11institutetext: Universidade de São Paulo, Instituto de Astronomia, Geofí
sica e Ciências Atmosféricas
R. do Matão, 1226, Cidade Universitária, 05508-090 São Paulo SP, Brazil
11email<EMAIL_ADDRESS>22institutetext: Universidade Federal do ABC
(UFABC), Centro de Ciências Naturais e Humanas (CCNH)
Avenida dos Estados, 5001, 09210-580, Santo André - SP, Brazil

# A light strange star in the remnant HESS J1731-347: minimal consistency
checks

J.E. Horvath 11 L.S. Rocha . L.M. de Sá 1111 P.H.R.S. Moraes 22 L. G. Barão 11
M.G.B. de Avellar 11 A. Bernardo 11 R.R.A. Bachega 11 ,

###### Abstract

Context. Recently, Doroshenko and collaborators reported a very low-mass
compact star, a Central Compact Object named XMMU J173203.3-344518 inside the
supernova remnant HESS J1731-347. Its tiny mass is at odds with all
calculations of minimum masses of neutron stars generated by iron cores,
therefore (and even if not compellingly) it has been suggested to be a strange
star. Besides the mass, a radius and surface temperature have been extracted
from data, and the whole body of information should ultimately reveal if this
object is truly consistent with an exotic composition.

Aims. To understand the status of the compact object XMMU J173203.3-344518 in
HESS J1731-347 within the existing models of strange stars, including its
prompt formation.

Methods. The information obtained on the mass, radius and surface temperature
are compared to theoretical calculations performed within usual theoretical
models using General Relativity as the assumed theory of gravitation and a
handful of cooling scenarios. A qualitative discussion showing the consistency
of the strange-matter driven supernova scenario with a low-mass compact star
is provided.

Results. We found that the object HESS J1731-347 fits within the same quark
star models recently employed to explain the masses and radii of the NICER
objects PSR J040+6620 and PSR J0030+0451, in which both quantities were
simultaneously determined. It is also remarkable to find that a simple cooling
scenario devised $30\text{ yr}$ ago with superconducting quarks provides an
overall good explanation of the surface temperature.

Conclusions. We conclude that XMMU J173203.3-344518 in the remnant HESS
J1731-347 fits into a “strange star” scenario that is also consistent with
heavier compact stars, which can also belong to the same class and constitute
an homogeneous type of self-bound objects produced in Nature.

##  1 Introduction

The quest for the state of matter in its ultimate dense version (inside
compact stars) has been going on for decades. Since it was clear from the
early days that densities above the saturation value $\rho_{0}\sim 2.4\times
10^{14}\text{ g cm}^{-3}$ were involved, all sorts of approaches have been
devised to reliably describe the behavior of degrees of freedom in that
regime. However, and in spite of abundant data from heavy-ion colliders
(Bratkovskaya et al., 2013), and extensive work on compact star physics
(Weber, 1999), we still have only a basic knowledge about matter under these
extreme conditions.

One of the most fundamental questions is the very type of degrees of freedom
present at the highest densities. Work on known nuclear species (hyperons,
resonances, etc.) occupies most of the literature, but also the exotic
composition has been amply explored. In addition to the liberation of quarks
and gluons degrees of freedom in heavy ion collisions (McLerran, 2009), quark
matter has been explored indirectly as an important potential component of
compact star composition (Ivanenko & Kurdgelaidze, 1969). In the most extreme
proposal, a cold variety of this matter was suggested to be absolutely stable,
and therefore a candidate to compose all but the outermost crust of a compact
star (Bodmer, 1971; Terazawa, Akama & Chikashige, 1978; Itoh, 1970; Witten,
1984), and because of the large abundance and fundamental role of the $s$
quark in the mixture, called strange quark matter (SQM).

The search for evidence of a SQM composition of compact stars has been
extensive, with some reports and arguments being withdrawn, and others still
being considered (for example, Di Salvo et al. (2019)). The recent report by
Doroshenko et al. (2022) of a low mass for the Compact Central Object XMMU
J173203.3-344518 in the remnant HESS J1731-347 adds interest to this quest.
The observations revealed a mass below the $1\,M_{\odot}$ scale, the formation
of which is thought to be not viable in standard Stellar Evolution (Suwa et
al., 2020). The measurement of the radius is also interesting, because it came
out slightly smaller than the most reliable currently known ones. This was
possible by means of a spectral modeling and the use of Gaia data to refine
the distance estimate. Therefore, this compact object became a candidate to a
strange star, as stated by the authors, although a “normal” composition is by
no means excluded. We shall show below that the XMMU J173203.3-344518
candidate passes the minimal requirements to be considered a strange star, and
discuss possible reasons for its very low mass.

##  2 Mass and radius of XMMU J173203.3-344518 in HESS J1731-347 within
existing models

A whole class of quark matter models were investigated in the last $20\text{
yr}$ or so, after the recognition of the importance of quark-quark paring in
dense matter. The pairing was perturbatively discussed earlier (Bailin & Love,
1984), but the new calculations were performed within a non-perturbative
approach and suggested a much larger value of the gap $\Delta$ (Alford et al.,
2001). For large densities the so-called Color-Flavor Locked phase (CFL) in
which all quarks pair, discussed in Alford et al. (2001) is expected to be the
ground state. Even though the densities inside compact stars should not be so
extreme, the asymptotic CFL description was adopted as a plausible model. In
fact, the pairing has been shown to considerably enlarge the “stability
window” (Lugones & Horvath, 2002) for CFL strange matter. Other pairing
patterns, for example, the 2SC that allows only the lighter quarks to pair,
have been explored Alford et al. (2001); Rjagopal (1999) and may be relevant
if the actual density inside compact objects is not too high. Ultimately,
these theoretical possibilities must be related to actual observations.

The equation of state (EoS) including pairing effects has been discussed in
the literature. In the case of the CFL state, the simplest version of it
(Alford et al., 2001; Lugones & Horvath, 2002) reads

$P_{r}=\frac{1}{3}\rho+\frac{2\Phi}{\pi}\rho^{1/2}-\left(\frac{3\Phi^{2}}{\pi^{2}}+\frac{4}{3}B\right),$
(1)

with $\Phi$ defined by

$\Phi=-\frac{m_{s}^{2}}{6}+\frac{2\Delta^{2}}{3},$ (2)

where $m_{s}$ is the mass of the strange quark, $\Delta$ the pairing gap
(usually treated as a parameter) and $B$ is the bag constant. In the limit
$m_{s}\rightarrow 0$ and non-interacting (unpaired) quarks, a MIT bag model-
like EoS is restored (Alcock, Fahri & Olinto, 1986).

Variants of this self-bound EoS have been considered over the years. In a
recent work (Horvath & Moraes, 2021) we have collected a few examples that
allow an explanation of the masses and radius of the most reliable
simultaneous measurements available, those presented in Coleman Miller et al.
(2019) for PSR J0030+0451 and Riley et al. (2021) for PSR J0740+6620. The
compact object in the GW merger GW190814 (Abbott et al., 2020) (although no
estimate for the radius is available) is a candidate to the heaviest neutron
star known (Horvath et al., 2022), and the stellar sequences may be arguably
required to go up to this high value. It is important to remark that the
analysis of the existing sample of galactic neutron stars (Horvath et al.,
2022) favors a large maximum mass, and also that a similar value for it was
found in an analysis of the GW sample (Ye & Fishbach, 2022). The maximum mass
should be above $2.2\,M_{\odot}$ empirically (Romani et al., 2022; Ye &
Fishbach, 2022; Horvath et al., 2022).

The four EoS employed in Horvath & Moraes (2021) correspond to self-bound
versions of cold quark matter. The first is the NJL-CFL EoS with vector
interactions (Ferrer,, 2017) in which the self-bound condition is achieved by
introducing a vacuum term and repulsive vector interactions boost the maximum
mass of the stellar sequence upwards. The Mean Field Theory of QCD (MFTQCD)
(Franzon et al., 2012) with the parameter $\xi$ in the range of self-
boundness, and two versions of the CFL model with a self-bound EoS given in
Eqs. (1) and (2), solved within an exact anisotropic metric ansatz due to
Thirukkanesh-Ragel-Malaver (Thirukkanesh and Ragel, 2012; Malaver, 2014). This
last model was analytically solved in Rocha et al. (2020).

Since we have derived in Rocha et al. (2020) an exact solution, some features
of the models can be “read off” quite simply. For example, the two anisotropic
models presented in Horvath & Moraes (2021) were calculated using a large
value of the pairing gap $\Delta=100\,MeV$. Given that cooling suggests a very
small gap (see below), this may be seen as misleading. However, from the
anisotropic analytic solution we can readily check that even in the limit of
“zero” pairing, the sequences display a lower maximum mass than the ones
corresponding to large $\Delta$, but not dramatically so. The curves are not
too different mainly because of the presence of the first term $m_{s}^{2}$ in
the quantity $\Phi$ of eq.(2), which precludes a dramatic drop of $M_{max}$ in
the $\Delta\rightarrow 0$ limit. As a consequence, low-gap anisotropic CFL
would be viable models for all the objects addressed here. We have added one
of such models (CFL with ”zero gap” in the anisotropic ansatz) in Fig. 1 for
comparison.

All these self-bound EoS, with a zero-point pressure at finite density,
produced the sequences of quark stars seen in Fig. 1. Without any change, we
have plotted the new object HESS J1731-347
($M=0.77^{+0.20}_{-0.17}\,M_{\odot}$ and $R=10.4^{+0.86}_{-0.78}\text{ km}$)
considering a rectangle of $1\sigma$ and the result shows that the same
description attempted for the heavier objects is a very good model for the
low-mass compact star reported by Doroshenko et al. (2022). Larger contours,
beyond $1\sigma$, would make the agreement even more plausible for the HESS
object and the three other cases. Therefore, we consider this consistency
check as successful.

Figure 1: Mass-radius relation of the theoretical models vs. observed data.
The green curve corresponds to the MFTQCD quark EoS, with $\xi=0.003658\text{
MeV}^{-1}$ and $B_{QCD}=62\text{ MeV/fm}^{3}$. The blue curve depicts the
results for the NJL CFL+vector interactions EoS with $B_{0}=57.3\text{
MeV/fm}^{3}$ and $G_{V}/G_{S}=0.5$. Two anisotropic models corresponding to
the Thirukkanesh-Ragel-Malaver ansatz, both with
$\Delta=100\leavevmode\nobreak\ \mathrm{MeV}$ and
$m_{s}=150\leavevmode\nobreak\ \mathrm{MeV}$ appear in red (corresponding to
$B=57.5\leavevmode\nobreak\ \mathrm{MeV/fm^{3}}$) and black
($B=70\leavevmode\nobreak\ \mathrm{MeV/fm^{3}}$). The dotted sequence is the
$\Delta\rightarrow 0$ for the same anisotropic models with
$B=70\leavevmode\nobreak\ \mathrm{MeV/fm^{3}}$, showing that the downward
shift in the stellar sequence curves is not dramatic and may be viable as
models of the new object. Data rectangles are $1\sigma$ estimates for PSR
J0030+0451 (purple), PSR 0740+6620 (green) and the new reported HESS J1731-347
(blue).

##  3 Surface temperature of XMMU J173203.3-344518 in HESS J1731-347

Before the announcement of a low-mass value by Doroshenko et al. (2022), the
central object was known with the name XMMU J173203.3-344518 (H.E.S.S.
Collaboration, 2011), although there was considerable doubt concerning its
age. The high temperature reported by H.E.S.S. Collaboration (2011),
$T_{e}^{\infty}=153^{+4}_{-2}\text{ keV}$, was difficult to reconcile with a
$27\text{ kyr}$ age, although later a much younger age was found for the
shorter distance of the admissible range (Acero et al., 2015; Cui, Pühlhofer &
Santangelo, 2016; Maxted et al., 2018), which is now believed to lie in the
range $2-6\,kyr$.

The thermal history of compact stars has been modelled for many years (see,
for example, Nomoto & Tsuruta (1981)). In its basic form (ignoring gravity for
now), the cooling equation is just

$C_{V}{\frac{dT}{dt}}=-\sum_{i}L_{i},$ (3)

where $C_{V}$ is the heat capacity, dominated by the quark component, and
$L_{i}$ are the luminosities resulting from the processes that produce
neutrinos and photons. The early calculations of quark matter recognized that
a rapid cooling would ensue, when compared to hadronic compositions. However,
the pairing of quarks due to attractive interactions, recognized in the
perturbative calculations in the 1980’s (Bailin & Love, 1984), and later in a
non-perturbative fashion (Alford et al., 2001), changed things substantially.
As stated in Page, Geppert & Weber (2006), CFL pairing with large gaps would
render paired quarks essentially passive, since both $C_{V}$ and $L_{\nu}$
would be strongly suppressed on both sides of Eq. (3) for temperatures much
lower than the gap scale $\sim 100\text{ MeV}$. However, if the pairing gap is
much lower, of the order of the temperature $\Delta\leq 1\text{ MeV}$, the
cooling can mimic the slow history of hadronic composition, and the large drop
in $T_{e}^{\infty}$ does not occur promptly.

The first attempt to model the latter behavior (Horvath, Benvenuto & Vucetich,
1991) is useful to illustrate the situation, as shown in Fig. 2. The exact
numerical values of the theoretical curves should not be taken too seriously,
because many details were oversimplified or ignored in 1991. However, the
comparison is useful to state the consistency of the surface temperature with
the structure considerations derived from self-bound EoS, provided the pairing
gap is much smaller than the $10-100\text{ MeV}$ ballpark considered in CFL
models today. One of the suitable patterns of pairing discussed over the years
is the 2SC Grigorian, Blaschke & Voskresensky (2005); Blaschke and Berdermann
(2010); Sedrakian (2016); Berdermann et al. (2016); Noda et al. (2015),
expected to have small gaps and to be relevant at densities which are not
extreme.The specific case of cooling, linked to a low mass which prompted a
strange star suggestion, may be a good clue to go forward in these issues,
stimulating further calculations. Let us remark that rapid cooling was sought
as a signal of exotic composition over the years, whereas we now have a hint
that the opposite may be true, as shown in the first papers addressing paired
quark matter (Horvath, Benvenuto & Vucetich, 1991).

Figure 2: Surface temperature of HESS J1731-347 (H.E.S.S. Collaboration,
2011), marked with the red cross, contrasted with the early theoretical models
cooling with paired quarks (Horvath, Benvenuto & Vucetich, 1991) with small
$\Delta$ gaps, for three different initial temperatures. Refined scenarios
(Grigorian, Blaschke & Voskresensky, 2005) confirm the relatively slow
cooling, instead of a large drop in $T_{e}^{\infty}$, for up to $\sim
10^{5}\text{ yr}$ provided the gap is small (see text)

##  4 Formation scenario for HESS J1731-347: why is the mass so low?

The conversion of neutron matter into strange quark matter (SQM) is now a
venerable subject, started before 1990. While the first attempts to model the
propagation of the conversion from neutrons to SQM were made within a laminar
deflagration framework (Olinto, 1987; Olesen & Madsen, 1991), it soon became
clear that a much faster combustion mode should be considered (Horvath &
Benvenuto, 1988), prompted by the action of instabilities. Both Landau-
Darreius and Rayleigh-Taylor instabilities quickly wrinkle the flame front
within $\sim\text{ms}$ and set the stage for an acceleration. The burning
enters the so-called flamelet regime, in which the flame propagation is still
determined by diffusion, but the total burning rate is in turn controlled by
turbulence, in a turbulent region called the flame brush. In the flamelet
regime, for all scale $\gg l_{G}$ (the Gibson scale) (Horvath, 2010), the
turbulent velocity $u_{turb}$ and front width $l_{turb}$ are determined by the
turbulence spectrum at the largest scales (very likely of a Kolmogorov type).
From there, the possibility of the combustion entering the distributed regime,
in which turbulent eddies disrupt the flame and dominate the burning on
macroscopic and microscopic scales is substantial. In Horvath (2010) the
conditions for the combustion to “jump” to the detonation branch were
discussed, although the very mechanism behind such transitions is poorly known
and may be in fact multiple (Furusawa, Sanada & Yamada, 2016). Dedicated
simulations have not produced a transition to a detonation stage (Furusawa,
Sanada & Yamada, 2016; Niebergal, Ouyed, R. & Jaikumar, 2010), although this
does not seem definitive.

In the general case, the hypothesis that supernovae are aided by SQM
appearance (Benvenuto & Horvath, 1989) helps to visualize that the supernova
event provides a clue for its reported low mass. The suggestion is that the
burning of neutrons $n\rightarrow\text{SQM}$ has enough energy to blow out a
substantial mass from the outer core, provided the conversion happens in the
detonation mode. This is true even for cores more massive than the minima of
Suwa et al. (2020), because their density profile does not need to be sharply
decreasing (Burrows & Vartanyan, 2021). It is not known how the details would
work, because a numerical simulation of the process is not available for these
events, which would be the common case in Nature. The possibility that the
progenitor was a low mass star in the Main Sequence, even one undergoing an
electron-capture supernova first (and not an iron core collapse), would be
also possible, since the $\text{O}-\text{Mg}-\text{Ne}$ white dwarf-like core
collapses with a fixed baryonic “Chandrasekhar” mass of $1.37\,M_{\odot}$
(Horvath et al., 2022).

While there is in principle enough energy to blow away $0.3-0.4M_{\odot}$ and
leave a strange star with $M\leq 1\,M_{\odot}$, the final outcome depends on
many things, including the moment in which the SQM is nucleated inside the
collapsing core (Horvath, Benvenuto & Vucetich, 1992; Lugones & Benvenuto,
1998), the density profile at that time and other factors. We note that, even
considering the binding of sub-nuclear origin Bombaci & Datta (2000), the
smallest cores would not produce a final gravitational mass below
$1\,M_{\odot}$ (Di Clemente, Drago & Pagliara, 2022). Sweeping away a
considerable fraction of the mass of outer core seems thus inevitable in a
scenario in which the “strange” compact object is born in the very supernova
event.

It is interesting to consider the case in which a detonation does not happen,
that is, the velocity of a fast deflagration mode would still be a fraction of
$c$, meaning that the conversion of the star would take place within $\sim$
few ms. We can state that the neutrinosphere temperature would be suddenly
increased (the SQM hypothesis led to “reading” a neutrinosphere temperature of
$T_{\nu}\simeq 1.78^{+0.7}_{-0.5}\text{ MeV}$ associated to the SQM energy
release in the case of SN 1987A, Benvenuto & Horvath (1989)). At this stage
the outer core and envelope are loosely bound, and a suitable gain region for
the neutrino flux can produce the explosion with the ejection of
$\geq\,0.3\,M_{\odot}$ of the former. The suggestion is that a neutrino-driven
explosion around $\sim\text{s}$ after prompt shock stalling, due to a
refreshed neutrino flux, could be viable. Much in the same way that
establishing the basic current supernova picture took several years, a
scenario producing a strange star has to be explored with care.

An important point along the lines is related to the presence of a normal
crust. The simple qualitative scenario outlined here would predict a fallback
crust of a heavy composition, likely carbon or heavier. The thermal surface
emission has to be carefully explored, it may hold spectroscopic clues to
confirm or discard this expectation and would be important for a full
evaluation.

##  5 Conclusions

The report of a low mass $M=0.77^{+0.20}_{-0.17}\,M_{\odot}$ for the Central
Compact object XMMU J173203.3-344518 born in the supernova HESS J1731-347 is
one of the many interesting works seeking for the composition of superdense
compact objects in Nature. Its mass is truly unexpected within standard
current models of compact stellar remnants, independently of its ultimate
composition. On the other hand, an exotic nature of the former is entirely
possible, and as we have seen, tentatively consistent with all three direct
observations (mass, radius and surface temperature). A summary of these points
elaborated above using existing calculations are:

  1. 1.

The same quark EoS employed for “large” massive objects with reliable
determinations of mass and radius naturally accommodates XMMU
J173203.3-344518. Other EoS are also capable of explaining the whole set of
masses and radius, for example the ones presented by Bombaci et. al. (2020)
and Ferrer, de la Incera & Paulucci (2015). The point here is that there is a
hint of unification of description, although to pinpoint the “right” EoS is
still a distant goal.

  2. 2.

The observations indicating that the surface temperature is quite high, even
for the younger age estimate $2-6\text{ kyr}$ (Acero et al., 2015; Cui,
Pühlhofer & Santangelo, 2016; Maxted et al., 2018), exclude rapid cooling
within a quark EoS. Moreover, the simplest explanation would be achieved
within a “low gap” ($\Delta\leq\,1\text{ MeV}$) complete pairing of quarks. A
large value of the gap would be disfavored on these cooling grounds. A new
round of detailed calculations will surely shed light on this issue.

  3. 3.

The idea that strange matter is an ultimate cause of collapse supernovae
(Benvenuto & Horvath, 1989) can be tested in this case. The SQM theory has not
produced a supersonic detonation yet, but a very rapid subsonic mode has been
obtained. This energy release would need to unbind a large fraction of the
outer core to explain the low mass of XMMU J173203.3-344518, and a neutrino-
driven wind could be enough for that task.

  4. 4.

We believe that a classification of the supernova event could be attempted
using the light echo techniques and spectroscopic observations. A shell of
$\sim 2\,M_{\odot}$ is present around XMMU J173203.3-344518, and remains to be
fully characterized. The issue of the whole ejected mass is quite similar to
the case of the Crab, for which no definitive identification of a supernova
type has been confirmed (Horvath, 2022). Given that a so-called “canonical
mass” has been employed/imposed as certain over the years, and that we now
know that there are confirmed lower mass objects (Martinez et al., 2015;
Horvath et al., 2022), a reanalysis of the Crab case would be in order too.

We became aware of a preprint (Di Clemente, Drago & Pagliara, 2022) which
shares many of the points made above, not only the structure considerations,
but also issues related to the relatively hot temperature reported for the
estimated age $2-6\text{ kyr}$ of HESS J1731-347. We hope to have contributed
with additional considerations to the case.

Besides confirming all the reported data and analysis, and since the
extraction of parameters is always difficult and merits further scrutiny, we
can state that the compact object XMMU J173203.3-344518 born in the supernova
explosion which left behind the remnant HESS J1731-347 passes the initial
consistency checks and opens a window for theoretical and observational work
to study its nature and formation history in depth. It is premature to claim
any stronger conclusion, although this is an important case and other
detections could add to the whole picture. A recent work by Lin et al. (2023)
has reported another candidate with $M=0.98\pm\,0.03\,M_{\odot}$ that deserves
attention in connection to the XMMU J173203.3-344518 case.

###### Acknowledgements.

This work was supported by the FAPESP Agency (São Paulo State) under the grant
2020/08518-2 and the CNPq (Federal Government) for the award of a Research
Fellowship to JEH. L.M.S. acknowledges CNPq for financial support. The CAPES
Agency (Federal Government) and PRG-USP São Paulo are acknowledged for
financial support in the form of scholarships. We would like to acknowledge an
anonymous referee for valuable scientific suggestions that improved the
initial version of the manuscript.

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\Sigma_{q_0}\Delta_{i_Y}\Delta_G\ar[d]_{\eta_q}\ar[r]^{\eta_p}&\Sigma_{q_0}\Delta_{i_Y}\Delta_G\Delta_p\Sigma_p\ar[d]^{\eta_q}\ar@{=}[r]&\Sigma_{q_0}\Delta_{q_0}\Delta_{i_D}\Delta_F\Sigma_p\ar[r]^{\epsilon_{q_0}}&\color{orange}{\Delta_{i_D}\Delta_F\Sigma_p}\\
\Sigma_{q_0}\Delta_{q_0}\Delta_{i_D}\Sigma_q\Delta_G\ar[d]_{\epsilon_{q_0}}\ar[r]^{\eta_p}&\Sigma_{q_0}\Delta_{q_0}\Delta_{i_D}\Sigma_q\Delta_G\Delta_p\Sigma_p\ar[d]^{\epsilon_{q_0}}\ar@{=}[r]&\Sigma_{q_0}\Delta_{q_0}\Delta_{i_D}\Sigma_q\Delta_q\Delta_F\Sigma_p\ar[u]_{\epsilon_q}\ar[d]^{\epsilon_{q_0}}\\
\color{ForestGreen}{\Delta_{i_D}\Sigma_q\Delta_G}\ar[r]^{\eta_p}&\Delta_{i_D}\Sigma_q\Delta_G\Delta_p\Sigma_p\ar@{=}[r]&\Delta_{i_D}\Sigma_q\Delta_q\Delta_F\Sigma_p\ar@/_1pc/[uur]_{\epsilon_q}
Every square in the above diagram clearly commutes, completing the proof.

\end{proof}

\begin{proposition}[Typed distributivity]\label{prop:typed distributivity}

Suppose given the diagram of typed signatures
\Set&C_0\ar[l]_{\Gamma_C}\ar[r]^{i_C}\ar[d]_{u_0}\ullimit&C\ar[d]^u\\
such that $u$ is a discrete op-fibration and the square is a pullback. Note that $\ol{u}=(u,u_0)$ is $\Sigma$-ready and that $\ol{f}=(f,\id_{B_0})$ is $\Pi$-ready. Construct the {\em typed distributivity diagram}
as follows. First form the distributivity diagram as in Proposition \ref{prop:distributive}, which is the big rectangle to the right. Now take $N_0$ to be the pullback of either the top-left square, the big-left rectangle, or the big outer square --- all are equivalent. The functors $v$ and $e$ are discrete opfibrations. Note that $\ol{g}=(g,\id_{N_0})$ is $\Pi$-ready and $\ol{v}=(v,u_0\circ e_0)$ is $\Sigma$-ready.

Then there is an isomorphism
of functors $\ol{C}\inst\to\ol{A}\inst$.

\end{proposition}

\begin{proof}

By Proposition \ref{prop:distributive} we have a distributivity isomorphism
\begin{align}\label{dia:dist for typed}
\Sigma_v\Pi_g\Delta_e\To{\iso}\Pi_f\Sigma_u
\end{align}
of functors $C\inst\to A\inst$, but we want an isomorphism of functors $\ol{C}\inst\to\ol{A}\inst$.

Suppose given a $\ol{C}$-instance $(I,\delta\taking\Delta_{i_C}I\to\Gamma_C)$. This is equivalent to a natural transformation $\delta\taking I\to\Pi_{i_C}\Gamma_C$ or equivalently a functor $\delta\taking\int\Pi_{i_C}\Gamma_C\to\Set$. The two sides of isomorphism (\ref{dia:dist for typed}) give rise to the same instance $A\to\Set$, but not a priori the same typed instance. A typed $A$-instance is a functor $\int\Pi_f\Pi_{i_B}\Gamma_B\to\Set$. It may be useful to find $\int\Pi_{i_C}\Gamma_C$ (middle vertex of left-hand square) and $\int\Pi_f\Pi_{i_B}\Gamma_B$ (bottom right back vertex) in the diagram below, which we will presently describe:
\begin{align}\label{dia:typed distributivity helper}
\xymatrix@=15pt{
\int\Delta_f\Pi_f\partial u\ar[rrr]^(.6)g\ar[dd]_e&&&\int\Pi_f\partial u\ar[dddd]^(.3)v\\
\end{align}
The front square is the (untyped) distributivity diagram of Proposition \ref{prop:distributive}; note that it is a pullback. The bottom square is our interpretation of $\Pi_{\ol{f}}$ from Remark \ref{rem:interpretation of typed pi}; note that it is a pullback. In the left-hand square we begin by form the little square in the front portion (the part that includes $q,u,u', u''$) as a pullback; see Remark \ref{rem:interpretation of typed sigma}. Form the other little square as a pullback along $e$. Form the diagonal square as another distributivity diagram for $\Pi_f$ of $\partial u'$. Note that we can complete the right-hand square by the functoriality of $\Pi_f$ and that it is a pullback because $\Pi_f$ preserves limits. Finally, form the back square as a pullback. It is now easy to see that all six sides are pullbacks.

Our only task is to show that the top square is a distributivity square. It is already a pullback, so our interest is in the top right corner: we need to show that there is a natural isomorphism
$$\Pi_f\Pi_{i_C}\Gamma_C=\Pi_f\partial u'\iso^?\Pi_g\partial r=\Pi_g\Delta_e\Pi_{i_C}\Gamma_C.$$
The adjunction isomorphism for $\Pi_g\partial r$ says that for any discrete opfibration $X\to\int\Pi_{f}\partial u$ there is an isomorphism
$$\Hom_{/\int\Delta_f\Pi_{f}\partial u}(g^\m1X,\dispInt\Delta_e\Pi_{i_C}\Gamma_C)\iso\Hom_{/\int\Pi_f\partial u}(X,\dispInt\Pi_g\Delta_e\Pi_{i_C}\Gamma_C).$$
We thus need to show that there is a one-to-one correspondence between functors $X\to\dispInt\Pi_f\Pi_{i_C}\Gamma_C)$ over $\int\Pi_f\partial u$ and dotted arrows in the diagram
\int\Delta_e\Pi_{i_C}\Gamma_C\ar[r]_r&\int\Delta_f\Pi_f\partial u\ar[r]_g&\int\Pi_f\partial u

Given a map $X\to\dispInt\Pi_f\Pi_{i_C}\Gamma_C)$, we can pull it back along $g$ to get a map $g^\m1X\to\int\Delta_f\Pi_f\Pi_{i_C}\Gamma_C$ because the top square in (\ref{dia:typed distributivity helper}) is a pullback; this induces the required dotted arrow. Conversely, given a dotted arrow, we obtain the diagram
\xymatrix{
\int\Delta_e\Pi_{i_C}\Gamma_C\ar[r]_r\ar[d]&\int\Delta_f\Pi_f\partial u\ar[r]_g\ar[d]&\int\Pi_f\partial u\ar[d]\\
\int\Pi_{i_C}\Gamma_C\ar[r]_{u'}&B\ar[r]_f&A
and because the diagonal diagram in (\ref{dia:typed distributivity helper}) was constructed as a distributivity diagram, this induces a map $X\to\int\Pi_f\Pi_{i_C}\Gamma_C$ as desired.

Now that we have completed the aforementioned task, we are ready to prove the result. Our goal is as follows. We begin with an instance in the middle of the left square, a functor $\delta\taking\int\Pi_{i_C}\Gamma_C\to\Set$. We are interested in $\Pi_{\ol{f}}\Sigma_{\ol{u}}\delta$ and $\Sigma_{\ol{v}}\Pi_{\ol{g}}\Delta_{\ol{e}}\delta$; we consider them in this order.

By Remark \ref{rem:interpretation of typed sigma}, we have a natural isomorphism
Because $e$ is a discrete opfibration, Remarks \ref{rem:interpretation of typed delta}, \ref{rem:interpretation of typed pi}, and \ref{rem:interpretation of typed sigma}, we have a natural isomorphism
The result now follows from the usual distributive law, Proposition \ref{prop:distributive}.

\end{proof}

\begin{theorem}

Suppose that one has typed FQL queries $Q\taking\ol{S}\queryto\ol{T}$ and $Q'\taking\ol{T}\queryto \ol{U}$ as follows:
\begin{align}
\xymatrix{&\ol{B}\ar[r]^f\ar[dl]_s&\ol{A}\ar[dr]^t&&&\ol{D}\ar[r]^g\ar[dl]_u&\ol{C}\ar[dr]^v\\
\ol{S}\ar@{}[rrr]|Q&&&\ol{T}&\ol{T}\ar@{}[rrr]|{Q'}&&&\ol{U}}
\end{align}
Then there exists a typed FQL query $Q''\taking\ol{S}\queryto\ol{U}$ such that $\church{Q''}\iso\church{Q'}\circ\church{Q}$.

\end{theorem}

\begin{proof}

We will form $Q''$ by constructing the following diagram, which we will explain step by step:

\begin{align}
\xymatrix{
\ol{S}&&&&\ol{T}&&&\ol{U}}
\end{align}
We form the pullback $(i)$; note that since $\ol{t}$ was $\Sigma$-ready, so is $\ol{k}$. Now form the typed distributivity diagram $(ii)$ as in Proposition \ref{prop:typed distributivity}; note that $\ol{w}$ is $\Sigma$-ready and $\ol{q}$ is $\Pi$-ready. Now form the typed comma categories $(iii)$ and $(iv)$ as in Proposition \ref{prop:typed comparison Delta Pi}. Note that $\ol{r}$ and $\ol{p}$ are $\Pi$-ready. The result now follows from Proposition \ref{prop:pullback BC for typed Delta Sigma}, \ref{prop:typed distributivity}, and Corollary \ref{cor:comma BC for typed Delta Pi},

\end{proof}

\end{document}