# Spontaneous onset of three-dimensional motion with subsequent spatial and
temporal reduction in convective flow systems

Patrick J. Stofanak  Cheng-Nian Xiao  Inanc Senocak  Department of Mechanical
Engineering and Materials Science,
University of Pittsburgh, Pittsburgh, PA 15261, USA

###### Abstract

We study the spontaneous emergence of three-dimensional motion from a
quiescent, pure conduction state in stably stratified, convective flow within
a triangular enclosure, which eventually self-organizes into a two-dimensional
steady state. This phenomenon demonstrates that the optimal disturbance path
to reach the final state is more complex than the state itself, indicating the
“fastest” route involves a higher-dimensional intermediate state. This
provides a model for transient spatio-temporal chaos in nonlinear dynamical
systems and a challenge for classical hydrodynamic stability theory.

††preprint: APS/123-QED

#### Introduction —

Self-organization is the spontaneous formation of spatial, temporal, or
spatio-temporal structures in a complex, dissipative system driven away from
equilibrium. Self-organization in convective fluid systems encompass a broad
spectrum of significance, especially for geophysical and astrophysical
applications. Notable examples include galaxy formation and Jupiter’s red spot
[1], as well as pattern formation in Rayleigh-Bénard and Taylor-Couette flows
[2, 3].

In this letter, we identify a convective fluid system that exhibits
spontaneous three-dimensional (3D) growth from a zero-flow state and
subsequent self-organization into a steady two-dimensional (2D) state. We
concentrate on the distinctive path to self-organization, which we connect to
the system’s dynamical stability. We assess the system’s order based on the
spatial and temporal dimensionality of the solution. If an initially ordered
state is dynamically unstable, it is expected to progressively become more
disordered when perturbed [4]. This type of evolution is ubiquitous in fluid
systems, such as transition to turbulence through a sequence of instabilities
[5].

Self-organization in fluid systems is not uncommon. An example is 2D decaying
turbulence in bounded domains, which self organize into long-lived vortices
that decay over time [6, 7]. Another example is a two-layer fluid system with
a convective, turbulent bottom layer and a stably stratified upper layer [8].
Turbulent motions of the bottom layer generate internal waves in the upper
layer and these waves produce a mean flow that reverses direction on a much
longer timescale than the waves themselves. Other studies observe
relaminarization of turbulence due to an external change imposed on the
disordered state. Xu  _et al._ [9] study the case of Rayleigh-Bénard
convection with imposed wall shear and find that an initially turbulent state
can be relaminarized with an increase in wall shear, and that the subsequent
laminar state achieves more efficient heat transfer. Kühnen  _et al._ [10]
observe relaminarization of turbulent pipe flow in experiments and simulations
when the flow profile is flattened by adding perturbations in experiments and
adding a forcing term in the simulations. A similar effect was observed by
heating of the pipe walls as well [11].

Our example is distinct from the aforementioned examples. In our system, the
flow field self-organizes into a 2D steady state in the absence of any
external forcing on the initial configuration. Starting from a quiescent
conduction state, linear stability analysis (LSA) indicates a 3D instability,
hinting at a transition to a dynamically complex 3D flow state. However,
direct numerical simulations (DNS) of the Navier-Stokes equations reveal a 2D
steady state, with the “fastest” route from the initial state being a 3D path
marked by self-organization. Our discovery holds considerable significance for
two main reasons: first, our study presents a flow instability case where the
optimal initial disturbance exhibits greater complexity than the resultant
configuration of the instability. Such a disparity between the initial
disturbance, regarded as the “cause,” and its long-term “effect” on the flow
evolution underscores the limitations of predictions derived from hydrodynamic
stability theory. Second, our study demonstrates that the most natural
transition path can be counter-intuitive and involves self-organization, such
as going through a 3D path in order to reach a 2D steady state. This insight
could substantially inform our interpretations of the flow dynamics in
geophysical and astrophysical applications.

#### Background and Methods —

We examine a simple model of stratified flow within a V-shaped cavity heated
from below, as illustrated in Figure 1, which details key parameters and
boundary conditions. The V-shaped cross-section is oriented in the $x-y$
plane, featuring horizontal $u$ and vertical $v$ velocity components, while
the $w$ velocity component lies in the homogeneous $z$ direction. The walls of
the triangular enclosure are inclined at a slope angle $\alpha$ of
$30^{\circ}$. Buoyancy is defined by the equation
$b=(g/\Theta_{r})(\Theta-\Theta_{e}(y))$, where $\Theta$ represents the
potential temperature, $\Theta_{r}$ denotes a reference potential temperature,
and $\Theta_{e}$ signifies the potential temperature of the ambient
environment. We impose a constant stable stratification, characterized by the
buoyancy frequency $N=\sqrt{(g/\Theta_{r})\partial\Theta_{e}/\partial y}$.
Consequently, $b$ represents a perturbation from the linear background
stratification defined by the constant $N$.

The governing equations are represented by the Navier-Stokes equations,
formulated under the Oberbeck-Boussinesq approximation:

$\displaystyle\nabla\cdot\bm{u}=0,$ (1)
$\displaystyle\frac{\partial\bm{u}}{\partial
t}+\bm{u}\cdot\nabla\bm{u}=-\nabla p+\nu\nabla^{2}\bm{u}+b\bm{g},$ (2)
$\displaystyle\frac{\partial b}{\partial t}+\bm{u}\cdot\nabla
b=\beta\nabla^{2}b-N^{2}\bm{g}\cdot\bm{u},$ (3)

where $\nu$ is the kinematic viscosity, $\beta$ is the thermal diffusivity,
and $\bm{g}$ is the effective gravity vector $\bm{g}=[0,1,0]$. The cavity is
heated from the two sloping bottom walls with a constant buoyancy flux
$B_{s}=\beta\partial b/\partial n$ and no-slip velocity boundary conditions,
and a constant $b=0$ is imposed on the flat top boundary with free-slip
velocity conditions. Given these conditions, a quiescent state exists with an
exact solution of normalized buoyancy and pressure profiles given by
$P(y)=-0.5\left(y-1\right)^{2}$ and $B(y)=1-y$, where buoyancy and pressure
have been non-dimensionalized by the quantities
$b_{0}=B_{s}H/(\beta\cos\alpha)$ and $p_{0}=B_{s}H^{2}/(\beta\cos\alpha)$,
respectively. Additionally, we nondimensionalize velocity by
$u_{0}=\sqrt{B_{s}/N}$ and time by the diffusive timescale $t_{0}=H/\nu$.

Figure 1:  Schematic of the computational domain, buoyancy boundary
conditions, and key parameters, illustrated with vorticity and velocity
vectors of the 2D asymmetric steady state.

The flow is controlled by four dimensionless parameters: the stratification
perturbation parameter $\Pi_{s}=B_{s}/\beta N^{2}$, a measure of the strength
of surface buoyancy flux relative to the background stratification, the
buoyancy number $\Pi_{h}=NH^{2}/\beta$, a ratio of the thermal diffusion and
stratification time scales, as well as Prandtl number $Pr=\nu/\beta$, and
slope angle $\alpha$. In our results, we only vary $\Pi_{s}$ and $\Pi_{h}$,
setting the Prandtl to 7 and the slope angle to $30^{\circ}$.

We linearize the flow equations around the quiescent base state and apply LSA.
Additionally, we perform DNS of the Navier-Stokes equations to observe the
nonlinear evolution of the disturbances. All simulations are carried out with
the spectral/hp element code Nektar++ [12] with a homogeneous Fourier
expansion used in the $z$ direction.

#### Results —

LSA reveals a dominant 3D instability. The eigenvector is visualized in Figure
2 with the Q-criterion and consists of two symmetric circulation rolls
intertwined along the $z$-direction. The dependence of the growth rate on the
length in the homogeneous direction $L_{z}$ is investigated and the optimal
wavelength is found to be approximately 1.6, as shown in the Appendix. The
imaginary part of the eigenvalue is zero for all cases, indicating a non-
oscillatory instability.

Figure 2:  Visualization of Q-criterion of the 3D eigenfunction, with contour
of Q-criterion set to approximately 4% of the maximum value, and colored by
normalized $w$ velocity.

Our investigation of the stratified, heated triangular enclosure in 2D reveals
two primary instabilities from the zero flow state, each of which leads to a
different 2D steady-state profile. The first, which we refer to as the
symmetric state, consists of two equal and opposite circulation rolls on
either side of the cavity. The second, the asymmetric state, also consists of
two circulation rolls, but one has now become stronger and occupies the center
of the cavity. We find that the symmetric steady state is unstable and will
transition to the asymmetric state under a small perturbation. The asymmetric
state is depicted via the spanwise component of vorticity in Figure 1. The 3D
mode is more dominant than both 2D modes, as shown for $L_{z}=2$ in Table 1.
This is in contrast to the results for the Rayleigh-Bénard case, in which the
3D mode exhibits the same growth rate as the 2D mode. This is described in the
Appendix.

Table 1:  Growth rate for 3D, 2D symmetric, and 2D asymmetric modes from LSA with $L_{z}=2$. $(\Pi_{s},\Pi_{h})$ | 3D | 2D symmetric | 2D asymmetric
---|---|---|---
$(0.9,1500)$ | $6.23\times 10^{-2}$ | $4.70\times 10^{-2}$ | $5.49\times 10^{-2}$

We now perform DNS of the Navier-Stokes equations to investigate the behavior
of the 3D eigenmode found from LSA. Traditionally, we would expect the 3D
eigenmode to establish a 3D flow field, representing a step closer to
transition to turbulence. We perform DNS with $L_{z}=2$ and an initial
perturbation defined by a small multiple of the 3D eigenvector. The time
evolution of the volume averaged $w^{2}$ is shown in Figure 3(a).
Interestingly, we do not observe the expected evolution to a 3D steady state,
but rather we see that after the nonlinear saturation of the initial
exponential growth, the $w$ velocity begins to decay until the flow field
returns to a final 2D steady state. Visualization of the flow field at three
different points in the evolution are shown in Figure 3(b-d) with the
Q-criterion. At $t_{n}=1$, the flow is at the maximum three-dimensionality and
forms a strong vortex ring circulation structure, as depicted in Figure 3(b).
After the nonlinear saturation of the initial exponential growth of the
instability, there is a brief period of fast decay of $w$, followed by a long
period of slow decay, from approximately $t_{n}=5$ to $t_{n}=25$. During this
period, the flow field comes to resemble the 2D symmetric state with a small
and diminishing magnitude of $w$ velocity, as shown in Figure 3(c). Finally,
at later times, for $t_{n}>28$, the decay of the $w$ velocity suddenly
increases to the point of $w$ approaching zero. This is associated with the 2D
symmetric state transitioning to the 2D asymmetric state, as shown in Figure
3(d). This evolution represents an unstable 3D instability that spontaneously
self-organizes to a less complex 2D steady state.

Figure 3:  (a) Evolution of normalized, volume-averaged $w^{2}$ velocity over
time for $\Pi_{s}=0.9,\Pi_{h}=1500$, and $L_{z}=2$, along with growth/decay
rates predicted by LSA at three different points in the evolution shown with
colored dashed lines. The inset shows the evolution zoomed in on early times.
Vertical dashed lines represent the times at which the flow field is depicted
with the Q-criterion in (b), (c), and (d). Q-criterion contours are taken as
4% of the maximum, and are colored with $w$ velocity.

To unravel the reasons underlying this behavior, we separate the evolution
into four distinct phases. The first phase from the start of the simulations
to $t_{n}=1$ represents the phase of linear growth. This phase is shown in
more detail in the inset of Figure 3(a), in which the growth rate of the 3D
eigenmode is plotted alongside the initial exponential growth of the
simulation, and these growth rates match well. Therefore, this phase follows
the expected linear stability behavior, before any nonlinear effects become
significant. During this phase, the velocity profile evolves from the shape of
the eigenmode, shown in Figure 2(a), to the state with the maximum $w$
velocity shown in Figure 3(b).

The second phase represents the nonlinear saturation of the instability, and
is manifested by the fast decay of the $w$ velocity immediately after the peak
at $t_{n}=1$ until approximately $t_{n}=5$. In traditional transition to
turbulence theory, after the nonlinear saturation of the instability, the flow
field evolves into a new steady or oscillating state upon which secondary
instabilities can arise [5]. However, we see an initial fast decay of the $w$
velocity which significantly changes the structure of the flow field. We
unravel the cause of this decay through budget analysis of the disturbance
kinetic energy (KE) equation. We define the perturbation velocity
$\mathbf{u^{\prime}}$ and the perturbation buoyancy $b^{\prime}$ as the
difference between the flow field and the motionless base state. Following the
formulation of Joseph [13], the disturbance KE equation can be written as

$\frac{dE_{K}}{dt}=-\int_{V}\left(\mathbf{u}\cdot\mathbf{D}\cdot\mathbf{u}+\mathbf{g}\cdot\mathbf{u}b^{\prime}+\nu\nabla\mathbf{u}:\nabla\mathbf{u}\right)dV,$
(4)

where the kinetic energy
$E_{K}=0.5\int_{V}\left(\mathbf{u}\cdot\mathbf{u}\right)dV$, and $\mathbf{D}$
represents the strain rate tensor of the base flow. Because our base flow has
zero velocity and gravity acts only in the $y$ direction, we can simplify the
above equation to the following:

$\frac{dE_{K}}{dt}=\underbrace{-\int_{V}vb^{\prime}dV}_{\text{Buoyant
production}}\underbrace{-\int_{V}\nu\nabla\mathbf{u}:\nabla\mathbf{u}dV}_{\text{Viscous
dissipation}}.$ (5)

The budget analysis of the disturbance KE equation simplifies to the interplay
of two terms: the first signifies buoyant production, leading to an increase
in KE, while the second denotes KE dissipation due to viscous effects.

Figure 4 shows the disturbance KE budget terms for two different cases. The
first case is with $\Pi_{s}=0.9$ and $\Pi_{h}=1500$, which is the evolution
pictured in Figure 3 and exhibits the 3D instability self-organizing to the 2D
steady state. In order to contrast this with a more unstable case, we compare
it to a simulation with $\Pi_{s}=1.2$ and $\Pi_{h}=6000$, which remains 3D and
unsteady for the entire evolution. This can be seen in Figure 4(a) which shows
the evolution of disturbance KE for both cases.

Figure 4:  (a) Evolution of total disturbance kinetic energy (KE) over time
for two cases: $\Pi_{s}=0.9,\Pi_{h}=1500$ and $\Pi_{s}=1.2,\Pi_{h}=6000$.
$L_{z}=2$ in both cases. (b) Evolution of (absolute value of) buoyant
production and viscous dissipation terms (left axis) and the difference
between the curves (right axis) for $\Pi_{s}=0.9,\Pi_{h}=1500$ case. (c) shows
the same as (b) for the $\Pi_{s}=1.2,\Pi_{h}=6000$ case.

Figure 4(b) and 4(c) show the disturbance KE budget equation terms versus time
for the early evolution for the 0.9 $\Pi_{s}$ and 1.2 $\Pi_{s}$ cases,
respectively. Focusing first on the 0.9 $\Pi_{s}$ case, we see that the
absolute value of the buoyant production and viscous dissipation terms follow
a very similar trend. However, we note that there is a small difference
between the two curves that is critical for explaining the observed self
organization. The difference between the two curves is plotted on the right
vertical axis. When this difference is positive, it indicates that the buoyant
production exceeds the viscous dissipation, and $dE_{K}/dt$ increases. When
negative, the viscous dissipation exceeds the buoyant production and
$dE_{K}/dt$ decreases. From Figure 4(b), immediately before the peak in KE at
$t_{n}=1$, the buoyancy term dominates and causes the KE to increase. However,
at the peak of the KE at $t_{n}=1$, the viscous dissipation matches and then
exceeds the buoyant production term, causing the KE to decrease with time.
Thus, the driving mechanism of the initial exponential growth of the
instability is the buoyant production, and the nonlinear saturation of this
instability and subsequent decay is due to the increase of the viscous
dissipation as the velocity field continues to evolve. We also note that the
viscous dissipation aligns perfectly with the diffusion timescale, causing the
velocity to peak at a dimensionless time of exactly 1. While this is not
observed for all cases, it provides further evidence that the nonlinear
saturation of the instability depends upon the timescale of the diffusion to
“kick in” and cause the initial decay of the $w$ velocity.

The behavior of the 0.9 $\Pi_{s}$ case can be contrasted to the fully 3D,
unsteady 1.2 $\Pi_{s}$ case, shown in Figure 4(c), in which there is a
constant competition between the two budget terms, neither of which is seen to
dominate the other, leading to a 3D, unsteady flow state.

Returning to the evolution of the $w$ velocity over time shown in Figure 3(a),
after the nonlinear saturation of the instability and the initial quick decay
due to the effects of viscous dissipation, a slower decay of the $w$ velocity
begins, lasting from approximately $t_{n}=5$ to $t_{n}=25$. Additionally,
after a dimensionless time of approximately 28, the decay rate increases
substantially until the $w$ velocity vanishes in the final state. To explain
these decay rates, we perform secondary LSA for the 2D steady states.

The flow states during the initial slow exponential decay and the later faster
exponential decay are visualized with the Q-criterion at $t_{n}=20$ and
$t_{n}=34$ in Figures 3(c) and 3(d) respectively. We notice at $t_{n}=20$, the
flow resembles the steady, 2D symmetric state with the addition of a small 3D
perturbation. When 3D LSA is applied with the 2D symmetric steady state as the
base flow, we obtain a 3D eigenmode with a negative growth rate (or decay
rate) equal to $-3.54\times 10^{-4}$. This decay rate is depicted in Figure
3(a) by the red dashed line, and we find that this fairly accurately matches
the exponential decay in the $w$ velocity from a dimensionless time of
approximately 10 to 25. Similarly, Figure 3(d) closely matches the 2D
asymmetric steady state, and when we perform 3D LSA with the 2D asymmetric
steady state as the base flow, we find a 3D eigenmode with a negative growth
rate of $-3.25\times 10^{-3}$, and this decay rate is shown in Figure 3(a) by
the dashed green line. Additionally, we compare the shape of the two
eigenvectors with the 3D flow structure of our simulation at $t_{n}=20$ and
$t_{n}=34$, and we find good agreement in structures. This is described in
more detail in the Appendix. Thus, the decay of the $w$ velocity from a
dimensionless time of approximately 10 to the final 2D steady state can be
explained by the linear stability of the 2D steady states to 3D perturbations
caused by the initial 3D growth.

#### Conclusion —

We observe spontaneous growth of 3D motion in a simple convective flow system,
which, in the long-time evolution, self-organizes back to a 2D steady state
without a change of external parameters and any dependency on initial
perturbations. An additional case with random 3D perturbation in a domain with
a large spanwise extent, and an example of the same behavior in Rayleigh-
Bénard convection are included in the Appendix to further support our
discovery.

We attribute this self-organizing dynamics in convective systems to the
interplay between competing terms in the disturbance kinetic energy (KE), as
well as the linear stability of the eventual 2D steady-state. Initially,
viscous dissipation of KE lags behind buoyant production, but it eventually
dominates following the nonlinear saturation of the instability. This behavior
presents a tractable model to investigate similar phenomena such as transient
chaos in nonlinear dynamical systems [14]. Aside from DNS of the Navier-Stokes
equations, current methods in hydrodynamic stability theory are not capable to
identify this path to self organization.

## Appendix

### Secondary stability analysis of 2D steady states in the V-shaped
triangular cavity

To confirm the decay rates of the three-dimensionality in the simulation of
our valley geometry, we perform secondary linear stability analysis for the
two different two-dimensional (2D) steady-state profiles, the symmetric and
asymmetric states. These analyses reveal stable 3D eigenmodes to both 2D
steady states, with decay rate of $-3.54\times 10^{-4}$ for the symmetric
state and $-3.25\times 10^{-3}$ for the asymmetric state. This indicates that
both 2D steady-state profiles are stable to any three-dimensional (3D)
disturbance, and this explains the exponential decay of the $w$ velocity we
see at later times of the simulation, as well as the sharp increase of the
rate of decay when the symmetric state transitions to the asymmetric state.

Figure 5(a) shows the eigenvector to the 2D symmetric steady state, and Figure
5(b) shows the disturbance velocity field at $t_{n}=20$ relative to the 2D
symmetric steady state. Similarly, Figure 5(c) shows the eigenvector to the 2D
asymmetric steady state, while Figure 5(d) shows the disturbance velocity
field at $t_{n}=32$ relative to the 2D asymmetric steady state. We notice that
the 3D structures are almost identical between the eigenvectors and the
disturbance velocity fields, and this indicates that the behavior observed in
our simulation at later times is consistent with the results of the secondary
stability analysis, and that a large amount of the decay of $w$ velocity we
observe in our simulation is due to the linear stability of the two 2D steady-
state profiles.

Figure 5:  (a) Visualization of the Q-criterion of the 3D stable eigenmode to
the 2D symmetric base flow, (b) Q-criterion of the disturbance velocity field
from DNS results relative to the 2D symmetric base flow at $t_{n}=20$. (c)
$x$-vorticity contours of the 3D stable eigenmode to the 2D asymmetric base
flow, and (d) $x$-vorticity contours of the disturbance velocity field from
DNS results relative to the 2D asymmetric base flow at $t_{n}=32$. Contours
are colored by normalized $w$ velocity. Axes are labeled in (a).

### Results for a domain with a long span-wise extent and with initial random
disturbance

In Figure 6, we show the variation of the growth rate of the primary 3D
instability with the length of the homogeneous direction $L_{z}$, We can
observe that the optimal $L_{z}$ for the 3D instability is approximately 1.6.

Figure 6:  Growth rate of 3D instability versus $L_{z}$.

In order to ensure our results are not impacted by a limited extent in the $z$
direction, or the initial disturbance, we perform an additional simulation
with $L_{z}=16$, and an initial random perturbation generated by the Gaussian
noise generator implemented in Nektar++. We observe the same spontaneous
growth of a 3D instability before the 3D structures self-organize back to the
final 2D steady state. Visualizations of the flow field at six different times
in the evolution are shown in Figure 7. Movies of the full transition for both
$2L_{z}$ and $16L_{z}$ cases are attached with the Supplemental Material as
Movie 1 and 2. When comparing the flow structures pictured here to the flow
structures observed in the 2 $L_{z}$ case, while they are similar (for
example, compare Figure 7(b) shown here and Figure 3(b) of main paper), we see
that the structures in the 16 $L_{z}$ case are significantly more disordered,
which we suspect is a consequence of the initial random perturbation in this
case. However, this more clearly shows the self-organization behavior as this
relatively more disordered state transitions to the same ordered, 2D state as
in the 2 $L_{z}$ case. The results of this simulation give us confidence that
the spontaneous growth of 3D structures and subsequent decay of three-
dimensionality is not influenced by the extent of the $z$ direction in our
simulations or the initial perturbation.

Figure 7:  Visualization of the Q-criterion over time for DNS of the case
$\Pi_{s}=0.9$, $\Pi_{h}=1500$, and $L_{z}=16$. Initial conditions are given as
a small amplitude random noise, depicted in (a). Subsequent non-dimensional
times are (b) 1, (c) 1.5, (d) 2.5, (e) 4, and (f) 10. Contours of Q-criterion
are taken as 4% of the maximum, and are colored by the normalized $w$
velocity.

### Additional example from Rayleigh-Bénard convection

To further demonstrate that the spontaneous onset of 3D motion followed by a
self organization into a 2D steady state behavior is not confined to the
special problem that we presented in our Letter, we present similar results
obtained from the classical Rayleigh-Bénard convection. While both systems
display similar behavior, convection in a V-shaped cavity exhibits this
behavior more strongly, due to greater amplification of the initial
disturbance and having a 3D eigenmode as the dominant instability.

We consider the Rayleigh-Bénard convection problem of a layer of fluid between
two rigid walls with fixed temperature conditions, with a higher temperature
on the bottom wall, and periodic boundary conditions in the $x$ and $z$
directions. It is well-known that under these conditions the only stable
solution is the form of two-dimensional (2D) rolls with appropriate Rayleigh
number, $Ra$, and that the onset of secondary, oscillatory instabilities does
not occur until $Ra$ exceeds approximately 5500, depending on Prandtl number
and wavenumbers [15]. Here, we consider the case of $Ra=5000$ and $Pr=1$.

Similar to our V-shaped triangular cavity, we first apply linear stability
analysis (LSA) with the quiescent base state. Our results provide one 2D
eigenmode corresponding directly to the 2D rolls. However, as shown by
Chandrasekhar [16], because we have two homogeneous directions, any 2D
eigenmode can be transformed into a 3D eigenmode with the same growth rate.
This is what we observe in our LSA, as we also obtain a 3D eigenmode with
square circulation cells and the same growth rate as the 2D mode.

We perform 3D Navier-Stokes simulations of this case with the initial
disturbance defined by a small multiple of the 3D eigenmode. The results of
this simulation are shown in Figure 8. Similar to the case in the heated
triangular enclosure, we observe an initial exponential increase of the $w$
velocity, representing the three-dimensionality of the flow state. This
initial increase is expected from the linear theory, and leads to the 3D state
depicted in Figure 8(b) at $t=5$ seconds. However, after the nonlinear
saturation of the 3D instability, we observe the $w$ velocity begin to
exponentially decay until the flow state returns to the 2D rolls. Just like in
the triangular enclosure, this can be explained due to the fact that the 2D
rolls are stable to any 3D disturbances, and the 2D rolls are the only steady
state in this configuration at the given $Ra$. This can also be shown with the
disturbance kinetic energy (KE) budget terms, shown in Figure 9. Similar to
the valley case, the initial increase in KE is due to the dominance of the
buoyant production term at early times, but the viscous dissipation term
eventually catches up and exceeds the buoyant production, causing the decay of
KE.

Figure 8:  (a) Evolution of volume-averaged $w^{2}$ over time for Rayleigh-
Bénard convection with $Ra=5000$, $Pr=1$, and $L_{x}=L_{z}=2$. (b)
Visualization of Q-criterion at $t=5$ s, and (c) at $t=30$ s. Q-criterion
contours are approximately 4% of maximum value, and are colored by the $w$
velocity. Figure 9:  Evolution of disturbance kinetic energy budget terms over
time for the Rayleigh-Bénard case shown in Figure 8. Left axis shows buoyant
production and viscous dissipation terms, and right axis shows the difference
between these curves.

While this behavior is very similar to what we observe in the heated
triangular enclosure, there are also some important differences. Most
importantly, in the Rayleigh-Bénard case, the 3D eigenmode has the same growth
rate as the 2D eigenmode. This indicates that the path through the 3D state is
not necessarily preferred, and that with a small random disturbance, the
growth of the 2D mode is equally likely. We observe this in some of our
simulations. In contrast, in the valley geometry, the 3D mode has a larger
growth rate than either of the 2D modes. This means that the “fastest” path to
the 2D steady state is through the 3D eigenmode, and with a random 3D
perturbation, the 3D mode is chosen over either of the 2D modes. In this way,
the case in the valley geometry displays a “stronger” form of this behavior
than that observed in Rayleigh-Bénard convection.

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with $ m$ itself in $\mathcal{L}$. Then by <ref> we know that the diagonalization of the underlying pseudosquare $ (m,f, \alpha)$ in $\mathcal{C}$ can be used to produce the diagonalization in the pseudoslice $ \mathcal{C}$, with the 2-cell below it being the composite

\[\begin{tikzcd}
& A \\
{A_2} & B \\
&& C
\arrow[""{name=0, anchor=center, inner sep=0}, "{l_\alpha}", from=2-1, to=1-2]
\arrow["r"{description}, from=1-2, to=2-2]
\arrow["f"{description}, from=2-1, to=2-2]
\arrow["b"{description}, from=2-2, to=3-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "a", curve={height=-18pt}, from=1-2, to=3-3]
\arrow[""{name=2, anchor=center, inner sep=0}, "{a_2}"', curve={height=18pt}, from=2-1, to=3-3]
\arrow["{\rho_\alpha \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\arrow["{\lambda \atop \simeq}"{description}, Rightarrow, draw=none, from=1, to=2-2]
\arrow["{\beta \atop \simeq}"{description}, Rightarrow, draw=none, from=2-2, to=2]
\end{tikzcd}\]

but as $ \lambda_\alpha$ is invertible, we have that $ l_\alpha$ is in $\mathcal{L}$ by right cancellation. Hence the 2-cell above defines a left map in $\mathcal{L}_C$ in the pseudoslice $\mathcal{C}/C$. Moreover the 2-dimensional property of the generic morphisms follows from the 2-dimensional property of the diagonalization.

Finally, for the factorization is inherited in $\mathcal{C}/C$, we have, as seen in <ref>, that a morphism in the pseudoslice factorizes as below

\[\begin{tikzcd}[sep=large]
A & {A_f} & B \\
& C
\arrow["{l_f}", from=1-1, to=1-2]
\arrow["{r_f}", from=1-2, to=1-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "a"', from=1-1, to=2-2]
\arrow["{a_f}"{description}, from=1-2, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "b", from=1-3, to=2-2]
\arrow["{\lambda_{(f,\alpha)} \atop \simeq}"{description, pos=0.4}, Rightarrow, draw=none, from=1-2, to=0]
\arrow["{\rho_{(f,\alpha)}  \atop \simeq}"{description, pos=0.4}, Rightarrow, draw=none, from=1-2, to=1]
\end{tikzcd}\]

where $ l_f, r_f$ is the $(\mathcal{L},\mathcal{R})$ factorization of $f$ in $\mathcal{C}$. Hence if $ a$ is in $\mathcal{L}$, so is $ a_f$ by right cancellation, and $ \rho_{(f,\alpha)} : a_f \rightarrow b$ is dual to a generic morphism. This proves the inclusion above to be bistable.

§.§ Specification of left objects

In the examples listed at the end of this work, the bistable inclusion obtained from bifactorizations systems does not encode by themselves exactly what we would expect from a geometry: indeed, those bifactorizations systems are all part of a “connected-truncated" paradigm where the left maps code for some notion of connectedness rather than punctualness, and are somewhat too coarse to represent focal data. However, this can be corrected by choosing, amongst left maps, a subclass of left maps with a convenient cancellation property which will ensure that one can restrict to left objects with a specified left map as local objects, and arbitrary left maps between them. In particular, while left maps are seldom - but sometime, see below the hyperconnected-localic case - stable under bipullbacks, we can choose the specified left maps to be so to encode more local-like behavior.

Let be $ \mathcal{L}$ a class of 1-cell in a bicategory $ \mathcal{C}$ and $ \mathcal{L}' \subseteq \mathcal{L}$ a subclass. We say that $ \mathcal{L}'$ has right cancellation along $\mathcal{L}$-maps if for any invertible 2-cell as below

\[\begin{tikzcd}
A && B \\
& C
\arrow[""{name=0, anchor=center, inner sep=0}, "l", from=1-1, to=1-3]
\arrow["a"', from=1-1, to=2-2]
\arrow["f", from=1-3, to=2-2]
\arrow["{\alpha \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

where $ a$ is in $\mathcal{L}'$ and $ l$ is in $\mathcal{L}$ then $f$ is in $\mathcal{L}'$.

Again, in the presence of a biterminal object, we can define a class of $ \mathcal{L}'$-objects as those objects whose terminal map is in $\mathcal{L}'$, and also the relativized version over arbitrary objects. The key idea is that left cancellation along $\mathcal{L}$-maps encodes exactly the gliding condition required to have bistable factorization. In the following we denote as ${\mathcal{L}'\hy\textup{\textbf{Obj}}^{\mathcal{L}}}$ (resp. $\mathcal{L}'\hy\textup{\textbf{Obj}}^{\mathcal{L}}$) the (1,2)-full sub-bicategory of $ \mathcal{L}\hy\textbf{Obj}$ (resp. $ \mathcal{L}\hy\textbf{Obj}_C$) whose objects are $\mathcal{L}'$-objects (resp. $ \mathcal{L}'$-maps with codomain $C$).

Let be $ (\mathcal{L},\mathcal{R})$ a bifactorization system on a bicategory $\mathcal{C}$ and $\mathcal{L}' \subseteq \mathcal{L}$ with right cancellation along $\mathcal{L}$-maps. Then:

* if $\mathcal{C}$ has a biterminal object and $ \mathcal{L}'$ contains identities then we have a bistable inclusion

\[\begin{tikzcd}
({\mathcal{L}'\hy\textup{\textbf{Obj}}^{\mathcal{L}})^{\op}} & {\mathcal{C}^{\op}}
\arrow["{\iota_\mathcal{L}^{\op}}", hook, from=1-1, to=1-2]
\end{tikzcd}\]

* For each object $C$ we have a bistable inclusion

\[\begin{tikzcd}
{(\mathcal{L}'\hy\textup{\textbf{Obj}}^{\mathcal{L}_C}_C)^{\op}} & {\mathcal{C}/C^{\op}}
\arrow["{\iota_{\mathcal{L},C}^{\op}}", hook, from=1-1, to=1-2]
\end{tikzcd}\]

Again the first item is a specific case of the second item. The generic morphisms are now dual to invertible 2-cells

\[\begin{tikzcd}
A && B \\
& C
\arrow[""{name=0, anchor=center, inner sep=0}, "r", from=1-1, to=1-3]
\arrow["a"', from=1-1, to=2-2]
\arrow["b", from=1-3, to=2-2]
\arrow["{\alpha \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

with $ a$ in $\mathcal{L}'$ and $ r$ in $\mathcal{R}$: the genericness is a restriction of the genericness of right maps toward left objects. Now just use the right cancellation of $\mathcal{L}'$-objects along $ \mathcal{L}$-map to ensure that the $ (\mathcal{L}_C, \mathcal{R}_C)$ factorization the middle object is in $\mathcal{L}'$.

§ BIFACTORIZATION GEOMETRIES FOR GROTHENDIECK TOPOI

In this section, we apply <ref> to produce bistable sub-bicategories of the opposite bicategory of Grothendieck topoi - that are, bistable sub-bicategories of the bicategory of logoi.

We investigate examples for the following bifactorizations systems:

* $(\textbf{Hyp}, \textbf{Loc})$ in $\GTop$, the hyperconnected-localic bifactorization system on the bicategory of Grothendieck topoi and geometric morphisms

* $ (\textbf{Co}, \textbf{Et})$ in $\textbf{Loco}\GTop$, the connected-etale bifactorization system on the bicategory of locally connected Grothendieck topoi and locally connected geometric morphisms; this bifactorization system is to be used together with the following specifications of left maps

* $\textbf{Foc}$, the class of local geometric morphism

* $\textbf{Toc}$, the class of totally connected geometric morphisms

* $(\textbf{TCo}, \textbf{Et})$ on $\textbf{Loco}\GTop^{\textbf{Ess}}$, the terminally connected-etale bifactorization system on the bicategory of locally connected Grothendieck topoi and essential geometric morphisms; this bifactorization system generalizes the previous one and is to be used with the same specifications as above.

* Though we do not know yet a characterization of the right class, we prove there exists an orthogonal bifactorization in $ \GTop$ with terminally connected geometric morphisms on the left; this is also to be used with the local and totally connected specification.

* $(\textbf{Surj}, \textbf{Emb})$ in $\GTop$, the surjection-embedding factorization system in the bicategory of Grothendieck topoi.

The key idea of this section relies on the observation that the geometric properties of Grothendieck topoi are often encoded in their terminal map above $\Set$.

In all the following examples, we will end with bistable factorization from (opposite bicategories of) left objects and left maps - with eventual specification of left objects - with the right maps toward left objects as generic morphisms. However it is interesting to notify that in none of those examples, the right maps are lax-generic; in fact, those factorization systems all are instances of bifactorization systems with right-truncated lax-factorizations, so that in particular diagonalization of lax squares returns diagonal maps in the right class, not in the left class. This seems to be related to the fact that all those factorization systems are part of a “connected-truncated" scheme where the right class exhibits some level of truncation as opposed to the left class, so that 2-dimensional data are “expelled" on the left side in the factorization of globular 2-cells. We do not know currently a bifactorization systems for geometric morphisms such that the duals of right maps have the lax-generic property.

We also conjecture that the terminally connected-etale bifactorization generalizes to a bifactorization system on the category of Grothendieck topoi without restrictions on geometric morphisms, where the left maps will be part of a wider class of pro-etale geometric morphisms; this factorization would be probably related to the connected-disconnected factorization mentioned in <cit.>[3.2.15], and will certainly involve a generalization of Grothendieck-Verdier localization. However we do not know yet whether the local geometric morphisms have right cancellation along terminally connected geometric morphisms in the general case - nor for the totally connected geometric morphisms.

§.§ The hyperconnected-localic geometry

This is a situation where the left maps are both sufficient to encode the local objects and the map between them for they enjoy good properties. We first recall some generalities about those two classes of maps.

Let $\mathcal{E}$ be a topos; a subquotient of an object $E$ is an object $X$ related to $E$ by either a span or a cospan as follows

\[\begin{tikzcd}
S & X \\
\arrow["m"', tail, from=1-1, to=2-1]
\arrow["q", two heads, from=1-1, to=1-2]
\end{tikzcd} \hskip1cm
% https://q.uiver.app/?q=WzAsMyxbMSwwLCJYIl0sWzEsMSwiUSJdLFswLDEsIkUiXSxbMiwxLCJxIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzAsMSwibSIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV1d
\begin{tikzcd}
& X \\
E & Q
\arrow["q", two heads, from=2-1, to=2-2]
\arrow["m", tail, from=1-2, to=2-2]
\end{tikzcd}\]

with $m$ a mono and $q$ an epi.

Those two conditions are equivalent: if $X$ a subobject of a quotient of $E$, the pullback exhibits $X$ as a quotient of a subobject by stability of epi and mono under pullback in a topos, for topos are regular categories; if $X$ is a quotient of a subobject, the pushout exhibits $X$ as a subobject of a quotient by stability of mono and epi under pushout for topos are adhesive categories.

A geometric morphism $ f : \mathcal{E} \rightarrow \mathcal{F}$ is said to be

* hyperconnected if it is connected ($f^*$ is full and faithful) and the essential image of $ f^*$ is closed in $\mathcal{E}$ under subquotient

* localic if the essential image of $f^*$ generates $ \mathcal{E}$ under subquotient.

Localic morphisms are related to the notion of internal locale; we choose here not to define the notion of internal locale, as the internale version of the completenes under arbitrary suprema is tiresome to express; for this we refer to <cit.>[C.1]. For any Grothendieck topos $ \mathcal{E}$ we can define the locally posetal 2-category $ \textbf{Loc}[\mathcal{E}]$ of internal locales in $\mathcal{E}$; in particular internal locales in $\Set$ are ordinary locales. For any topos, the subobject classifier is in particular an internal locale. Similarly, it is well known that the direct image part of a geometric morphism lift to the 2-categories of internal locales.

A geometric morphism $f : \mathcal{E} \rightarrow \mathcal{F}$ is localic if and only if there exists an internal local $ L_f $ in $\textbf{Loc}[\mathcal{F}]$ such that one has a geometric equivalence together with an invertible 2-cell as below

\[\begin{tikzcd}
{\mathcal{E}} && {\Sh_\mathcal{F}(L_f)} \\
& {\mathcal{F}}
\arrow[""{name=0, anchor=center, inner sep=0}, "f"', from=1-1, to=2-2]
\arrow["\simeq"{description}, from=1-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{l_f}", from=1-3, to=2-2]
\arrow["\simeq", Rightarrow, draw=none, from=0, to=1]
\end{tikzcd}\]

In particular we have an equivalence of categories

\[ \textbf{Loc}/\mathcal{F} \simeq \textbf{Loc}[\mathcal{F}]  \]

Hyperconnected geometric morphisms are left bi-orthogonal to localic geometric morphisms.

Hyperconnected and localic geometric morphisms from a bifactorization system on $\GTop$: any geometric morphism factorizes in an essentially unique way as an hyperconnected geometric morphism followed by a localic geometric morphism.

The factorization can be alternatively described in two ways:

* from the point of view of the subquotients, the factorization is obtained from the factorization of the inverse image part in $\Cat$

\[\begin{tikzcd}
{\mathcal{E}} && {\mathcal{F}} \\
& {\overline{\textup{Im} f}}
\arrow[""{name=0, anchor=center, inner sep=0}, "{f^*}"', from=1-3, to=1-1]
\arrow["{l_f^*}", from=1-3, to=2-2]
\arrow["{h_f^*}", hook, from=2-2, to=1-1]
\arrow["{\alpha_f \atop \simeq}"{description}, Rightarrow, draw=none, from=2-2, to=0]
\end{tikzcd}\]

where $ \overline{\textup{Im} f}$ is the closure of the essential image of $ f^*$ in $\mathcal{E}$ under subquotient, which can be proven directly to be a Grothendieck topos.

* from the point of view of internal locales, the factorization is obtained from the direct image of the subobject classifier as

\[\begin{tikzcd}
{\mathcal{E}} && {\mathcal{F}} \\
& {\Sh_\mathcal{F}(f_*\Omega_\mathcal{E})}
\arrow[""{name=0, anchor=center, inner sep=0}, "f", from=1-1, to=1-3]
\arrow["{l_f}"', from=2-2, to=1-3]
\arrow["{h_f}"', from=1-1, to=2-2]
\arrow["{\alpha_f \atop \simeq}"{description}, Rightarrow, draw=none, from=2-2, to=0]
\end{tikzcd}\]

That is, $ f_*\Omega_\mathcal{E}$ is the desired internal local corresponding to the localic geometric morphism $l_f$.

A Grothendieck topoi will be said to be localic if it is of the form $ \Sh(L)$ for $L$ a locale - that is, an internal locale in $\Set$. In the following, we denote as $ \textbf{LocGTop}$ the bicategory of localic Grothendieck topoi and localic geometric morphisms between them.

In particular, observe that a Grothendieck topos is localic if and only if its terminal map $ !_\mathcal{E} : \mathcal{E} \rightarrow \Set$ is localic. For any topos, the global section functor $ \Gamma_\mathcal{E} = !_{\mathcal{E}*}$ sends the subobject classifier $\Omega_\mathcal{E}$ to a locale in $\Set$ which we also denote in the same way.

The following standard fact can be exhibited as a corollary of <ref> for localic topoi are the right objects for the hyperconnected-localic bifactorization system:

We have a (1,2)-full, bireflective, sub-bicategory

\[\begin{tikzcd}
{\textup{\textbf{LocGTop}}} && \GTop
\arrow[""{name=0, anchor=center, inner sep=0}, bend right=30, hook, from=1-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{\textbf{L}}"', bend right=30, from=1-3, to=1-1]
\arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=1, to=0]
\end{tikzcd}\]

where the pseudofunctor $ \textbf{L}$, known as the localic reflection, sends a Grothendieck topos $ \mathcal{E} $ on the localic topos $ \Sh(\Omega_\mathcal{E})$. This generalizes, for any Grothendieck topos $ \mathcal{E}$, as (1,2)-full bireflective sub-bicategory

\[\begin{tikzcd}
{\textup{\textbf{Loc}}/\mathcal{E}} && \GTop/\mathcal{E}
\arrow["\iota_\mathcal{E}"', ""{name=0, anchor=center, inner sep=0}, bend right=30, hook, from=1-1, to=1-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{\textbf{L}_\mathcal{E}}"', bend right=30, from=1-3, to=1-1]
\arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=1, to=0]
\end{tikzcd}\]

where the pseudofunctor $ \textbf{L}_\mathcal{E}$, known as the localic reflection, sends a geometric morphism $ f : \mathcal{F} \rightarrow \mathcal{E} $ on the localic geometric morphism $ \Sh_\mathcal{E}(f_*(\Omega_\mathcal{F}))$.

Now let us look to left objects:

A Grothendieck topos is said to be hyperconnected if its terminal map $ !_\mathcal{E} : \mathcal{E} \rightarrow \Set$ is hyperconnected.

A Grothendieck topos is hyperconnected if and only if its localic reflection is $ \Set$.

Beware that we do not claim however that any geometric morphism which is sent to an equivalence the localic reflection is hyperconnected. We do not know how to characterize the class of geometric morphism that are localized by the localic reflection - as characterizing the morphisms that are localized by a reflection can be a difficult problem even in the 1-dimensional case.

Now by the gliding property of left objects, we know that a Grothendieck topos which is the codomain of a hyperconnected geometric morphism with hyperconnected domain is itself hyperconnected.

We have a bistable inclusion

\[\begin{tikzcd}
{({\textbf{\textup{HypGTop}}^{\textbf{\textup{Hyp}}}})^{\op}} & {\GTop^{\op}}
\arrow[hook, from=1-1, to=1-2]
\end{tikzcd}\]

of the dual of the inclusion of the bicategory of hyperconnected Grothendieck topoi and hyperconnected geometric morphisms between them is dual to a bistable pseudofunctor

This suggests that hyperconnected topoi can also play the role of focal components of a geometry.

The hyperconnected-localic factorization system has oplax right truncated factorization of 2-cells.

Let be a globular 2-cell $ \sigma : f_1 \Rightarrow f_2$ in $\GTop$: this corresponds to a pair of adjoint 2-cell between the inverse and direct image parts, respectively $ \sigma^\flat : f_1^* \Rightarrow f^*_2$ and $ \sigma^\sharp : f_{2*} \Rightarrow f_{1*}$; in particular the direct image part $ \sigma^\sharp $ lifts to the induced direct image part functors for internal locales, producing a morphism $ \sigma^\sharp_{\Omega_\mathcal{E}} : f_{2*}(\Omega_\mathcal{E}) \rightarrow f_{1*}(\Omega_\mathcal{E})$ in $\textbf{Loc}[\mathcal{F}]$, which is sent by the localic reflection to a morphism in the pseudoslice over $\mathcal{F}$

\[\begin{tikzcd}
{\Sh(f_{2*}(\Omega_\mathcal{E}))} && {\Sh(f_{1*}(\Omega_\mathcal{E}))} \\
& {\mathcal{F}}
\arrow["{l_{f_2}}"', from=1-1, to=2-2]
\arrow["{l_{f_1}}", from=1-3, to=2-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{\Sh(\sigma^\sharp_{\Omega_\mathcal{E}})}", from=1-1, to=1-3]
\arrow["\alpha_{\sigma^\sharp_{\Omega_{\mathcal{E}}}} \atop \simeq"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

§.§ Connected-Etale bifactorization of locally connected geometric morphisms

Here we investigate a restricted version of what should actually be a more general geometry related to the construction of spectra.

A geometric morphism $ f : \mathcal{E} \rightarrow \mathcal{F}$ is said to be locally connected if for any $ F$ in $\mathcal{F}$ the slice of the inverse image part $ (f/A)^* : \mathcal{F}/F \rightarrow \mathcal{E}/f^*F$ is cartesian closed. Equivalently, this means that $f^* $ has a further left adjoint $f_!$ defining a $\mathcal{F}$-indexed functor. A Grothendieck topos $\mathcal{E}$ is locally connected if its terminal map is locally connected. In the following we denote as $ \textbf{LocoGTop}$ the 2-full sub-bicategory of locally connected Grothendieck topoi and locally connected geometric morphisms.

It can be shown that a Grothendieck topos is locally connected if its terminal map is essential. This condition is also equivalent to require that any object $ E$ is a coproduct of connected objects indexed by the set $f_!(E)$ of connected components.

A geometric morphism $f : \mathcal{E} \rightarrow \mathcal{F}$ is said to be

* connected if $ f^*$ is full and faithful

* etale if there exists an object $ F$ in $\mathcal{F} $ such that one has an invertible 2-cell

\[\begin{tikzcd}
{\mathcal{E}} && {\mathcal{F}/F} \\
& {\mathcal{F}}
\arrow["f"', from=1-1, to=2-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "\simeq", from=1-1, to=1-3]
\arrow["{\pi_F}", from=1-3, to=2-2]
\arrow["\simeq"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

As localic morphisms were indexed by internal locales, etale geometric morphisms are indexed by objects, that is, we have an equivalence of categories

\[ \textbf{Et}/\mathcal{E} \simeq \mathcal{E} \]

In fact etale geometric morphisms are locally connected; however beware that not all connected geometric morphisms are locally connected.

Connected geometric morphisms are left biorthogonal to etale geometric morphisms.

Beware that not any geometric morphism that is right biorthogonal to connected geometric morphism has to be etale: for instance, 2-cofiltered limits of etale geometric morphisms are so, though they are not etale, not even locally connected.

It is well known that the bicategory of locally connected topoi and geometric morphisms enjoy the following factorization system:

Any locally connected geometric morphism factorizes uniquely as a connected, locally connected geometric morphism followed by an etale geometric morphism as follows:

\[\begin{tikzcd}
{\mathcal{E}} && {\mathcal{F}} \\
& {\mathcal{F}/f_!(1)}
\arrow["{c_f}"', from=1-1, to=2-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "f", from=1-1, to=1-3]
\arrow["{\pi_{f!(1)}}"', from=2-2, to=1-3]
\arrow["\simeq"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

where $ (c_f)_!$ sends $ E $ to the image of the terminal map $ f_!(!_E) : f_!(E) \rightarrow f_!(1)$.

The connected-etale factorization on $\textup{\textbf{LocoGTop}}$ has pseudofunctorial oplax right truncated factorization of 2-cells.

Let be $ \sigma : f_1 \Rightarrow f_2$ a 2-cell between locally connected geometric morphisms. Then the inverse image part $ \sigma^\flat : f_1^* \Rightarrow f_2^*$ has a mate $ \sigma_! : f_{2!} \Rightarrow f_{1*}$, whose component $ \sigma_{!1_\mathcal{E}} : f_{2!}(1_\mathcal{E}) \rightarrow f_{1!}(1_\mathcal{E})$ at $ 1_\mathcal{E}$ is sent to an invertible 2-cell between etale geometric morphism over $\mathcal{F}$

\[\begin{tikzcd}
{\mathcal{F}/f_{2!}(1_\mathcal{E})} && {\mathcal{F}/f_{1!}(1_\mathcal{E})} \\
& {\mathcal{F}}
\arrow["{\pi_{f_{2!}(1_\mathcal{E})}}"', from=1-1, to=2-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathcal{F}/\sigma_{!1_\mathcal{E}}}", from=1-1, to=1-3]
\arrow["{\pi_{f_{1!}(1_\mathcal{E})}}", from=1-3, to=2-2]
\arrow["\simeq"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

where the intermediate geometric morphism has postcomposition with $ f_!(!_E)$ as left adjoint to the inverse image part. Now we have by naturality of $\sigma_!$ a square in $\mathcal{F}$ for each $E$ in $\mathcal{E}$

\[\begin{tikzcd}
{f_{2!}(E)} & {f_{2!}(1_\mathcal{E})} \\
{f_{1!}(E)} & {f_{1!}(1_\mathcal{E})}
\arrow["{\sigma_{!E}}"', from=1-1, to=2-1]
\arrow["{\sigma_{!1_{\mathcal{E}}}}", from=1-2, to=2-2]
\arrow["{f_{2!}(!_E)}", from=1-1, to=1-2]
\arrow["{f_{1!}(!_E)}"', from=2-1, to=2-2]
\end{tikzcd}\]

which provides us with a 2-cell

\[\begin{tikzcd}
& {\mathcal{E}} \\
{\mathcal{F}/f_{2!}(1_\mathcal{E})} && {\mathcal{F}/f_{1!}(1_\mathcal{E})}
\arrow["{c_{f_2}}"', from=1-2, to=2-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathcal{F}/\sigma_{!1_\mathcal{E}}}"', from=2-1, to=2-3]
\arrow["{c_{f_1}}", from=1-2, to=2-3]
\arrow["{\lambda_\sigma \atop \Leftarrow}"{description}, Rightarrow, draw=none, from=0, to=1-2]
\end{tikzcd}\]

Those 2-cells can be shown to provide with a decomposition of $\sigma$, by observing that their adjoint part provide such an adjoint decomposition of $\sigma_!$. Similarly, pseudofunctoriality processes from pseudofunctoriality of the externalization functor.

We have a bistable inclusion

\[\begin{tikzcd}
{(\textup{\textbf{CoLocoGTop}}^{\textup{\textbf{Co}}})^{\op}} & {\textup{\textbf{Loco}}\GTop^{\op}}
\arrow[hook, from=1-1, to=1-2]
\end{tikzcd}\]

of the dual of the inclusion of the bicategory of connected, locally connected topoi and connected, locally connected geometric morphisms between them into the bicategory of locally connected Grothendieck topoi and locally connected geometric morphisms between them.

Beware that we must restrict also the objects to locally connected topoi in order to preserve terminalness of $\Set$.

However, connected geometric morphisms are not stable under bipullback, and cannot be seen as a notion of focal objects as lacking a focal point. However, there are two classes of geometric morphisms that are known to have right cancellation relatively to connected geometric morphisms: local and totally connected ones.

§.§ Terminally connected-Etale bifactorization of essential geometric morphisms

A generalization of the previous bifactorization system was recently described in <cit.>[4.7], where one relax the condition of local connectedness to essentialness: there one can still use the further left adjoint to the inverse image part to produce a factorization, but the left maps are in a wider class generalizing connected geometric morphisms.

A geometric morphism $ f : \mathcal{E} \rightarrow \mathcal{F}$ is essential if $f^*$ has a further left adjoint $ f_!$.

One can show that the terminal map of a Grothendieck topos $!_\mathcal{E} : \mathcal{E} \rightarrow \Set$ is automatically locally connected whenever it is essential: hence over set, there is no distinction between locally connected topoi and essential topoi.

A geometric morphism $f : \mathcal{E} \rightarrow \mathcal{F}$ is sais to be terminally connected if its inverse image $ f^*$ lifts uniquely global elements, that is, if one has for each $ F$ a natural equivalence

\[  \mathcal{E}[1_\mathcal{E}, f^*(F)] \simeq \mathcal{F}[1_\mathcal{F}, F]  \]

The following observation is immediate by adjunction:

An essential geometric morphism is terminally connected if and only if one has $ f_!(1_\mathcal{E}) \simeq 1_\mathcal{F}$.

A connected geometric morphism is terminally connected ; beware that despite the terminology, terminal connectedness is a weaker condition that connectedness, as it requires only full faithfulness relatively to global elements and not arbitrary generalized elements.

Now we should give a few remarks about etale geometric morphisms. Being locally connected, they are always essential, with the further left adjoint of the projection $ \pi_{E} : \mathcal{E}/E \rightarrow \mathcal{E}$ given by postcomposition with the terminal map $ !_E : E \rightarrow 1_\mathcal{E}$. Moreover, we should give here an interpretation of what encodes a factorization through an etale geometric morphism on the right. Suppose we have an invertible 2-cell

\[\begin{tikzcd}
{\mathcal{F}} && {\mathcal{E}} \\
& {\mathcal{E}/E}
\arrow["a"', from=1-1, to=2-2]
\arrow["{\pi_E}"', from=2-2, to=1-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "f", from=1-1, to=1-3]
\arrow["\simeq"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

Then we have a factorization of $a$ through the base change

\[\begin{tikzcd}
{\mathcal{F}} \\
& {\mathcal{F}/f^*E} & {\mathcal{E}/E} \\
& {\mathcal{F}} & {\mathcal{E}}
\arrow["{\pi_E}", from=2-3, to=3-3]
\arrow["f"', from=3-2, to=3-3]
\arrow[from=2-2, to=2-3]
\arrow[curve={height=12pt}, Rightarrow, no head, from=1-1, to=3-2]
\arrow["{\pi_{f^*E}}"{description}, from=2-2, to=3-2]
\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3]
\arrow["a"{description}, curve={height=-12pt}, from=1-1, to=2-3]
\arrow[dashed, from=1-1, to=2-2]
\end{tikzcd}\]

which also defines a global section of $ \pi_{f^*E}$, which is the name of a global element $ a : 1_\mathcal{F} \rightarrow f^*E$: that is

\[ \GTop/\mathcal{E}[f,\pi_E] \simeq \mathcal{F}[1_\mathcal{F},f^*E] \]

Terminally connected geometric morphisms are left orthogonal to etale geometric morphisms.

Let be an invertible square as below

\[\begin{tikzcd}
{\mathcal{G}} & {\mathcal{E}/E} \\
{\mathcal{F}} & {\mathcal{E}}
\arrow["t"', from=1-1, to=2-1]
\arrow["f"', from=2-1, to=2-2]
\arrow["a", from=1-1, to=1-2]
\arrow["{\pi_{E}}", from=1-2, to=2-2]
\arrow["\simeq"{description}, draw=none, from=1-1, to=2-2]
\end{tikzcd}\]

corresponding to a global element $ a : 1_\mathcal{G} \rightarrow t^*f^*E  $ in $\mathcal{G}$. If $ t$ is terminally connected, then this element comes uniquely from a global element $ \overline{a} : 1_\mathcal{F} \rightarrow f^*E$ in $\mathcal{F}$, whose name is a diagonalization of the pseudosquare above $ \mathcal{F} \rightarrow \mathcal{E}/E$. As any other possible diagonalization would name another antecedent of $a$, this diagonalization has to be unique. The converse is exactly the converse reasoning.

Then the connected-etale factorization of locally connected geometric morphisms generalizes to essential geometric morphisms as below - this is <cit.>[Proposition 4.62], where the bifactorization is obtained exactly in the same way:

Any essential geometric morphism factorizes uniquely as a terminally connected geometric morphism followed by an etale geometric morphism: that is, we have a bifactorization system $ (\textbf{TCoEss}, \textbf{Et})$ on $\GTop^{\textbf{Ess}}$.

We have a bistable inclusion

\[\begin{tikzcd}
{(\textup{\textbf{CoLocoGTop}}^{\textup{\textbf{TCo}}})^{\op}} & {(\textup{\textbf{Loco}}\GTop^{\textup{\textbf{Ess}}})^{\op}}
\arrow[hook, from=1-1, to=1-2]
\end{tikzcd}\]

of the dual of the inclusion of the bicategory of connected, locally connected Grothendieck topoi and terminally connected essential geometric morphisms between them into the bicategory of locally connected Grothendieck topoi and essential geometric morphisms between them.

Again, we must restrict to locally connected topoi to ensure terminalness of $\Set$, though the bifactorization still exists if we only restrict to essential geometric morphisms.

§.§ Local topoi as specified left objects

In the connected-etale and its generalization, the left map encode something a bit “coarse"; for connected - nor terminally connected - geometric morphisms are stable under bipullback, one might not want them to be local objects; however we can specify two kinds of left objects: local and totally connected topoi, the first specification being of special interest.

A geometric morphism $ f : \mathcal{F} \rightarrow \mathcal{E}$ is said to be localgeometric morphism!local if its inverse image part $f^*$ is full and faithful and moreover the direct image part $ f_*$ has a further right adjoint $ f^!$.

In particular, a Grothendieck topos $\mathcal{E}$ is said to be localtopos!local if the global section functor $ !_{\mathcal{E}*} : \mathcal{E} \rightarrow \mathcal{S}$ has a further right adjoint.

Observe that, for a Grothendieck topos to be local, the full-and-faithfulness condition is automatic as soon as the existence of the further left adjoint is established: indeed, for $\Set$ is the terminal topos, the pair $ \gamma_{\mathcal{E}*} \dashv !_{\mathcal{E}}^!$ must form a retract of $ !_\mathcal{E}$ in the bicategory of Grothendieck topoi: hence we must automatically have natural isomorphisms $ !_{\mathcal{E}*}!_{\mathcal{E}}^* \simeq \textup{id}_{\Set}$ and $ !_{\mathcal{E}*}!_{\mathcal{E}}^! \simeq \textup{id}_{\Set} \simeq !_{\mathcal{E}*}!_{\mathcal{E}}^*$, which force both $ !_{\mathcal{E}}^*$ and $!_{\mathcal{E}}^!$ to be full and faithful.

Now we have the following property over $\Set$ expressing why local topoi can serve as a specification of left objects:

Local topoi have the gliding property along terminally connected geometric morphisms.

Suppose we have a triangle as below in $\GTop$

\[\begin{tikzcd}
{\mathcal{L}} && {\mathcal{E}} \\
& \Set
\arrow["{!_\mathcal{L}}"', from=1-1, to=2-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "t", from=1-1, to=1-3]
\arrow["{!_{\mathcal{E}}}", from=1-3, to=2-2]
\arrow["\simeq"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

where $ !_\mathcal{L}$ is local and $t$ is terminally connected. By what precedes, it suffices to prove that $ !_{\mathcal{E}*}$ has a further right adjoint. Recall that the direct image part of the terminal map is the global section functor returning the set of global elements. But as $t$ is terminally connected one has for any $E$ in $\mathcal{E}$

\begin{align*}
!_{\mathcal{L}*}(t^*E) &\simeq \mathcal{L}[1_\mathcal{L}, t^*E] \\
&\simeq \mathcal{E}[1_\mathcal{E}, E] \\
&\simeq !_{\mathcal{E}*}(E)
\end{align*}

Hence for any $E$ in $\mathcal{E}$ and any $X$ in $\Set$ we have the following isomorphisms of homsets

\begin{align*}
\Set[!_{\mathcal{E}*}(E), X] &\simeq \Set[!_{\mathcal{L}*}(t^*E), X] \\
&\simeq \mathcal{L}[t^*E, !_{\mathcal{L}}^!(X)] \\
&\simeq \mathcal{E}[E, t_*!_{\mathcal{L}}^!(X)]
\end{align*}

Hence one can choose the composite $t_*!_{\mathcal{L}}^!$ as a further right adjoint to $ !_{\mathcal{E}*}$. As $!_\mathcal{E}$ is moreover connected by what precedes, it is actually local.

We have a bistable inclusion

\[\begin{tikzcd}
{(\textup{\textbf{LocoFocGTop}}^{\textup{\textbf{TCoEss}}})^{\op}} & {(\textup{\textbf{Loco}}\GTop^{\textup{\textbf{Ess}}})^{\op}}
\arrow[hook, from=1-1, to=1-2]
\end{tikzcd}\]

of the dual of the inclusion of the bicategory of locally connected local Grothendieck topoi and terminally connected essential geometric morphisms between them into the bicategory of locally connected Grothendieck topoi and essential geometric morphisms between them.

This restricts to a bistable inclusion

\[\begin{tikzcd}
{(\textup{\textbf{LocoFocGTop}}^{\textup{\textbf{Co}}})^{\op}} & {(\textup{\textbf{Loco}}\GTop)^{\op}}
\arrow[hook, from=1-1, to=1-2]
\end{tikzcd}\]

of the dual of the inclusion of the bicategory of local, locally connected Grothendieck topoi and connected, locally connected geometric morphisms between them into the bicategory of locally connected Grothendieck topoi and locally connected geometric morphisms.

Generic morphisms for this bistable inclusion are duals of the etale geometric morphisms $\pi_E : \mathcal{E}/E \rightarrow \mathcal{E}$ such that $ \mathcal{E}/E$ is local. One can characterize objects $E$ in $\mathcal{E}$ with this properties as those that are connected projective. Recall that $E$ is projective if for any epimorphism $ e : F \twoheadrightarrow G$ and $ a :E \rightarrow G$ one has a lift

\[\begin{tikzcd}
& F \\
E & G
\arrow["e",two heads, from=1-2, to=2-2]
\arrow["a"', from=2-1, to=2-2]
\arrow[dashed, from=2-1, to=1-2]
\end{tikzcd}\]

so that $\mathcal{E}[E,e]$ is epic, while $E$ is connected if for any family $(E_i)_{i \in I}$ one has

\[ \mathcal{E}[ E, \coprod_{i \in I}E_i] \simeq \coprod_{i \in I} \mathcal{E}[E,E_i]  \]

If $ E$ is then both connected and projective, then the hom functor $ \mathcal{E}[E,-] : \mathcal{E} \rightarrow \Set$ preserves colimits. But now for colimit in $\mathcal{E}/E$ are computed in $\mathcal{E}$, we have that $ 1_\mathcal{E}$ is connected-projective in $\mathcal{E}/E$. Then $ \mathcal{E}/E$ is local.

The prototypical example of local geometric morphisms is the universal domain map $\partial_0 : \mathcal{E}^2 \rightarrow \mathcal{E}$ of a topos. Now for any point $ p : \mathcal{S} \rightarrow \mathcal{E}$, we can consider the Grothendieck Verdier localization at $p$Grothendieck-Verdier localization, which is defined as the pseudopullback
\[\begin{tikzcd}
{\mathcal{E}_p} & {\mathcal{E}^2} \\
{\mathcal{S}} & {\mathcal{E}}
\arrow["{\partial_0}", from=1-2, to=2-2]
\arrow["p"', from=2-1, to=2-2]
\arrow["{p^*\partial_0}"', from=1-1, to=2-1]
\arrow[from=1-1, to=1-2]
\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2]
\end{tikzcd}\]

Its universal property is that for any Grotendieck topos $ \mathcal{F}$, we have an equivalence with the cocomma category

\[ \GTop[\mathcal{F}, \mathcal{E}_p] \simeq p \; !_\mathcal{F} \! \downarrow \GTop[\mathcal{F}, \mathcal{E}] \]

In particular, when $ \mathcal{F}$ is $\mathcal{S}$, we have an equivalence of category

\[ \pt(\mathcal{E}_p) \simeq p\downarrow \pt(\mathcal{E})  \]

From <cit.>[Theorem 3.7] we know that if $ \mathcal{E}$ has $ (\mathcal{C},J)$ as a lex site of definition, then $ \mathcal{E}_p$ can be expressed as a cofiltered pseudolimit of etales geometric morphism

\[ \mathcal{E}_p \simeq \underset{(C,a) \in \int p^*}{\bilim} \mathcal{E}/\mathfrak{a}_J\hirayo^*_C  \]

where $ \int p^*$ is the cofiltered category of elements of the $J$-flat functor $ p^*: \mathcal{C} \rightarrow \mathcal{S}$.

Now we discuss how the terminally connected-etale factorization generalizes to the bicategory of Grothendieck topoi and all geometric morphisms between them. While the left class will be unchanged, the right class has to be generalized; though we do not know the best characterization of it, we can still tell something about.

A geometric morphism will be called pro-etale if it decomposes as a cofiltered bilimit of a etale geometric morphisms over its codomain.

Pro-etale geometric morphisms are those of the form

\[\begin{tikzcd}
{\underset{i \in I}{\bilim} \, \mathcal{E}/E_i} && {\mathcal{E}/E_i} \\
& {\mathcal{E}}
\arrow[""{name=0, anchor=center, inner sep=0}, "{p_i}", from=1-1, to=1-3]
\arrow["{\underset{i \in I}{\bilim} \, \pi_{E_i}}"', from=1-1, to=2-2]
\arrow["{\pi_{E_i}}", from=1-3, to=2-2]
\arrow["{\alpha \atop \simeq}"{description}, Rightarrow, draw=none, from=0, to=2-2]
\end{tikzcd}\]

with $ I$ a small cofiltered category. Observe that the induced map $ \bilim_{i \in I}f_i$ really is the bilimit in $\GTop/\mathcal{E}$ for the domain projection $\GTop/\mathcal{E} \rightarrow \GTop$ must preserve cofiltered bilimits.

From <cit.>[Theorem 2.4 and 2.5] we know that a cofiltered bilimit in $\GTop$ of a diagram of geometric morphisms induced by morphisms of sites between them is computed as the corresponding filtered bicolimits of the underlying sites, that is, the filtered pseudocolimit of the underlying categories with the induced topology. If $ \mathcal{E}$ has $(\mathcal{C},J)$ as a small standard site of definition, then for each object $ C$ of $\mathcal{C}$, a site for $\mathcal{E}/\hirayo_C$ can be constructed by $ (\mathcal{C}/C, J_C)$ where $ J_C$ is the relativised topology.
Suppose now that the objects $E_i$ are chosen as arising from representable $ \hirayo_{C_i}$; for etale geometric morphisms have left cancellation, all the transition morphisms in the diagram must be etale, but then they correspond to morphisms in $\mathcal{E}$ between the corresponding object, which, by full faithfulness of the Yoneda embedding, come uniquely from morphisms $ C_i \rightarrow C_j$ in $\mathcal{C}$. Then a the cofiltered bilimit $ \bilim_{i \in I} \mathcal{E}/E_i$ has the following presentation

\[ \underset{i \in I}{\bilim} \, \mathcal{E}/E_i \simeq \Sh(\underset{i \in I}{\bicolim} \, \mathcal{C}/C_i, \langle \iota_i (J_{C_i}) \rangle )  \]

where $\bicolim_{i \in I} \mathcal{C}/C_i $ is computed in $\Cat$ where one can chose the pseudocolimit, and $\langle \iota_i (J_{C_i}) \rangle$ is the Grothendieck topology induced by the bicolimit inclusions.

The following fact is an immediate consequence of closure properties of right classes under limits in the pseudoslices:

Pro-etale geometric morphisms are right biorthogonal to terminally connected geometric morphisms.

Any geometric morphism factorizes in an essentially unique way as a terminally connected geometric morphism followed by a pro-etale geometric morphism.

Let be $ f : \mathcal{E} \rightarrow \mathcal{F}$ be a geometric morphism and $ (\mathcal{C},J)$ a small standard site for $\mathcal{F}$. Then consider the category of global elements of the inverse image part $ f^*$ restricted along the Yoneda embedding $ \mathcal{C} \hookrightarrow \mathcal{F}$ (which we denote as $f^*$), that is, the comma category $ 1_\mathcal{E}\downarrow f^*$. Then we have an equivalence of categories

\[  1_\mathcal{E} \downarrow f^* \simeq f \downarrow \textbf{Et}/\mathcal{E}  \]

(where the latter is a 1-category for etale geometric morphisms are discrete) as global elements of $f^*$ are in correspondence with factorization of $f^*$ along etale geometric morphisms. Moreover, $f^*$ defines a representably flat functor $ f^* : \mathcal{C} \rightarrow \mathcal{E}$ so in particular $1_\mathcal{E} \downarrow f^*$ is cofiltered. Hence we have a unique factorization as below

\[\begin{tikzcd}
{\mathcal{E}} && {\mathcal{F}} \\
& {\underset{(C,a) \in 1_{\mathcal{E}}\downarrow f^*}{\bilim} \, \mathcal{F}/\hirayo_C}
\arrow["{l_c}"', from=1-1, to=2-2]
\arrow["{\underset{(C,a) \in 1_{\mathcal{E}}\downarrow f^*}{\bilim} \, \pi_{\hirayo_C}}"', from=2-2, to=1-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "f", from=1-1, to=1-3]
\arrow["{\alpha_f \atop \simeq}"{description}, Rightarrow, draw=none, from=2-2, to=0]
\end{tikzcd}\]

with $\bilim_{(C,a) \in 1_\mathcal{E} \downarrow f^*} \pi_{\hirayo_C}$ pro-etale, and moreover this factorization is initial amongst all factorization of $f$ through a pro-etale geometric morphism on the right.

Now we must prove $ l_f$ to be terminally connected. From <ref>, as $ 1_\mathcal{E}\downarrow f^*$ we know $\bilim_{(C,a) \in 1_\mathcal{E} \downarrow f^*} \mathcal{F}/\hirayo_C$ to have $ (\bicolim_{(C,a) \in 1_\mathcal{E} \downarrow f^*} \mathcal{C}/C_i, \langle \iota_{(C,a)} (J_C) \rangle_{(C,a) \in 1_\mathcal{E} \downarrow f^*}) $ as presentation site, and the inverse image part $ l_f^*$ is the map induced by the universal property of the bicolimit as below

\[\begin{tikzcd}
{\mathcal{E}} && {\mathcal{C}} \\
{\underset{(C,a) \in 1_\mathcal{E} \downarrow f^*}{\pscolim} \, \mathcal{C}/C} && {\mathcal{C}/C}
\arrow["{l_f^*}", from=2-1, to=1-1]
\arrow["{f^*}"', from=1-3, to=1-1]
\arrow["{\pi_C^*}", from=1-3, to=2-3]
\arrow["{\iota_{(C,a)}}", hook', from=2-3, to=2-1]
\arrow[draw=none, from=2-3, to=1-1]
\arrow["{a^*}"{description}, from=2-3, to=1-1]
\end{tikzcd}\]

where $ a^*$ is the inverse image part of the name $ \mathcal{E} \rightarrow \mathcal{F}/\hirayo_C$ of the global element $ a: 1_\mathcal{E} \rightarrow f^*(C)$. But recall that objects of $ \pscolim_{(C,a) \in 1_\mathcal{E} \downarrow f^*} \mathcal{C}/C$ is a pair $ ((C,a), h)$ with $ {(C,a) \in 1_\mathcal{E} \downarrow f^*}$ and $ h : D \rightarrow C$ an object of $ \mathcal{C}/C$, and as the map induced by the bicolimit property, one has $ l_f^*((C,a),h) = a^*(h)$.

Let us describe the action of $a^*$ in term of global sections, that is of the functor $ \mathcal{E}[1_\mathcal{E}, a^*]$: this latter sends an object $ h : D \rightarrow C$ of the slice $ \mathcal{C}/C$ to the set of global elements $ b : 1_\mathcal{E} \rightarrow f^*(D)$ such that we have a factorization

\[\begin{tikzcd}
{1_\mathcal{E}} & {f^*(D)} \\
& {f^*(C)}
\arrow["b", from=1-1, to=1-2]
\arrow["a"', from=1-1, to=2-2]
\arrow["{f^*(h)}", from=1-2, to=2-2]
\end{tikzcd}\]

But from $ l_f^*((C,a),h) = a^*(h)$, we know such a factorization is exactly what a global element $ b : 1 \rightarrow l_f^*((C,a),h) $ codes for. We want to prove that any such global element comes from a unique global element in the pseudocolimit.

We must first understand how are global elements in $ \pscolim_{(C,a) \in 1_\mathcal{E} \downarrow f^*} \mathcal{C}/C$, which require a description of its terminal object. The terminal object in the oplax colimit $\oplaxcolim_{(C,a) \in 1_\mathcal{E} \downarrow f^*} \mathcal{C}/C $ is the triple $((1_\mathcal{C},1_\mathcal{E}), 1_{1_\mathcal{C}})$. Now recall that the pseudocolimit in $\Cat$ is obtained as the localization of the oplax colimit at the cartesian morphisms. Hence here we know that for any $ (D,b)$ in $1_\mathcal{E}\downarrow f^*$, the triple $ ((D,b), 1_D)$ is a representant for the terminal object as one always has the commutation

\[\begin{tikzcd}
{1_\mathcal{E}} & {f^*(D)} \\
& {f^*(1_\mathcal{C})}
\arrow["b", from=1-1, to=1-2]
\arrow["{!}", from=1-2, to=2-2]
\arrow[Rightarrow, no head, from=1-1, to=2-2]
\end{tikzcd}\]

for $ f^*$ preserves the terminal element and $ f^*(1_\mathcal{C})$ has only one global element, while the vertical  is related by a pullback square

\[\begin{tikzcd}
D & {1_\mathcal{C}} \\
D & {1_\mathcal{C}}
\arrow[Rightarrow, no head, from=1-2, to=2-2]
\arrow[Rightarrow, no head, from=1-1, to=2-1]
\arrow["{!}"', from=2-1, to=2-2]
\arrow["{!}", from=1-1, to=1-2]
\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2]
\end{tikzcd}\]

which codes for a cartesian morphism $ ((D,b), 1_D) \rightarrow ((1_\mathcal{C}, 1_\mathcal{E}), 1_{1_\mathcal{C}})$ identifying them in the pseudocolimit.

But then we see that a global element $1 \rightarrow l_f^*((C,a),h) $ defines uniquely a morphism

\[\begin{tikzcd}
{((D,b), 1_D)} & {((C,a),h)}
\arrow["{(h, 1_D)}", from=1-1, to=1-2]
\end{tikzcd}\]

in the oplax colimit, which induces a global element

\[\begin{tikzcd}
{((1_\mathcal{C},1_\mathcal{E}), 1_{1_\mathcal{C}})} & {((C,a),h)}
\arrow["{[(h, 1_D)]_\sim}", from=1-1, to=1-2]
\end{tikzcd}\]

after the localization to the pseudocolimit. This achieves to prove that $ l^*$ is terminally connected.

Hence, though we cannot yet characterize exactly the right maps, we have an biorthogonal bifactorization in $\GTop$. Moreover, we can use the gliding property of local topoi along terminally connected geometric morphisms to get the following:

We have a bistable inclusion

\[\begin{tikzcd}
{(\textup{\textbf{FocGTop}}^{\textup{\textbf{TCo}}})^{\op}} & {\GTop^{\op}}
\arrow[hook, from=1-1, to=1-2]
\end{tikzcd}\]

dual to the inclusion of the bicategory of local topoi and terminally connected geometric morphisms between them into the bicategory of Grothendieck topoi.

§.§ Totally connected topoi as specified left objects

A geometric morphism $f$ is said to be totally connected if the following equivalent conditions are fulfilled:

* $f^*$ has cartesian left adjoint $ f_!$;

* $ f$ has a right adjoint in the 2-category of Grothendieck toposes;

* $ f$ has a terminal section.

Morally, a topos is totally connected if it has a terminal point, dually to local toposes who have an initial point. In particular, the universal codomain morphism $ \partial_1$ of a topos always is totally connected (as the universal domain morphism $ \partial_0$ always is local). Totally connected geometric morphisms are stable under pullback (cf. Lemma C3.6.18 <cit.>). Observe that totally connected geometric morphisms are both connected and locally connected.

The following is well known, see <cit.>[C3.6.18]:

Totally connected geometric morphisms have right cancellation along connected geometric morphisms.

We have a bistable inclusion

\[\begin{tikzcd}
{(\textup{\textbf{LocoTocGTop}}^{\textup{\textbf{Co}}})^{\op}} & {(\textup{\textbf{Loco}}\GTop)^{\op}}
\arrow[hook, from=1-1, to=1-2]
\end{tikzcd}\]

of totally connected topoi and connected, locally connected geometric morphisms between them into the bicategory of locally connected Grothendieck topoi and locally connected geometric morphisms.

For now it is not clear whether totally connected geometric morphisms, though being themselves terminally connected, have right cancellation along terminally connected geometric morphisms.

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

titleGrothendieck topologies from unique factorisation systems

journaltitleJournal of Pure and Applied Algebra
titleSheaf Representation for Topoi

journaltitleAdvances in Mathematics
titlevan Kampen theorems for toposes

titleDenseness conditions, morphisms and equivalences of toposes

Paris 6 et Centre universitaire de Valenciennes et du Hainaut Cambrésis

titleCatégories localisables

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

titleA construction of 2-cofiltered bilimits of topoi

journaltitleJournal of Pure and Applied Algebra
titleProper factorization systems in 2-categories

journaltitleCahiers de topologie et geometrie differentielle categoriques
titleMore on orthogonality in locally presentable categories

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

booktitleThe L. E. J. Brouwer Centenary Symposium
seriesStudies in Logic and the Foundations of Mathematics
titleTwo Sheaf Representations of Elementary Toposes

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

titleThe Trace Factorisation of Stable Functors

titleLax familial representability and lax generic factorizations

journaltitleTheory Appl. Categ
titleFamilial 2-functors and parametric right adjoints

journaltitleTheory Appl. Categ
titleGeneric morphisms, parametric representations and weakly Cartesian monads