# On the nature of polariton transport in a Fabry-Perot Cavity

Zeyu Zhou  Department of Chemistry, University of Pennsylvania, 231 South 34th
Street, Philadelphia, Pennsylvania 19104, United States  Hsing-Ta Chen
Department of Chemistry, University of Pennsylvania, 231 South 34th Street,
Philadelphia, Pennsylvania 19104, United States  Department of Chemistry and
Biochemistry, 251 Nieuwland Science Hall, Notre Dame, Indiana 46556, United
States  Maxim Sukharev  Department of Physics, Arizona State University,
Tempe, Arizona 85287, United States  College of Integrative Sciences and Arts,
Arizona State University, Mesa, Arizona 85212, United States  Joseph E.
Subotnik  Department of Chemistry, University of Pennsylvania, 231 South 34th
Street, Philadelphia, Pennsylvania 19104, United States  Abraham Nitzan
Department of Chemistry, University of Pennsylvania, 231 South 34th Street,
Philadelphia, Pennsylvania 19104, United States  Department of Physical
Chemistry, School of Chemistry, The Raymond and Beverly Sackler Faculty of
Exact Sciences and The Sackler Center for computational Molecular and
Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel

###### Abstract

Fabry-Pérot microcavities can strongly enhance interactions between light and
molecules, leading to the formation of hybrid light-matter states known as
polaritons. Polaritons possess much smaller effective masses and much larger
group velocities when the molecules are resonant with cavity modes that have
finite (non-zero) in-plane wavevectors, giving rise to the possibilities of
long-range and ultrafast ballistic transport. In this paper, we present
results of numerical simulations of the ultrafast ballistic transport
phenomenon in real space and time during and after initialization with a
short, spatially localized pulse. We find that the transport of the molecular
excitons as induced by the external light field is synchronized with the
evolution of the enhanced and localized electromagnetic field inside the
cavity. Moreover, the synchronized transport rate is in good agreement with
the group velocities predicted from a calculated dispersion relation across a
wide range of frequencies. These simulations provide an intuitive tool for
understanding the collective motion of light and excitons and helps to better
understand how experimental observations of polaritons should be interpreted.

##  I Introduction

Understanding energy transfer in materials is an active and important subject,
ranging from photochemistry to solar energy harvesting. Because of the defects
and disorder in materials that arise at finite temperatures, long-range
ballistic energy transport is often suppressed and transport becomes a short-
range diffusive process. Recently, however, with advances in microcavity
engineering and the introduction of high quality one and two-dimensional
photonic crystals, the inherent light-matter coupling in materials can be
drastically enhanced, leading to hybrid light-matter excitations and the
formation of long-lived polaritons.[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14] Experiments have shown that many material properties can be modified by
the formation of polaritons. Furthermore, since polaritons contain photonic
components, they are able to accelerate energy transfer and extend the length
scale of the ballistic energy transport.[15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25]

To numerically investigate polaritonic motion, one can envision different
approaches with different levels of simulation details. One approach is based
on a model Hamiltonian comprising a limited number of molecular and optical
degrees of freedom that is assumed to represent the hybrid light-matter system
and numerically integrate the equations of motion associated with this
Hamiltonian, so as to capture the ultrafast transport rate of polaritons along
the in-plane direction of the cavity[5, 14, 24, 21, 23]. One caveat of this
approach is the choice of initial conditions (which in turn determines the
subsequent dynamics) is not obvious. In Ref. 24, the molecules were modeled as
2-level emitters with initial conditions chosen using Monte Carlo sampling to
rotate a set of cavity QED eigenstates (within a certain energy window) in
order to minimize the final standard deviation in real space. In so doing, the
authors were able to construct a non-equilibrium localized initial
wavefunction that was transported in time.

In Ref 23, an atomistic molecular model was used and, again, the
electromagnetic (EM) field was represented using just a few modes. The
simulations were initialized by constructing a localized gaussian of matter
excitations and propagation was done using semiclassical Ehrenfest dynamics to
follow the motion of the wavefront of molecular electronic excitations. In
both calculations, the focus is more on molecular excitation than on the
radiation field. A different approach combines a numerical solution of the
Maxwell equations for the classical electromagnetic field on a real space grid
with a quantum, mean-field description of the matter part. It has been
demonstrated that with this level of calculation, one can capture a wide range
of interesting cavity effects and important electromagnetic observables in
experiments.[26, 27, 28, 29, 30, 31, 32, 33, 34, 35]

In this work, we follow the latter approach. We will perform mixed quantum-
classical calculations on a 2-dimensional grid integrating the Maxwell-Bloch
equations using the finite-difference time-domain (FDTD) methodology [26] and
capture the responses of the system to an actual spatially-focused long pulse.
Our choice of the system is a Fabry-Pérot cavity constructed using two
dielectric Bragg reflectors made of two types of dielectric materials. The
bare cavity yields a series of transverse magnetic (TM) modes for different
incident angles $\theta$ (or equivalently, in-plane wavevectors
$k_{\parallel}$) and the dispersion relation between $\omega$ and
$k_{\parallel}$ for these cavity modes can be directly extracted from the FDTD
calculations as well as the standard transfer matrix method (TMM).[36, 9] For
the polariton propagation simulations, we place a single layer of molecules
near the center of the cavity and use an external focused long pulse to
initiate polariton excitation. Subsequently, we monitor the evolution of the
excited state population and extract a ballistic transport rate for each
choice of incident frequency of the exciting pulse. We find that the
calculated ballistic transport rates are in good agreement with the group
velocities obtained from the dispersion relation of polaritons, and our
simulations provide a new interpretation of that transport.

This manuscript is arranged as follows. In section II, we present our system
composed of one layer of 3-level molecules and a cavity formed by two
dielectric Bragg reflectors under investigation. In section III, we first
present the dispersion relation obtained by the transfer matrix method and
Maxwell-Bloch calculations and demonstrate that they agree very well. Second,
we investigate the molecular excited-states population dynamics for these
systems following incident spatially localized pulse excitation using
numerical solution of the corresponding Maxwell-Bloch calculations. Third, we
extract the transport rates obtained from Maxwell-Bloch calculations and show
that they recover the group velocities predicted from the dispersion relation.
In section IV, we analyze the underlying meaning of the group velocities and
the Hopfield coefficients. In section V, we conclude and discuss possible
future directions.

##  II Model and Methods

###  II.1 System geometry

Figure 1: A schematic setup of the 2-dimensional system under investigation.
The actual simulation box is $16\times 16\ \mu m^{2}$. A focused light source
(a red dotted arrow) is placed near the edge of the simulation box and
incident from the bottom of the figure. The black dashed line along the
diagonal of the box represents the very thin molecular layer ($40/\sqrt{2}$ nm
thickness) and therefore, we may assume that the spatial variation of the
field in the direction normal to the cavity axis may be disregarded. The red
and blue layers represent two distributed Bragg reflectors (DBRs) that compose
the cavity. The white regions represent vacuum.

In this work, we consider the two dimensional system shown in Fig 1. Here, a
cavity made of two distributed Bragg reflectors (each represented by $3$ pairs
of red and blue stripes) is placed along the diagonal of the simulation box.
This orientation ($45$ degrees with respect to the $x$ or $y$ axis) is chosen
for optimal grid size, as the effective grid size is $dl=\sqrt{2}dx$. Both
types of layers have standard quarter-wave thicknesses (depending on their
refractive indices $n_{1}$ and $n_{2}$, respectively) according to the central
energy wavelength (chosen to be $2$ eV, $\lambda_{c}=620$ nm)

$\displaystyle L_{1,2}=\frac{\lambda_{c}}{4n_{1,2}}$ (1)

to ensure maximal transmission for a normal incident cavity mode. By this two-
dimensional setup, we focus on the transverse magnetic (TM) modes of the
cavity.[36, 9]

A single molecular layer (black dashed line) is placed at the center of the
cavity with thickness $40/\sqrt{2}$ nm. Because this layer is so thin, the
long-wavelength approximation holds and the layer will collectively respond to
the external light source along the $y$ direction.

###  II.2 Maxwell-Bloch equations

The symmetry of the simulated system indicates that meaningful results can be
obtained using a 2-dimensional calculation. The optical field is modeled using
the three variables ${\cal E}_{x},{\cal E}_{y}$ and ${\cal B}_{z}$, and the
FDTD solver is applied to the relevant EM equations:

$\displaystyle\frac{\partial{\cal B}_{z}}{\partial t}$
$\displaystyle=\frac{\partial{\cal E}_{x}}{\partial{y}}-\frac{\partial{\cal
E}_{y}}{\partial{x}}$ (2) $\displaystyle\epsilon\frac{\partial{\cal
E}_{x}}{\partial t}$ $\displaystyle=\frac{\partial{\cal
B}_{z}}{\mu_{0}\partial{y}}-J_{x}$ (3)
$\displaystyle\epsilon\frac{\partial{\cal E}_{y}}{\partial t}$
$\displaystyle=-\frac{\partial{\cal B}_{z}}{\mu_{0}\partial{x}}-J_{y}$ (4)

Here, $z$ is the direction perpendicular to the two-dimensional system,
$\epsilon=\epsilon_{0}n^{2}$ is the local dielectric constant and
$\vec{J}=(J_{x},J_{y})$ are the polarization currents along the $x$ and $y$
directions

$\displaystyle\vec{J}=\frac{d\vec{P}}{dt}=n_{0}\frac{d(\text{Tr}(\hat{\rho}\vec{\hat{\mu}}))}{dt}$
(5)

In eq 5, $\vec{P}(x,y)$ is the local polarization and $n_{0}$ is the number
density of the molecular layer,
$\vec{\hat{\mu}}=(\hat{\mu}_{x},\hat{\mu}_{y})$ are two matrices of transition
dipole moments between the molecular states in the $x$ and $y$ directions. The
molecular layers are modeled as a set of 3-level systems represented by the
Hamiltonian:

$\displaystyle\hat{H}=\sum_{a=0}^{2}E_{a}\left|a\right\rangle\left\langle
a\right|+\sum_{a=1}^{2}V_{a}(t)(\left|0\right\rangle\left\langle
a\right|+\left|a\right\rangle\left\langle 0\right|)$ (6)

Here the molecular subsystem is modeled by a two-dimensional Hydrogen-like
atom where the ground state $\left|0\right\rangle$ corresponds to the $1s$
orbital and the two excited states $\left|1\right\rangle$ and
$\left|2\right\rangle$ are degenerate, i.e. $E_{1}=E_{2}$, corresponding to
the $2p_{x}$, $2p_{y}$ orbitals.[37] The coupling to the EM field within the
cavity takes the standard form under dipole approximation:

$\displaystyle V_{a}(t)=\vec{\mu}_{0a}\cdot\vec{{\cal E}}(t),a=1,2$ (7)

Note that the Hamiltonians for all 3-level molecules are explicitly time-
dependent, because the electric field $(\vec{{\cal E}(t)}=\bigl{(}{\cal
E}_{x}(x,y;t),{\cal E}_{y}(x,y;t)\bigr{)})$ enters the Hamiltonian in the
coupling between the ground state and the doubly-degenerate excited states.
The spatial position $(x,y)$ is described on a numerical grid, and each grid
point in the molecular layer is taken to contain a molecule whose internal
3-state dynamics is described by a $3\times 3$ density matrix $\hat{\rho}(t)$.
These molecular density matrices are propagated in time following the
Liouville equation

$\displaystyle i\hbar\frac{d}{dt}\hat{\rho}$
$\displaystyle=[\hat{H}(t),\hat{\rho}]$ (8)

###  II.3 Initial condition and the incident pulse

At time $t=0$, all molecules are assumed to be in the ground state
($\hat{\rho}=\left|0\right\rangle\left\langle 0\right|$, respectively). Near
the edge of the simulation domain (e.g., far outside the cavity), we generate
an incident pulse by applying an external EM field along a line of grid points
(parallel to the $x$ axis at $y=y_{0}$) with a Gaussian spatial distribution

$\displaystyle{\cal E}_{x}(x,y=y_{0},t)={\cal
E}_{0}\exp{(-x^{2}/2\sigma_{x}^{2})}\times F(t).$ (9)

and a temporal pulse with a sine envelope

$\displaystyle F(t)=\begin{cases}\sin(\pi t/\tau)\sin\omega_{0}t&0<t<\tau\\\
0&t>\tau\end{cases}$ (10)

Here, the central frequency of the incident pulse is $\hbar\omega_{0}=2.41$eV,
the initial spatial size of the external field is $\sigma_{x}=200$, and the
pulse duration is $\tau=100$ fs (which determines the pulse shape along $y$
direction). Note that the choices of the spatial size $\sigma_{x}$ and pulse
duration $\tau$ do not affect the polariton transport rate. The incident EM
field comprises two components propagating in the $\pm y$ directions
respectively. The $+y$ component propagates into the cavity and the $-y$
component is absorbed by the perfectly matched layer boundaries (PML) that is
routinely placed at the edge of the simulation cell.[26] Note that we choose
the pulse duration sufficiently long (corresponding to narrow linewidth) so
that, when the $+y$ component reaches the cavity, the spatial distribution
along the $x$ direction remains focused. After the pulse reach the cavity, the
subsequent time evolution includes the molecular excitations followed by
polariton propagation and the non-radiative decay. For a collection of all
parameters and their values in this paper, please see Appendix C, Table 1.

##  III Results

In this section, we present our main results. First, we vary the incident
angle (or equivalently, the in-plane wavevector $k_{\parallel}$) and use
conventional transfer matrix method (TMM)[36, 9] to obtain the dispersion
relationships for $(i)$ the bare cavity photon modes (i.e. when the cavity is
empty) and $(ii)$ the polaritonic modes (i.e. when the cavity is filled with a
complex-valued dielectric). Second, after the incident pulse hits the DBR
mirror, we investigate how the spatial motion of the EM field is captured by
the cavity and how collective transport occurs within the molecular layer
inside the cavity. Finally, we compare the group velocities for different
$k_{\parallel}$ (or equivalently, $\omega_{0}$) obtained from the dispersion
relation of the lower polariton branch and the transport rates obtained by
numerically integrating the Maxwell-Bloch equations.

###  III.1 Dispersion relation of the cavity

We begin by studying the bare cavity photon mode energy as a function of
incident angle, i.e. the dispersion relation of the bare cavity modes.

As is well-known, the velocity of transport of any quasi particle is
proportional to the derivative of its dispersion relation line. Moreover, we
are aware of that the velocity of light (one component of polariton) is
dominated by the dielectric constant of the system. Therefore, the geometry of
the cavity (as discussed in Section II A) and the refractive indices ($n_{1}$
and $n_{2}$) of the two dielectric layers determine the dispersion relation of
bare cavity modes. Without loss of generality, we assume $n_{1}<n_{2}$, and we
choose $n_{1}=1.005$ (close to air) and $n_{2}=2.52$ (TiO2), such that we have
very large group velocities, especially at large incident angles. Note that we
have designed parameters here to give a large cavity mode group velocity, so
that the dispersion curve of the lower polariton will be sensitive to the
central frequencies of the incident pulse and the Hopfield coefficients; see
Fig 4 (b) and Fig 5 (a) below.

Figure 2: Angle-resolved transmission spectrum as a function of energy and
incident angles of (a) the empty cavity (without the molecular layer) modes
and (b) the polaritons formed (with molecular layer) near $45$ degrees. Both
spectra are obtained by transfer matrix method. (c) Magnified dispersion
relation (Energy vs wavevector) of polaritons near the avoided crossing at
$2.5$ eV. The dispersion lines of polaritons and bare cavity modes are
obtained by finding the maximal transmission signal for each $k_{\parallel}$
(see eq 12). We also use Maxwell-Bloch equations to obtain the transmission
spectrum at $45$ degrees as the incident angle. The transmission spectrum
agrees with the results obtained from the transfer matrix method.

Once we have fixed the refractive indices of the two dielectric layers and the
central wavelength, angle-resolved transmission spectra can be obtained by
applying the standard transfer matrix method (TMM)[36, 9] for the system made
of two 3-pair distributed Bragg reflectors (see Fig 1), as shown in Fig 2
(a-b). In Fig 2 (c), we further extract the dispersion relation of lower/upper
polaritons ($E_{LP/UP}$) and the bare cavity photonic modes ($E_{cav}$)
predicted from the TMM method by numerically finding the maximal transmission
value. (The brightest energetic values of the lines on the angular-resolved
transmission spectra for each incident angle) We can extract the group
velocity by evaluating the derivative of the energy with respect to the in-
plane wavevector ($k_{\parallel}$) along the dispersion relation in Fig 2 (c):

$\displaystyle
v_{\parallel}^{j}=\frac{1}{\hbar}\frac{dE_{j}}{dk_{\parallel}}\qquad
j=\text{cav/LP/UP}$ (11)

Here, $k_{\parallel}$ is defined by

$\displaystyle k_{\parallel}^{j}=\frac{\hbar}{E_{j}c_{0}}\sin{\theta}\qquad
j=\text{cav/LP/UP}$ (12)

where $\theta$ is the incident angle and $c_{0}$ is the speed of light in
vacuum.

Note that in Fig 2 (a-b), for normal incidence ($k_{\parallel}\approx 0$), the
group velocity of bare cavity photon mode ($E_{cav}=2eV$) is nearly $0$. In
other words, as one would expect, when the light beam enters the cavity at the
normal incidence, the refracted and reflected beams remain completely
perpendicular to the cavity dielectric layers and the light does not propagate
along an in-plane direction.

To observe polaritons transport, we must apply the EM field at a finite
incident angle (chosen to be $45$ degrees for numerical simplicity). Thus, we
choose the molecular excitation energy to be
$E_{2}-E_{0}=E_{1}-E_{0}=2.5\text{ eV}$. Following this parameter choice, we
can obtain the dispersion line with TMM using the dielectric constant (see eq
41 in ref31, it is the Drude model for a two-level system). As shown in Fig
2(b), the avoided crossings confirm that polaritons are indeed formed near
$2.5\text{ eV}$. The transmission spectrum (with two polariton peaks) at $45$
degrees incidence has also been verified using Maxwell-Bloch equations with a
short-time light pulse (same functional form as in eqs 9 and 10, but with
$\tau=10\text{ fs}$). These dispersion lines are later used to determine the
light-matter interaction and the Hopfield coefficients (see Fig 5 and Section
IV).

Finally, as a side note, we want to remind the reader of one limitation of our
light source. In theory, the molecular layer with excitation energy $2.5$ eV
is strictly resonant with only the $45$ degree bare cavity mode. However, the
molecular layer does formally interact with a continuum of cavity modes that
are close to $2.5$ eV. Moreover, because the incident light is spatially
localized (in contrast to completely coherent and parallel light source), the
envelope of the incident light will spread along the $x$ direction when
traveling towards the positive $y$ direction. Thus, in our FDTD simulation,
the effective incident angle has a natural spread around $45$ degrees. Below,
we will use this imperfection of the light source to our advantage, insofar as
the spread in angles will allow us to pump the lower polariton states (which
have a spread in energy between 2.41 and 2.47 eV) by simply varying the center
frequency $\omega_{0}$ (as in eq 9) and not worrying about matching
wavevectors (i.e. there is no need to change the incident angle).

Figure 3: Excited state population of (a) bare molecular layer and (b)
molecular layer inside a DBR cavity as in Fig 1, plotted as a function of time
and position along the direction of the cavity mirror/molecular layer. For the
case outside the cavity, because the molecular layer is along the diagonal of
the simulation box while the incoming light field is moving along $y$
direction, the excited state population is not symmetric with respect to the
diagonal direction. Although the excitation spot slightly leans towards right
side, the population map does not show obvious transport behavior. For the
case inside the cavity in (b), we can clearly observe that after the pulse
create an excitation spot near the center, the excited state population is
transported in time to the right side of the box, as time ($y$ axis) flies.

###  III.2 Polariton transport

Having confirmed the existence of polaritonic states, we will now proceed to
study the polaritonic transport. In the context of a Maxwell-Bloch treatment,
because the EM field is treated in the position (and not mode) representation
, we cannot easily compute the population of the cavity mode. Instead, we
focus on the evolution of the molecular excitations in real space, assuming
that at any point in space, the molecular excitation synchronise with the
local EM field intensity. This serves as a way to effectively capture the
polariton propagation. Fig 3 shows the evolution of the molecular excitation
density following excitation by a nearly monochromatic pulse centered near
lower polariton resonance ($\hbar\omega_{0}=2.41$eV) of duration $100$ fs.
Both figures in Fig 3 show a heat map of molecular excitation (which starts as
a spot). Outside the cavity, no tail appears around the spot. In contrast, as
shown in Fig 3(b), for the case inside a cavity, after the excited state
population reaches a maximum near $50$ fs, the spot moves towards positive $x$
direction and shows a long tail. We can extract the transport rate of
polaritons by numerically calculating the slope of this tail.

###  III.3 Comparing the TMM Group Velocity and the Maxwell-Bloch Transport
Rates

Fig 4(a) reproduces the results of Fig. 2(c) above. When the polaritons are
formed, the avoided crossing near the molecular resonance strongly distorts
the dispersion lines, and the group velocities of the lower and upper
polaritons are drastically changed from the original group velocities of the
bare cavity photon (as explicitly expressed by eq 14 below).

Figure 4: (a) The dispersion relation of the lower (cyan dashed-dotted line)
and upper (blue dash line) polaritons, bare exciton (magenta dotted line) and
bare cavity photon modes (red solid line). (b)Observed transport velocity
calculated from the dispersion relation (blue solid line) and Maxwell-Bloch
calculations (red dotted line with crosses) as a function of central frequency
of the incoming pulse. The vacuum speed of light (black dashed line,
$300\mu\text{m/ps}$) is much larger than the polariton transport velocities
inside the cavity. The dispersion relation group velocity is obtained
according to eq 11 and by calculating the first derivative of the lower
polaritons dispersion line.

In blue in Fig 4(b), we plot the corresponding group velocity associated with
the lower polariton as calculated in the energy range of $2.41-2.47$ eV from
eq 11 using the dispersion obtained from the TMM calculation. The value
compared well with the speed (red dotted line) calculated from the slope of
the propagation tail seen in Fig 3(b) that was extracted from the Maxwell-
Bloch simulations. The small discrepancy may arise from the imperfect light
source. The calculated speed is of the order of $1/3$ of the speed of light in
vacuum (dashed black line), but still represents a considerable speed for
propagation involving molecular excitation. We note that the group velocity of
the cavity photon, as calculated from the cavity photon dispersion is
approximately $160\mu\text{m/ps}$. While the calculated number is smaller than
that reported in Ref 25, we note that the speed is nearly the largest we could
get with this combination of cavity geometry and excitation process, by using
$n_{1}=1.005\approx 1.00$. If we choose more realistic parameters, for
example, $n_{1}=1.48$ as silica, the group velocities will become much
smaller. Finally, we briefly discuss the effects of $n_{2}$ (quality factor)
and show the corresponding numerical results in Appendix A.

##  IV Discussion

The polariton states are considered as linear combinations of molecular
excitation and cavity photon states.

$\displaystyle\left|LP/UP\right\rangle=\alpha_{ex}^{LP/UP}\left|ex\right\rangle+\alpha_{cav}^{LP/UP}\left|cav\right\rangle$
(13)

A simple consideration of the dispersion relation that would be obtained from
these equations leads to the group velocities and their relation to the
Hopfield coefficients for varying incident angles, or equivalently, in-plane
wavevectors ($k_{\parallel}$). Explicitly,

$\displaystyle
v^{LP/UP}_{g}=|\alpha_{cav}^{LP/UP}|^{2}v^{cav}_{g}\pm\frac{d|V|}{dk_{\parallel}}|\alpha_{ex}^{LP/UP}\alpha_{cav}^{LP/UP}|$
(14)

The full derivation is presented in Appendix B. The first term is intuitively
easy to understand, the group velocities of the lower and upper polaritons
depend on their portion of photonic components. In other words, the more
photonic component a polariton state contains, the faster the group velocity
is. For instance, when the bare exciton and the bare cavity photon mode are
resonant at $k_{\parallel}\approx 1.415\mu m^{-1}$ (incident angle $\theta=45$
degrees), both polaritons possess $50\%$ photon component and thus, are
expected to transport with the same velocity, which is that of a half of the
group velocity of bare cavity photon mode. Moreover, when the energy of bare
cavity photon mode is very close to that of the lower polariton, i.e.,
$k_{\parallel}<1.35\mu m^{-1}$, the group velocity of the lower polariton is
approximately the same as that of the bare cavity photon, and the group
velocity of the upper polariton is approximately $0$ (as shown in Fig 4(a),
the cyan line is very close to the red line, and the blue line is very close
to the magenta line, respectively). More generally, the fraction of the
photon/exciton occupation in the polariton state is determined by the Hopfield
coefficients $|\alpha_{cav/ex}|^{2}$.

If we assume that the magnitude of light-matter interaction $|V|$ is weakly
dependent on the in-plane wavevector $k_{\parallel}$ (in contrast to the
cavity mode energy $E_{cav}$) so that the second term is small, the first term
agrees with the data previously presented: the transport rate is primarily
dominated by the bare cavity photon transport rate ($v^{cav}_{g}$) and the
Hopfield coefficient $|\alpha_{cav}^{LP/UP}|^{2}$. This allows for a simple
classical picture of the polariton transport. Namely, for the initial near-
monochromatic excitation that matches a given polariton energy, the excitation
induced in the system appears partly in the EM field inside the cavity and
partly in molecular excitation, divided according to their corresponding
Hopfield coefficients. In the strong coupling regime, the two portions reaches
equilibrium much faster than the effective transport and cavity leakage rates,
hence the observed rate is a weighted average of the group velocity of the
cavity photon modes (which is solely governed by the mirror properties). The
molecular excitation effectively does not move on the timescale of
observation. Indeed, in Fig 5(a), we plot polariton speed as a function of
$|\alpha_{cav}|^{2}$. We find that the result is basically a straight line,
because within the very small energy window $2.41-2.47$ eV, the group velocity
of the bare photon does not significantly change.

Figure 5: (a)Observed transport velocity from the dispersion relation (blue
solid line) and Maxwell-Bloch calculations (red dotted line with crosses) as a
function of Hopfield coefficients of lower polaritons and (b) effective light-
matter interaction $|V(k_{\parallel})|$ obtained from TMM calculations.

Note that the second term on the right side of eq 14 is a product of the
derivative of the light-matter coupling term ($V$) with respect to the in-
plane wavevector $k_{\parallel}$ multiplied by the coherence between the
cavity mode and the quantum molecular subsystem. The magnitude of $V$ can be
estimated from a TMM calculation by insisting that the eigenvalues of the
matrix in eq 25 (see Appendix B) are consistent with the values of
$E_{cav},E_{ex},E_{LP}$ and $E_{UP}$ in Fig. 4(a). In Fig 5 (b), we plot $|V|$
as a function of $k_{\parallel}$. Clearly, $|V|$ is not a constant. However,
the derivative of $V$ is less than $1/10$ of the group velocity of the bare
cavity photon and so this term appears to be small.

This analysis has important implications for understanding experiments. In
many cases, the assumption that the light-matter coupling $|V|$ is constant –
or equivalently, that the Rabi splitting $\Omega_{R}$ is constant – with
respect to small incident angle/energy changes is a valid one. However, in the
context of polariton transport, this assumption obscures the quantum nature of
polaritons. In principle, polaritons need not function as simple averages of
the cavity photon mode and molecular excited states, especially in the strong
collective coupling regime ($|V|$ being large), and we hope that the present
analysis will trigger further investigations on polariton transport where both
terms in eq 14 contribute to the transport.

##  V Conclusion and Outlook

In conclusion, we have performed calculations based on a Maxwell-Bloch
formalism within a mean field approximation to simulate polariton transport
phenomena for a two dimensional system. The polariton transport is represented
as the synchronized motion of molecular excitations and the EM field inside a
cavity. The transport rate is in excellent agreement with the group velocity
as obtained from the transfer matrix method (and we attribute the small errors
to the finite system size and a non-ideal light source). Moreover, the
Maxwell-Bloch equations formalism gives us a very intuitive picture of
polariton transport, as one can directly monitor the such transport by
quantifying either the molecular excitations or the EM field intensities in
the real space. In the present case, we find that the transport has a very
simple functional form, whereby the velocity is simply the hopfield
coefficient multiplied by the speed of light in a raw cavity. As a side note,
we mention that the calculated timescale for polariton transport phenomena is
less than $100fs$, the total length of our simulation; this timescale is much
faster than the timescale ($\approx 1ns$) for Förster resonance energy
transfer (FRET) between a donor and acceptor, even when enhanced by a
cavity[38].

Looking forward, one can directly introduce different types of disorder to
reduce the polariton transport rate and hopefully observe simultaneously
ballistic transport and diffusive behaviors. In this work, when working with
two dimensional calculations, we could focus only on the transverse magnetic
(TM) cavity modes. In the future, though it would be more computationally
expensive, one could perform the calculation also on a three dimensional
system and observe the interplay between transverse electric (TE) and
transverse magnetic (TM), where the results would be most interesting if the
molecules had non-trivial polarizability tensors and were able to change light
polarization. Extending the present work to more complicated materials, with
more geometric degrees of freedom, will be an important next step for further
understanding cavity-promoted polariton transport phenomena from first
principles.

## Acknowledgements

This work has been supported by the U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences, under Award No.DE-SC0019397
(J.E.S.); the U.S. National Science Foundation under Grant No.CHE1953701
(A.N.). MS acknowledges support by the Air Force Office of Scientific Research
under Grant No. FA9550-22-1-0175.

##  Appendix A Effects of quality factor

As discussed in section III, the quality factor of the cavity is governed by
the larger refractive index ($n_{2}$). This fact can be easily understood by
noting that the reflection coefficient near resonant mode at normal incidence
of a $N$-pair DBR mirror with refractive indices $n_{1}$ and $n_{2}$ is

$\displaystyle
r_{DBR}=\frac{n_{2}^{2N}-n_{1}^{2N}}{n_{1}^{2N}+n_{2}^{2N}}=1-\frac{2n_{1}^{2N}}{n_{1}^{2N}+n_{2}^{2N}}$
(15)

Thus, the EM field trapped inside the cavity becomes more difficult to
transmit the mirrors. In other word, the cavity generates a larger field-
amplification factor and a longer lifetime. For a detailed derivation of eq 15
and an overview of the standard transfer matrix method for any incident angle,
there are many great textbooks on optics, for example, see ref 36, 9.

Figure 6: Excited state population of molecular layer inside a cavity with (a)
small $n_{2}=2.5$ and (b) large $n_{2}=7.5$ contrast between dielectric
layers. Fig 6(a) is identical to Fig 3(b) but with a colorbar showing the
actual populations. Obviously the transport rates extracted from the two
scenarios are identical. However, the signal in Fig 6(b) lasts longer.

We can minimally distort our simulation by simply increase $n_{2}$ and
decrease layer thickness $L_{2}$ according to eq 1. Again, this alteration
does not affect the slope of the dispersion relation and therefore, does not
affect the transport rate observed from Maxwell-Bloch calculations. As shown
in Fig 6, on the one hand, as the quality factor increases from scenario (a)
to (b), the signal intensity decreases ($4$ times smaller), because the EM
field enters the cavity with less efficiency. On the other hand, once the EM
field enters the cavity, it remains trapped and transport for a longer
distance. Therefore, refining quality factor increases the fidelity of the
signals.

##  Appendix B Derivation of the relation between polariton group velocities
and bare cavity mode group velocities

In this appendix, we present the detailed derivation in eq 14 For a system
with one bare excitation exciton energy ($E_{ex}$), cavity mode energy
($E_{cav}$), and an approximate collective light-matter coupling
$V=\Omega_{R}/2$ ($\Omega_{R}$ is the Rabi splitting), the standard $2\times
2$ Hamiltonian is:

$\displaystyle\hat{H}$ $\displaystyle=\left[\begin{array}[]{c c}E_{ex}&V\\\
V^{*}&E_{cav}\\\ \end{array}\right]$ (18)

The lower and upper polaritons energies $E_{LP/UP}$ are the eigenvalues of
this Hamiltonian, with eigenvectors
$\left|LP\right\rangle,\left|UP\right\rangle$. Let us focus on the lower
polariton state $\left|LP\right\rangle$

$\displaystyle\left[\begin{array}[]{c c}E_{ex}&V\\\ V^{*}&E_{cav}\\\
\end{array}\right]\left[\begin{array}[]{c}\alpha_{ex}^{LP}\\\
\alpha_{cav}^{LP}\\\ \end{array}\right]$
$\displaystyle=E_{LP}\left[\begin{array}[]{c}\alpha_{ex}^{LP}\\\
\alpha_{cav}^{LP}\\\ \end{array}\right]$ (25)

which clearly satisfies:

$\displaystyle\frac{\alpha_{ex}^{LP}}{\alpha_{cav}^{LP}}=\frac{V}{E_{LP}-E_{ex}}=\frac{E_{LP}-E_{cav}}{V^{*}}$
(26)

Clearly, both $\frac{\alpha_{ex}^{LP}}{V\alpha_{cav}^{LP}}$ and
$\frac{V^{*}\alpha_{ex}^{LP}}{\alpha_{cav}^{LP}}$ are real-valued. Now, the
standard characteristic equation is:

$\displaystyle(E_{LP}-E_{cav})(E_{LP}-E_{ex})-|V|^{2}=0$ (27)

If we take derivative of this equation with respect to $k_{\parallel}$, we
obtain

$\displaystyle(\frac{dE_{LP}}{dk_{\parallel}}-\frac{dE_{cav}}{dk_{\parallel}})(E_{LP}-E_{ex})+(\frac{dE_{LP}}{dk_{\parallel}})(E_{LP}-E_{cav})-V^{*}\frac{dV}{dk_{\parallel}}-V\frac{dV^{*}}{dk_{\parallel}}=0$
(28)

Here, we naturally assume that the exciton energy is independent of the in-
plane wavevector $k_{\parallel}$ of the incident photon. Therefore, we can
combine eqs 28 and 26 to obtain:

$\displaystyle(\frac{dE_{LP}}{dk_{\parallel}}-\frac{dE_{cav}}{dk_{\parallel}})(\frac{\alpha_{cav}^{LP}}{\alpha_{ex}^{LP}}V+\frac{\alpha_{cav}^{LP*}}{\alpha_{ex}^{LP*}}V^{*})/2+(\frac{dE_{LP}}{dk_{\parallel}})(\frac{\alpha_{ex}^{LP}}{\alpha_{cav}^{LP}}V^{*}+\frac{\alpha_{ex}^{LP*}}{\alpha_{cav}^{LP*}}V)/2$
(29)
$\displaystyle-V^{*}\frac{dV}{dk_{\parallel}}-V\frac{dV^{*}}{dk_{\parallel}}=0$

which simplifies to (using
$|\alpha_{ex}^{LP}|^{2}+|\alpha_{cav}^{LP}|^{2}=1$):

$\displaystyle\frac{dE_{LP}}{dk_{\parallel}}(\frac{V}{\alpha_{cav}^{LP*}\alpha_{ex}^{LP}}+\frac{V^{*}}{\alpha_{cav}^{LP}\alpha_{ex}^{LP*}})-\frac{dE_{cav}}{dk_{\parallel}}(\frac{\alpha_{cav}^{LP}}{\alpha_{ex}^{LP}}V+\frac{\alpha_{cav}^{LP*}}{\alpha_{ex}^{LP*}}V^{*})=2V^{*}\frac{dV}{dk_{\parallel}}+2V\frac{dV^{*}}{dk_{\parallel}}$
(30)

or even better,

$\displaystyle\frac{dE_{LP}}{dk_{\parallel}}-|\alpha_{cav}^{LP}|^{2}\frac{dE_{cav}}{dk_{\parallel}}=\frac{\frac{d|V|^{2}}{dk_{\parallel}}}{\frac{E_{LP}-E_{ex}}{|\alpha_{cav}^{LP}|^{2}}+\frac{E_{LP}-E_{cav}}{|\alpha_{ex}^{LP}|^{2}}}=\frac{\frac{d|V|^{2}}{dk_{\parallel}}}{\frac{V}{\alpha_{cav}^{LP*}\alpha_{ex}^{LP}}+\frac{V^{*}}{\alpha_{ex}^{LP*}\alpha_{cav}^{LP}}}$
(31)

Note that, without loss of generality, one is free to choose the overall phase
of the $\left|LP\right\rangle$ eigenvector (eq 13), such that we can assume
that $\alpha_{ex}^{LP}=\alpha_{ex}^{LP*}$ is real-valued. From eq 26, it then
follows that
$V/\alpha_{cav}^{LP*}=V^{*}/\alpha_{cav}^{LP}=|V|/|\alpha_{cav}^{LP*}|$ is
also real-valued. And so, in the end, we can further simplify eq 31 according
to eq 26

$\displaystyle\frac{dE_{LP}}{dk_{\parallel}}=|\alpha_{cav}^{LP}|^{2}\frac{dE_{cav}}{dk_{\parallel}}+\frac{d|V|}{dk_{\parallel}}|\alpha_{ex}^{LP}\alpha_{cav}^{LP}|$
(32)

##  Appendix C Parameters for Maxwell-Bloch simulation

In this appendix, we list all parameters in our simulation.

Table 1: Parameters for Maxwell-Bloch simulation. Name | Value$\dagger$
---|---
Grid resolution ($dx$) |  $4$ nm
Time step (dt) | $dx/1.5/c_{0}$
Thickness of Molecular layer |  $40/\sqrt{2}$ nm
Number of pairs of dielectric materials |  $3$ pairs
Thickness of layer 1 |  $154$ nm
Thickness of layer 2 |  $248$ nm
Distance between inner surface of the two mirrors |  $156$ nm
Transition dipole moment |  $\mu_{x}^{01}=\mu_{y}^{02}=10/\sqrt{3}$ Debye
Pulse duration |  $\tau=100$ fs

## References

  * Freixanet  _et al._ [2000] T. Freixanet, B. Sermage, A. Tiberj, and V. Thierry-Mieg, “Propagation of excitonic polaritons in a microcavity,” physica status solidi (a) 178, 133–138 (2000).
  * Bayer  _et al._ [2001] M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. Wasilewski, O. Stern, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
  * Walther  _et al._ [2006] H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Reports on Progress in Physics 69, 1325 (2006).
  * Hutchison  _et al._ [2012] J. A. Hutchison, T. Schwartz, C. Genet, E. Devaux, and T. W. Ebbesen, “Modifying chemical landscapes by coupling to vacuum fields,” Angewandte Chemie International Edition 51, 1592–1596 (2012).
  * Feist and Garcia-Vidal [2015] J. Feist and F. J. Garcia-Vidal, “Extraordinary exciton conductance induced by strong coupling,” Physical review letters 114, 196402 (2015).
  * Schachenmayer  _et al._ [2015] J. Schachenmayer, C. Genes, E. Tignone, and G. Pupillo, “Cavity-enhanced transport of excitons,” Physical review letters 114, 196403 (2015).
  * Ebbesen [2016] T. W. Ebbesen, “Hybrid light–matter states in a molecular and material science perspective,” Accounts of chemical research 49, 2403–2412 (2016).
  * Herrera and Spano [2016] F. Herrera and F. C. Spano, “Cavity-controlled chemistry in molecular ensembles,” Physical Review Letters 116, 238301 (2016).
  * Kavokin  _et al._ [2017] A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy,  _Microcavities_ , Vol. 21 (Oxford university press, 2017).
  * Ribeiro  _et al._ [2018] R. F. Ribeiro, L. A. Martínez-Martínez, M. Du, J. Campos-Gonzalez-Angulo, and J. Yuen-Zhou, “Polariton chemistry: controlling molecular dynamics with optical cavities,” Chemical science 9, 6325–6339 (2018).
  * Frisk Kockum  _et al._ [2019] A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, “Ultrastrong coupling between light and matter,” Nature Reviews Physics 1, 19–40 (2019).
  * Hertzog  _et al._ [2019] M. Hertzog, M. Wang, J. Mony, and K. Börjesson, “Strong light–matter interactions: a new direction within chemistry,” Chemical Society Reviews 48, 937–961 (2019).
  * Garcia-Vidal, Ciuti, and Ebbesen [2021] F. J. Garcia-Vidal, C. Ciuti, and T. W. Ebbesen, “Manipulating matter by strong coupling to vacuum fields,” Science 373, eabd0336 (2021).
  * Chávez  _et al._ [2021] N. C. Chávez, F. Mattiotti, J. Méndez-Bermúdez, F. Borgonovi, and G. L. Celardo, “Disorder-enhanced and disorder-independent transport with long-range hopping: Application to molecular chains in optical cavities,” Physical Review Letters 126, 153201 (2021).
  * Orgiu  _et al._ [2015] E. Orgiu, J. George, J. Hutchison, E. Devaux, J. Dayen, B. Doudin, F. Stellacci, C. Genet, J. Schachenmayer, C. Genes, _et al._ , “Conductivity in organic semiconductors hybridized with the vacuum field,” Nature Materials 14, 1123–1129 (2015).
  * Lerario  _et al._ [2017] G. Lerario, D. Ballarini, A. Fieramosca, A. Cannavale, A. Genco, F. Mangione, S. Gambino, L. Dominici, M. De Giorgi, G. Gigli, _et al._ , “High-speed flow of interacting organic polaritons,” Light: Science & Applications 6, e16212–e16212 (2017).
  * Rozenman  _et al._ [2018] G. G. Rozenman, K. Akulov, A. Golombek, and T. Schwartz, “Long-range transport of organic exciton-polaritons revealed by ultrafast microscopy,” ACS Photonics 5, 105–110 (2018).
  * Zakharko  _et al._ [2018] Y. Zakharko, M. Rother, A. Graf, B. Hähnlein, M. Brohmann, J. Pezoldt, and J. Zaumseil, “Radiative pumping and propagation of plexcitons in diffractive plasmonic crystals,” Nano Letters 18, 4927–4933 (2018).
  * Nagarajan  _et al._ [2020] K. Nagarajan, J. George, A. Thomas, E. Devaux, T. Chervy, S. Azzini, K. Joseph, A. Jouaiti, M. W. Hosseini, A. Kumar, _et al._ , “Conductivity and photoconductivity of a p-type organic semiconductor under ultrastrong coupling,” ACS nano 14, 10219–10225 (2020).
  * Hou  _et al._ [2020] S. Hou, M. Khatoniar, K. Ding, Y. Qu, A. Napolov, V. M. Menon, and S. R. Forrest, “Ultralong-range energy transport in a disordered organic semiconductor at room temperature via coherent exciton-polariton propagation,” Advanced Materials 32, 2002127 (2020).
  * Tichauer, Feist, and Groenhof [2021] R. H. Tichauer, J. Feist, and G. Groenhof, “Multi-scale dynamics simulations of molecular polaritons: The effect of multiple cavity modes on polariton relaxation,” The Journal of Chemical Physics 154, 104112 (2021).
  * Pandya  _et al._ [2022] R. Pandya, A. Ashoka, K. Georgiou, J. Sung, R. Jayaprakash, S. Renken, L. Gai, Z. Shen, A. Rao, and A. J. Musser, “Tuning the coherent propagation of organic exciton-polaritons through dark state delocalization,” Advanced Science 9, 2105569 (2022).
  * Sokolovskii  _et al._ [2022] I. Sokolovskii, R. H. Tichauer, J. Feist, and G. Groenhof, “Enhanced excitation energy transfer under strong light-matter coupling: Insights from multiscale molecular dynamics simulations,” arXiv preprint arXiv:2209.07309 (2022).
  * Xu  _et al._ [2022] D. Xu, A. Mandal, J. M. Baxter, S.-W. Cheng, I. Lee, H. Su, S. Liu, D. R. Reichman, and M. Delor, “Ultrafast imaging of coherent polariton propagation and interactions,” arXiv preprint arXiv:2205.01176 (2022).
  * Balasubrahmaniyam  _et al._ [2023] M. Balasubrahmaniyam, A. Simkhovich, A. Golombek, G. Sandik, G. Ankonina, and T. Schwartz, “From enhanced diffusion to ultrafast ballistic motion of hybrid light–matter excitations,” Nature Materials , 1–7 (2023).
  * Taflove, Hagness, and Piket-May [2005] A. Taflove, S. C. Hagness, and M. Piket-May, “Computational electromagnetics: the finite-difference time-domain method,” The Electrical Engineering Handbook 3, 629–670 (2005).
  * Teixeira [2007] F. L. Teixeira, “Fdtd/fetd methods: a review on some recent advances and selected applications,” Journal of Microwaves, Optoelectronics and Electromagnetic Applications (JMOe) 6, 83–95 (2007).
  * McMahon  _et al._ [2007] J. M. McMahon, J. Henzie, T. W. Odom, G. C. Schatz, and S. K. Gray, “Tailoring the sensing capabilities of nanohole arrays in gold films with rayleigh anomaly-surface plasmon polaritons,” Optics express 15, 18119–18129 (2007).
  * Zhao  _et al._ [2008] J. Zhao, A. O. Pinchuk, J. M. McMahon, S. Li, L. K. Ausman, A. L. Atkinson, and G. C. Schatz, “Methods for describing the electromagnetic properties of silver and gold nanoparticles,” Accounts of chemical research 41, 1710–1720 (2008).
  * Puthumpally-Joseph  _et al._ [2014] R. Puthumpally-Joseph, M. Sukharev, O. Atabek, and E. Charron, “Dipole-induced electromagnetic transparency,” Physical review letters 113, 163603 (2014).
  * Sukharev and Nitzan [2017] M. Sukharev and A. Nitzan, “Optics of exciton-plasmon nanomaterials,” Journal of Physics: Condensed Matter 29, 443003 (2017).
  * You  _et al._ [2019] J. You, S. Bongu, Q. Bao, and N. Panoiu, “Nonlinear optical properties and applications of 2d materials: theoretical and experimental aspects,” Nanophotonics 8, 63–97 (2019).
  * Sidler  _et al._ [2020] D. Sidler, C. Schäfer, M. Ruggenthaler, and A. Rubio, “Polaritonic chemistry: Collective strong coupling implies strong local modification of chemical properties,” The journal of physical chemistry letters 12, 508–516 (2020).
  * Tancogne-Dejean  _et al._ [2020] N. Tancogne-Dejean, M. J. Oliveira, X. Andrade, H. Appel, C. H. Borca, G. Le Breton, F. Buchholz, A. Castro, S. Corni, A. A. Correa, _et al._ , “Octopus, a computational framework for exploring light-driven phenomena and quantum dynamics in extended and finite systems,” The Journal of chemical physics 152, 124119 (2020).
  * Sukharev, Salomon, and Zyss [2021] M. Sukharev, A. Salomon, and J. Zyss, “Second harmonic generation by strongly coupled exciton–plasmons: The role of polaritonic states in nonlinear dynamics,” The Journal of Chemical Physics 154, 244701 (2021).
  * Yeh [2005] P. Yeh,  _Optical Waves in Layered Media_ (John Wiley & Sons, Inc., 2005).
  * Sukharev and Nitzan [2011] M. Sukharev and A. Nitzan, “Numerical studies of the interaction of an atomic sample with the electromagnetic field in two dimensions,” Physical Review A 84, 043802 (2011).
  * Andrew and Barnes [2000] P. Andrew and W. L. Barnes, “Forster energy transfer in an optical microcavity,” Science 290, 785–788 (2000).


for every $S_{1},S_{2}\in\mathscr{S}$, a bi-additive functor

$-\boxtimes-=-\boxtimes_{S_{1},S_{2}}-:\mathscr{C}(S_{1})\times\mathscr{C}(S_{2})\rightarrow\mathscr{C}(S_{1}\times
S_{2}),$

called the external tensor product functor over $(S_{1},S_{2})$,

     * •

for every choice of morphisms $f_{i}:T_{i}\rightarrow S_{i}$ in $\mathscr{S}$,
$i=1,2$, a natural isomorphism of functors
$\mathscr{C}(S_{1})\times\mathscr{C}(S_{2})\rightarrow\mathscr{C}(T_{1}\times
T_{2})$

$m=m_{f_{1},f_{2}}:f_{1}^{*}A_{1}\boxtimes
f_{2}^{*}A_{2}\xrightarrow{\sim}(f_{1}\times f_{2})^{*}(A_{1}\boxtimes
A_{2}),$

called the external monoidality isomorphism along $(f_{1},f_{2})$

satisfying the following condition:

    1. ($m$ETS)

For every choice of composable morphisms $f_{i}:T_{i}\rightarrow S_{i}$,
$g_{i}:S_{i}\rightarrow V_{i}$, $i=1,2$, the diagram of functors
$\mathscr{C}(V_{1})\times\mathscr{C}(V_{2})\rightarrow\mathscr{C}(T_{1}\times
T_{2})$

${(g_{1}f_{1})^{*}C_{1}\boxtimes(g_{2}f_{2})^{*}C_{2}}$${(g_{1}f_{1}\times
g_{2}f_{2})^{*}(C_{1}\boxtimes C_{2})}$${f_{1}^{*}g_{1}^{*}C_{1}\boxtimes
f_{2}^{*}g_{2}^{*}C_{2}}$${(f_{1}\times f_{2})^{*}(g_{1}^{*}C_{1}\boxtimes
g_{2}^{*}C_{2})}$${(f_{1}\times f_{2})^{*}(g_{1}\times
g_{2})^{*}(C_{1}\boxtimes
C_{2})}$$\scriptstyle{m}$$\scriptstyle{m}$$\scriptstyle{m}$

is commutative.

  2. (2)

In case the $\mathscr{S}$-fibered category $\mathscr{C}$ is abelian, we say
that an external tensor structure $(\boxtimes,m)$ on $\mathscr{C}$ is exact if
all external tensor product functors $-\boxtimes_{S}-$ are bi-exact.

In order to extend the notion of external tensor structure to
$\mathscr{S}$-fibered representations, we also need to recall the notion of
external tensor structure on a morphism of $\mathscr{S}$-fibered categories
from [Ter23T, § 8], adapted to the additive setting:

######  Definition 6.2.

Let $R:\mathscr{C}\rightarrow\mathscr{E}$ be a $1$-morphism of (additive)
$\mathscr{S}$-fibered categories. An external tensor structure on $R$ (with
respect to $(\boxtimes_{\mathscr{C}},m_{\mathscr{C}})$ and
$(\boxtimes_{\mathscr{E}},m_{\mathscr{E}})$) is the datum of

  * •

for every $S_{1},S_{2}\in\mathscr{S}$, a natural isomorphism of functors
$\mathscr{C}(S_{1})\times\mathscr{C}(S_{2})\rightarrow\mathscr{E}(S_{1}\times
S_{2})$

$\rho=\rho_{S_{1},S_{2}}:R_{S_{1}}(C_{1})\boxtimes_{\mathscr{E}}R_{S_{2}}(C_{2})\xrightarrow{\sim}R_{S_{1}\times
S_{2}}(C_{1}\boxtimes_{\mathscr{C}}C_{2}),$

called the external $R$-monoidality isomorphism at $(S_{1},S_{2})$,

satisfying the following condition:

  1. (mor-ETS)

For every two morphisms $f_{i}:T_{i}\rightarrow S_{i}$ in $\mathscr{S}$,
$i=1,2$, the natural diagram of functors
$\mathbb{H}_{1}(S_{1})\times\mathbb{H}_{1}(S_{2})\rightarrow\mathbb{H}_{2}(T_{1}\times
T_{2})$

${f_{1}^{*}R_{S_{1}}(C_{1})\boxtimes_{\mathscr{E}}f_{2}^{*}R_{S_{2}}(C_{2})}$${(f_{1}\times
f_{2})^{*}(R_{S_{1}}(C_{1})\boxtimes_{\mathscr{E}}R_{S_{2}}(C_{2}))}$${(f_{1}\times
f_{2})^{*}R_{S_{1}\times
S_{2}}(C_{1}\boxtimes_{\mathscr{C}}C_{2})}$${R_{T_{1}}(f_{1}^{*}C_{1})\boxtimes_{\mathscr{E}}R_{T_{2}}(f_{2}^{*}C_{2})}$${R_{T_{1}\times
T_{2}}(f_{1}^{*}C_{1}\boxtimes_{\mathscr{C}}f_{2}^{*}C_{2})}$${R_{T_{1}\times
T_{2}}((f_{1}\times
f_{2})^{*}(C_{1}\boxtimes_{\mathscr{C}}C_{2}))}$$\scriptstyle{m_{2}}$$\scriptstyle{\theta}$$\scriptstyle{\rho}$$\scriptstyle{\theta}$$\scriptstyle{\rho}$$\scriptstyle{m_{1}}$

is commutative.

With these notions at our disposal, we can define external tensor structures
on additive $\mathscr{S}$-fibered representations in a natural way as follows:

######  Definition 6.3.

Let $(\mathscr{C},\mathscr{E};\beta)$ be an (additive) $\mathscr{S}$-fibered
representation.

  1. (1)

An external tensor structure on $(\mathscr{C},\mathscr{E};\beta)$ is the datum
of

     * •

an external tensor structure $(\boxtimes_{\mathscr{C}},m_{\mathscr{C}})$ on
$\mathscr{C}$,

     * •

an external tensor structure $(\boxtimes_{\mathscr{E}},m_{\mathscr{E}})$ on
$\mathscr{E}$,

     * •

an external tensor structure $\rho$ on the morphism of $\mathscr{S}$-fibered
categories $\beta:\mathscr{C}\rightarrow\mathscr{E}$.

We write it as a triple
$(\boxtimes_{\mathscr{C}},\boxtimes_{\mathscr{E}};\rho)$ where, for
convenience, we leave the symbols $m_{\mathscr{C}}$ and $m_{\mathscr{E}}$
unexplicit.

  2. (2)

In case the $\mathscr{S}$-fibered representation
$(\mathscr{C},\mathscr{E};\beta)$ is abelian, we say that an external tensor
structure on $(\mathscr{C},\mathscr{E};\beta)$ is exact if the external tensor
structure $(\boxtimes_{\mathscr{E}},m_{\mathscr{E}})$ on the abelian
$\mathscr{S}$-fibered category $\mathscr{E}$ is exact.

In [Ter23T, §2] we explained how to axiomatize external tensor structures on
$\mathscr{S}$-fibered categories as suitable
$(\mathscr{S}^{2}\times\mathscr{I})$-fibered categories, where $\mathscr{I}$
denotes the category with two objects and exactly one non-trivial morphism
between them.

The result that we discuss now bears a strong resemblance to those concerning
abelian fibered categories. Note that, strictly speaking, we are not allowed
to deduce them directly from the previous results on fibered categories via
the constructions of [Ter23T, §2]: the reason is that the external tensor
product functors are not exact on the product abelian category but rather
separately exact on each variable. However, the proof method is essentially
the same as before.

######  Proposition 6.4.

Let $(\mathscr{D},\mathscr{A};\beta)$ be an abelian $\mathscr{S}$-fibered
representation, and suppose that we are given an exact external tensor
structure $(\boxtimes_{\mathscr{D}},\boxtimes_{\mathscr{A}};\rho)$ on it.
Then:

  1. (1)

Associating

     * •

to every $S_{1},S_{2}\in\mathscr{S}$, the bi-exact functor

$-\overline{\boxtimes}_{S_{1},S_{2}}-=-\overline{\boxtimes}-:\mathbf{A}(\beta_{S_{1}})\times\mathbf{A}(\beta_{S_{2}})\rightarrow\mathbf{A}(\beta_{S_{1}\times
S_{2}})$

obtained via Proposition 4.3(1),

     * •

to every two morphisms $f_{i}:T_{i}\rightarrow S_{i}$, $i=1,2$, the natural
transformation of exact functors
$\mathbf{A}(\beta_{S_{1}})\times\mathbf{A}(\beta_{S_{2}})\rightarrow\mathbf{A}(\beta_{T_{1}\times
T_{2}})$

(33)
$\tilde{m}=\tilde{m}_{f_{1},f_{2}}:f_{1}^{*}M_{1}\overline{\boxtimes}f_{2}^{*}M_{2}\xrightarrow{\sim}(f_{1}\times
f_{2})^{*}(M_{1}\overline{\boxtimes}M_{2})$

obtained via Proposition 4.6

defines an exact external tensor structure $(\overline{\boxtimes},\tilde{m})$
on the abelian $\mathscr{S}$-fibered category $\mathbf{A}(\beta)$.

  2. (2)

Associating

     * •

to every $S_{1},S_{2}\in\mathscr{S}$, the natural isomorphism of functors
$\mathbf{A}(\beta_{S_{1}})\times\mathbf{A}(\beta_{S_{2}})\rightarrow\mathscr{A}(S_{1}\times
S_{2})$

(34)
$\tilde{\rho}=\tilde{\rho}_{S_{1},S_{2}}:\iota_{S_{1}}(M_{1})\boxtimes\iota_{S_{2}}(M_{2})\xrightarrow{\sim}\iota_{S_{1}\times
S_{2}}(M_{1}\overline{\boxtimes}M_{2})$

obtained via Proposition 4.3(2)

defines an external tensor structure on the morphism of $\mathscr{S}$-fibered
categories $\iota:\mathbf{A}(\beta)\hookrightarrow\mathscr{A}$ of Proposition
5.10 (with respect to the external tensor structure obtained in the previous
point).

###### Proof.

Arguing as in the proofs of the previous results, we see that the natural
isomorphisms (33) necessarily satisfy condition ($m$-ETS) of Definition
6.1(1), which implies (1). Similarly, the natural isomorphisms (34)
necessarily satisfy condition (mor-ETS) of Definition 6.1(2), which implies
(2). ∎

###  6.2. Lifting associativity and commutativity constraints

In [Ter23T, §§ 3,4] we explained how to translate the usual notions of
associativity and commutativity constraints for monoidal $\mathscr{S}$-fibered
categories in the setting of external tensor structures. We now want to
discuss briefly how to lift these constraints to universal abelian
factorizations. As usual, we start by recalling the relevant definitions from
[Ter23T], adapted to the additive setting:

######  Definition 6.5.

Let $\mathscr{C}$ be an (additive) $\mathscr{S}$-fibered category endowed with
an (additive) external tensor structure $(\boxtimes,m)$.

  1. (1)

An external associativity constraint $a$ on $(\boxtimes,m)$ is the datum of

     * •

for every $S_{1},S_{2},S_{3}\in\mathscr{S}$, a natural isomorphism of functors
$\mathscr{C}(S_{1})\times\mathscr{C}(S_{2})\times\mathscr{C}(S_{3})\rightarrow\mathscr{C}(S_{1}\times
S_{2}\times S_{3})$

$a=a_{S_{1},S_{2},S_{3}}:(A_{1}\boxtimes A_{2})\boxtimes
A_{3}\xrightarrow{\sim}A_{1}\boxtimes(A_{2}\boxtimes A_{3})$

satisfying the following conditions:

    1. ($a$ETS-1)

For every choice of $S_{1},S_{2},S_{3},S_{4}\in\mathscr{S}$, the diagram of
functors
$\mathscr{C}(S_{1})\times\mathscr{C}(S_{2})\times\mathscr{C}(S_{3})\times\mathscr{C}(S_{4})\rightarrow\mathscr{C}(S_{1}\times
S_{2}\times S_{3}\times S_{4})$

${((C_{1}\boxtimes C_{2})\boxtimes C_{3})\boxtimes C_{4}}$${(C_{1}\boxtimes
C_{2})\boxtimes(C_{3}\boxtimes C_{4})}$${(C_{1}\boxtimes(C_{2}\boxtimes
C_{3}))\boxtimes C_{4}}$${C_{1}\boxtimes((C_{2}\boxtimes C_{3})\boxtimes
C_{4})}$${C_{1}\boxtimes(C_{2}\boxtimes(C_{3}\boxtimes
C_{4}))}$$\scriptstyle{a}$$\scriptstyle{a}$$\scriptstyle{a}$$\scriptstyle{a}$$\scriptstyle{a}$

is commutative.

    2. ($a$ETS-2)

For every three morphisms $f_{i}:T_{i}\rightarrow S_{i}$ in $\mathscr{S}$,
$i=1,2,3$, the diagram of functors
$\mathscr{C}(S_{1})\times\mathscr{C}(S_{2})\times\mathscr{C}(S_{3})\rightarrow\mathscr{C}(T_{1}\times
T_{2}\times T_{3})$

${(f_{1}^{*}C_{1}\boxtimes f_{2}^{*}C_{2})\boxtimes
f_{3}^{*}C_{3}}$${(f_{1}\times f_{2})^{*}(C_{1}\boxtimes C_{2})\boxtimes
f_{3}^{*}C_{3}}$${(f_{1}\times f_{2}\times f_{3})^{*}((C_{1}\boxtimes
C_{2})\boxtimes C_{3})}$${f_{1}^{*}C_{1}\boxtimes(f_{2}^{*}C_{2}\boxtimes
f_{3}^{*}C_{3})}$${f_{1}^{*}C_{1}\boxtimes(f_{2}\times
f_{3})^{*}(C_{2}\boxtimes C_{3})}$${(f_{1}\times f_{2}\times
f_{3})^{*}(C_{1}\boxtimes(C_{2}\boxtimes
C_{3}))}$$\scriptstyle{m}$$\scriptstyle{a}$$\scriptstyle{m}$$\scriptstyle{a}$$\scriptstyle{m}$$\scriptstyle{m}$

is commutative.

  2. (2)

An external commutativity constraint $c$ on $(\boxtimes,m)$ is the datum of

     * •

for every $S_{1},S_{2}\in\mathscr{S}$, a natural isomorphism of functors
$\mathscr{C}(S_{1})\times\mathscr{C}(S_{2})\rightarrow\mathscr{C}(S_{1}\times
S_{2})$

$c=c_{S_{1},S_{2}}:A_{1}\boxtimes
A_{2}\xrightarrow{\sim}\tau^{*}(A_{2}\boxtimes A_{1}),$

(where $\tau:S_{1}\times S_{2}\rightarrow S_{2}\times S_{1}$ denotes the
permutation isomorphism)

satisfying the following conditions:

    1. ($c$ETS-1)

For every $S_{1},S_{2}\in\mathscr{S}$, the diagram of functors
$\mathscr{C}(S_{1})\times\mathscr{C}(S_{2})\rightarrow\mathscr{C}(S_{1}\times
S_{2})$

${C_{1}\boxtimes C_{2}}$${\tau^{*}(C_{2}\boxtimes
C_{1})}$${\tau^{*}\tau^{*}(C_{1}\boxtimes
C_{2})}$$\scriptstyle{c}$$\scriptstyle{c}$

is commutative.

    2. ($c$ETS-2)

For every choice of of morphisms $f_{i}:T_{i}\rightarrow S_{i}$ in
$\mathscr{S}$, $i=1,2$, the diagram of functors
$\mathscr{C}(S_{1})\times\mathscr{C}(S_{2})\rightarrow\mathscr{C}(T_{2}\times
T_{1})$

${f_{1}^{*}C_{1}\boxtimes f_{2}^{*}C_{2}}$${\tau^{*}(f_{2}^{*}C_{2}\boxtimes
f_{1}^{*}C_{1})}$${(f_{1}\times f_{2})^{*}(C_{1}\boxtimes
C_{2})}$${(f_{1}\times f_{2})^{*}\tau^{*}(C_{2}\boxtimes
C_{1})}$${\tau^{*}(f_{2}\times f_{1})^{*}(C_{2}\boxtimes
C_{1})}$$\scriptstyle{c}$$\scriptstyle{m}$$\scriptstyle{m}$$\scriptstyle{c}$

(where $\tau:S_{1}\times S_{2}\xrightarrow{\sim}S_{2}\times S_{1}$ and
$\tau:T_{1}\times T_{2}\xrightarrow{\sim}T_{2}\times T_{1}$ denote the
permutation isomorphisms) is commutative.

In order to define reasonable notions of associativity and commutativity
constraints in the setting of $\mathscr{S}$-fibered representations, we also
need to recall the notions of associativity and symmetry for external tensor
structures on morphisms of $\mathscr{S}$-fibered categories from [Ter23T, §
8], adapted to the additive setting:

######  Definition 6.6.

Let $\mathscr{C}$ and $\mathscr{E}$ be two (additive) $\mathscr{S}$-fibered
categories endowed with additive external tensor structures
$(\boxtimes_{\mathscr{C}},m_{\mathscr{C}})$ and
$(\boxtimes_{\mathscr{E}},m_{\mathscr{E}})$, respectively; in addition, let
$R:\mathscr{C}\rightarrow\mathscr{E}$ be an additive morphism of
$\mathscr{S}$-fibered categories, and let $\rho$ be an external tensor
structure on $R$ (with respect to $(\boxtimes_{\mathscr{C}},m_{\mathscr{C}})$
and $(\boxtimes_{\mathscr{E}},m_{\mathscr{E}})$).

  1. (1)

Let $a_{\mathscr{C}}$ and $a_{\mathscr{E}}$ be external associativity
constraints on $(\boxtimes_{\mathscr{C}},m_{\mathscr{C}})$ and
$(\boxtimes_{\mathscr{E}},m_{\mathscr{E}})$, respectively. We say that $\rho$
is associative (with respect to $a_{\mathscr{C}}$ and $a_{\mathscr{E}}$) if it
satisfies the following additional condition:

    1. (mor-$a$ETS)

For every $S_{1},S_{2},S_{3}\in\mathscr{S}$, the diagram of functors
$\mathscr{C}(S_{1})\times\mathscr{C}(S_{2})\times\mathscr{C}(S_{3})\rightarrow\mathscr{E}(S_{1}\times
S_{2}\times S_{3})$

${(R_{S_{1}}(C_{1})\boxtimes_{\mathscr{E}}R_{S_{2}}(C_{2}))\boxtimes_{\mathscr{E}}R_{S_{3}}(C_{3})}$${R_{S_{1}\times
S_{2}}(C_{1}\boxtimes_{\mathscr{C}}C_{2})\boxtimes_{\mathscr{E}}R_{S_{3}}(C_{3})}$${R_{S_{1}\times
S_{2}\times
S_{3}}((C_{1}\boxtimes_{\mathscr{C}}C_{2})\boxtimes_{\mathscr{C}}C_{3})}$${R_{S_{1}}(C_{1})\boxtimes_{\mathscr{E}}(R_{S_{2}}(C_{2})\boxtimes_{\mathscr{E}}R_{S_{3}}(C_{3}))}$${R_{S_{1}}(C_{1})\boxtimes_{\mathscr{E}}R_{S_{2}\times
S_{2}}(C_{2}\boxtimes_{\mathscr{C}}C_{3})}$${R_{S_{1}\times S_{2}\times
S_{3}}(C_{1}\boxtimes_{\mathscr{C}}(C_{2}\boxtimes_{\mathscr{C}}C_{3}))}$$\scriptstyle{\rho}$$\scriptstyle{a_{2}}$$\scriptstyle{\rho}$$\scriptstyle{a_{1}}$$\scriptstyle{\rho}$$\scriptstyle{\rho}$

is commutative.

  2. (2)

Let $c_{\mathscr{C}}$ and $c_{\mathscr{E}}$ be external commutativity
constraints on $(\boxtimes_{\mathscr{C}},m_{\mathscr{C}})$ and
$(\boxtimes_{\mathscr{E}},m_{\mathscr{E}})$, respectively. We say that $\rho$
is symmetric (with respect to $c_{\mathscr{C}}$ and $c_{\mathscr{E}}$) if it
satisfies the following additional condition:

    1. (mor-$c$ETS)

For every $S_{1},S_{2}\in\mathscr{S}$, the diagram of functors
$\mathscr{C}_{1}(S_{1})\times\mathscr{C}_{1}(S_{2})\rightarrow\mathscr{C}_{2}(S_{1}\times
S_{2})$

${R_{S_{1}}(C_{1})\boxtimes_{\mathscr{E}}R_{S_{2}}(C_{2})}$${R_{S_{1}\times
S_{2}}(C_{1}\boxtimes_{\mathscr{C}}C_{2})}$${\tau^{*}(R_{S_{2}}(C_{2})\boxtimes_{\mathscr{E}}R_{S_{1}}(C_{1}))}$${\tau^{*}R_{S_{2}\times
S_{1}}(C_{2}\boxtimes_{\mathscr{C}}C_{1})}$${R_{S_{1}\times
S_{2}}(\tau^{*}(C_{2}\boxtimes_{\mathscr{C}}C_{1}))}$$\scriptstyle{\rho}$$\scriptstyle{c_{\mathscr{E}}}$$\scriptstyle{c_{\mathscr{C}}}$$\scriptstyle{\rho}$$\scriptstyle{\theta}$

(where $\tau:S_{1}\times S_{2}\xrightarrow{\sim}S_{2}\times S_{1}$ denotes the
permutation isomorphism) is commutative.

With these notions at hand, we can define external associativity and
commutativity constraints on additive $\mathscr{S}$-fibered representations in
a natural way as follows:

######  Definition 6.7.

Let $(\mathscr{C},\mathscr{E};\beta)$ be an (additive) $\mathscr{S}$-fibered
representation endowed with an (additive) external tensor structure
$(\boxtimes_{\mathscr{C}},\boxtimes_{\mathscr{E}};\delta)$.

  1. (1)

An external associativity constraint $(a_{\mathscr{C}},a_{\mathscr{E}})$ on
$(\boxtimes_{\mathscr{C}},\boxtimes_{\mathscr{E}};\delta)$ is the datum of

     * •

an external associativity constraint $a_{\mathscr{C}}$ on
$(\boxtimes_{\mathscr{C}},m_{\mathscr{C}})$,

     * •

an external associativity constraint $a_{\mathscr{E}}$ on
$(\boxtimes_{\mathscr{E}},m_{\mathscr{E}})$

such that the external tensor structure $\nu$ on $\beta$ is associative in the
sense of Definition 6.6(1).

  2. (2)

An external commutativity constraint $(c_{\mathscr{C}},c_{\mathscr{E}})$ on
$(\boxtimes_{\mathscr{C}},\boxtimes_{\mathscr{E}};\delta)$ is the datum of

     * •

an external commutativity constraint $c_{\mathscr{C}}$ on
$(\boxtimes_{\mathscr{C}},m_{\mathscr{C}})$,

     * •

an external commutativity constraint $c_{\mathscr{E}}$ on
$(\boxtimes_{\mathscr{E}},m_{\mathscr{E}})$

such that the external tensor structure $\nu$ on $\beta$ is symmetric in the
sense of Definition 6.6(2).

We now show that external associativity and commutativity constraints of
abelian $\mathscr{S}$-fibered representations lift to universal abelian
factorizations in the expected way:

######  Lemma 6.8.

Let $(\mathscr{D},\mathscr{A};\beta)$ be an abelian $\mathscr{S}$-fibered
representation endowed with an exact external tensor structure
$(\boxtimes_{\mathscr{D}},\boxtimes_{\mathscr{A}};\nu)$; let
$(\overline{\boxtimes},\bar{m})$ be the exact external tensor structure on
$\mathbf{A}(\beta)$ obtained via Proposition 6.4. Then the following
statements hold:

  1. (1)

Suppose that we are given an external associativity constraint
$(a_{\mathscr{C}},a_{\mathscr{E}})$ on
$(\boxtimes_{\mathscr{D}},\boxtimes_{\mathscr{A}};\delta)$. Then, associating

     * •

to every $S_{1},S_{2},S_{3}\in\mathscr{S}$, the natural isomorphisms of
functors
$\mathbf{A}(\beta_{S_{1}})\times\mathbf{A}(\beta_{S_{2}})\times\mathbf{A}(\beta_{S_{3}})\rightarrow\mathbf{A}(\beta_{S_{1}\times
S_{2}\times S_{3}})$

(35)
$\overline{a}_{S_{1},S_{2},S_{3}}:(M_{1}\overline{\boxtimes}M_{2})\overline{\boxtimes}M_{3}\xrightarrow{\sim}M_{1}\overline{\boxtimes}(M_{2}\overline{\boxtimes}M_{3})$

obtained via Proposition 4.6

defines an external associativity constraint on
$(\overline{\boxtimes},\bar{m})$, in such a way that the external tensor
structures $(\boxtimes_{\mathscr{D}},\overline{\boxtimes};\textup{id})$ and
$(\overline{\boxtimes},\boxtimes_{\mathscr{A}};\tilde{\nu})$ are associative.

  2. (2)

Suppose that we are given an external commutativity constraint
$(a_{\mathscr{C}},a_{\mathscr{E}})$ on
$(\boxtimes_{\mathscr{D}},\boxtimes_{\mathscr{A}};\delta)$. Then, associating

     * •

to every $S_{1},S_{2}\in\mathscr{S}$, the natural isomorphisms of functors
$\mathbf{A}(\beta_{S_{1}})\times\mathbf{A}(\beta_{S_{2}})\rightarrow\mathbf{A}(\beta_{S_{1}\times
S_{2}})$

(36)
$\overline{c}_{S_{1},S_{2}}:M_{1}\overline{\boxtimes}M_{2}\xrightarrow{\sim}\overline{\tau^{*}}(M_{2}\overline{\boxtimes}M_{1})$

obtained via Proposition 4.6

defines an external associativity constraint on
$(\overline{\boxtimes},\bar{m})$, in such a way that the external tensor
structures $(\boxtimes_{\mathscr{D}},\overline{\boxtimes};\textup{id})$ and
$(\overline{\boxtimes},\boxtimes_{\mathscr{A}};\tilde{\nu})$ are symmetric.

###### Proof.

  1. (1)

In order to show that the natural isomorphisms (35) define an external
associativity constraint, we need to check that they satisfy condition
($a$ETS-1) and ($a$ETS-2) of Definition 6.5(1). In view of Remark 4.7(2), this
follows from the fact that the same condition is satisfied by the
corresponding natural isomorphisms on the $\mathscr{S}$-fibered category
$\mathscr{D}$. Moreover, saying that the external tensor structures id on
$\pi$ and $\tilde{\nu}$ on $\iota$ are associative is the same as saying that,
for every $S_{1},S_{2},S_{3}\in\mathscr{S}$, the pairs
$(a_{\mathscr{D};S_{1},S_{2},S_{3}},\overline{a}_{S_{1},S_{2},S_{3}})$ and
$(\overline{a}_{S_{1},S_{2},S_{3}},a_{\mathscr{A};S_{1},S_{2},S_{3}})$ define
$2$-morphisms of multi-linear representations, which is indeed the case by
Proposition 4.6.

  2. (2)

In order to show that the natural isomorphisms (36) define an external
commutativity constraint, we need to show that they satisfy condition ($c$ETS)
of Definition 6.5(2). In view of Remark 4.7(2), this follows from the fact
that the same condition is satisfied by the corresponding natural isomorphisms
on the $\mathscr{S}$-fibered category $\mathscr{D}$. Moreover, saying that the
external tensor structures id on $\pi$ and $\tilde{\nu}$ on $\iota$ are
associative is the same as saying that, for every $S_{1},S_{2}\in\mathscr{S}$,
the pairs $(c_{\mathscr{D};S_{1},S_{2}},\overline{c}_{S_{1},S_{2}})$ and
$(\overline{c}_{S_{1},S_{2}},c_{\mathscr{A};S_{1},S_{2}})$ define
$2$-morphisms of multi-linear representations, which is indeed the case by
Proposition 4.6.

∎

Finally, recall that in [Ter23T, § 6] we explained how to translate the
natural compatibility condition between associativity and commutativity in the
setting of external tensor structures. We want to lift this to universal
abelian factorizations as well. To this end, it suffices to recall one more
definition:

######  Definition 6.9.

Let $\mathscr{C}$ be an (additive) $\mathscr{S}$-fibered category endowed with
an (additive) external tensor structure $(\boxtimes,m)$; in addition, let $a$
and $c$ be an external associativity constraint and an external commutativity
constraint on $(\boxtimes,m)$, respectively.

We say that the constraints $a$ and $c$ are compatible if they satisfy the
following condition:

  1. ($ac$ETS)

For every $S_{1},S_{2},S_{3}\in\mathscr{S}$, the diagram of functors
$\mathscr{C}(S_{1})\times\mathscr{C}(S_{2})\times\mathscr{C}(S_{3})\rightarrow\mathscr{C}(S_{1}\times
S_{2}\times S_{3})$

${(C_{1}\boxtimes C_{2})\boxtimes C_{3}}$${\tau_{(12)}^{*}(C_{2}\boxtimes
C_{1})\boxtimes C_{3}}$${\tau_{(12)(3)}^{*}((C_{2}\boxtimes C_{1})\boxtimes
C_{3})}$${\tau_{(12)(3)}^{*}(C_{2}\boxtimes(C_{1}\boxtimes
C_{3}))}$${\tau_{(12)(3)}^{*}(C_{2}\boxtimes\tau_{(13)}^{*}(C_{3}\boxtimes
C_{1}))}$${C_{1}\boxtimes(C_{2}\boxtimes
C_{3})}$${\tau_{(132)}^{*}((C_{2}\boxtimes C_{3})\boxtimes
C_{1})}$${\tau_{(132)}^{*}(C_{2}\boxtimes(C_{3}\boxtimes
C_{1}))}$${\tau_{(12)(3)}^{*}\tau_{(2)(13)}^{*}(C_{2}\boxtimes(C_{3}\boxtimes
C_{1}))}$$\scriptstyle{c}$$\scriptstyle{a}$$\scriptstyle{m}$$\scriptstyle{a}$$\scriptstyle{c}$$\scriptstyle{m}$$\scriptstyle{c}$$\scriptstyle{a}$

is commutative.

######  Lemma 6.10.

Let $(\mathscr{D},\mathscr{A};\beta)$ be an abelian $\mathscr{S}$-fibered
representation endowed with an exact external tensor structure
$(\boxtimes_{\mathscr{D}},\boxtimes_{\mathscr{A}};\nu)$; let
$(\overline{\boxtimes},\bar{m})$ be the exact external tensor structure on
$\mathbf{A}(\beta)$ obtained via Proposition 6.4.

Suppose that we are given an external associativity constraint
$(a_{\mathscr{C}},a_{\mathscr{E}})$ and an external commutativity constraint
$(c_{\mathscr{D}},c_{\mathscr{A}})$ on
$(\boxtimes_{\mathscr{D}},\boxtimes_{\mathscr{A}};\delta)$ such that
$a_{\mathscr{D}}$ and $c_{\mathscr{D}}$ are compatible. Then the external
associativity and commutativity constraints $\bar{a}$ and $\bar{c}$ on
$(\overline{\boxtimes},\overline{m})$ obtained via 6.8 are compatible as well.

###### Proof.

We need to check that the external associativity constraint $\overline{a}$ and
the external commutativity constraint $\overline{c}$ satisfy condition
($ac$ETS) of Definition 6.9. In view of Remark 4.7(2), this follows from the
fact that the same condition is satisfied by the corresponding natural
isomorphisms on the $\mathscr{S}$-fibered category $\mathscr{D}$. ∎

######  Remark 6.11.

In a similar way, it is possible to lift external tensor structures on
morphisms of $\mathscr{S}$-fibered categories (see Definition 6.2) to the
level of universal abelian factorizations; the same applies to the
associativity and symmetry properties of tensor structures on morphisms (see
Definition 6.7). However, we refrain from stating these results precisely
since they do not seem to be relevant for our applications; the interested
reader will find the correct statements without difficulty.

## References

  * [SGA1] Revêtements étales et groupe fondamental. Séminaire de Géometrie Algébrique du Bois-Marie, dirigé par A. Grothendieck, augmenté de deux exposés de Mme M. Raynaud, Lecture Notes in Math., 224, Springer-Verlag, Berlin, 1964.
  * [Ara13] D. Arapura, An abelian category of motivic sheaves. Adv. Math. 233, 2013, 135–195.
  * [Ara19] D. Arapura, Motivic sheaves revisited. Joural of Pure and Applied Algebra, vol. 227, Issue 8, 2019.
  * [BVCL18] L. Barbieri-Viale, O. Caramello, L. Lafforgue, Syntactic categories for Nori motives. Sel. Math. 24(4), 3619–3648 (2018).
  * [BVP18] L. Barbieri-Viale, M. Prest, Definable categories and $\mathbb{T}$-motives. Rend. Sem. Mat. Univ. Padova 139 (2018), pp. 205–224.
  * [BVHP20] L. Barbieri-Viale, A. Huber, M. Prest, Tensor structure for Nori motives. Pacific Journal of Mathematics 206 (2020), pp. 1–30.
  * [Del01] P. Deligne, Voevodsky’s lectures on cross functors. Motivic Homotopy Theory program, IAS Princeton, Fall 2001.
  * [Fre65] P. Freyd, Representations in abelian categories. 1966 Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) pp. 95–120 Springer, New York.
  * [HMS17] A. Huber, S. Müller-Stachs, Periods and Nori motives. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 65, Springer, Cham, 2017, With contributions by Benjamin Friedrich and Jonas von Wangenheim.
  * [Ivo17] F. Ivorra, Perverse Nori motives. Math. Res. Lett. 24 (2017), no. 4, pp. 1097–1131.
  * [IM19] F. Ivorra, S. Morel, The four operations on perverse motives. Preprint, available at https://arxiv.org/abs/1901.02096.
  * [Mac63] S. Mac Lane, Natural associativity and commutativity. Rice University Studies 49, 1963, no. 4, pp. 28–46.
  * [Nee01] A. Neeman, Triangulated categories. Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001.
  * [Ter23T] L. Terenzi, Tensor structures on fibered categories. Preprint, available at https://sites.google.com/view/luca-terenzi/research.
  * [Ter23N] L. Terenzi, Tensor structure on perverse Nori motives. Preprint, available at https://sites.google.com/view/luca-terenzi/research.