11institutetext: School of Arts and Sciences, Ahmedabad University,Ahmedabad
380009,India

# Lorentz violating inflation and the Swampland

Oem Trivedi<EMAIL_ADDRESS>
(Received: date / Revised version: date)

###### Abstract

The swampland conjectures from String theory have had very interesting
implications for cosmology and particularly for Inflation. It has been shown
that the single field inflationary models in a GR based cosmology are in
unavoidable tensions with these conjectures, while these single field models
can still be consistent in certain non-trivial inflationary regimes. So it
becomes interesting to see whether there is a way to overcome the issues of
the swampland and single field inflation in an essentially GR based cosmology.
We show that this can indeed be the case and for this, we consider a certain
type of Lorentz violating inflationary scenario. We work out the swampland
bounds for Lorentz violating inflationary models after which show that
inflationary models which would have had otherwise serious tensions with these
conjectures in a usual GR based scenario, can be very tranquil with the
criterion in this regime. For this we take examples of Higgs inflation, radion
gauge inflation and Spontaneous symmetry breaking inflation and show that the
bounds imposed by the swampland on the Lorentz violating parameter can be
easily satisfied in these models.

##  1 Introduction

The concept of cosmic inflation has been extremely effective in explaining
different properties of the early universe starobinskii1979spectrum ;
sato1981first ; guth1981inflationary ; linde1983chaotic ; linde1995quantum  .
Various satellite observations have consistently confirmed inflationary
estimates for the early universe, and the most recent evidence from the Planck
experiment continues this trend aghanim2020planck ; akrami2020planck1 ;
aghanim2018planck ; akrami2018planck  . Furthermore observational evidence
supports a wide range of inflationary models, a lot of which are inspired by
radically different contexts ranging from modified gravity theories to quantum
gravitational martin2014encyclopaedia ; martin2014best ; wands2008multiple ;
berera1995warm ; dvali1999brane ; alexander2013chern ; kanti2015gauss ;
oikonomou2021generalizing ; Odintsov:2019clh  . However, traditional single
field models (which some refer to as ”supercooled inflationary models
berera1995warm ”) have a lot of empirical support and are still widely used in
theoretical studies.

In recent years, a lot of effort has gone into developing a “Theory of
Everything,” and String Theory is probably the most well-known candidate for
such a paradigm. Since string theory is presented in such a vivid way, it is
reasonable to expect it to have far-reaching consequences for cosmology. As a
result, there is a large and diverse body of literature that has looked into
the cosmological consequences of string theory. The extremely high number of
possible vacua states that string theory makes, which can go as high as
$\mathcal{O}(10^{500})$ , is one of the many cosmologically interesting
aspects of the theory and all these possible vacua constitute the ”Landscape”
of string theory. The question of which class of low energy EFT’s are actually
compatible with string theory then becomes a very interesting one. In order to
address this issue, Vafa coined the word ”Swampland,” which refers to a class
of low-energy effective field theories that could not be have a consistent UV
completion with regards to the string theoretic paradigm. Furthermore, a
number of field theoretic UV completion criterion from string theory known as
the ”swampland conjectures” have been proposed in recent years to classify
whether a given regime is in the swampland or not. Many people consider string
theory to be a feasible quantum gravity framework, so if a low-energy EFT
meets these criteria, it might also get along with a self-consistent quantum
gravity theory. Although quite a number of swampland criterion have been
proposed in recent times, like the completeness polchinski2004monopoles  and
cobordism mcnamara2019cobordism  conjectures,the distance and the dS
conjectures have been shown to quite clearly have the most striking
implications for cosmology of all these conjectures. These two conjectures can
be described as follows:

$1$ : Swampland Distance Conjecture : This conjecture limits the field space
of validity of any effective field theory ooguri2016non  . This sets a maximum
range traversable by the scalar fields in an EFT as

$\frac{\Delta\phi}{m_{p}}\leq d\sim\mathcal{O}(1)$ (1)

where $m_{p}$ is the reduced Planck’s constant, d is some constant of
$\mathcal{O}(1)$ , and $\phi$ is the Scalar Field of the EFT.

$2$ Swampland De Sitter Conjecture : This Conjecture states that it is not
possible to create dS Vacua in String Theory obied2018sitter . The conjecture
is a result of the observation that it has been very hard to generate dS Vacua
in String Theory dasgupta2019sitter ; danielsson2018if ( While it has been
shown that creating dS Vacua in String Theory is possible in some schemes
,like the KKLT Construction kachru2003sitter ). The Conjecture sets a lower
bound on the gradient of Scalar Potentials in an EFT ,

$m_{p}\frac{|V^{\prime}|}{V}\geq c\sim\mathcal{O}(1)$ (2)

where c is some constant of $\mathcal{O}(1)$ , and V is the scalar Field
Potential. A ” refined ” form of the Swampland De Sitter Conjecture places
constraints on the hessian of the scalar potential (a finding which first
appeared in garg2019bounds  and later in ooguri2019distance  ) and is given by

$m_{p}^{2}\frac{V^{\prime\prime}}{V}\leq-c^{\prime}\sim\mathcal{O}(1)$ (3)

These criterion have intriguing implications for cosmology, especially single-
field inflation. Considering the data on inflation, it was shown that single
field inflation in a GR based cosmology is incompatible with these conjectures
kinney2019zoo . A lot of effort has gone into resolving this dispute
geng2020potential ; scalisi2019swampland ; ashoorioon2019rescuing . However,
it has been shown that if the background cosmology for single field inflation
is not GR based, then this inflationary regime can still satisfy the swampland
criterionlin2019chaotic ; odintsov2020swampland ; yi2019gauss ;
trivedi2020swampland ; oikonomou2021rescaled . Multi field inflationary models
have also been shown to be quite consistent with the conjectures even in GR
based paradigms bravo2020tip . It’s also worth noting that in both GR and non-
GR based cosmologies, the warm inflation paradigm has been shown to be very
compatible with the swampland criterion even for single field models
motaharfar2016warm ; motaharfar2019warm ; das2019warm ; das2020runaway  The
recently proposed “Trans Planckian Censorship Conjecture”
(TCC)bedroya2019trans  is another swampland conjecture that has sparked a lot
of interest in inflationary cosmology . Models with tachyonic scalar fields as
the inflaton can also be consistent with the conjectures mohammadi2020warm ;
trivedi2021rejuvenating . Single field GR based inflationary models can only
be compatible with the TCC if they are extremely fine tuned bedroya2020trans ,
which is ironic given that inflation was invented to solve the fine tuning
problem of traditional big bang cosmology. Much further work has been done to
clarify the TCC’s problems with single field inflation, with the overall
picture appearing to be that the TCC is more comfortable with non-standard
inflationary regimes than usual single field models bernardo2020trans ;
mizuno2020universal ; brahma2020trans ; dhuria2019trans ;
brandenberger2020strengthening ; schmitz2020trans ; guleryuz2021trans . What
one can hence understand from the current literature on the swampland and
inflation is that one is not well placed to expect conventional(cold) single
field models to be on amicable terms with these conjectures. Hence, it would
optimistic for one to think that non-trivial modifications to cold single
field models based in a GR cosmology could lead to these paradigms being
consistent with the conjectures.

One such non trivial modification to single field inflationary models can be
found by investigating such regimes in a Lorentz violating background. Several
inflationary regimes in Lorentz-violating cosmological scenarios have been
studied in the recent times gasperini1985inflation ; lim2005can ;
zuntz2008constraining ; armendariz2010primordial ; kanno2006Lorentz ;
avelino2009impact ; donnelly2010coupling ; almeida2017cosmology . Gasperini
gasperini1985inflation  proposed that the primordial period of rapid expansion
of the Universe may be achieved if the gravitational interactions were
characterised by a non locally Lorentz invariant theory at some extremely
early epoch.The authors in donnelly2010coupling  examined a Lorentz-violating
inflationary theory based on an Einstein-aether theory and a scalar field
Lagrangian. Such scenarios have also been used to understand dark energy,
where it has been demonstrated that violating the Lorentz invariance creates
Lagrangians capable of driving the universe’s current acceleration
blas2011technically ; audren2013cosmological . The former and latter theories,
as a result, deal with Lorentz symmetry violation at small and large
distances, respectively jacobson2008einstein . In almeida2017cosmology , it
was shown that a time-like Lorentz violating background can produce sufficient
inflation whilst also providing an explanation for the current dark energy
epoch. Another interesting point with Lorentz violations is that Lorentz
violations can be found in certain quantum gravity constructions
Jacobson:2004qt ; collins2006lorentz ; Mavromatos:2007xe ; li2011background .
Lorentz violation during inflation generates some non standard changes in the
usual inflationary dynamics and could hence be an interesting regime with
regards to the swampland.We also want to emphasize an important point, which
is that the swampland conjectures have been aimed to be criterion for the
underlying quantum gravity as a whole, even though it is based in string
theory. Hence we are not under any restriction to only consider some
particular form of Lorentz violating cosmology which is suited or based in
some specific quantum gravity approach. The goal of this paper here is to
understand whether Lorentz violations during inflation (and more generally in
the early universe ) could possibly be appreciated by the underlying theory of
quantum gravity and for achieving that, we use the swampland criterion.

In the next section,we briefly describe about a particular kind of Lorentz
violating cosmology, while in section III we discuss the swampland status quo
of (cold) single field inflation in that regime. In section IV, we will
consider 3 different inflationary models and discuss which of them could be
consistent with the swampland on the basis of the groundwork made in Section
III. We summarize our work in section V with some concluding remarks about the
overall picture of the early universe painted by the swampland so far.

##  2 Basics of the Lorentz violating cosmology

We will be working in a time-like Lorentz violating cosmological background
here.A purely time-like background can be defined by a tiny subset of Lorentz
invariant violating operators that preserve rotational invariance, which is
why one would use a time-like Lorentz operator here passos2017Lorentz  .
Furthermore as Kostelecky and Mewes describe in kostelecky2009electrodynamics
, the Cosmic Microwave Background becomes a natural choice of a preferred
frame in this scenario. We start off with a Lagrangian in a 3+1 dimensional
background describing a scalar field ( which will eventually be the inflaton)
coupled to a Lorentz violating background (similar to the one here
almeida2017cosmology )

$\mathcal{L}=\Bigg{(}\frac{R}{2\zeta^{2}}-\frac{1}{2}\left(g{{{}_{\mu}}{{}_{\nu}}}+\xi_{1}k^{1}{{{}_{\mu}}{{}_{\nu}}}\right)\partial^{\mu}\phi\partial^{\nu}\phi-V(\phi)\Bigg{)}\sqrt{-g}$
(4)

where $\zeta^{2}=8\pi G$ and $V(\phi)$ is the inflaton potential while
$k^{i}{{{}_{\mu}}{{}_{\nu}}}$ are time-like tensors which can even couple to
other fields in a more general Lagrangian. It does so in a way such that the
only non-zero component is $k^{i}{{{}_{0}}{{}_{0}}}$ , which means

$k^{i}{{{}_{\mu}}{{}_{\nu}}}=\begin{pmatrix}-\beta_{i}&0\\\ 0&0\end{pmatrix}$
(5)

where the tensor couples to the scalar field via the coupling $\xi_{1}>0$.
This kind of a theory has been touted to be able to explain both the current
expansion of the universe as well the inflationary phase almeida2017cosmology
. It can do so if $\beta_{1}\to 0$ for short distances, which allows for all
the inflationary dynamics to be controlled only by the inflaton field while if
$\beta_{1}\to-1$ for large distances, the Lorentz violating regime can also
explain dark energy. We will, however, just be discussing about the
inflationary phase in our analysis in the next section. For studying the
cosmological properties of the Lagrangian described above, we can take the
metric to be of the FLRW form

$ds^{2}=-dt^{2}+a(t)^{2}d\boldsymbol{x}^{2}=g{{{}_{\mu}}{{}_{\nu}}}dx^{\mu}dx^{\nu}$
(6)

The background fields responsible for the Lorentz violating factor would
change the metric as

$d\overline{s}^{2}=\overline{g}{{{}_{\mu}}{{}_{\nu}}}dx^{\mu}dx^{\nu}=\left(g{{{}_{\mu}}{{}_{\nu}}}+\xi_{1}k^{1}{{{}_{\mu}}{{}_{\nu}}}\right)dx^{\mu}dx^{\nu}=\\\
-\left(1+\xi_{1}\beta_{1}\right)dt^{2}+a(t)^{2}d\boldsymbol{x}^{2}$ (7)

here one can then go on to write an effective velocity for the scalar field as

$v=\sqrt{1+\xi_{1}\beta_{1}}c$ (8)

where c is the usual speed of light. One can then also define a relationship
between the Lorentz violating parameter and the redshift z as
almeida2017cosmology

$\xi_{1}\beta_{1}=-\frac{C_{1}}{z(z+2)+C_{1}}$ (9)

where $C_{1}$ is a constant. This relation becomes very important when one
wants to realize both the inflationary phase and the dark energy phase with
the same inflation field but as we are currently interested only in the former
phase we will not explore the redshift relation further. If one considers a
General Relativity based cosmological scenario, then the Friedmann equation
will be remain the same as they are in the usual GR sense. Hence, the
Friedmann equation remains as

$H^{2}=\frac{\rho}{3m_{p}^{2}}$ (10)

where we are working in $m_{p}=\sqrt{\frac{1}{8\pi G}}$ units. However, the
energy and pressure densities will not remain as they usually are in GR. The
energy-momentum tensor can be written here as

$T{{{}_{\mu}}{{}_{\nu}}}=\partial_{\mu}\phi\partial_{\nu}\phi+g{{{}_{\mu}}{{}_{\nu}}}\mathcal{L}_{\phi}$
(11)

where
$\mathcal{L}_{\phi}=-\frac{1}{2}\left(g{{{}_{\mu}}{{}_{\nu}}}+\xi_{1}k^{1}{{{}_{\mu}}{{}_{\nu}}}\right)\partial^{\mu}\phi\partial^{\nu}\phi-V(\phi)$.
Now we can write down the pressure and energy densities, given that our field
configurations are homogeneous and so $\phi=\phi(t)$, we have the energy and
pressure densities as

$\rho_{\phi}=\frac{1}{2}(1+\xi_{1}\beta_{1})\dot{\phi}^{2}+V(\phi)$ (12)
$p_{\phi}=\frac{1}{2}(1-\xi_{1}\beta_{1})\dot{\phi}^{2}-V(\phi)$ (13)

Finally, the equation of motion for the scalar field is found out to be

$\ddot{\phi}+3H\dot{\phi}+\frac{V^{\prime}(\phi)}{1+\xi_{1}\beta_{1}}=0$ (14)

This completes our groundwork for the formulation of inflation in this
scenario. In the next section we will be discussing about the different
inflationary parameters in this scenario and consider the possibility of this
regime being swampland consistent.

##  3 Swampland implications for Lorentz violating inflation

In the inflationary phase, the energy density $\rho$ is dominated by the
inflaton energy density $\rho_{\phi}$ and so one can take
$\rho\sim\rho_{\phi}$ during inflation. Hence, the Friedmann equation (10)
during inflation takes the form

$H^{2}=\frac{1}{3m_{p}^{2}}\left(\frac{1}{2}(1+\xi_{1}\beta_{1})\dot{\phi}^{2}+V(\phi)\right)$
(15)

From now on wards in the paper, we define the Lorentz violating parameter to
be $\kappa=\xi_{1}\beta_{1}$. Now during inflation, $\dot{\phi}^{2}<<V(\phi)$
which corresponds to the slow roll condition. Applying this criterion here, we
find the Friedmann equation to be

$H^{2}=\frac{V(\phi)}{3m_{p}^{2}}$ (16)

One can note that this is the same form that the Friedmann equation has for
all sorts of inflationary models in a GR based cosmology, which are Lorentz-
abiding regimes. Hence, the Friedmann equation remains unchanged even after
factoring in Lorentz violations in this case. Now, we cast our attention to
the field equation for the inflaton (14). Another slow roll criterion during
inflation is that $\ddot{\phi}<<3H\dot{\phi}$. Considering this, we can write
the field equation to be

$3H\dot{\phi}+\frac{V^{\prime}(\phi)}{1+\kappa}\approx 0$ (17)

Now we begin to see effects of Lorentz violation in the inflationary dynamics,
as the Lorentz violating parameter plays an important role in the field
equation during inflation. Now lets find expressions for an important part of
the inflationary setup, the slow roll parameters. The $\epsilon$ slow roll
parameter is given by

$\epsilon=-\frac{\dot{H}}{H^{2}}$ (18)

Using the Friedmann equation (16) and the Field equation (17), we can write
the $\epsilon$ potential slow roll parameter from its usual definition as

$\epsilon=\frac{m_{p}^{2}}{2(1+\kappa)}\left(\frac{V^{\prime}}{V}\right)^{2}$
(19)

And further, we can find the $\eta$ potential slow roll parameter similarly as

$\eta=\frac{m_{p}^{2}}{1+\kappa}\frac{V^{\prime\prime}(\phi)}{V(\phi)}$ (20)

After this, one can then find out the e-fold number during Inflation as

$N=\int_{t_{i}}^{t_{f}}Hdt$ (21)

This can then be written in terms of $\phi$ as

$N=\int_{\phi(t_{i})}^{\phi(t_{f})}\frac{(1+\kappa)}{m_{p}^{2}}\frac{V(\phi)}{V^{\prime}(\phi)}d\phi$
(22)

where $\phi(t_{i})$ and $\phi(t_{f})$ are the values of the inflaton field at
the time of Horizon crossing and at the end of inflation, respectively. At
this point, we can start to discuss the implications of the swampland
conjectures on this inflationary regime. A point to note is that for the dS
conjecture is that if a low energy EFT scenario is consistent with either the
original dS conjecture (2) or it’s refined form (3), then it can have a
consistent UV completion with regards to the dS criterion. Hence one can use
only one of these conjectures in this regard if they like to, and we will be
using the original dS conjecture (2) instead of the refined form in our
analysis. The issues of the swampland conjectures with single field inflation
were firstly discussed in kinney2019zoo , where the background cosmology was
also general relativistic. Two of the very strong disagreements between the
conjectures and inflation concerned the bounds on the $\epsilon$ parameter
during inflation and the number of e-folds. In their work, they showed that
the dS conjecture forbids the $\epsilon$ parameter to satisfy it’s usual bound
needed for sufficient inflation, which is $\epsilon<<1$. It is easy to see how
this is the case, as this $\epsilon$ parameter bound for single field
inflation in a GR based cosmology is given as

$\epsilon=\frac{m_{p}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2}\geq\frac{c^{2}}{2}$
(23)

where c is an $\mathcal{O}(1)$ parameter in the original dS conjecture (2).
The above equation makes it clear that it would be problematic to achieve
$\epsilon<<1$ during inflation considering $c\sim\mathcal{O}(1)$. From this
definition, they were further able to work out some bounds on the scalar
spectral index and they then found out that there is a disagreement between
what the data on single field inflation and string theory itself predicts
about the order estimates of the c parameter. Namely, the c parameter should
be an $\mathcal{O}(0.1)$ term for it to be consistent with the data on
inflation instead of $\mathcal{O}(1)$ as estimated from String theory. The
next serious issue is concerns the e-fold number during Inflation. If one
considers both the distance (1) and the dS conjectures (2) to hold true, then
one finds out that these conjectures induce a fatal bound on the number of
e-folds during inflation. This can also be readily shown here, as the number
of e-folds for single field (in a GR based cosmology) can be roughly written
as

$N\simeq\frac{\Delta\phi}{m_{p}}\frac{1}{m_{p}\frac{V^{\prime}}{V}}$ (24)

Applying both the distance ($\frac{\Delta\phi}{m_{p}}\leq d\sim\mathcal{O}(1)$
) and dS conjecture( $m_{p}\frac{|V^{\prime}|}{V}\geq c\sim\mathcal{O}(1)$)
here , one finds out that the e-fold number is constrained to be less than
unity by these criterion. This is an incredibly distressing prediction, as the
latest observational bounds akrami2020planck1  require at least around 60
e-folds in order for Inflation to explain the problems of big bang cosmology
which it was originally brought in to rectify. Hence these explorations seem
to suggest that all kinds of single field inflationary models in a GR based
cosmology lie in the swampland and so, possibly, would not be viable with
quantum gravity eventually.

However, now we make the case that single field inflation can be consistent
with the swampland conjectures even when the background cosmology is described
by GR, albeit with some Lorentz violations. We can now tackle both the prime
concerns raised in kinney2019zoo  and find bounds on the parameter( $\kappa$
due to Lorentz violation) which would allow us to rectify these issues.
Considering the dS conjecture (2), one can write the following bound on the
$\epsilon$ parameter (19) as

$\epsilon\geq\frac{c^{2}}{2(1+\kappa)}$

One can clearly notice that the dS conjecture bound for this $\epsilon$
parameter is similar to the one for the usual single field model encountered
in (23), with the difference between both being the $(1+\kappa)$ term in the
denominator here. As $c\sim\mathcal{O}(1)$, in order for $\epsilon<<1$ one
requires that

$2(1+\kappa)>>1\ $ (25)

This provides us a lower bound on the Lorentz violating parameter in order for
the inflationary regime to be consistent with the swampland. Furthermore, we
can roughly write the e-fold number(22) during inflation in this case as

$N\simeq(1+\kappa)\frac{\Delta\phi}{m_{p}}\frac{1}{m_{p}\frac{V^{\prime}}{V}}$
(26)

Again applying the dS and distance conjectures together here, we find that the
term $\frac{\Delta\phi}{m_{p}}\frac{1}{m_{p}\frac{V^{\prime}}{V}}$ has to be
less than 1. Hence, in order to have sufficient inflation one requires

$(1+\kappa)>>1$ (27)

Hence considering the conditions in (25) and (27), the final bound on $\kappa$
is

$\boxed{\kappa>>1}$ (28)

Hence in order for inflation to be consistent with the swampland conjectures
in this regime, the minimum requirement that needs to be satisfied is
$\kappa>1$. This will get us rid of the issues of both the $\epsilon$
parameter and the e-fold number and hence this lower limit on $\kappa$ is what
we need in order to have swampland consistent inflation in this regime.

A question one might wonder about now is, how would we actually get to know
whether or not this inequality is satisfied in a particular inflationary model
? The answer to that lies in the perturbation parameters for inflation, in
particular the scalar spectral index. The definition of the scalar spectral
index remains the same here as it is for usual inflationary models
almeida2017cosmology ; baumann2009tasi

$n_{s}-1=2\eta-6\epsilon$ (29)

One can even further work out other perturbation parameters like the tensor-
to-scalar ratio, the tensor spectral index etc. but the scalar spectral index
is all we need here to constrain $\kappa$ in this case using the latest
observational data from the Planck experiment akrami2020planck1 . In the next
section we will now constrain the Lorentz violating parameter in 3 different
inflationary models and try to ascertain whether the constraints from the
swampland conjectures can be satisfied by a significant amount of potentials.

##  4 Analysing the swampland consistency of some models in the Lorentz
violating regime

In this section, we will work out the Lorentz violating parameter in the
Higgs, Radion Gauge and Spontaneous Symmetry breaking inflationary models and
check whether these models are swampland consistent in a Lorentz violating
regime.

###  4.1 Higgs Inflation

Higgs Inflation is an inflationary regime of particular phenomenological
interest bezrukov2008standard ; bezrukov2009standard ; bezrukov2009standard1 ;
garcia2011higgs . As the name suggests, here the well known Higgs field h
itself is considered to be playing the role of the inflaton field. The well
known inflaton potential in this case is bezrukov2008standard ;
bezrukov2009standard ; garcia2011higgs ; martin2014encyclopaedia

$V(\phi)=M^{4}(1-e^{-\sqrt{2/3}\phi/m_{p}})^{2}$ (30)

where M is a mass scale (in fact this will remain the convention for mass
scale depiction for the later potentials in this work as well). One
interesting thing to note here is that the Higgs inflation potential is an
example of an inflationary model with no free parameters. In order to obtain
the $\kappa$ values allowed by this model, we first need to find the slow roll
parameters $\epsilon$ and $\eta$ ( as defined in (19) and (20) ) for the
potential (30),. Although this model has been shown to be consistent with
observational data, it has been shown that even the higgs potential can be in
tensions with the swampland conjectures in the usual GR inflationary regime
denef2018sitter . Hence it would be interesting to see whether considering a
Lorentz violating regime for inflation would help alleviate these issues.

For this potential, one can find these parameters to be

$\epsilon=\frac{4}{3(1+\kappa)\left(e^{\sqrt{2/3}\phi/m_{p}}-1\right)^{2}}$
(31)
$\eta=\frac{4\left(2-e^{\sqrt{2/3}\phi/m_{p}}\right)}{3(1+\kappa)\left(e^{\sqrt{2/3}\phi/m_{p}}-1\right)^{2}}$
(32)

After being powered with the knowledge of these parameters, we can now find
the scalar spectral index $n_{s}$ after some algebra as

$n_{s}-1=-\frac{4(-1+\coth[\phi/(\sqrt{6}m_{p})])\coth[\phi/(\sqrt{6}m_{p})]}{3(1+\kappa)}$
(33)

The latest observational constraints akrami2020planck1  on the scalar spectral
index estimate that $n_{s}=0.9649\pm 0.0042$ at the time of Horizon crossing,
hence for our purposes here we can take $n_{s}\simeq 0.9649$ and be assured of
considerable precision in our analysis. Using this value of the scalar
spectral index, we can arrive at the following value for the Lorentz violating
parameter

$\kappa\simeq
40\left(\coth\left(\frac{\phi}{\sqrt{6}m_{p}}\right)-1\right)\coth\left(\frac{\phi}{\sqrt{6}m_{p}}\right)-1$
(34)

We consider the $\phi$ value at the time of horizon crossing
$\phi=\phi(t_{i})$ to be around the reduced planck mass

$\phi\simeq m_{p}$ (35)

(We will be justifying this choice in the end of this subsection)Inserting
this into the above equation, we finally find the Lorentz violating parameter
to be

$\kappa\simeq 154.495$ (36)

This is well above the minimum bound implied by the swampland conjectures on
inflation in the Lorentz violating regime , which is $\kappa>1$(28). To now
show that one is not wrong to take $\phi\simeq m_{p}$ at the time of horizon
crossing, we take the help of the e-folding number(22) for which we first need
to find $\phi$ at the end of inflation. This can be achieved quite easily
using the fact that $\epsilon=1$ at the end of inflation, which from (22) can
be written as

$\frac{4}{3(1+\kappa)\left(e^{\sqrt{2/3}\phi(t_{f})/m_{p}}-1\right)^{2}}=1$
(37)

This is quite straightforward to solve and one gets

$\phi(t_{f})\simeq 0.1084m_{p}$ (38)

Now plugging the value of $\kappa$ (36) into this $\phi$ at the end of
inflation and using our estimate $\phi(t_{i})\simeq m_{p}$, we can compute the
integral(22) for the Higgs Potential (30) to get the e-folding number as

$N=\frac{1+\kappa}{m_{p}^{2}}\int_{\phi(t_{f})}^{\phi(t_{i})}\frac{1}{2}\sqrt{\frac{3}{2}}m_{p}\left(e^{\frac{\sqrt{\frac{2}{3}}\phi}{m_{p}}}-1\right)\simeq
52$ (39)

This is well within the usual range between 50-70 for the e-fold number which
is required for inflation to solve the problems of conventional big bang
cosmology and be consistent with the latest observational data
akrami2020planck1 ; aghanim2018planck  and so the estimate $\phi\simeq m_{p}$
is justified and is not in any kind of tensions with the data. This analysis
tells us that Higgs Inflation is consistent with the swampland conjectures
even in a GR based cosmology when one considers Lorentz violations in the
cosmological background. It is also interesting to note that this is a zero
parameter model (in the sense that the potential has no free parameters to
tune accordingly) but is still consistent with the swampland conjectures.

###  4.2 Radion Gauge Inflation

Radion Gauge inflation was first studied in fairbairn2003radion  and is an
extension of the gauge inflation scenario in which the radius modulus field
around which the Wilson loop is wrapped assists inflation as it shrinks
freese1990natural . The potential in this scenario can be written as
martin2014encyclopaedia

$V(\phi)=\frac{M^{4}(\phi^{2}/m_{p}^{2})}{\alpha+(\phi^{2}/m_{p}^{2})}$ (40)

where M is again a mass scale and $\alpha$ is a positive dimensionless
parameter, which would act as the only free parameter of the model. This
potential can also be found in S-dual superstring models de1996inflation
Although Radion gauge inflation has never been studied in the context the
swampland for GR based single field inflation, the issues pointed out for GR
based inflation in kinney2019zoo  are so strong that they would generally rule
all kinds of GR based potentials in the swampland and hence, it would be
interesting to see how this model stands with these criterion in a Lorentz
violating regime.

We proceed similar to the way we did for Higgs Inflation in the subsection
4.1, by firstly finding out the slow roll parameters in this regime. One can
find that the parameters, after some algebra, take the forms

$\epsilon=\frac{2\alpha^{2}m_{p}^{6}}{(1+\kappa)\left(\alpha
m_{p}^{2}\phi+\phi^{3}\right)^{2}}$ (41) $\eta=\frac{2\alpha
m_{p}^{4}\left(\alpha m_{p}^{2}-3\phi^{2}\right)}{(1+\kappa)\left(\alpha
m_{p}^{2}\phi+\phi^{3}\right)^{2}}$ (42)

After finding the slow roll parameters, it is straightforward to write the
scalar spectral index using its definition (29) and one finds the index to be

$n_{s}-1=-\frac{4\alpha m_{p}^{4}\left(2\alpha
m_{p}^{2}+3\phi^{2}\right)}{(1+\kappa)\left(\alpha
m_{p}^{2}\phi+\phi^{3}\right)^{2}}$ (43)

Again using the estimate $n_{s}\simeq 0.9649$ and the field value at Horizon
crossing to be $\phi\simeq m_{p}$ (we will again justify this towards the end
of this as in subsection 4.1), one finds the following relation for the
Lorentz violating parameter in terms of $\alpha$ as

$\kappa=\frac{\alpha(227.92\alpha+341.88)}{(\alpha+1)^{2}}-1$ (44)

In order to be consistent with the swampland conjectures, one requires the
minimum bound $\kappa>1$ on the Lorentz violating parameter and this implies

$\kappa=\frac{\alpha(227.92\alpha+341.88)}{(\alpha+1)^{2}}-1>1\implies\alpha>0.0058$
(45)

The requirement imposed on the free parameter $\alpha$ in order for this model
to be swampland consistent is quite minimal and extremely in line with the
basic view that $\alpha$ is a positive parameter. Before concluding, we again
justify the choice we made for $\phi(t_{i})\simeq m_{p}$ by computing the
e-fold number. As in subsection 4.1, we firstly find the $\phi$ value at the
end of inflation using $\epsilon=1$, which gives us

$\frac{2\alpha^{2}m_{p}^{6}}{(1+\kappa)\left(\alpha
m_{p}^{2}\phi+\phi^{3}\right)^{2}}=1$ (46)

Solving this analytically for $\phi$ can be quite cumbersome but fortunately,
we can do this in an easier way. Given the definition of $\kappa$ (44) in
terms of $\alpha$, we can plug some particular values of $\alpha$ , thereby
obtaining the corresponding $\kappa$, and then use those values to obtain the
$\phi$ value at the end of inflation and compute the Number of e-folds. For
simplicity, let’s take $\alpha=1$, which would result in $\kappa=141.45$.
Using these values in (46), one can find the $\phi$ value at the end of
inflation to be

$\frac{0.01404m_{p}^{6}}{\phi(t_{f})^{2}\left(m_{p}^{2}+\phi(t_{f})^{2}\right)^{2}}=1\implies\phi(t_{f})\simeq
0.1143m_{p}$ (47)

Using the $\phi(t_{f})$ to be the above value and $\phi(t_{i})\simeq m_{p}$
one finds the e-folding number to be

$N=\int_{\phi(t_{f})}^{\phi(t_{i})}\frac{71.225\phi\left(m_{p}^{2}+\phi^{2}\right)}{m_{p}^{4}}d\phi\simeq
53$ (48)

We see that even for an elementary value of $\alpha=1$, one can sufficient
number of e-folds of Inflation needed for solving the conventional problems of
big bang cosmology and being consistent with the latest observational data 111
We understand that a reader might be tempted to tend towards a more graphical
approach here, which is that one might want to make a graph of N vs $\alpha$
and would want to look for the required e-fold consistent $\alpha$ by doing
that rather than plugging values on their own. Even we would have preferred to
go that route but as it turns out, analytically finding N as a function of
$\alpha$ for arbitrary values of the parameter is not quite feasible, partly
because changing the $\alpha$ values also changes the $\phi(t_{f})$ value
significantly(which again is quite hard to solve for arbitrary values) and
this also, hence, changes the integration limits for computing the e-folding
number. This was the reason we decided to plug in $\alpha$ values ourselves to
do this analysis, and we were still able to show that this model can
nevertheless be easily consistent with the observational requirements for the
e-folding number.  akrami2020planck1 ; aghanim2018planck . Hence considering
$\phi(t_{i})\simeq m_{p}$ would not be a wrong choice in this scenario and is
completely consistent with both the swampland conjectures and the data. This
analysis finally allows us to conclude that Radion gauge inflation can be
easily viable with the swampland conjectures in a Lorentz violating regime
even when the background cosmology is general relativistic.

###  4.3 Spontaneous Symmetry Breaking Inflation

Spontaneous symmetry breaking inflation, in the sense that we are going to be
discussing here, was firstly worked on by Moss in moss1985primordial . Moss
considered the forthcoming potential in the framework of models with
spontaneous symmetry breaking where $\phi$ represents one of the components of
a Higgs field. The potential in this inflationary regime takes the form

$V(\phi)=M^{4}\left(1+\frac{\alpha\phi^{2}}{m_{p}^{2}}+\frac{\gamma\phi^{4}}{m_{p}^{4}}\right)$
(49)

where $\alpha$ and $\gamma$ are constant dimensionless parameters and M is
again a mass scale. This model is different from the others we have discussed
so far in that it has 2 free parameters in $\alpha$ and $\gamma$ instead of
just one (like the radion gauge model) or none (like Higgs Inflation). It is
interesting to mention that besides spontaneous symmetry breaking, this type
of potential also finds place in certain SUSY and Spontaneous D-Parity
Breaking inflationary scenariosdine1997inflaton ; riotto1998inflation ;
gong2008inflation . It also becomes interesting to see the consistency of this
model with the swampland as spontaneous symmetry breaking is the way in which
Lorentz violating effects are introduced in the Standard Model extension
kostelecky1989spontaneous , which is an EFT framework which contains the
Standard Model, General Relativity and other possible operators which can
break Lorentz symmetry.

The slow roll parameters for this potential take the form

$\epsilon=\frac{2\left(\alpha m_{p}^{3}\phi+2\gamma
m_{p}\phi^{3}\right)^{2}}{(1+\kappa)\left(\alpha
m_{p}^{2}\phi^{2}+\gamma\phi^{4}+m_{p}^{4}\right)^{2}}$ (50)
$\eta=\frac{2m_{p}^{2}\left(\alpha
m_{p}^{2}+6\gamma\phi^{2}\right)}{(1+\kappa)\left(\alpha
m_{p}^{2}\phi^{2}+\gamma\phi^{4}+m_{p}^{4}\right)}$ (51)

This allows us to find the scalar spectral index (29) to be

$n_{s}-1=\frac{4m_{p}^{2}\left(-2m_{p}^{4}\phi^{2}\left(\alpha^{2}-3\gamma\right)\right)}{(1+\kappa)\left(\alpha
m_{p}^{2}\phi^{2}+\gamma\phi^{4}+m_{p}^{4}\right)^{2}}-\frac{5\alpha\gamma
m_{p}^{2}\phi^{4}-\alpha m_{p}^{6}+6\gamma^{2}\phi^{6}\
}{(1+\kappa)\left(\alpha
m_{p}^{2}\phi^{2}+\gamma\phi^{4}+m_{p}^{4}\right)^{2}}$ (52)

We again take note of the fact that the index is being evaluated at horizon
crossing, and consider $\phi=m_{p}$ which for this particular potential can be
considered pretty natural as spontaneous symmetry breaking inflationary models
lie in the class of small field models and the field value doesn’t take values
much higher than the Planck mass but we will still be justifying this estimate
later on. Using the estimate $n_{s}\simeq 0.9649$ again and imposing the
minimum requirement for swampland consistency for Lorentz violating inflation,
$\kappa>1$, we get the following inequality

$\kappa=\frac{683.76\gamma^{2}+\gamma(569.8\alpha-683.76)+\alpha(227.9\alpha-113.96)}{(\gamma+\alpha+1)^{2}}-1>1$
(53)

One can check with the help of any mathematical computation tool that this
inequality can be satisfied for a huge range of $\alpha$ and $\gamma$, where
both of them can be either positive or negative. Just for an example, we
highlight one possible solution of the above inequality which puts the
following simple bounds on $\alpha$ and $\gamma$,

$\alpha\leq-1,\gamma<0$ (54)

These restrictions on $\alpha$ and $\gamma$ do not represent a great deal of
fine tuning and are just one of the many possible range of values which can be
attained by these parameters in order for satisfy the swampland constraint for
Lorentz violating inflation. In order to compute the e-folding number in this
case, we can again go the route 222We again choose this method instead of
doing the forthcoming analysis in a fully analytical sense for arbitrary
values of the free parameters for similar reasons as to why we chose a similar
route for doing the e-folding analysis for radion gauge inflation in
subsection 4.2  that we explored in the previous subsection and start by
getting a particular value for $\kappa$ .In this case that can be obtained by
choosing any $\alpha$ and $\gamma$ that satisfy (54), where just for
simplicity we choose $\alpha=\gamma=-10$ . To get $\phi(t_{f})$ we again set
$\epsilon=1$, which gives us

$\frac{0.441\left(m_{p}^{3}\phi(t_{f})+2m_{p}\phi(t_{f})^{3}\right)^{2}}{\left(m_{p}^{4}-10m_{p}^{2}\phi(t_{f})^{2}-10\phi(t_{f})^{4}\right)^{2}}=1$
(55)

The equation above can be solved to obtain $\phi(t_{f})\simeq 0.271041m_{p}$.
Using this value and our estimate $\phi(t_{i})\simeq m_{p}$(alongside our
chosen values for $\alpha,\gamma$ and $\kappa$), we get the number of e-folds
for the potential (49) by the integral (22) as

$N=\int_{\phi(t_{f})}^{\phi(t_{i})}\frac{+216.24m_{p}^{2}\phi^{2}+216.24\phi^{4}-21.624m_{p}^{4}}{m_{p}^{4}\phi+2m_{p}^{2}\phi^{3}}\simeq
60$ (56)

The number of e-folds of inflation produced in this case are again enough to
solve the conventional problems of big bang cosmology and be consistent with
the latest observational data akrami2020planck1 ; aghanim2018planck . Hence
our horizon crossing field estimate $\phi(t_{i})\simeq m_{p}$ produces
adequate inflation for such elementary values for the parameters $\alpha$ and
$\gamma$ whilst still being consistent with the data. This analysis finally
allows us to conclude single field inflation can still be consistent with the
swampland conjectures in an essentially GR based cosmology in a Lorentz
violating regime.

##  5 Concluding remarks and discussion

To summarize, in this work we attempted to find a way in which single field
inflationary models can be consistent with the swampland conjectures even when
the cosmological background in essentially general relativistic. Although
there has been significant work on this in recent times, the novelty of our
current work lies in the fact that we achieve this goal in quite a trouble-
free way by considering a time-like Lorentz violating background. We start off
our work by firstly reflecting on the problems between the swampland
conjectures and single field inflation, which are particularly in unavoidable
loggerheads in a general relativistic cosmology. We then discuss some crucial
aspects of the Lorentz violating cosmology we have considered in our work,
showing how the inflationary dynamics is affected by considering the Lorentz
violations. We then show what is the requirement for Lorentz violating single
field models to be consistent with the swampland conjectures, which eventually
turns out to be a minimum bound on the value of the Lorentz violating. We then
briefly touch upon the fact that quadratic inflation is not consistent with
the swampland bound on the Lorentz violating parameter, building on previous
work on the same model in this regime. After this we consider 3 inflationary
models of deep phenomenological interest in the form of Higgs inflation,
Radion gauge inflation and Spontaneous symmetry breaking inflation and show
that all three of these models(which would otherwise have faced the same
difficulties with the swampland in a simple GR based cosmology as their other
compatriots) are quite easily consistent with the swampland criterion in our
Lorentz violating GR based cosmology.

The main takeaway from this work is that there can still be ways for (cold
)single field inflation to be consistent with the swampland conjectures even
when the background cosmology is essentially GR based. Another interesting
outcome idea that can be pondered upon from our work here concerns the
significance of Lorentz violations in the Early Universe. Lorentz symmetry is
a cornerstone idea of Relativity and if essentially GR based inflationary
regimes, which is in quite unavoidable tension with the swampland criterion,
becomes rather easily consistent with these conjectures only by considering a
certain form of Lorentz violation in the background cosmology then it could
possibly have interesting implications from a quantum gravity point of view.
The premise of the swampland conjectures is that these criterion are
supposedly necessary conditions in order for low energy EFT’s to have
consistent UV completion, so the fact that a Lorentz violating cosmology makes
it more tranquil for inflationary regimes to be consistent with the swampland
could really give heed to the notion that quantum gravity points towards
Lorentz violations being more significant in the early universe than what one
presumes till now . Also, it would be interesting to see in future works about
the implications of a Lorentz violating scenario on warm inflation, non-GR
based inflation and multi-field inflation in the context of the swampland and
any corresponding implications it could have on more late universe scenarios
like dark energy, dark matter and the Hubble tension.

##  6 acknowledgments

The author would like to thank the Prof. Ralf Lehnert and all the organizers
of the $4^{th}$ IUCSS Summer School on Lorentz- and CPT- Violating Standard
Model extensions, which was hosted by Indiana University, Bloomington. The
ideas for this work developed during the summer school and the author would
like to express his deepest gratitude to the school organizers for putting
together such an intellectually enriching event.

##  7 Data availability statement

There are no new data associated with this work.

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\[ \mathcal{V}_F = \underset{c \in \mathcal{C}_\mathcal{F}}{\oplaxcolim} \; \mathcal{V}_{F(c)} \]

But now, at each $ c $ in $\mathcal{C}$, the fiber $ \mathcal{V}_{F(c)}$ is itself equiped with a topology $ J_{F(c)}$. Moreover we have the following:

The transition morphisms $F(s)_*$ are cover preserving

For a family $(m'_i : n' \rightarrow n'_i)$ in $ \mathcal{V}_{F(c')}$ obtained as a pushout along some $f$ of a covering family $ (k_i : K \rightarrow K_i)$ of finite presentation under $ n$ as visualised below

\[
\begin{tikzcd}
F(c) \arrow[]{dd}[swap]{F(f)_*n'_i} \arrow[]{dr}{F(f)_*n'} & & F(c') \arrow[]{ll}[swap]{F(s)} \arrow[]{rd}{n'} \arrow[]{dd}[near start]{n'_i} & & \\
& F(s)_*B_{c',n'} \arrow[phantom, very near start]{ru}{\urcorner} \arrow[]{ld}[swap]{F(s)_*(m_i)} & & \arrow[]{ll}{} B_{c',n'} \arrow[]{ld}[swap]{m_i} & \arrow[]{l}[swap]{f} K \arrow[]{ld}{k_i} \arrow[phantom, very near end]{lld}{\urcorner} \\
F(s)_*B_{c',l_i'} & &  \arrow[]{ll}{} B_{c',l_i'} & K_i \arrow[]{l}{} &
\end{tikzcd}
\]

where the front and back square both are pushouts, so that, by left cancellation of pushouts, the bottom square also is. Then the composite of the two bottom pushouts still is a pushout:
\[ F(s)_*(m'_i) = (l_*F(s))_*(k_i)\]

and, as a pushout of finitely presented covering family, it is a covering family of $F(s)_*(n')$ in $ \mathcal{V}_{F(c)}$.

Hence we have a fibered lex site $ \mathcal{V}_{F(-)}^{\op} : \mathcal{C}^{\op} \rightarrow \Lex $; denote as

\[\begin{tikzcd}
{\displaystyle\int \mathcal{V}_{F(-)}^{\op} } & { \mathcal{C}}
\arrow["{p_F}", from=1-1, to=1-2]
\end{tikzcd}\]
the associated fibered site. Now recall that each of the sites $ (\mathcal{V}^{\op}_{F(c)}), J_{F(c)})$ are the spectra site of $F(c)$, that is, $ \Spec(F(c)) = \Sh(\mathcal{V}^{\op}_{F(c)}), J_{F(c)})$. Then the associated fibered topos has the spectrum of the $F(c)$:
as fiber, that is, we have a fibered topos

\[\begin{tikzcd}
{\displaystyle\int \Spec(F(-))} & { \mathcal{C}}
\arrow["{\overline{p_F}}", from=1-1, to=1-2]
\end{tikzcd}\]

whose transition morphisms are the inverse image $ \Spec(F(s))^* : \Spec(F(c_2)) \rightarrow \Spec(F(c_1))$ for $ s : c_1 \rightarrow c_2$. We can consider its associated direct fibration

\[\begin{tikzcd}
{\displaystyle\int \Spec(F(-))'} & { \mathcal{C}}
\arrow["{\overline{p_F}'}", from=1-1, to=1-2]
\end{tikzcd}\]

Recognize now in the definition of the topology $J_F$ that it is generated from the horizontal and vertical families as done in the previous section. Applying <ref> we have the following:

The spectrum of $F$ is the topos of continuous sections of the direct fibration associated to the fibered spectral topos:

\[  \Spec(F) \simeq \Gamma_J(\overline{p_F}') \]

We have a cloven fibration of sites defined by the projection $\mathcal{V}_F^{op}  \rightarrow \mathcal{C}_\mathcal{F} $, which is cover reflecting, and its right adjoint given by the inclusion of $ \mathcal{C}_\mathcal{F}$ which is cover preserving. Hence there is a geometric surjection
\[  \Spec(F) \twoheadrightarrow \mathcal{F} \]

We address the fibrational aspect (where cocartesiannes is an easy consequence of the use of the pushout) of this in the next remarks; first just observe that the projection $\mathcal{V}_F^{op}  \rightarrow \mathcal{C}_\mathcal{F} $ is cover reflecting because of the very existence of the adjoint inclusion as for any $ (c,l)$ in $ \mathcal{V}_F$ any $J_\mathcal{F}$-cover $(c_i \rightarrow c)$ in $\mathcal{C}_\mathcal{F}$ is lifted to a cover $ ((c_i, 1_{F(c_i)}) \rightarrow (c,1_{F(c)}))$ in $ \mathcal{V}_F$.

On the other side, taking the etale part of the composite of an etale map with the restriction map should define some kind of adjoint of those transition functors:

\[
\begin{tikzcd}
F(c) \arrow[]{d}[swap]{l} & \arrow[]{l}[swap]{F(s)} \arrow[]{d}{n_{lF(s)}} F(c') \\
B_{c,l} & \arrow[]{l}{u_{lF(s)}} B_{lF(s)}
\end{tikzcd}
\]

However the etale part $ n_{lF(s)}$ in such factorization needs not be a finitely presented map even if $l$ was. This can be fixed at the level of the indcompletion of the underlying site in which live all such factorizations. Actually we have the following adjunction for each $s : c \rightarrow c'$

\[ \begin{tikzcd}
\Ind(\mathcal{V}_{F(c')}) \arrow[bend left=20]{rr}{F(s)_*(-)} & \perp & \Ind(\mathcal{V}_{F(c)}) \arrow[bend left=20]{ll}{n_{(-)\circ F(s)}}
\end{tikzcd} \]

Where the unit and counit are defined as

\[ \begin{tikzcd}
F(c) \arrow[]{d}[swap]{F(f)_*l'} & F(c') \arrow[]{l}[swap]{F(s)} \arrow[]{d}{l'} \arrow[bend left=20]{rdd}{n_{F(s)_*l'}} & \\
F(s)_*B_{c',l'} \arrow[phantom, very near start]{ru}{\urcorner} & \arrow[]{l}{} B_{c',l'} \arrow[dashed]{rd}{\eta_{l'}} & \\
& & B_{F(s)_*l'} \arrow[bend left=20]{llu}{u_{F(s)_*l'}}
\end{tikzcd} \quad  \begin{tikzcd}
& F(c) \arrow[bend right=20]{ldd}[swap]{l} \arrow[]{d}{F(s)_*n_{lF(s)}} & \arrow[]{l}[swap]{F(s)} \arrow[]{d}{n_{lF(s)}} F(c') \\
& F(s)_*B_{lF(s)} \arrow[dashed]{ld}[swap]{\epsilon_l}& \arrow[bend left=20]{lld}{u_{lF(s)}} B_{lF(s)} \arrow[]{l}{}\\
B_{c,l} &&
\end{tikzcd} \]

where $ \eta_l'$ is induced from the etale/locale factorization being the terminal one amongst those with an etale map on the left, while the counit is induced by property of pushout. Beware that this adjunction will be actually flipped upside down between the actual underlying sites of the spectrum.

Observe that the right adjoint $n_{(-)\circ F(s)}$ is finitary as its left adjoints send finitely presented object into finitely presented objects. It is in fact the morphism of locally finitely presentable categories in induced by $ F(s)_*^{op}$ seen as a left exact functor $ \mathcal{V}^{op}_{F(c')} \rightarrow \mathcal{V}^{op}_{F(c)}$

We have a cloven bifibration defined by the projection $\Ind(\mathcal{V}_F^{op})  \rightarrow \mathcal{C}_\mathcal{F} $ and its right adjoint given by the inclusion of $ \mathcal{C}_\mathcal{F}$.

We have to prove the cartesianness and cocartesiannes of the projection. First, for a given $(c,l) $ and an arrow $ c \rightarrow c'$, the cleavage comes from the factorization by taking the etale part of $lF(s)$ as pullback object $ s^*(c,l) = (c', n_{lF(s)})$ and the local part as the lifting. This choice is cocartesian: indeed, for any situation as the following left diagram:

\[ \begin{tikzcd}[row sep=small, column sep=large]
F(c) \arrow[]{dd}[swap]{l} & \arrow[]{l}[swap]{F(s)} F(c') \arrow[]{dd}[swap, near end]{n_{lF(s)}} & \\
& & \arrow[]{lu}[swap]{F(t)} \arrow[]{dd}{l''} \arrow[bend left=20, crossing over]{llu}[near start]{F(x)} F(c'')\\
B_{c,l} & \arrow[]{l}{u_{lF(s)}} B_{lF(s)} & \\
& & \arrow[bend left=20]{llu}{h} B_{c'',l''}\\
c \arrow[bend right=20]{rrd}[swap]{x} \arrow[]{r}{s} & c' \arrow[]{rd}{t} & \\
& & c''
\end{tikzcd} \quad
\begin{tikzcd}[row sep=small]
& F(c) \arrow[]{ld}[swap]{F(t)} \arrow[]{dd}[swap, near start]{F(s)_*l'} & F(c') \arrow[phantom]{ldd}[very near end]{\urcorner} \arrow[]{l}[swap]{F(s)} \arrow[]{dd}{l'} \arrow[bend left=20, crossing over]{lld}[near start]{F(x)}\\
F(c'') \arrow[]{dd}[swap]{l''}   & & \\
& F(s)_*B_{c',l'} \arrow[dashed]{ld}{\exists}  & \arrow[]{l}{} B_{c',l'} \arrow[bend left=20]{lld}{h} \\
B_{c'',l''} & & \\
& c \arrow[]{r}{s} & c \\
c'' \arrow[]{ru}{t} \arrow[bend right=20]{rru}{x}
\end{tikzcd} \]

we get two factorizations of $lF(s)F(t)$: one with an etale part on the left $hl''$ and one with a local part on the right $u_{lF(s)}n_{lF(s)}F(t)$ so by the property of admissibility there exists a composite arrow between them

\[ B_{c'',l''} \rightarrow B_{lF(s)F(t)} \rightarrow B_{lF(s)} \]

providing the desired lifting. On the other hand, cartesianness it obvious from the use of pushout as depicted in the right diagram.

Recall that the 2-functor $\pt$ is representable, hence preserves any sort of limit, so we have an accessible embedding:

\[ \pt \Spec(F) \hookrightarrow \underset{c \in \mathcal{C}}{oplaxlim}\, \pt \Spec(F(c)) \]

where $ {oplaxlim}_{c \in\mathcal{C}}\, \pt \Spec(F(c))$ has as points families $ ((x_c)_{c \in \mathcal{C}}, (\phi_s)_{s \in \overrightarrow{\mathcal{C}}})$ with

Concerning the structural sheaf, observe that one has the following restrictions at the lever of the structural presheaf

\[ \begin{tikzcd}
\mathcal{V}^{op}_F \arrow[]{r}{\overline{F}} & \mathcal{B} \\ \mathcal{V}^{op}_{F(c)} \arrow[hook]{u}{i_c} \arrow[]{ru}[swap]{\overline{F(c)}}
\end{tikzcd} \]

as the codomain functor has to be applied fiberwise in the indexed site. Hence $ i_c^*\overline{F} = \overline{F(c)}$. But now inverse image commute with sheafification, so that one has

\[ i_c^*\widetilde{F} = \widetilde{F(c)} \]

That is, for any $(c,n) \in \mathcal{V}_F$ one has

\[ \widetilde{F}(c,n) = \widetilde{F(c)}(n) \]

while at a point $(p,x)$ of $\Spec(F)$, that is a local form $ \xi : x^*F \rightarrow A_\xi $ with $x : \mathcal{S} \rightarrow \mathcal{F}$, one has

\[ \widetilde{F}\mid_{(x, \xi)} \simeq A_x \]

Now let us turn to the geometric characteristics of spectra of local objects. To this end, we recall the following notion. Recall that a local topos is a Grothendieck topos $ \mathcal{E}$ such that the global section functor $ {!_\mathcal{E}}_*$ has a further right adjoint
\[\begin{tikzcd}[column sep= huge]
\mathcal{E} \arrow["\Gamma" description]{r} & \arrow[bend right =20]{l}[swap]{disc \textrm{ ff, lex}} \arrow[bend left=20]{l}{codisc \textrm{ ff}} \mathcal{S}et
\end{tikzcd}\]

In particular, it can be shown that a sheaf topos $ \Sh(\mathcal{C},J)$ is local if and only if the terminal object of $ \mathcal{C}$ is $J$-local.

Let $A$ be in $ \mathbb{T}_J[\mathcal{S}]$. Then $\Spec(A)$ is a local topos.

Spectra of local object are local over the base topos
Indeed we have that $ \Spec (A)$

§.§ Ionadic approach to Diers spectrum

In <cit.> was introduced a point-set construction of the spectrum from a right multi-adjoint $ U : \mathcal{A} \rightarrow \mathcal{B}$ satisfying some additional assumptions. In this approach, the spectrum of an object $B$ is a topological space constituted of the set indexing the local unit of $B$, equipped with the topology generated by diagonally universal morphisms of finite presentation under $B$; those would define a basis of compact opens. A points $x$ of the spectrum lies in the compact open if the corresponding diagonally universal morphism factorizes the local unit $ n_x : B \rightarrow U(A_x)$. Then Diers constructs a structural sheaf from the codomain functor.

This point-set construction, while encompassing a wide array of example, present two slight limitations:

* Since Diers consider only right multi-adjoints, the spectrum of an object has a small set of points. This is true for a lot of examples, for instance in many rings-like or lattice-like situations where the points of the spectrum of an object are in correspondence with a kind of ideals of this objects, which form a small set. In contrast, in the topos theoretic approach we saw above, points of the spectrum were in particular points of a topos, hence formed an accessible category, hence had a small generator, but could be large itself. Hence the associated inclusion of local objects and local map produces a local right adjoint, but not necessarily a right multi-adjoint.

* In some prominent examples (the etale spectrum of a commutative ring for instance), the spectrum as constructed above fails to be spatial, hence may have a category of points rather than a poset. This invite to substitute poset of points in Diers construction with a category of points: this is the purpose of the notion of ionad, we are going to use here. The reader may find some prerequisite on the notion of ionad in appendix[].

In this section we get back to Diers approach, though with some slight modification to take account of size issues. In the following we consider a local right adjoint $ U : \mathcal{A} \rightarrow \mathcal{B}$ as in section 1.3. We moreover make the assumption that any candidate $ n : B \rightarrow U(A)$ is the filtered colimit of all finitely presentable diagonally universal morphisms above it.

For each object $B$, consider the full subcategory $ \mathcal{D}_B \hookrightarrow B \downarrow \mathcal{B}$ whose objects are diagonally universal morphisms of finite presentation under $B$. By left cancellation of diagonally universal morphisms as a class of left maps, in any triangle

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRRkAjFVqMWbAMIB9Ed14gM2PASIjSY6vWatEIOQCZu4mFADm8IqABmAJwgBbJGRA4ISIzsn7wskx47RxdENw8kTQk9NjB5EGoGOgAjGAYABX41IRB7LAsACxxFYOdI6gjEb2ipAwYEkCTUjKzBNjzC4q4KLiA
\begin{tikzcd}
B \arrow[rd, "n_2"] \arrow[d, "n_1"'] &     \\
C_1 \arrow[r, "l"']                   & C_2
\end{tikzcd} \]

with $n_1$, $n_2$ diagonally universal, the middle arrow $ l$ also must be diagonally universal. Moreover, this is a small category closed under finite colimit in $ B \downarrow \mathcal{B}$. Hence its opposite category $ \mathcal{D}_B^{op}$ is small with finite limits.

On the other hand, consider the large subcategory $ I_B \hookrightarrow B\downarrow \mathcal{B} $ whose objects are local units $ n_x : B \rightarrow U(A_x)$, and arrows between them are triangle of the form

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACEQBfU9TXfIRRkAjFVqMWbAKoAKAIIB9EQEpuvEBmx4CREaTHV6zVohBylAJjVdxMKAHN4RUADMAThAC2SMiBwQSJZGkqbgisAAHoqWXOpunj6IfgFI+hImbGAR0SJx1Ax0AEYwDAAK-DpCIO5YDgAWOPEgHt5p1KmIwRlSZgwgBcWlFdqCbLUNTbZcQA
\begin{tikzcd}
B \arrow[rd, "n_{x_2}"] \arrow[d, "n_{x_1}"'] &        \\
U(A_1) \arrow[r, "l"']                        & U(A_2)
\end{tikzcd}\]

where again $ l$ must be diagonally universal. By Diers hypothesis we require that any local unit is the filtered colimit of all diagonally universal morphisms of finite presentation under it, that is, that any local unit is an object of $ Ind(\mathcal{D}_B)$.

We define Diers ionad of $B$ as the functor

\[ \mathcal{D}_B^{op} \rightarrow \mathcal{S}et^{I_B} \]

sending any $ n : B \rightarrow C$ to the functor

\[ Ind(\mathcal{D}_B)[n,-] : I_B \rightarrow \mathcal{S}et \]

sending an local unit $ n_x : B \rightarrow U(A_x)$ to the homset $Ind(\mathcal{D}_B)[n,n_x] $, whose elements are all diagonally universal morphisms $ l : C \rightarrow U(A_x)$ such that $ n_x= ln$, and any arrow $ m : n_{x_1} \rightarrow n _{x_2}$ to the map
\[ m \circ (-) :Ind(\mathcal{D}_B)[n,n_{x_1}] \rightarrow Ind(\mathcal{D}_B)[n,n_{x_2}]   \]
At the level of arrows, any arrow $ l : n_1 \rightarrow n_2$ in $\mathcal{D}_B$ is sent to the natural transformation
\[ (-) \circ l : Ind(\mathcal{D}_B)[n_2,-] \rightarrow Ind(\mathcal{D}_B)[n_1,-]\]

Beware that the category $\mathcal{S}et^{I_B}$ of small presheaves over $I_B$ is not a Grothendieck topos.

Functoriality : for a morphism $ B_1 \rightarrow B_2$, the corresponding morphism of ionad...

Sheaf on a ionad ?

Now we apply construction of appendix...

Consider the Yoneda extension $ \Phi_B^* = \lan_ \hirayo \Phi_B$ as below

\[  % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYABEAvgH0AQgD1gENGJBjS6TLnyEUZAIxVajFm049+QhsADKYmDjkneg4eOlKVa7HgJEd5ffWZWRA5uR3MrGztgAElXZX0YKABzeCJQADMAJwguJDIQHAgkACZqAKNgzgAFPixpZVUQLJy86kKkXwNA43ZazLoATyLqBjoAIxgGKvUvLRBMrCS+HBA2uiwGNj4ICABrBozs3MRO9sRSroqQ0bAJTj7BiAACaqw+epHxyenPTTYFpYrMQUMRAA
\begin{tikzcd}
\mathcal{D}_B^{op} \arrow[r, "\Phi_B"] \arrow[d, "\hirayo"', hook] & \mathcal{S}et^{I_B} \\
\mathcal{S}et^{\mathcal{D}_B} \arrow[ru, "\lan_\hirayo \Phi_B"']   &
\end{tikzcd}\]

This functor is left exact and has a right adjoint $ \Phi_* = \mathcal{S}et^{I_B}[\Phi_B, -]$ sending a functor $ X : I_B \rightarrow \mathcal{S}et$ to the functor $ \mathcal{S}et^{I_B}[ \Phi_B(-),B] : \mathcal{D}_B^{op} \rightarrow \mathcal{S}et $

The topos of opens of this ionad is the middle term of the factorization

§ EXAMPLES

§.§ Structured Stone duality

In general Stone duality is considered from the point of view of concrete dualities. However it is possible to provide a spectral account of it, or at least to reconstruct the Stone dual of a distributive lattice as the spectra of a certain geometry. But in this process the stone dual is also endowed with a structural sheaf which is not considered in classical Stone duality, and the spectrum will be adjoint to the global section functor rather than the “compact open" functor, which, in Stone duality, returns the basis of compact open set of Stone spaces to which it is hence restricted.

As for Stone duality, this construction allows can be done in two manners, equipping the Stone dual with either the Zariski or the coZariski topology: this depends on the way we define the admissibility structure.

We recall here the admissibility structure for Zariski. Our ambient locally finitely presentable category is $ \rm{DLat}$, the category of bounded distributive lattices. Recall that distributive lattices are not 1-regular, that is, for any given ideal there are several congruences whose class in 1 is this ideal. For etale maps one can choose 1-minimal quotients: those are morphisms $ A \twoheadrightarrow A/\theta$ with $ \theta$ minimal amongst congruences whose class in 1 is $ [ 1]_\theta$. One can easily prove this class is closed by composition and colimits, contains iso and is left-cancellative. For a lattice $ D$ finitely presented 1-minimal quotients are of the form $ D \rightarrow D/\theta_{(a,1)}$

Then it can be shown that one has a factorization system $(MinQuo_1, 1-Cons)$ on $ \mathcal{DL}at$, where $1-Cons$ is the class of maps that reflect the top element, that is those $ f: D \rightarrow D'$ such that $ f^{-1}(1) = \{ 1 \}$.

Then define the category $ \mathcal{L}oc\mathcal{DL}at^{1-cons}  $ having:

* as objects local distributive lattices, where $\{ 1 \} $ is prime filter

* as morphisms 1-conservative morphisms $f$.

Then $ \mathcal{L}oc\mathcal{DL}at^{1-cons} \hookrightarrow \mathcal{DL}at $ is a multireflection. But we can also axiomatize the category of local lattices as follows:
define $ J_1$ on $ \mathcal{DL}at_{fp}^{op}$ generated by $(f_i : D \twoheadrightarrow D/\theta_i)  $  such that $ \bigcap_{i \in I} \theta_i = diag_D  $. Now observe that a distributive lattice $D$ is $ J_1$-local if an only if $\{ 1 \}$ is a prime ideal, that is, $D $ has a minimal point $ L \rightarrow 2$ sending any $a \neq 1$ on 0. Local lattices are the points of the topos $  \mathcal{S}h(\mathcal{DL}at_{fp}^{\op}, J_1)$.

The associated spectrum for $D$ is

\[ (\Spec(D) = (\mathcal{F}^{Prime}_D, \tau^{Zariski}_D), \widetilde{D}) \]

with $\widetilde{D} $ defined on the basis as $ \widetilde{D}(U^{coZar}_a) = D/\theta_{(a,0)} $ for any $a \in D$. Then we have an adjunction

\[ \begin{tikzcd}
\mathcal{DL}at^{op} \arrow[bend right = 20]{rr}[swap]{Spec^{Zar}} & \quad \quad  \perp & \quad \mathcal{DL}at^*-\mathcal{S}paces \arrow[bend right = 20 ]{ll}[swap]{\Gamma}
\end{tikzcd} \]

Then one recovers the Stone spaces as the underlying spaces of affine $\mathcal{DL}at$-spaces.

The spectral site of a distributive lattice $D$ is $(Zar_D^{op},J_1(D))  $ where $ Zar_D$ consists of finitely presented 1-minimal quotients of $ D$; in particular for a filter $ F$ of $D$, a factorization as below
$ D \twoheadrightarrow D/\theta_{(a,1)}$, and a factorization

\[ % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRABEQBfU9TXfIRQBGclVqMWbdgHoAOnJwALGDjoB9AGLdeIDNjwEiZYePrNWiDvMUq164AAo6pYQEou3cTCgBzeESgAGYAThAAtkhkIDgQSABM1Cp0UGw4AO4QyVAI1OZSVgCO6nQg1Ax0AEYwDAAK-IZCICFYvko4OsFhkYjRsUiiINlpmdm5EhZsxdo8XREJ1P2Ig8NWGVkwKeMV1XUNgmwtbR15kpYgdAAEClhgl9rlt+dQdHAqqVwUXEA
\begin{tikzcd}[row sep=small]
D \arrow[d, "q_a"', two heads] \arrow[r, "q_F", two heads]   & D/\theta_F \\
{D/\theta_{(a,1)}} \arrow[ru, two heads, dashed] &
\end{tikzcd} \]

expresses the fact that $ a $ lies in $F$.

Now, at a distributive lattice $D$, the induced topology $J_1(D)$ consists of finite families $ (D\twoheadrightarrow D/\theta_{(a_i,1)})_{i \in I}  $ with $ \bigvee a_i = 1 $. Being made of epi, $Zar_D$ is a poset and $ Zar_D \simeq D^{\op}$ and we have $ D \hookrightarrow \tau_{Zar}$. $ J_1(D)$ coincides with the coherent topology on $D$. The spectrum is spatial and is equiped with the Zariski topology which is the frame of filters $ \mathcal{F}_D$.

Opens of Zariski topology form the frame $ \tau_{Zar} = \Sh(Zar_D^{op}, J_1(D)) = \Sh(I_D)$: Zariski opens correspond to ideals of $ D$ and $ D \hookrightarrow I_D$ is a base of compact open of Zariski topology. On the other side, $D \hookrightarrow (\mathcal{F}_D)^{op}$, but a filter $F$ of $D$ defines a filtered diagram whose colimit is the 1-minimal quotient at $F$

\[S \twoheadrightarrow S/\theta^{min}_F = \underset{a \in F}{colimfilt}\; S/\theta(a,1) \]

Those filters are saturated compact  of Zariski topology. A prime filter $ x$ corresponds to the 1-quotient $ D \twoheadrightarrow D/\theta_x$, which is the saturated compact $ \uparrow x$, the focal component in $x$.

However Zariski geometry is not the only way to retrieve Stone duality. One could have either defined the factorization system $ (0-minQuo, 0-cons)$, whith $ 0-minQuo$ as the minimal quotient with a fixed ideal, and $0-cons$ the morphisms $f$ such that $f^{-1}(0)=\{ 0 \}$, and could have taken as local objects those distributive lattices  with $ \{0\}$ prime.

The CoZariski site would have been $(coZar_D^{op}, J_0(D)) $ with $coZar_D$ made of the minimal quotients $ D \twoheadrightarrow D/\theta_{(a,0)} $ and $J_0(D)$ defined by $ (a_i)_{i \in I}$ such that $\bigwedge_{i \in I} a_i = 0$.

Then $ D \simeq CoZar_D$, so that $ D^{op} \hookrightarrow \tau_{coZar} = Sh(coZar_D^{op}, J_0(D)) \simeq (\mathcal{F}_D)^{op} $. Then filters are the closed subsets of coZariski topology. On their side ideals $ I_D$ define filtered colimits of finitely presented minimal quotients maps in $coZar_D$, hence correspond to saturated compacts.

We recover Hochster duality  from Isbell duality at a lattice seen as a category.

\[\begin{tikzcd}
& {D} \\
\\
{\tau_{Zar} \simeq I_D} && {\tau_{coZar}^{\op}\simeq (\mathcal{F}_D)^{\op}} \\
\\
{[D^{\op},2]} && {[D,2]}
\arrow[from=3-1, to=5-1]
\arrow[from=1-2, to=3-1, curve={height=12pt}]
\arrow[from=1-2, to=3-3, curve={height=-12pt}]
\arrow[from=3-3, to=5-3]
\arrow[""{name=0, inner sep=0}, from=3-1, to=3-3, curve={height=18pt}]
\arrow[""{name=1, inner sep=0}, from=3-3, to=3-1, curve={height=18pt}]
\arrow[""{name=2, inner sep=0}, from=5-1, to=5-3, curve={height=18pt}]
\arrow[""{name=3, inner sep=0}, from=5-3, to=5-1, curve={height=18pt}]
\arrow["\dashv"{rotate=-90}, from=1, to=0, phantom]
\arrow["\dashv"{rotate=-90}, from=3, to=2, phantom]
\end{tikzcd}\]

§.§ Jipsen-Moshier duality

Jipsen-Moshier duality is an example of an geometry without specification of local objects, so that any object is actually local.

In a space $X$ with specialization order $ \sqsubseteq$, a compact open filter is an upset $ F$ for $ \sqsubseteq$ which is both open and compact. For $ X$ denote $ \mathcal{KOF}(X)$ its set of compact open filters. A point $x$ is basic compact open if $ \uparrow$ is a compact open filter. Hofmann-Mislove-Stralka spaces are sober spaces $X$ such that $\mathcal{KOF}(X)$ is a basis closed under finite intersection.

Denote $ \mathcal{HMS}$ the category of HMS spaces with continuous maps $f : X \rightarrow Y $ such that $ f^{-1}$ restricts to $ \mathcal{KOF}(Y)  \rightarrow \mathcal{KOF}(X) $. Any compact open filter of a HMS space has a focal point. Moreover, in a HMS space, any point is a directed join of basic compact open points. The specialization order makes $ (X, \sqsubseteq)$ a complete lattice, and there are both an initial and terminal points in such a $X$.

Then one can recover the spectra of Jispen and Moshier duality for $\wedge$-semilattices with unit:

\[ \wedge-\mathcal{SL}at_1^{op} \simeq \mathcal{HMS}  \]

Defines $Spec(S) = (\mathcal{F}_S, \downarrow S)$. For $X$ HMS, $ \mathcal{KOF}(X)$ is a $ \wedge$-semilattice. Moreover, for a semillatice one has $ S \simeq \mathcal{KOF}(Spec(S))$, while for a HMS space, one has $ X = \Spec(\mathcal{KOF}(X))$.

If $ S$ is a $\wedge$-semilattice, $ \mathcal{F}_S \simeq (\mathcal{I}^{prime}_S)^{op}$ is a complete lattice. Moreover any filter of a $\wedge$-semilattice is trivially prime. This says that $ \Spec(S) = \wedge-Slat[S, 2]$.

Then the geometry associated to Jispen and Moshier duality can be constructed as follows: take $(1-MinQuo, \; 1-Cons) $ also is a factorization system on $ \wedge-\mathcal{SL}at$, and no topology. Finitely presented etale maps under a $ \wedge$-semilattice $S$ are principal 1-minimal quotients
\[ S \twoheadrightarrow S/\theta(a,1)\]
and they always define a basic compact open point. Conversely for any filter $F$ one has a minimal quotient
\[S \twoheadrightarrow S/\theta^{min}_F = \underset{a \in F}{colimfilt}\; S/\theta(a,1) \]

where $\theta^{min}_F$ is the congruence in $F$ given as $ \theta^{min}_F = \bigcap \{ \theta \mid [1]_\theta = F \}$. This defines a point of $Spec(S)$, and any saturated compact actually has a focal point. Observe that this is a situation of the purely factorization geometries as in <ref>,which is induced from the stable inclusion of the right class of a factorization system $ 1-Cons \hookrightarrow \wedge-Slat$.

§.§ Zariski geometry for commutative rings

Example of local rings
$ \mathbb{T}_{LocRing} = \mathbb{T}_{CRing} \cup \{ x \neq 0 \vdash \exists y(xy=1) \vee \exists y'((1-x)y'=1)\}$

§.§ Etale geometry for commutative rings

§.§ Domains geometry for commutative rings

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

journaltitleAnn. of Math. Stud., to appear, a rXiv
titleTopos co-évanescents et généralisations

Cambridge University Press

titleLocally presentable and accessible categories

titleGrothendieck topologies from unique factorisation systems

journaltitleJournal of Pure and Applied Algebra
titleSheaf Representation for Topoi

journaltitleRevue Roumaine de Mathematiques Pures et Appliquees
titleSur la duale de la categorie des treillis

journaltitleJournal of Pure and Applied Algebra
titleA representation theorem for integral rigs and its applications to residuated lattices

* Entry 'MenniRig' (SurveySpectra.bib): Invalid format 'October 2016' of date field 'date' - ignoring

journaltitleReprints in Theory and Applications of Categories
titleThe bicategory of topoi and spectra

booktitleApplications of Sheaves
titleLocalisation, spectra and sheaf representation

journaltitleAdvances in Mathematics
titleSigma limits in 2-categories and flat pseudofunctors

booktitleCategory Theory
titleUn critere de representabilite par sections continues de faisceaux

journaltitleCommunication in Algebra
titleUne construction universelle des spectres, topologies spectrales et faisceaux structuraux

titleRepresentation theory of MV-algebras

journaltitleJournal of pure and applied Algebra
titleAxiomatic etal maps and a theory of spectrum

titleA construction of 2-cofiltered bilimits of topoi

abstractStarting from known examples of factorization systems in 2-categories, we discuss possible definitions of proper factorization system in a 2-category. We focus our attention on the construction of the free proper factorization system on a given 2-category.
journaltitleJournal of Pure and Applied Algebra
titleProper factorization systems in 2-categories

journaltitleJournal of Pure and Applied Algebra
noteSpecial Issue devoted to the International Conference in Category Theory `CT2010'

Berlin, Heidelberg

Springer Berlin Heidelberg

booktitleToposes, Algebraic Geometry and Logic
titleClassifying topos

titleTopos annelés et schémas relatifs

Courier Corporation

titleTopos theory

Dunod éditeur, publié avec le concours du CNRS

journaltitleCahiers de Topologie et Géométrie Différentielle Catégoriques
titleFactorization theorems for geometric morphisms, I

Oxford University press

seriesOxford logic guides
titleSketches of an Elephant: A Topos Theory Compendium

journaltitleProceedings of the London Mathematical Society
titleLocal maps of toposes

Cambridge University Press

booktitleNew Spaces in Mathematics and Physics

titleDerived Algebraic Geometry V: Structured Spaces
maclane   moerdijkbook

Springer Science & Business Media

titleSheaves in geometry and logic: A first introduction to topos theory

American Mathematical Soc.

titleProper maps of toposes

titleOn Diers theory of Spectrum I : Stable functors and right multi-adjoints

titleOn Diers theory of Spectrum II : Geometries and dualities

journaltitleLecture Notes in Mathematics vols
titleGrothendieck et al., Théorie des topos et cohomologie étale des schémas, SGA 4

titleLifting PIE limits with strict projections

titleThe Trace Factorisation of Stable Functors