a
conjecture on the connection between global hyperbolicity of a manifold and
the geodesic completeness of its corresponding space-like surfaces.

###### Contents

  1. 1 Introduction
  2. 2 Some Preliminaries
    1. 2.1 Elements of Domain Theory and General Topology
    2. 2.2 Causal Relations in a Spacetime
    3. 2.3 Weighted Riemannian Manifolds and all that
  3. 3 Globally Hyperbolic Spacetimes
  4. 4 Different Candidates for a Topology: which is the most "fruitful" choice?
  5. 5 The Klein-Gordon Equation in globally hyperbolic Manifolds

##  1 Introduction

While in Riemannian geometry it is natural to consider geodesic completeness,
in Lorentzian geometry – and in particular in spacetime geometry – it is more
sensible to consider global hyperbolicity as a condition that two events, that
are chronologically related, can be joined by a maximal timelike geodesic.
This is due to the validity of the Hopf-Rinow theorem, whose naive analogue
fails to exist in Lorentzian geometry (see [9]).

The advantage of global hyperbolicity is its multifaceted perspectives. For
example, it can be examined from an order-theoretic and topological
perspective, as the strongest of the causal conditions on a spacetime (the
causal diamonds are compact - see [35]) or from a purely geometric viewpoint
(existence of a Cauchy surface - see [12]). The equivalence of such seemingly
different statements is what supplies mathematical physics with mathematical
tools and physical insights. The importance of such an interdisciplinary work
cannot be overstated; in mathematical physics this is significant in order to
construct quantum field theories in curved spacetimes in a mathematically
rigorous fashion. In particular, the input that comes from a topological
direction is fruitful towards a more abstract formulation. From a mathematical
point of view, it is important to see the applications of general theorems and
to examine and push the scope of their validity.

It is worth mentioning that there is a theoretical background behind the
notion of globally hyperbolic spacetimes that uses tools from domain theory,
and interconnects mathematical physics with the foundations of mathematics. A
recent example is reference [16], where the authors showed that globally
hyperbolic spacetimes belong to a category444In the frame of the field of
mathematics called Category Theory; see, for example, [25]., which is
equivalent to integral domains. Integral domains are partially ordered sets
which carry, intrinsically, notions of completeness and approximation, and
were used in theoretical computer science (see [37]). The equivalence of
causality between events to an order on regions of a spacetime suggests that
questions about spacetime can be translated to questions on domain theory.
This seems of great interest, because the type of domains called
$\omega$-continuous are the ideal completions of countable abstract bases, so
that a spacetime can be reconstructed (in a purely order-theoretical manner)
from a dense discrete set. In particular, this suggests that a globally
hyperbolic spacetime is linked to a discrete object.

Strongly causal spacetimes, and thus globally hyperbolic ones which are
strongly causal by definition, are conformal to spacetimes in which all null
geodesics are complete, that is, their null geodesics can be extended to
infinite values of their affine parameters, as the following theorem suggests
(see [4]).

######  Theorem 1.1.

If $M$ is a strongly causal spacetime with metric $g$, then there is a
$C^{\infty}$-function $\Omega$, such that null geodesics with respect to
$\Omega^{2}g$ are complete.

As for timelike geodesic completeness, even if there is no obvious relation to
global hyperbolicity, there has been shown that under particular sectional or
Ricci curvature conditions, if a spacelike hypersurface is future-timelike
geodesically complete, then it is globally hyperbolic; see [9]. Next, we turn
to the spacelike geodesic completeness of the Cauchy surface of a globally
hyperbolic manifold.

An interesting result relating spacelike geodesic completeness to global
hyperbolicity was given in [18, Proposition 5.3]. The author proved that an
ultra-static spacetime $(M,g)$ is globally hyperbolic if and only if the
global Cauchy surface is geodesically complete. The physical advantage of
working in globally hyperbolic manifolds is that the wave equation (in what
follows referred to as the Klein-Gordon equation), i.e. the differential
equation describing the dynamics of a spin zero particle, is well-posed. In
particular, given smooth initial data on the Cauchy hypersurface, there exists
a corresponding global smooth solution. For these manifolds, it is in general
possible to rewrite the second order differential equation as a first order
system of equations; this is referred to as the Hamiltonian formulation. The
generator of the time evolution can then be written as a $2\times 2$ matrix
differential operator. In order to define a corresponding quantum field theory
using the Hamiltonian formulation, a necessary requirement is the essential
self-adjointness of weighted Laplace-Beltrami operators stemming from the
Klein-Gordon equation. The essential self-adjointness of these operators
depends on the spacelike geodesic completeness of the Cauchy surfaces. In what
follows, we expand more on the obtained results and a still open conjecture.

##  2 Some Preliminaries

###  2.1 Elements of Domain Theory and General Topology

In this Section we list some definitions, which we need in section 3, where we
discuss a link between a certain type of partially ordered sets (posets) with
globally hyperbolic spacetimes.

Throughout the text, the power set of a set $X$ is denoted by
$\mathcal{P}(X)$, and it is defined as the set of all subsets of $X$.

######  Definition 2.1.

If $(A,\prec)$ is a poset, then:

  1. 1.

a nonempty set $D\in\mathcal{P}(A)$ is called directed, if for every $x,y\in
D$, there exists $z\in D$, such that both $x\prec z$ and $y\prec z$.

  2. 2.

(dually to (1)) a nonempty set $F\in\mathcal{P}(A)$ is called filtered, if for
every $x,y\in F$, there exists $z\in F$, such that $z\prec x$ and $z\prec y$.

  3. 3.

a nonempty set $L\in\mathcal{P}(A)$ is called lower set, if for every $x\in A$
and for every $y\in L$, $x\prec y$ implies that $x\in L$.

  4. 4.

$(A,\prec)$ is a dcpo (directed, complete poset), if every directed set has a
supremum, that is, if $D\in\mathcal{P}(A)$ is directed, then $D$ has a least
upper bound.

  5. 5.

$(A,\prec)$ is continuous, if there exists $B\in\mathcal{P}(A)$, such that
$B\cap\downarrow x$ contains a directed set with supremum, for all $x\in A$,
where $\downarrow x=\\{a\in A:a\ll x\\}$ and where $\ll$ is defined as $x\ll
y$, if for all directed sets $D$ with supremum, $y\prec\sup D$ implies that
there exists $d\in D$, such that $x\prec d$.

  6. 6.

$(A,\prec)$ is bicontinuous, if it is continuous and for every $x,y\in A$,
$x\ll y$ if for every filtered set $F\in\mathcal{A}$ with infimum, $\inf
F\prec x$ implies that there exists $f\in F$, such that $f\prec y$ and also,
for every $x\in A$, $\uparrow x$ is filtered, with infimum $x$ (and where,
dually to $\downarrow x$, $\uparrow x=\\{a\in A:x\ll a\\}$).

  7. 7.

if $(A,\prec)$ is bicontinuous, then its interval topology is the topology
that has a basis that consists of the intervals $(a,b):=\\{x\in A:a\ll x\ll
b\\}$.

  8. 8.

$(A,\prec)$ is globally hyperbolic, if it is bicontinuous and the closure of
the intervals $(a,b)$, that is $\overline{(a,b)}:=[a,b]=\\{x:a\preceq x\preceq
b\\}$, is compact in the interval topology on $A$.

  9. 9.

A domain is a continuous dcpo. An example of an interval domain is the domain
of compact intervals of the real line.

  10. 10.

Let $\Uparrow x=\\{a\in D:x\ll a\\}$, where $D$ is a dcpo. Then the class
$\\{\Uparrow x:x\in D\\}$ forms a base for the Scott topology on a continuous
poset.

###  2.2 Causal Relations in a Spacetime

It is standard (see [35]) in a spacetime to consider two partial orders, the
chronological order $\ll$, which is considered to be irreflexive if one wants
to avoid closed timelike curves, and the causal order $\prec$ which is
reflexive, and are defined, respectively, as follows:

  1. 1.

$x\ll y$ if and only if $y\in C_{+}^{T}(x)$ and

  2. 2.

$x\prec y$ if and only if $y\in C_{+}^{T}(x)\cup C_{+}^{L}(x)$

where $C_{+}^{T}(x)$ denotes the future time-cone of an event $x$,
$C_{+}^{L}(x)$ its future light-cone and $C_{+}^{T}(L)\cup C_{+}^{L}(x)$ its
future causal-cone (for an analytical exposition of time-coordinates, space-
coordinates and the spacetime “metric”, that leads to the definition of these
cones see [12]; for a detailed exposition of these definitions on Minkowski
space see [2]).

In addition, the reflexive relation horismos $\rightarrow$ is defined as
$x\rightarrow y\textrm{ iff }x\prec y\textrm{ but not }x\ll y$. Naturally, in
a spacetime $M$ one can define the sets $I^{+}(x)=\\{y\in M:x\ll y\\}$ and
$J^{+}(x)=\\{y\in M:x\prec y\\}$; $I^{-}(x)$ and $J^{-}(x)$ are defined
dually.

In Section 3, we will examine the relation between $\ll$ in Definition 2.1 (5)
and the chronological order $\ll$ that will be defined in 2.2 (1).
Furthermore, the interval topology of Definition 2.1 (7), defined for
bicontinuous posets, will be called the Alexandrov topology $T_{A}$ for a
spacetime $M$, whenever $\ll$ is the chronological order on $M$.

###  2.3 Weighted Riemannian Manifolds and all that

In the following we define a weighted manifold, [48, Chapter 3.6, Definition
3.17].

######  Definition 2.2.

A triple $(\Sigma,g_{\Sigma},\mu)$ is called a _weighted manifold_ if
$(\Sigma,g_{\Sigma})$ is a Riemannian manifold and $\mu$ is a measure on
$\Sigma$ with a smooth and everywhere positive density function $\rho$, i.e.,
$d\mu=\rho\,d\Sigma$. A _weighted Hilbert space_ , denoted by
$L^{2}(\Sigma,\mu)$, is given as the space of all square-integrable functions
on the manifold $\Sigma$ with respect to the measure $\mu$. The corresponding
_weighted Laplace-Beltrami operator_ (also called the Dirichlet-Laplace
operator), denoted by $\Delta_{g_{\Sigma},\mu}$, is given by

$\Delta_{g_{\Sigma},\mu}=\frac{1}{\rho\sqrt{|g_{\Sigma}|}}\partial_{i}(\rho\sqrt{|g_{\Sigma}|}\,g_{\Sigma}^{ij}\partial_{j}).$
(2.1)

In regards to the upcoming discussion on essential self-adjointness, we need
an essential result that was given in [39, Theorem 1].

######  Theorem 2.3.

Let the Riemannian manifold $(\Sigma,g_{\Sigma})$ be complete, $\mu$ be a
measure on $\Sigma$ with a smooth and everywhere positive density function
$\rho$ and suppose that the potential $Y\in L^{2}_{loc}(\Sigma)$ point-wise.
Furthermore, let $A\in\Lambda^{1}_{(1)}(\Sigma)$ and let the operator
$L=-{\Delta}^{A}_{{g_{\Sigma},\mu}}+Y$, where the minimally coupled Laplace
Beltrami operator is defined by

$\displaystyle-{\Delta}^{A}_{{g_{\Sigma},\mu}}:=-(\rho\sqrt{|g_{\Sigma}|})^{-1}(\partial-A)^{*}_{i}(\sqrt{|g_{\Sigma}|}\,\rho\,g^{ij}_{\Sigma}\,(\partial-A)_{j})$

be semi-bounded from below. Then, the operator $L$ is an essentially self-
adjoint operator on $C_{0}^{\infty}(\Sigma)$.

Here we write $f\in L^{2}_{loc}(\Sigma)$ for a local $L^{2}(\Sigma)$ function
$f$ that is an element of the Hilbert space $L^{2}(\Sigma)$ on every compact
subset of the manifold $\Sigma$, and we denote $\Lambda^{p}_{(k)}(\Sigma)$ as
the set of all $k$-smooth (i.e. of the class $C^{k}$) complex-valued $p$-forms
on $\Sigma$. Note that the above result is (apart from the potential term and
the gauge field) a reformulation of the result of [1] for weighted Riemannian
manifolds.

##  3 Globally Hyperbolic Spacetimes

Classically, when we talk about global hyperbolicity, in terms of the causal
structure of a spacetime, we consider the following definition (see [35]).

######  Definition 3.1.

A spacetime $M$ is globally hyperbolic, if and only if $M$ is strongly causal
and every set $J^{+}(x)\cap J^{-}(y)$ is compact, for some $x,y\in M$.

In [46] it was proven that any globally hyperbolic spacetime admits a _smooth_
foliation into Cauchy surfaces [46, Theorem 1.1]. Moreover, the induced metric
of such a globally hyperbolic spacetime admits a specific form [47, Theorem
1.1].

######  Theorem 3.2.

Let $(M,{g})$ be an $(n+1)$-dimensional globally hyperbolic spacetime. Then it
is isometric to the smooth product manifold $\mathbb{R}\times\Sigma$ with a
metric $g$, i.e.,

$g=-N^{2}dt^{2}+h_{ij}dx^{i}dx^{j},$ (3.1)

where $\Sigma$ is a smooth $n$-manifold,
$t:\mathbb{R}\times\Sigma\mapsto\mathbb{R}$ is the natural projection,
$N:\mathbb{R}\times\Sigma\mapsto(0,\infty)$ a smooth function, and
$\mathbf{h}$ a $2$-covariant symmetric tensor field on
$\mathbb{R}\times\Sigma$, satisfying the following condition: Each
hypersurface $\Sigma_{t}\subset M$ at constant $t$ is a Cauchy surface, and
the restriction $\mathbf{h}(t)$ of $\mathbf{h}$ to such a $\Sigma_{t}$ is a
Riemannian metric (i.e. $\Sigma_{t}$ is spacelike).

Hounnonkpe and Minguzzi proved that the (strong-)causality condition is
unnecessary in the definition of global hyperbolicity, for a reasonable
(meaning non-compact) spacetime of dimension strictly greater than $2$ (see
[16]).

######  Theorem 3.3 (Hounnonkpe-Minguzzi).

A non-compact spacetime of dimension strictly greater than $2$ is globally
hyperbolic, if and only if the causal diamonds are compact in the topology,
$T_{A}$.

Recently, there has been a generous step towards an understanding of the
nature of globally hyperbolic spacetimes down to the most fundamental level.
In [26], it has been shown that the structure of such spacetimes is equivalent
to the structure of interval domains, which are purely order-theoretic
objects. Let us have a closer look at the main results.

######  Definition 3.4.

A set $B$ equipped with a transitive relation $\ll$ is an abstract basis, if
$\ll$ is $-$-interpolative, that is for all $S\in\mathcal{P}(B)$, such that
$S$ is finite, if $S\ll x$, then there exists $y\in B$, such that $S\ll y\ll
x$, where by $S\ll x$ one means that $y\ll x$, for all $y\in S$.

By $int(C)$ one denotes the set $\\{(a,b):a\ll b\\}$, where $(a,b)\ll(c,d)$ if
and only if $a\ll c$ and $d\ll b$. It is easy to see that, if $(B,\ll)$ is an
abstract basis which is $-$\- and $+$-interpolative (where $+$-interpolation
is the dual to $-$-interpolation), then the set $(int(C),\ll)$ is an abstract
basis.

######  Definition 3.5.

An ideal of an abstract basis $(B,\ll)$ is a nonempty set
$I\in\mathcal{P}(B)$, which is lower and directed. The set
$(\overline{B},\subset)$ of all ideals of the abstract basis $(B,\ll)$ is a
poset and is called the ideal completion of $B$.

Before introducing the main theorems of [26], we remind that since spacetimes
are considered naturally as being second countable and second countability
implies separability, a spacetime admits a countable dense subset. In the
theorem that follows a countable set, which is equipped with a causal
relation, determines the entire space.

######  Theorem 3.6 (Martin-Panangaden).

If $C$ is a countable dense subset of a globally hyperbolic spacetime $M$,
where $\ll$ denotes chronology, then $\max IC\simeq M$, where the maximal
elements are equipped with the Scott topology.

Theorem 3.6 can be written more generally as follows:

######  Theorem 3.7.

If $C$ is a countable dense subset of a globally hyperbolic set $M$, then
$\max IC\simeq M$.

The proof of Theorem 3.7 is the same as that of Theorem 3.6. Since $M$ is
bicontinuous by definition, $(C,\ll)$ will be an abstract basis and so
$(int(C),\ll)$ will be an abstract basis for $IM$. Since ideal completions for
bases of $IM$ are isomorphic to $IM$, the result follows immediately. The
result of Theorem 3.6 follows from the fact that the relation $\ll$ of
Definition 2.1 (5), which reads $x\ll y$ as “$x$ approximates $y$”, coincides
with the chronological partial order $\ll$ in a globally hyperbolic spacetime
(see [27], Proposition 4.4).

######  Theorem 3.8 (Martin-Panangaden).

The category of globally hyperbolic posets is equivalent to the category of
interval domains.

The proof of Theorem 3.8 depends on several technical results that precede it
in [26], but the result on its own is powerful; within a globally hyperbolic
spacetime, one can convert questions of a physical significance, to questions
in domain theory (and vice versa).

Question 1: Consider a globally hyperbolic poset $(A,\prec)$. Does the
compactness of the intervals $[a,b]$, under the interval topology, imply the
bicontinuity of $(A,\prec)$, in a similar fashion that the compactness in the
closed diamonds in a spacetime $M$, under the Alexandrov topology $T_{A}$,
implies the (strong) causality of $M$ (for dimensions strictly greater than
$2$)? This is a reasonable question, given the results in [16] as well as the
main result of [26], which guarantee the equivalence (up to category theory)
of interval domains and globally hyperbolic spacetimes. The only problem,
towards [16], of considering a globally hyperbolic poset, is that we are left
only with an order-theoretic structure; what would be, if there is such, the
equivalent condition in [16] of “dimension strictly greater than $2$”? Is
there a possibility that the condition on the dimension applies only to
globally hyperbolic spacetimes, while the proposed conjecture holds in
globally hyperbolic posets without any restriction?

In the section that follows, we consider a spacetime, and we discuss how
topology can affect the way that we look at it.

##  4 Different Candidates for a Topology: which is the most "fruitful"
choice?

When one considers a spacetime manifold $(M,g)$, the spacetime topology is
traditionally taken to be the manifold topology $T_{M}$, a topology which does
not incorporate the causal-structure of $M$, a structure which is linked to
the Lorentzian metric $g$. This problem was first addressed by Zeeman in his
papers [43] and [44], where he restricted the discussion to Minkowski space.
It was then extended to curved spacetimes by Göbel in [10] and by Hawking-
King-McCarthy in [13], and the discussion was taken into a further level by
several authors, including an important feedback by Low in [23] (for a recent
survey on the topologization of spacetime, see [34]). In particular, Low
proved that the Limit Curve Theorem (LCT) does not hold under the Path
topology of Hawking-King-McCarthy, and claimed that this fact gives the right
for the manifold topology to be called a fruitful one, against any other known
candidates. Given that the LCT plays an important role in building
contradictions in proofs in singularity theorems in general relativity (see,
for example, [8] and [4]), the authors of [33] posed the question whether the
singularity problem is a purely topological one, depending largely on the
topology that one chooses to equip the spacetime with. For example, if one
considers the space of timelike paths, the notion of convergence stays
unaffected, if choosing either the Path topology $\mathcal{P}$ or the manifold
topology $T_{M}$, while convergence in the space of causal paths under $T_{M}$
does not imply convergence in the space of causal paths under $\mathcal{P}$
(see [23]). Consider the causal cone of an event $x$ in a spacetime $M$, that
is, consider the union of the time-cone $C^{T}(x)$ with the light-cone
$C^{L}(x)$. Consider now an open ball $B_{\epsilon}^{d}(x)$ of radius
$\epsilon$ w.r.t. the manifold topology $T_{M}$ (where the distance is defined
via an induced Riemann metric $d$). Consider the intersection $A=(C^{T}(x)\cup
C^{L}(x))\cap B_{\epsilon}^{d}(x)$; the sets $A$ form a local basis for a
topology $Z^{LT}$ on $M$, where convergence under $Z^{LT}$ and under $T_{M}$
stays unaffected in the space of causal paths (see [33]). Obviously, this
statement cannot hold if $Z^{LT}$ is considered in the space of timelike
paths.

It is clear that the path topology $\mathcal{P}$ in [13] has several
advantages over the manifold topology on a spacetime, $\mathcal{P}$
incorporates the causal, differential and smooth conformal structure of
spacetime and, most importantly, the group of homeomorphisms of $\mathcal{P}$
is the conformal group. According to Göbel (see [10]), the reason for
considering the Euclidean topology as a “natural” topology for the Minkowski
spacetime (and, more generally, the manifold topology for a spacetime
manifold) was that people were mostly concerned with Riemannian spaces and not
with spaces equipped with a Lorentzian metric; it really seems that the blind
use of the manifold topology, while proving theorems in spacetime geometry,
was due to ignorance (in the words of Göbel)!

In the setting of globally hyperbolic spacetimes, Low introduced a list of
interesting topological results, including the following two (see [24]).

######  Proposition 4.1.

A strongly causal spacetime $M$ is globally hyperbolic if and ony if the space
of smooth inextendible causal curves $\mathcal{C}$ is Hausdorff.

######  Proposition 4.2.

$M$ is globally hyperbolic if and only if $\mathcal{C}$ is metrizable.

The topologization of the space of smooth inextendible causal curves
$\mathcal{C}$ is important, because it affects the topology induced on $C$,
the space of causal geodesics. $C$ need not be a manifold but, in particular,
if the spacetime $M$ is strongly causal, then $C$ is a smooth manifold with
boundary, with a smooth structure inherited from the homogeneous tangent
bundle $UM$555The quotient of the tangent bundle minus the zero section, is
called the homogenous tangent bundle over $M$. (see [24]). The canonical lift
of a smooth causal curve from $M$ to $CM$ (where $CM$ is the bundle of causal
directions) and the lift from $C$ to $CM$ results in a foliation of $CM$. The
space of leaves of this foliation is a topological space, with the quotient
topology coming from $UM$. The topology of this space need not be Hausdorff
even if $M$ is strongly causal. In the case that $C$, under this topology, is
non-Hausdorff, then $M$ will be nakedly singular (a TIP lies inside a PIP; see
[24] and for a more general exposition see [35]).

Low suggests a topology $T^{0}$ on $\mathcal{C}$ which can be described as
follows. Given a smooth inextendible causal curve $\gamma$ in $\mathcal{C}$,
let $\Gamma$ be the corresponding curve in $M$. Let $x\in\Gamma$ and $U$ be an
open neighbourhood of $x$ in $M$, (w.r.t. the manifold topology $T_{M}$). Let
$\mathcal{U}\subset\mathcal{C}$ be the set consisting of all (smooth
inextendible causal-) curves which pass through $U$. Let $T^{0}$ be the
topology generated by $\mathcal{U}$. Then $T^{0}$ is a pointwise convergent
topology. It follows, that if a spacetime $M$ has a closed timelike curve
(CTC), then $(M,T^{0})$ cannot satisfy the $T_{1}$-separation axiom and, if
$M$ is totally vicious666For the definition and properties of (non-) totally
vicious spacetimes, see [28]., then there exists $x\in\mathcal{C}$ such that
$x$ is dense in $\mathcal{C}$ w.r.t. $T^{0}$. For the proof of Proposition
4.2, Low constructs a metric which induces $T^{0}$ and remarked that one can
find metrics which induce the topologies of convergence to any degree of
smoothness (by means of the slicing of the jet bundles over the spacetime
manifold $M$).

Next, consider a spacetime $(M,g)$ of the form $M=\mathcal{I}\times\Sigma$,
with $\mathcal{I}$ being an interval in $\mathbb{R}$ and $\Sigma$ a smooth
$n$-dimensional manifold. The spatial slices $\Sigma_{t}=\Sigma\times\\{t\\}$
are spacelike submanifolds. We call such a spacetime a _sliced space_. In
regards to the foliation, in [5], sliced spaces were considered to have
uniformly bounded lapse, shift and spatial metric, in order to achieve the
equivalence of global hyperbolicity of $(M,g)$ with the completeness of the
slice $(\Sigma,g_{\Sigma})$ (Theorem 2.1). Being motivated by this result, the
authors of [22] consider global topological conditions, for showing the
equivalence of the global hyperbolicity of $(M,g)$ with the existence of a
foliation $(\Sigma_{t},g_{\Sigma_{t}})$ being $T_{A}$-complete. Theorem 4.3,
below, differs from Theorem 2.1 in [5] in that the slices are complete
Riemannian manifolds (with uniformly bounded spatial metric, lapse and shift
functions) while the slices are $T_{A}$-complete in [22, Theorem 2.1.].

######  Theorem 4.3.

Let $(M,g)$ be a sliced space, equipped with its natural product topology
$T_{P}$, where $M=\mathbb{R}\times\Sigma$, $\Sigma$ is an $n$-dimensional
manifold ($n\geq 2$)and $g$ the $n+1$-Lorentz “metric” on $M$. Let also
$T_{A}$ be the Alexandrov spacetime topology on $M$. Then, the following
statements are equivalent:

  1. 1.

$(M,g)$ is globally hyperbolic.

  2. 2.

For every basic-open set $D\in T_{A}$, there exists a basic-open set $B\in
T_{P}$, such that $D\subset B$.

  3. 3.

$(M_{t},g_{t})$ is complete with respect to $T_{A}$.

Question 2: As we have seen in this section, there are many different
candidates for spacetime topologies is, other than the manifold one. Is there
a fruitful777Fruitful in the sense of [44], [10] and [13].  and physically
meaningful spacetime topology that was not considered so far? What are the
criteria for choosing it? For example, how should such a topology be related
to the foliation of a spacetime? These questions certainly need a more
systematic work, because even if they can touch important problems (for
example the singularity problem in relativity theory as well as Penrose’s
cosmic censorship conjecture), they seem underestimated until now.

Question 3: In the light of Theorem 3.8, we ask whether the topology of a
spacetime can be reconstructed, in some technical and rigorous way, from the
topology of an interval domain. The spacetime topologies that we have
mentioned in this section, like for example the Zeeman fine topology or its
generalization by Göbel, or the one by Hawking-King-McCarthy are not
metrizable; does this give us any insight on the appropriate candidate for a
spacetime topology (other than the manifold topology), given that interval
domains are order-theoretic objects?

##  5 The Klein-Gordon Equation in globally hyperbolic Manifolds

In what follows, we consider the Klein-Gordon operator on a Lorentzian
manifold $(M,g)$ minimally coupled to an electromagnetic potential $A$ and
with a scalar potential $Y$,

$\displaystyle
K\phi:=\left(({\sqrt{|g|}})^{-1}D^{A}_{\mu}(\sqrt{|g|}g^{{\mu}{\nu}}D^{A}_{\nu})+Y\right)\phi=0$
(5.1)

where ${|g|}=\text{det}(g_{\mu\nu})$ and $D^{A}_{\mu}=(i\partial-A)_{\mu}$.
Next, we insert the form of the metric given in Theorem 3.2 and, instead of
working with the operator $K$ (from (5.1)), it is more convenient to work with
the operator

$\tilde{K}:=N\,K\,N$

with $N$ being as before the Lapse function and where $\tilde{K}$ can be
expressed as

$\tilde{K}=-(D_{t}+W^{*})(D_{t}+W)+L$ (5.2)

with

$\displaystyle
W:=-A_{0}-\frac{1}{2}(N\,|g_{\Sigma}|^{1/2})^{-1}D_{t}\,(N\,|g_{\Sigma}|^{1/2}),$

and $L(t)$ is the spatial Klein-Gordon operator given by

$L(t)=-N\,(\sqrt{|g_{\Sigma}|})^{-1}(D-A)^{*}_{i}(\sqrt{|g_{\Sigma}|}\,Ng^{ij}_{\Sigma}\,(D-A)_{j})+N^{2}\,Y.$
(5.3)

Next, we define the operator $B$ by

$\displaystyle
B(t)=\begin{matrix}\left(\begin{array}[]{rrrr}W(t)&\mathbb{I}\\\
L(t)&W(t)\end{array}\right).\end{matrix}$

Let $u_{1}(t)=u(t)$ and $u_{2}(t)=-(D_{t}+W(t))\,u(t)$ be the Cauchy data for
$u$ at time $t$, then

$\displaystyle\left(\partial_{t}+iB(t)\right)\begin{matrix}\left(\begin{array}[]{r}u_{1}(t)\\\
u_{2}(t)\end{array}\right)=0\end{matrix}$

only if $u$ is a (weak) solution of the Klein-Gordon equation $\tilde{K}u=0$.
Then the Hamiltonian is given by the multiplication of $B$ with the matrix $Q$

$\displaystyle Q=\begin{matrix}\left(\begin{array}[]{rrrr}0&\mathbb{I}\\\
\mathbb{I}&0\end{array}\right)\end{matrix}$

i.e. $H(t)=Q\,B(t)$. Since $W(t)$ is simply a function the problem of proving
self-adjointness of the Hamiltonian reduces to proving self-adjointness of
$L(t)$ for all $t$.

In order to simplify the nature of the problem, let us assume that the
electromagnetic potential $A$ is equal to zero, the lapse function $N$ is
equal to one and the metric $g_{\Sigma}$ is time independent. Let us further
assume time independence and positivity for the scalar potential $Y$. By
taking the assumptions into account, the operator $L$ simply is the standard
Laplace-Beltrami operator w.r.t. the Riemannian manifold
$(\Sigma,g_{\Sigma})$. The results by [1] then state that the geodesic
completeness of the Riemannian manifold imply the essential self-adjointness
of the Laplace-Beltrami operator $L$. Since the Riemannian manifold stems from
a globally hyperbolic spacetime it is a priori not clear why it should be
geodesically complete. At this point let us state [18, Proposition 5.3].

######  Proposition 5.1.

Let $(M,g)$ be a Lorentzian manifold with metric tensor

$g=-dt^{2}+g_{\Sigma,ij}(\vec{x})dx^{i}dx^{j}.$ (5.4)

Then the manifold $(M,g)$ is globally hyperbolic if and only if the Riemannian
manifold $(\Sigma,g_{\Sigma})$ is complete.

Next, it follows from Theorem 2.3 that the simplified Laplace-Beltrami
operator we considered is an essentially self-adjoint operator on
$C_{0}^{\infty}(\Sigma)$.

In order to prove essential self-adjointness for the case when $N$ is not
equal to one, in the absence of an electromagnetic potential, we used
techniques from weighted manifolds and reduced the problem to the following
theorem [31, Theorem 4.1].

######  Theorem 5.2.

Let the Riemannian manifold $(\Sigma,N^{-2}g_{\Sigma})$ be geodesically
complete and let the scaled potential $N^{2}V\in L^{2}_{loc}(\Sigma)$ point-
wise. Furthermore, let the operator $L$ (from (5.3)) be semi-bounded from
below. Then the operator $L$ is essentially self-adjoint on
$C_{0}^{\infty}(\Sigma)$.

###### Proof.

For the proof see [31]. ∎

Next, we generalize the former result to the case of nonvanishing
electromagnetic potential and general lapse function.

######  Theorem 5.3.

Let the Riemannian manifold $(\Sigma,\tilde{g}_{\Sigma}:=N^{-2}g_{\Sigma})$ be
geodesically complete and let the scaled potential $N^{2}V\in
L^{2}_{loc}(\Sigma)$ point-wise. Furthermore, let
$A\in\Lambda^{1}_{(1)}(\Sigma)$ and let the operator

$L(t)=-N\,(\sqrt{|g_{\Sigma}|})^{-1}(D-A)^{*}_{i}(\sqrt{|g_{\Sigma}|}\,Ng^{ij}_{\Sigma}\,(D-A)_{j})+N^{2}\,Y.$

be semi-bounded from below. Then the operator $L$ (from (5.3)) is essentially
self-adjoint on $C_{0}^{\infty}(\Sigma)\subset L^{2}(\Sigma,\,\tilde{\mu})$.

###### Proof.

We rewrite the operator $L$ as a weighted Laplace-Beltrami operator multiplied
by the Lapse function, i.e.

$\displaystyle L=-N^{2}{\Delta}^{A}_{{g}_{\Sigma},\mu}+N^{2}Y.$

By redefining the metric and measure $\mu$ as

$\displaystyle\tilde{g}_{\Sigma}:=N^{-2}g_{\Sigma},\qquad
d\tilde{\mu}=N^{-2}d\mu,$

the operator $L$ reads (for the proof see [31, Proposition 3.1])

$\displaystyle L=-{\Delta}^{A}_{\tilde{g}_{\Sigma},\tilde{\mu}}+N^{2}\,Y.$

By rewriting the operator as a minimally coupled, weighted Laplace-Beltrami
operator, the condition of positivity of the potential and Theorem 2.3 imply
essential self-adjointness. ∎

Hence, _the problem of proving essential self-adjointess of $L$ is reduced to
proving geodesic completeness of the Riemannian manifold
$(\Sigma,N^{-2}g_{\Sigma})$_. For all globally hyperbolic static cases this
can be done by the use of Proposition 5.1. Theorem 5.2 was generalized to the
case of stationary spacetimes as well, see [45].

In the case of nonstatic spacetimes however this is still an open problem. In
the following, we simplify the problem. Since any globally hyperbolic
spacetime $(M,g)$ can be brought into the following form,

$g=-N^{2}(t,\vec{x})dt^{2}+g_{\Sigma,ij}(t,\vec{x})dx^{i}dx^{j},$

and keeping in mind that conformal transformations do not change the causal
structure of the manifold, we can consider the conformally transformed
globally hyperbolic spacetime $(M,\tilde{g})$ with

$\displaystyle\tilde{g}$
$\displaystyle=-dt^{2}+N^{-2}g_{\Sigma,ij}(t,\vec{x})dx^{i}dx^{j}$
$\displaystyle=-dt^{2}+\tilde{g}_{\Sigma,ij}(t,\vec{x})\,dx^{i}dx^{j}.$

Our strategical reason for the conformal transformation is our interest in the
connection of global hyperbolicity and the geodesic completeness of the
Riemannian manifold $(\Sigma_{t},\tilde{g}_{\Sigma_{t}})$ for all
$t\in\mathbb{R}$. For three large classes of Lorentzian manifolds, the
condition of global hyperbolicity is equivalent to the geodesic completenes:

  * •

Static case. Lapse function and the spatial metric are time-independent and
thus Proposition 5.1 leads to the equivalence.

  * •

The case of warped manifolds $M=\mathbb{R}\times_{f}\Sigma$, see [50, Theorem
3.66.] or [3, Lemma A.5.14.] where the following result holds.

######  Theorem 5.4.

Let $(\Sigma,g_{\Sigma})$ be a Riemannian manifold and let $I=(a,b)$ with
$-\infty\leq a<b\leq+\infty$. Furthermore, let $f:I\mapsto(0,\infty)$ be a
smooth function and the metric $g$ be given by the warped product

$g=-dt^{2}+f(t)\,g_{\Sigma}.$

Then the Lorentzian warped product $(I\times_{f}\Sigma,g)$ is globally
hyperbolic iff (H, h) is complete.

  * •

Sliced spaces. Assume that the metric $g_{\Sigma,ij}(\vec{x},t)$ is uniformly
bounded by the metric $g_{\Sigma,ij}(\vec{x},0)$ for all $t\in\mathbb{R}$ and
tangent vectors $u\in T\Sigma$, i.e. that there exist constants
$A,D\in\mathbb{R}>0$ such that

$\displaystyle A\,g_{\Sigma,ij}(\vec{x},0)u^{i}\,u^{j}\leq
g_{\Sigma,ij}(\vec{x},t)u^{i}\,u^{j}\leq
D\,g_{\Sigma,ij}(\vec{x},0)u^{i}\,u^{j}.$

Then by [5, Theorem 2.1] or Theorem 4.3, above, the equivalence of globally
hyperbolicity and geodesic completeness follows.

Due to these various cases we are led to the following conjecture.

######  Conjecture 5.5.

A Lorentzian manifold $(M=\mathbb{R}\times\Sigma,g)$ with metric tensor

$g=-dt^{2}+g_{\Sigma,ij}(\vec{x},t)dx^{i}dx^{j}$ (5.5)

is globally hyperbolic if and only if the Riemannian manifold
$(\Sigma_{t},g_{\Sigma_{t}})$ is complete for all $t$.

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