*natbibCitation

11institutetext: Lund Observatory, Division of Astronomy and Theoretical
Physics, Lund University, Box 43, 221 00 Lund, Sweden
11email<EMAIL_ADDRESS>22institutetext: Universitäts-Sternwarte
München, Fakultät für Physik, Ludwig-Maximilians-Universität München,
Scheinerstrasse 1, 81679 München, Germany 33institutetext: University of Bern,
Center for Space and Habitability, Gesellschaftsstrasse 6, CH-3012, Bern,
Switzerland 44institutetext: Research Centre for Astronomy, Astrophysics and
Astrophotonics, Macquarie University, Balaclava Road, Sydney, NSW 2109
Australia 55institutetext: University Observatory, Faculty of Physics, Ludwig
Maximilian University, Scheinerstrasse 1, D-81679 Munich, Germany
66institutetext: ARTORG Center for Biomedical Engineering Research,
Murtenstrasse 50, CH-3008, Bern, Switzerland 77institutetext: Astronomy &
Astrophysics Group, Department of Physics, University of Warwick, Coventry CV4
7AL, United Kingdom

# Small but mighty: High-resolution spectroscopy of ultra-hot Jupiter
atmospheres with compact telescopes

KELT-9 b’s transmission spectrum with Wendelstein’s FOCES Spectrograph

N. W. Borsato 1144 H. J. Hoeijmakers 11 D. Cont 22 D. Kitzmann 33 J. Ehrhardt
22 C. Gössl 22 C. Ries 22 B. Prinoth 11 K. Molaverdikhani 22 B. Ercolano 22 H.
Kellerman 22 Kevin Heng 556677

(Received August 9, 2023; accepted )

When observing transmission spectra produced by atmospheres of ultra-hot
Jupiters, large telescopes are typically the instrument of choice due to the
very weak signal of the planet’s atmosphere. This study aims to alleviate the
desire for large telescopes by illustrating that the same science is possible
with smaller telescope classes. We use the cross-correlation technique to
showcase the potential of the high-resolution spectrograph FOCES at
Wendelstein Observatory and demonstrate its potential to resolve the
atmosphere of the ultra-hot Jupiter, KELT-9 b. A performance comparison is
conducted between FOCES and HARPS-N spectrographs, considering both single
transit and combined observations over three nights. With FOCES, we have
detected seven species in KELT-9 b’s atmosphere: Ti II, Fe I, Fe II, Na I, Mg
I, Na II, Cr II, Sc II. Although HARPS-N surpasses FOCES in performance, our
results reveal that smaller telescope classes are capable of resolving ultra-
hot Jupiter atmospheres. This broadens the scope of potential studies,
allowing for investigations into phenomena like temporal variations in
atmospheric signals and the atmospheric loss characteristics of these close-in
planets.

###### Key Words.:

Planets and satellites: atmospheres – Planets and satellites: gaseous planets
– Techniques: spectroscopic

##  1 Introduction

In ground-based high-resolution spectrographic transit observations of
exoplanet atmospheres, the telescope’s size often sets the scale for
scientific aims, with larger telescopes typically favoured for these types of
observations. Atmospheric observations of exoplanets suffer from photon
scarcity, and as such, larger mirrors prove indispensable, collecting more
photons and compensating for the feeble planet signal, which can be as low as
a few parts per million relative to the host star’s flux (Brogi et al., 2012).
This scarcity often results in the exoplanet signal’s subtle absorption
profile becoming indistinguishable within the noise of the observation
(Stangret et al., 2020). These challenges limit detailed observational studies
to certain sub-classes of exoplanets which include the hottest exoplanets
known to date, Ultra-hot Jupiters (UHJs) (Parmentier et al., 2018).

UHJs orbit close to their host stars and serve as excellent targets for
atmospheric studies thanks to their typically inflated atmospheres resulting
in large transit depths (Baraffe et al., 2010; Merritt et al., 2021). The
extreme irradiation these planets undergo, as a result of their proximity to
their host star, leads to unique atmospheric effects, such as molecular
dissociation (Kitzmann et al., 2018; Arcangeli et al., 2018; Parmentier et
al., 2018), cloud-free day-sides (Helling et al., 2021), equatorial jets
moving at velocities of kilometres per second (Showman & Polvani, 2011), and
mass-loss through photoevaporation (Lowson et al., 2023).

Owing to their unique atmospheric properties, UHJs serve as laboratories for
exoplanet science, as their readily observable and simple atmospheres
facilitate the interpretation of results of observations and can be modelled
under the assumption of equilibrium chemistry (Heng & Kitzmann, 2017). In
fact, they are the ideal test targets for exploring the limits of telescope
size for atmospheric studies. Small telescopes are the workhorses of
astronomy, enabling vital research which may be high-risk or have long-term
payoffs. The lower subscription factors of these classes of telescope also
enable the possibility of multi-epoch observations of the same target which
may produce the same quality of scientific result when combining multiple
epoch observations, but would also enable studies of the time evolution of the
atmospheres of these exoplanets.

There is a growing list of telescopes which have successfully conducted high-
resolution spectrographic observations of UHJs including the Nordic Optical
Telescope (with a mirror diameter of 2.5m Bello-Arufe et al., 2022), as well
as the Fred Lawrence Whipple Observatory (Lowson et al., 2023) both telescopes
have produced detections within a single transit. In this work, we aim to add
another 2m class telescope to the pool which can produce an atmospheric
detection of a UHJ atmosphere in a single transit with the FOCES spectrograph
on the 2m Wendelstein telescope (Pfeiffer et al., 1998a). The target of choice
is KELT-9 b, an ultra-hot Jupiter with an equilibrium temperature of
approximately 4000 K, which has proven to be a highly observable target for
benchmark cases like this (e.g., Hoeijmakers et al., 2018; Yan et al., 2019;
Bello-Arufe et al., 2022; Lowson et al., 2023).

Furthermore, we aim to compare the performance of FOCES to the HARPS-North
(HARPS-N) high-resolution spectrograph on the Telescopio Nazionale Galileo
(TNG), a standard instrument used in the study of exoplanet atmospheres (e.g.,
Casasayas-Barris et al., 2019; Stangret et al., 2020, 2021; Cont et al., 2021;
Borsa et al., 2022; Cont et al., 2022; Pino et al., 2022). Our results will
shed light on the potential for atmospheric studies with 2m class telescopes.

##  2 Observations

We used FOCES to observe three transit observations of KELT-9 b on the
following dates 18th July 2022, 2nd September 2022, and 5th September 2022
(which is only a partial transit). Each transit was observed in time series
taking continual exposures of 300 s in length throughout the transit, plus
some additional exposures out of transit to obtain a stellar baseline without
the planet signal. The other data set used is a single transit observation
taken using the HARPS-N spectrograph on the TNG, which is available on the
data archive and was initially published in Hoeijmakers et al. (2018). The
transit data was obtained 31st July 2017, with an integration time of 600s per
exposure frame. The observation encompassed the entire transit and baseline
measurements. Table 1 provides a summary of the observations taken with each
spectrograph, including the relevant characteristics of the instruments and
the relative amount of stellar baseline which was taken compared to the length
of the transit, while the airmass plots for each transit are shown in Fig. 5
depicting where on the sky the transit occur and during what epoch of the
night.

Using the approach described by Borsato et al. (2023), we measure the average
quality of the data by taking the median signal-to-noise (S/N) over all the
echelle orders this allows us to track the time progression of the data
quality throughout the transit. The time progression of each observation S/N
is shown in Fig. 1 as a function of orbital phase, with the beginning and end
of the transit marked. The figure gives a good overall picture of how the
observation progressed throughout each night. To make the observations
comparable we divide the S/N values for the HARPS-N night by a factor of
$\sqrt{2}$ (assuming a Poisson noise distribution), to correct for the time
difference in the exposures.

Among the three FOCES nights, the first exhibited the highest S/N, which later
dropped by a factor of two towards the end of the observation. The night began
with an overcast sky of cirrus clouds. These clouds, however, began to
dissipate just before observations began. Approximately midway through
observations the signal quality dropped due to a slow build-up of clouds
towards the end of the night. The second night showed significant variations
in the S/N throughout the entire transit. The entire night was characterised
by the presence of cirrus clouds of varying opacity. These clouds, however,
did not completely obscure the sky enabling measurements to partially be
obtained. Throughout the night, the signal dropped steadily and oscillated
periodically. The third night was mainly clear, with occasional thin and
narrow cirrus clouds passing by, but due to twilight timing constraints only a
partial transit could be obtained. As the night progressed, there were slight
signal drops during the exposure times, likely due to fast-moving clouds
partially obscuring the observations. However, these clouds were transient,
leading to a patchwork drop in night flux rather than a consistent decrease.
The HARPS-N spectrograph outperforms all three observations by an approximate
factor of two, and is not noted to have any significant problems with the
weather.

##  3 Data processing

FOCES spans a wavelength range between 383 and 885 nm at a spectral resolution
of R $\sim$ 70,000 (Pfeiffer et al., 1998b; Brucalassi et al., 2012; Wang et
al., 2017). The primary science products derived from the spectrograph’s
pipeline include two-dimensional echelle data, recorded errors, and blaze
profiles. We stitched the overlapping pixels of the echelle orders by
averaging the overlapping flux values and using the recorded errors as
weights, thereby generating a one-dimensional spectrum of each exposure. The
HARPS-N spectrograph spans a wavelength range from 387 to 690 nm with a
resolution of R $\sim$ 115,000. The scientific products available from these
observations include the two-dimensional echelle orders, and the blaze-
corrected, stitched, and resampled one-dimensional spectra.

The gathered data necessitated the removal of systematic signals extraneous to
the planet. We adopted the methodology outlined in Hoeijmakers et al. (2020a)
for this process, which entailed the elimination of atmospheric telluric
contamination, velocity offsets, the interstellar medium (ISM), the host star
signal, and residual contamination in the continuum. To remove the telluric
lines produced by water and carbon dioxide, we used the one-dimensional
spectra. We were able to model the telluric lines using the software package
molecfit (Smette et al., 2015; Kausch et al., 2015), thereby creating an
atmospheric telluric model across the full wavelength range of the data
analysed. After modelling, we divided the model out of the spectra to
eradicate the telluric contamination.

Furthermore, there are two velocity offsets which require correcting, these
are caused by two Doppler shifts, those arising from the Earth-Sun barycentric
velocity and the radial velocity of the star produced by the planet’s
gravitational pull. The correction leaves the spectra at a constant radial
velocity shift set by the systemic velocity of approximately -18
$\textrm{km\>s}^{-1}$(Gaudi et al., 2017). The barycentric velocity correction
is calculated using the python library astropy, while the radial velocity
offset is available from the discovery paper (Gaudi et al., 2017).

To remove the host star’s signal, we compute the average spectrum over all the
exposures of KELT-9 b, and divide each exposure, this removes the static
components of the spectrum while maintaining the time transient properties,
like the planet’s signal. Dividing by the average produces a residual
continuum shape which requires flattening using a median filter.

Finally, any bad pixels remaining are flagged as NaNs using a running standard
deviation, with any values exceeding 5$\sigma$ flagged. The data is further
cleaned by applying a Gaussian high-pass filter with a bandwidth of 80 km/s,
and once again marking any outliers beyond the 5$\sigma$ limit. Finally, any
residual ISM effects are manually flagged as NaNs. The NaNs generated from
this process create an outlier mask that enables us to bypass problematic
pixels during the cross-correlation process. All these steps are applied
through the cross-correlation software
tayph111https://github.com/Hoeijmakers/tayph.

Figure 1: The time trend of the median S/N for each transit observation of
KELT-9 b as a function of orbital phase. The HARPS-N S/N values have been
scaled to values that would be expected to be produced for the exposure times
taken by the FOCES spectrograph. The HARPS-N transit covers the entire
planet’s transit, and a substantial amount of baseline. Additionally, the S/N
is consistently higher throughout the entire observation compared to the
observations taken by FOCES. Signal quality also drops off for all the FOCES
nights towards the end of each night. Table 1: Summary information of the
transit observations used in this study

Date |  Spectrograph |  Wavelength Range [nm] |  Resolution |  S/N [300s] |  Exp-Time [s] |  Exp-Number |  Baseline Fraction %
---|---|---|---|---|---|---|---
31-07-2017 |  HARPS-N |  387 - 690 |  115 000 |  56 |  600 |  49 |  57
18-07-2022 |  FOCES |  380 - 750 |  70 000 |  34 |  300 |  41 |  18
02-09-2022 |  FOCES |  380 - 750 |  70 000 |  26 |  300 |  41 |  33
05-09-2022 |  FOCES |  380 - 750 |  70 000 |  30 |  300 |  39 |  17

222The table summarises the details of the four observations used in this
study. Columns denote the date of observation, the spectrograph used, the
operational wavelength range in nm, the spectral resolution, the signal-to-
noise ratio per 300 seconds of exposure time, with the HARPS-N scaled through
Poisson noise assumptions, the exposure time in seconds, the total number of
exposures, and the percentage of observations outside of transit.

##  4 The cross-correlation technique

To search for atmospheric signals, we applied the cross-correlation method.
This innovative approach was first introduced by Snellen et al. (2010),
enabling the detection of CO in the transmission spectrum of HD 209458 b. The
essence of this method is to combine the flux contributions of specific
spectral lines within a planet’s transmission spectrum, thereby effectively
generating a weighted average of the spectral line profile for a given
species, based on its known line positions.

The variable radial velocity of the planet, however, introduces a shift in the
spectral line positions within the wavelength space. To accommodate this
variability, the technique employs Doppler shifting to adjust the known line
positions to different radial velocity values. This step allows for the
continuous scanning of the spectrum for these shifted lines.

The real strength of this method becomes evident when the predicted line
positions align with the planet’s absorption spectrum. This alignment leads to
a correlation peak, which signifies the successful detection of a species
within the planet’s atmosphere.

In order to carry out this method, it is necessary to construct a spectral
template, $T$. This template calculates the transmission spectrum of either
the entire planet or an individual species. It is then used to generate a set
of weights, assigned to the anticipated line positions of the species under
investigation:

$\hat{T}_{i}(v)=\frac{T_{i}(v)}{\Sigma T_{i}(v)}$ (1)

Here, the template function, $T_{i}(v)$, is dependent on the radial velocity,
where $v$ denotes the radial velocity of the template. The weights are
structured so that the total sum of the templates is equal to one, with large
weight values allocated to the line positions. This weighted average is then
employed to calculate the mean line depth, as illustrated in the following
equation:

$c(v)=\sum_{i=0}^{N}x_{i}\hat{T}_{i}(v)$ (2)

In this equation, $c(v)$ represents the computed mean depth of absorption
lines, where each line is weighted relative to its position at a specific
Doppler shift. The dependence on the velocity term implies that the final
value obtained from the sum will change depending on the Doppler shift applied
to the calculation. The total quantity of evaluated wavelength bins is
represented by $N$. Each $x_{i}$ corresponds to the spectral data that aligns
with the same wavelength segment $i$ as the reference template.

The cross-correlation function (CCF) leverages the fact that a planet’s radial
velocity changes throughout transit, leading to a shift in the line positions
of the transmission spectrum. By applying Eq. 2, a peak will form at the
correct velocity position if the species is present in the planet’s
atmosphere. This process results in an absorption trace, observable in the top
panel of Fig. 2, on a two-dimensional grid that plots time against planetary
radial velocity.

After applying the CCF, further processing steps are necessary to generate an
atmospheric signal. The process consists of several sequential actions.
Initially, residual broadband variations stemming from the removal of the
stellar signal are eliminated. Subsequently, the Rossiter-McLaughlin (RM)
effect (Rossiter, 1924; McLaughlin, 1924), a phenomenon caused by the planet
temporarily distorting the line-of-sight velocity of the star by obscuring
part of the rotating star’s surface (commonly known as the Doppler-shadow), is
modelled and removed. The methodology adopted for broadband variation removal
and Rossiter-McLaughlin effect compensation is inspired by Hoeijmakers et al.
(2020b), who successfully conducted a chemical inventory of WASP-121 b.
Finally, we remove any residual stellar aliases by applying the same vertical
detrending adapted from Prinoth et al. (2022) and implemented by Borsato et
al. (2023) to remove residual cross-correlation signatures which come directly
from the star’s spectra.

It is also possible to enhance the S/N ratio of a cross-correlation function
by co-adding all pixels along the planet’s trace. This is achieved by shifting
all exposures into the time-independent rest frame of the planet using its
orbital phase and inclination. This process can be executed with the following
equation:

$v_{rv}=v_{\rm{orb}}\sin{2\pi\phi}\sin{i}+v_{\rm{sys}}$ (3)

In the provided equation, $v_{orb}$ indicates the exoplanet’s orbital velocity
around its host star, while $\phi$ represents the exoplanet’s orbital phase.
The symbol $i$ refers to the system’s inclination relative to the observer’s
line of sight, and $v_{sys}$ corresponds to the host star’s systemic velocity.

We shift the radial velocity grid according to the equations for a presumed
semi-major amplitude ($K_{p}$) by applying the known orbital phase of the
planet. To find the optimal alignment, we sample a range of $K_{p}$ values.
Each one is then collapsed along the time axis by computing a vertical
average. When the shift leads to a vertical alignment of the cross-correlation
function, a peak becomes evident. However, if the alignment isn’t vertical,
the cross-correlation functions will destructively interfere with each other.
As a result, we expect to see a peak at the correct $K_{p}$ value.

We construct a grid, or a $K_{p}V_{sys}\ $-map (Brogi et al., 2012), from all
the collapsed cross-correlation functions corresponding to the various shifts.
This map is identified by a peak at the correct $K_{p}$ and $v_{sys}$ values.
The row corresponding to this peak provides an optimised one-dimensional
cross-correlation function, where all exposures are combined to yield the most
robust signal. For this study, we developed a grid of semi-major amplitudes
spanning from 0 to 400 $\textrm{km\>s}^{-1}$, incremented in steps of 1
$\textrm{km\>s}^{-1}$.

Finally, we extract the one-dimensional cross-correlation function by
selecting the $K_{p}$ that best aligns the CCF functions. In order to
determine the amplitude, width, and centre locations of the signal, we employ
a least-squares fit to the spectra, under the assumption of a Gaussian
profile. Uncertainties in the amplitude, width, and centre locations are
derived via a bootstrap method, which involves sampling the noise floor of the
one-dimensional CCF.

This process starts with the removal of the region where the peak is located,
followed by the computation of the standard deviation. Assuming a Gaussian
noise profile, we randomly adjust each data point in the CCF and apply the
same fitting procedure, recording the measured parameters each time. After
repeating this process for 5,000 iterations, we calculate the standard
deviation of the fitted values, treating these as their uncertainties. The
statistical significance of the signal is then evaluated as the ratio of the
measured amplitude to its corresponding uncertainty.

Figure 2: The cross-correlation results for searching for Fe I in a single
transit. On the left are the results for the HARPS-N spectrograph on the TNG,
on the right we have the results for the FOCES spectrograph. Top Panel: The
cross-correlation maps for Fe I. The pixels covering the planet’s trace have
been outlined to help guide the eye on where to expect the signal. Middle
panel: The $K_{p}V_{sys}\ $-maps for each Fe I detection, the detections are
the bright feature in the centre of the plots. The dashed crosshairs show the
row where the one-dimensional cross-correlation function (bottom panel) has
been extracted and the expected systemic velocity of KELT-9 b (-20
$\textrm{km\>s}^{-1}$). Bottom Panels: The one-dimensional cross-correlation
functions for the single transits, showing the average line depth for the
given signal. The dashed line shows the expected systemic velocity of KELT-9
b, which is the same in $K_{p}V_{sys}\ $-map.  Figure 3: Comparison of 1D CCFs
for the Fe I signal between different datasets extracted from the peak of each
$K_{p}V_{sys}\ $plot. Each panel represents the data from a different source:
HARPS-N, FOCES Night 1, and FOCES Nights 1, 2, and 3 combined respectively.
Each plot shows the 1D CCF, Gaussian fit, and model prediction. There is a
small discrepancy in the model peak height for each result, which is driven by
the differences in resolution of the instrument, HARPS-N being of higher
resolution has resolved a deeper peak. When the spectral resolution is lower,
the spectral lines tend to smear resulting in a decrease in line depth. The
model-predicted Cross-Correlation Function (CCF) displays values slightly
below zero near the peak of the function. This observation is attributed to
cross-correlation aliases, a phenomenon explained in Borsato et al. (2023).
Table 2: Comparative Statistics of CCF from the HARPS-N and FOCES
Observations, for the Fe I detection

Parameter | HARPS-N | FOCES | Combined FOCES
---|---|---|---
Amplitude [ppm] | $132.7\pm 6.9$ | $142.1\pm 24.4$ | $123.1\pm 17.2$
Center [$\textrm{km\>s}^{-1}$] | $-21.3\pm 0.5$ | $-20.4\pm 1.6$ | $-16.4\pm 1.7$
Width [$\textrm{km\>s}^{-1}$] | $7.0\pm 0.4$ | $7.9\pm 1.5$ | $8.2\pm 1.2$
Detection Significance | $19.2$ | $5.8$ | $7.2$
Model Discrepancy | $0.3\pm 0.04$ | $0.1\pm 0.1$ | $0.3\pm 0.1$
Discrepancy Significance | $7.5$ | $1.0$ | $3.0$
333This tables shows the summary statistics from the bootstrap fits of the
one-dimensional cross-correlation function for Fe I. Detection significance is
the ratio of the amplitude fits with its measured standard deviation. Model
discrepancy refers to the ratio of the one minus the amplitude fit divided by
the height of the model peak. The discrepancy significance is this value
divided by the fit standard deviation.

###  4.1 Combining the FOCES Nights

To amalgamate the three FOCES observations into a single $K_{p}V_{sys}\ $-map,
we employed an adapted approach from Borsato et al. (2023), which involves
computing a weighted average of the $K_{p}V_{sys}\ $-maps. Given that only one
spectrograph is under consideration, we do not need to factor in the
instrument’s wavelength range which is part of the process used in that
publication. Thus, we proceed to construct appropriate weights to combine the
nights.

Firstly, we mask out the planet’s signal in the $K_{p}V_{sys}\ $-map. Our mask
was devised to screen all data values ranging from $-50$
$\textrm{km\>s}^{-1}$to $10$ $\textrm{km\>s}^{-1}$for the systemic velocity,
and $100$ $\textrm{km\>s}^{-1}$to $300$ $\textrm{km\>s}^{-1}$for the orbital
velocity. Upon applying the mask, we calculate the standard deviation of the
remaining pixels, providing a measure of any residual noise and systematics
affecting the overall noise floor.

Next, we invert the mask, thereby isolating the region where we anticipate the
signal. Any signal within this location means that the region would yield a
higher average value compared to the background. By taking the ratio of this
mean to the calculated standard deviation, we generate a pseudo-S/N ratio,
which offers a general assessment of the night’s quality and can then be
converted into the desired weight.

Lastly, we normalise the pseudo-S/N values for all observations so that they
add up to one, using these resultant values as the weights.

###  4.2 Cross-correlation templates

In this study, we utilised the standard templates provided by the Mantis
network (Kitzmann et al., 2021), which comprises more than 750 high-resolution
templates for various species across a grid of different atmospheric
temperatures. These templates are publicly
available444https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/669/A113 and are
computed using a forward modelling approach, taking into account the target
planet’s transmission spectrum and the absorption lines of the species under
scrutiny. The computation process includes opacity functions, continuum
opacity contributions, and assumptions about metal abundance and temperature
profiles. These templates provide a standardised and efficient alternative to
creating new templates for each individual study. Mantis templates have
successfully aided in the detection of a range of species using the ESPRESSO
spectrograph on the VLT (Azevedo Silva et al., 2022). Our search only focused
on the species reported in Hoeijmakers et al. (2019) with an assumed
equilibrium temperature of 4000 K, to simplify the comparison between
telescopes.

###  4.3 Model injection and comparing instruments

In order to make a comparative analysis of the data from FOCES and HARPS-N,
our initial step was to create a forward model of KELT-9 b’s transmission
spectrum. This process, modelled after similar studies (e.g. Hoeijmakers et
al., 2019). We used the same model as described by Kitzmann et al. (2023) to
compute the transmission spectrum but changed the planet’s radius and surface
gravity to that of KELT-9 b. The temperature profile of the atmosphere was
assumed to be isothermal, with a value of 4000 K. We used all opacity species
listed in Table 2 of Kitzmann et al. (2023). The corresponding line absorption
coefficients were calculated with the
HELIOS-K555https://github.com/exoclime/HELIOS-K opacity calculator (Grimm &
Heng, 2015; Grimm et al., 2021). The chemical composition of the atmosphere
was obtained by using the equilibrium chemistry code FastChem
2666https://github.com/exoclime/FastChem (Stock et al., 2018, 2022), assuming
solar element abundances. The resultant transmission spectrum, representative
of the observed atmospheric characteristics, is shown in Fig. 6.

Following the methodology outlined in Hoeijmakers et al. (2015) (which
searched for TiO in HD 209458 b, and used model injection to explain its non-
detection), we take the predicted transmission spectrum, inject it into the
pipeline-reduced spectra obtained from the observations of both instruments,
and rotationally broaden the modelled lines to account for the planet’s
rotation. Subsequently, we convolve through Gaussian convolution to match the
spectral resolution of each spectrograph. To align with the spectrograph’s
grid, the wavelength axis is interpolated. Finally, we perform cross-
correlation on the spectra a second time. This approach facilitates a
comparison between the expected cross-correlation signal and the actual signal
generated by the spectrograph. By subtracting the injected case cross-
correlation results from the cross-correlation profile without an injected
signal, we produce a residual signature, which illustrates the predicted line
depth and shape of the models.

##  5 Results

We initiate our analysis by presenting a comparative visual representation of
the Fe I detection from a single night using HARPS-N and the first night with
FOCES. We illustrate these results in Fig. 2, in the upper portion of these
plots we show the cross-correlation function before conversion to a
$K_{p}V_{sys}\ $-map. The planets trace is clearly visible for the HARPS-N
spectrograph, but is barely visible for FOCES output. In the centre of these
plots we have the $K_{p}V_{sys}\ $-map, which shows the signal of Fe I for
both spectrographs, though the HARPS-N signal is much stronger. Finally, in
the lower portion of these figures, we have the extracted one-dimensional CCF
which showing the the idealised signal produced from the cross-correlation
process. We are able to detect atomic iron using both spectrographs, as
reflected in the $K_{p}V_{sys}\ $plots and the one-dimensional cross-
correlation functions. However, a prominent difference emerges in the trace of
the planet within the cross-correlation function: HARPS-N data clearly reveals
the trace, whereas it is only partially discernible in the FOCES output. A
closer look at the one-dimensional function illustrates the dominance of noise
in the FOCES data for that night. Despite this, the emergence of the signal
reaffirms the feasibility of resolving KELT-9 b’s atmosphere with a single
transit, even when baseline data is minimal.

In Fig. 3, we compare the performance of the FOCES to that of the HARPS-N
spectrograph for a single night of observation, and when all three nights are
combined. We do this by plotting the one-dimensional CCFs with their
corresponding Gaussian fits, and the signal predicted by the model. The values
and statistics for each fit are provided for comparison in Table 3, including
how the model compares to the actual data. At first glance, it is clear that
HARPS-N provides superior signal quality even when combining the transits.
However, the line depths are comparable across all three nights. When
comparing to the model injection with the model depicted in Fig. 3, the
discrepancy, computed as the difference between the signal and the model, is
approximately equivalent for both the HARPS-N observation and the combined
FOCES observations, although the uncertainties associated with the FOCES
discrepancy are nearly twice as large. This implies that, when juxtaposed with
our model, the signals derived from both spectrographs diverge from the model
to a similar extent.

The $K_{p}V_{sys}\ $-maps and one-dimensional cross-correlation functions for
all eight detections are displayed in Fig. 4 and the amplitude, width, centre
locations, and detection significance, are detailed in Table 7. Our method
successfully replicated seven out of the eight detections previously mentioned
in Hoeijmakers et al. (2019). The only exception was Y II, which did not
surface in our results. This is consistent with the difficulties encountered
in re-analysing the HARPS-N data to confirm the presence of Y II, as noted by
(Borsato et al., 2023). If Y II is indeed present, its relatively weak signal
is likely due to its low abundance in the planet’s atmosphere. More definitive
conclusions about the presence of Y II would require further observations,
either through combined efforts or with the use of a larger telescope.

Among the detected signals, Fe II appeared to be the strongest. However, we
also noted substantial detections of Fe I and Ti II, along with weaker
detections of Na I, Mg I, Na II, Cr II, and Sc II. The central peaks of these
signals were measured using a least squares Gaussian fit, which estimates the
best fit for the centre, width, and amplitude of the cross-correlation peak.
The derived parameters aligned with the findings of Hoeijmakers et al. (2019),
with the uncertainties of our measurements overlapping with those they
reported. The tentative Mg I detection, however, merits particular attention,
as it exhibits considerable smearing and is significantly delocalised from its
actual signal. This implies there may be additional issues with the tentative
Mg I detection, thereby warranting further exploration.

Figure 4: The cross-correlation results for the eight species investigated.
Ordered left to right, then top to bottom are: Na I Mg I, Sc II, Ti II, Cr II,
Fe II, Fe I and Y II. The top panels of each detection contain the
$K_{p}V_{sys}\ $-maps. The crosshairs locate the points where the signal peak
has been extracted, and the bottom plots show what the one-dimensional cross-
correlation slice looks like with the rows where the peak is extracted. We
have managed to successfully recover all seven out of the eight detections
reported in that paper at least partially. Only Y II has not been recovered,
but it is known to be a weak signal and difficult for the HARPS-N spectrograph
to even resolve. While the signals are present in most plots they exhibit a
strong degree of noise indicative of using a 2m class telescope.

##  6 Discussion

The HARPS-N spectrograph on the TNG, with a single transit observation, can
resolve the atmosphere of KELT-9 b to an impressively high fidelity. However,
our results shown in Fig. 2 demonstrate that FOCES can also resolve the
atmosphere of this source on a telescope half the size of the TNG. Hence, we
propose that by utilising multiple transits, results comparable to those
achieved with HARPS-N can also be realised using moderately-sized telescopes.

The HARPS-N spectrograph outperforms the FOCES spectrograph by a factor of two
to three, considering all three combined nights of FOCES observation. This
disparity is larger than what we would anticipate based solely on telescope
size; we would expect that four nights should yield detection statistics
comparable to one HARPS-N night. Therefore, there are other factors degrading
the quality of these observations.

The prevailing weather conditions, particularly cloud cover, were likely the
most substantial factors influencing the results. The cloud cover during all
FOCES observations likely had a substantial impact on the detection statistics
used for the comparison. Given that clouds were present at some point during
each of the three observations, it hindered the process of drawing a
definitive comparison between the two telescopes and their respective
spectrographs.

An alternative explanation for the deteriorated data quality could be the
airmass constraints of the observations. However, when comparing all the
observations in Fig. 5, we observe that all were conducted at low airmass,
suggesting this variable likely did not contribute significantly to the
discrepancy in the results. A portion of all the three transits observed by
FOCES took place during astronomical twilight, potentially introducing
contamination from diffuse background light. However, as this contamination
would not consistently affect all exposures, its overall impact is likely to
be minimal.

Other factors might have played a smaller role, such as the limited baseline
measurements of KELT-9, or the lower resolution of FOCES. The single night of
HARPS-N has baseline observations that almost match the total baseline
duration of all three FOCES observations. Despite the absence of additional
planetary flux, this increase in the measurement of the stellar flux could
permit a more precise constraint on the stellar component subtracted during
the cross-correlation process. The lower resolution operation of FOCES may
result in increased line blending in the resultant transmission spectrum,
thereby reducing the efficiency of the cross-correlation method (Borsato et
al., 2023). To draw more definitive conclusions, additional controls are
necessary, including consistent baseline timings, similar observing windows,
and comparable weather conditions.

Despite these issues, our findings attest that FOCES at Wendelstein
Observatory can proficiently detect signals from a multitude of chemical
species present in KELT-9 b’s atmosphere. As evidenced in Table 7 and Fig. 4,
we successfully retrieved spectral signals of seven atomic and ionised
species, each with a detection significance exceeding 3$\sigma$. Moreover, the
enhancements in detection significance, indicated in Table 3 and Fig. 3,
suggest that multiple transit observations could produce results commensurate
with those from HARPS-N. These results insinuate that smaller telescopes could
play a pivotal role in broadening the scientific scope of exoplanet
atmospheric studies.

The distinctive advantage that small telescopes offer lies in the abundance of
their observational time. These telescopes possess the capability to revisit
targets repeatedly, not only placing them in a competitive position with
larger classes of telescopes, but also allowing them to track the temporal
evolution of these planetary atmospheres. Larger telescopes, due to intense
competition for observation time and the drive for unique discoveries, tend to
impose limitations on the scope of these scientific explorations. This
insinuates that small telescopes could play a crucial role in the systematic
monitoring of such planetary atmospheres, especially considering their known
dynamic evolution. For example, current observations hint that KELT-9 b is
gradually losing its atmosphere (Wyttenbach et al., 2020). Therefore, the
persistent and repetitive observational capabilities of small telescopes could
provide invaluable insights into the evolving atmospheres of exoplanets.

Leveraging the successful detections achieved with FOCES, we have compiled a
roster of potential observation targets for the spectrograph situated at the
Wendelstein observatory. This list is encapsulated in Table 4. With a visual
magnitude of approximately 7.5, KELT-9 b serves as a reference point. Drawing
from Fig. 1, we propose an optimal signal-to-noise ratio (SNR) of 40 is
attainable during the most lucid nights. Assuming this ratio diminishes by a
square root factor of two with each increasing magnitude, we could set an
absolute observational limit at a magnitude of around 11, thus presenting an
SNR of 10. To elevate the SNR to 40 under these conditions would necessitate
16 transit observations—a formidable undertaking for a large telescope, yet
plausible given the ample observation time available at Wendelstein.

Additionally, we’ve chosen targets exhibiting a sufficiently high atmospheric
equilibrium temperature to allow molecular dissociation, thereby enabling us
to observe atomic and ionised line transitions at optical wavelengths. This
criterion largely restricts our scope to ultra-hot Jupiters. Utilising these
criteria, we’ve refined our list to 10 promising targets, with KELT-20 b
(MASCARA-2 b), TOI-1431 b (MASCARA-5 b), and WASP-189 b standing out. While
WASP-189 b is typically a southern target, it is observable from the
Wendelstein telescope during specific times of the year.

Should an observation programme be launched, most targets would necessitate
several transits, barring the primary targets mentioned above. Given the
generous availability of time on the two-meter telescope, this endeavour is
certainly feasible. Admittedly, unforeseeable factors, such as weather, may
necessitate additional observations to offset lost nights. Nevertheless, with
perseverance, FOCES exhibits a robust potential to observe an extensive range
of ultra-hot Jupiter atmospheres.

While FOCES delivers high performance, its scientific reach is primarily
limited to ultra-hot Jupiters due to its optical wavelength range. That said,
the feasibility of high-resolution near-infrared observations for exoplanets
with lower equilibrium temperatures, such as hot Jupiters, has been
convincingly demonstrated by the GIANO-B spectrograph on the TNG (Brogi et
al., 2018; Guilluy et al., 2019). Additionally, an instrument tailored for the
near-infrared could potentially probe the helium triplet at 1083 nm, proven to
be a reliable indicator of atmospheric evaporation in transitioning sub-
Neptunes to super-Earths (Oklopčić & Hirata, 2018; Nortmann et al., 2018). The
capability for long-term baseline measurements facilitated by small telescopes
could offer in-depth insights into this evaporation process, potentially
shedding light on competing theories of planetary evaporation, such as
photoevaporation (Hu et al., 2015; Owen & Wu, 2017) versus core-powered mass
loss (Ginzburg et al., 2018; Gupta & Schlichting, 2019, 2020). Given FOCES’
comparable performance to HARPS-N it is plausible that a similar near-infrared
spectrograph like GIANO-B could achieve comparable results.

Large telescopes often excel in scientific investigations that demand
precision and reduced noise levels, like resolving individual lines (e.g.
Seidel et al., 2022; Pai Asnodkar et al., 2022), analysing atmospheric
dynamics (Ehrenreich et al., 2020; Prinoth et al., 2022), or hunting for
fainter signals from heavier elements (Jiang et al., 2023). However, a common
challenge with large telescopes is their over-subscription. It is assumed that
over-subscription leads to a prioritization of the best-qualified projects
(Castro & Yáñez, 2003), which while generally accurate, damages the scope for
creative experimentation and diverse observational strategies. This likely
explains why the first exoplanet discovery, 51 Pegasi b, was made on a 2m
class telescope (Mayor & Queloz, 1995), despite the availability of larger
telescopes. Our study demonstrates that when it comes to studying exoplanet
atmospheres, smaller telescopes can deliver comparable performance to their
larger counterparts, enabling further experimentation and possibly heralding
new discoveries.

##  7 Conclusion

In this study, we successfully resolved the atmosphere of KELT-9 b in a single
transit, and detected seven of the eight detections presented in Hoeijmakers
et al. (2019). Additionally, we compared the performances of the high-
resolution spectrograph FOCES at the Wendelstein Observatory to the HARPS-N
spectrograph on the TNG and found promising evidence that given more
observations FOCES can produce results on a comparable level of quality for
HARPS-N. UHJs can be viewed as testing grounds for observational concepts
related to atmospheres. With our results it can be argued that smaller
telescopes carry the potential to explore new ideas and reframe the scientific
questions that we should pursue with larger telescopes. Our results also
advocate for the maintenance and ongoing performance of smaller, high-
performance telescopes, and potentially even expand the instrumentation they
are equipped with for studying UHJ atmospheres. These results show that two-
meter telescopes can still deliver pioneering science.

###### Acknowledgements.

N.W.B acknowledge partial financial support from The Fund of the Walter
Gyllenberg Foundation. BE and KM are supported by the Excellence Cluster
ORIGINS which is funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - EXC-2094-390783311.

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##  Appendix A Detections

We present the measured parameters of each cross-correlation peak for each
signal recovered when combining the three FOCES observations. Table 7 outlines
the measured amplitude, centre, width, and detection significance of each
cross-correlation peak. All detection significances, except for Y II, are
detected above a 3 $\sigma$ threshold.

Table 3: Summary of the detection parameters and significance for the each of
the eight species in the transmission spectrum of KELT-9 b.

Element | Amplitude [ppm] | Centre [$\textrm{km\>s}^{-1}$] | Width [$\textrm{km\>s}^{-1}$] | Detection Significance
---|---|---|---|---
Na I | $1118\pm 242$ | $-17.1\pm 2.49$ | $5.83\pm 2.31$ | 4.6
Mg I | $403\pm 112$ | $-26.8\pm 2.68$ | $7.30\pm 1.60$ | 3.6
Sc II | $1281\pm 191$ | $-21.7\pm 1.15$ | $5.29\pm 1.06$ | 6.7
Ti II | $976\pm 183$ | $-19.1\pm 1.64$ | $5.06\pm 2.75$ | 5.3
Cr II | $986\pm 322$ | $-25.9\pm 1.57$ | $3.02\pm 1.56$ | 3.1
Fe II | $1387\pm 121$ | $-17.4\pm 0.81$ | $7.65\pm 0.87$ | 11.4
Fe I | $147\pm 21$ | $-16.9\pm 1.25$ | $6.62\pm 1.09$ | 7.1
Y II | $20\pm 1080$ | $-20.2\pm 5.48$ | $12.4\pm 232$ | 0.02
777The table lists the amplitude, centre, width, and detection significance
for each detected element

##  Appendix B Air-mass plots

Here, we present the air-mass plots for each night of observation,
highlighting where the transit occurs.

(a)  (b)  (c)  (d)

Figure 5: Airmass plots for KELT-9 b observations. Each data point represents
an individual exposure frame. In-transit, ingress/egress, and out-of-transit
exposures are indicated with red, orange, and blue colour. The civil,
nautical, and astronomical twilight phases are indicated by the blue shaded
regions. All transits were observed at low airmass. Parts of the transits
observed with FOCES occurred during astronomical twilight given the short
duration of the nights in summer.

##  Appendix C KELT-9 b’s Atmospheric Model

In this section, we present the atmospheric model utilised for the injection
procedure of this study, shown in Fig. 6. The model, on average, exhibits a
relative transit depth on the order of a few tenths of a percent for the
deepest lines. The high transit depth highlights why this planet is so
conducive to atmospheric detections, which is a critical factor in
WENDELSTEIN’s success in detecting the planet’s atmosphere in a single
transit.

Figure 6: The radiative transfer model, employed for model injection in this
research, spans the wavelength range of FOCES. Line depth varies, typically on
the order of a few tenths of a percent.

##  Appendix D Targets Observable for Wendelstein

We have included a table that lists potential targets observable using the
Wendelstein Telescope with FOCES, summarizing the orbital and physical
characteristics of each planet. Given the mirror size of the Wendelstein
Telescope, it is likely that multiple transit observations will be necessary
to resolve the atmospheres of these targets. However, the task of resolving
these atmospheres remains achievable.

Table 4: Summary of the orbital and physical characteristics targets which
could be observable with FOCES

Name |  RA (deg) |  Dec (deg) |  $V_{\text{mag}}$ |  $T_{\text{eq}}$ (K) |  Period (days) |  Semi-major axis (AU) |  Mass ($M_{\text{Jup}}$) |  Inclination (deg)
---|---|---|---|---|---|---|---|---
KELT-20 b |  294.663 |  31.219 |  3.0 |  2230.0 |  3.5 |  0.054 |  3.5 |  86.2
WASP-189 b |  225.687 |  -3.031 |  6.6 |  3353.0 |  2.7 |  0.051 |  2.0 |  84.03
KELT-9 b |  307.86 |  39.939 |  7.6 |  3921.0 |  1.5 |  0.034 |  2.9 |  87.2
TOI-1431 b |  316.204 |  55.588 |  8.0 |  2370.0 |  2.7 |  0.047 |  3.1 |  80.4
KOI-13 b |  286.971 |  46.868 |  10.0 |  2550.0 |  1.8 |  0.036 |  9.3 |  86.77
TOI-2109 b |  253.188 |  16.58 |  10.0 |  3631.0 |  0.7 |  0.018 |  5.0 |  70.74
HAT-P-49 b |  305.442 |  26.693 |  10.2 |  2131.0 |  2.7 |  0.044 |  1.7 |  86.2
KELT-18 b |  216.525 |  59.444 |  10.1 |  2120.0 |  2.9 |  0.046 |  1.2 |  88.86
HAT-P-57 b |  282.242 |  10.597 |  10.5 |  2200.0 |  2.5 |  0.041 |  1.8 |  88.26
HAT-P-7 b |  292.246 |  47.969 |  10.5 |  2121.0 |  2.2 |  0.038 |  1.7 |  81.1

888The table includes right ascension (RA), declination (Dec), visual
magnitude ($V_{\text{mag}}$), equilibrium temperature ($T_{\text{eq}}$),
orbital period, semi-major axis, mass (in terms of Jupiter mass
$M_{\text{Jup}}$), and orbital inclination.


# Commutative graded monads and localisable monads

Rowan Poklewski-Koziell University of Cape Town, South Africa

(November 2021)

##  1 Background

Let us examine the following diagram of “specialisations” and
“generalisations”:

Commutative monadsMonads Commutative graded monads Graded monads

This is explained as follows: as originally observed by J. Benábou [1], a
monad in a bicategory $\mathcal{B}$ is nothing but a lax 2-functor
$\mathbf{1}\longrightarrow\mathcal{B}$. This is the “Monads” node on the left.
Specialising to “Commutative monads”, these are exactly lax 2-functors
$\mathbf{1}\longrightarrow\mathsf{MonCat}$, where $\mathsf{MonCat}$ is the
$2$-category of monoidal categories, lax monoidal functors and monoidal
natural transformations: they are monoidal monads (see also, for example,
[2]).

We may generalise monads to “Graded monads” over arbitrary bicategories
$\mathcal{B}$ as follows: for a monoidal category $\mathsf{M}$, an
$\mathsf{M}$-graded monad over $\mathcal{B}$ is a lax 2-functor
$\Sigma(\mathsf{M})\longrightarrow\mathcal{B}$, where $\Sigma(\mathsf{M})$ is
the suspension of $\mathsf{M}$. As we show below (and as has been remarked in
[3]), in the specific case of $\mathcal{B}=\mathsf{Cat}$ such graded monads
coincide with those described in [4]. In the latter paper, two notions of
algebra for a graded monad over $\mathsf{Cat}$ are introduced. While these
algebras satisfy respective universal properties, it is not yet clear if they
may be described as lax (or colax) limits of the lax 2-functors corresponding
to the graded monad. Additionally, examples of the algebras of [4] have been
presented as graded theories in [5, Section 3.3]. Graded monads over
$\mathsf{Cat}$ have also been called parametric monads [6, 7].

The purpose of the present work is the following: as has been shown in, for
example, [8], that the that the category of Kleisli algebras for a monoidal
monad carries a monoidal structure, we aim to specialise those graded monads
over $\mathsf{Cat}$ whose category of “Kleisli algebras” carry a monoidal
structure. This is the node on the right in our diagram above. We closely
imitate and adapt several of the results and definitions of [9].

Let $\mathcal{D}$ be a 2-category with finite products of 0-cells. In such a
2-category $\mathcal{D}$, we can talk about monoidal objects, (op)lax monoidal
1-cells, and monoidal 2-cells, as we talk about monoidal categories, (op)lax
monoidal functors, and monoidal natural transformations in the 2-category
$\mathsf{Cat}$.

######  Definition 1.

A monoidal object in a 2-category $\mathcal{D}$ with finite products of
0-cells is a tuple $(C,\otimes_{C},I_{C},\alpha_{C},\lambda_{C},\rho_{C})$ in
which $C$ is a 0-cell

$\displaystyle\otimes_{C}:C\times C\longrightarrow C\qquad\text{and}\qquad
I_{C}:1_{\mathcal{D}}\longrightarrow C$

(where $1_{\mathcal{D}}$ is the terminal object of $\mathcal{D}$) are 1-cells,
and

$\displaystyle\alpha_{C}:\otimes_{C}\cdot(1_{C}\times\otimes_{C})\longrightarrow$
$\displaystyle\otimes_{C}\cdot(\otimes_{C}\cdot
1_{C})\qquad\qquad\lambda_{C}:\otimes_{C}\cdot\langle
I_{C},1_{C}\rangle\longrightarrow 1_{C}$
$\displaystyle\rho_{C}:\otimes_{C}\cdot\langle
1_{C},I_{C}\rangle\longrightarrow 1_{C}$

are 2-cells which together make obvious pentagon and triangular diagrams
commute. A lax monoidal morphism of monoidal objects consists of a triple
$\big{(}F,\phi,\overline{\phi}\big{)}:(C,\otimes_{C},I_{C},\alpha_{C},\lambda_{C},\rho_{C})\longrightarrow(C^{\prime},\otimes_{C^{\prime}},I_{C^{\prime}},\alpha_{C}{{}^{\prime}},\lambda_{C^{\prime}},\rho_{C^{\prime}})$
in which:

  * •

$F:C\longrightarrow C^{\prime}$ is a 1-cell;

  * •

$\phi:\otimes_{C^{\prime}}\cdot(F\times F)\longrightarrow F\cdot\otimes_{C}$
and $\overline{\phi}:I_{C^{\prime}}\longrightarrow F\cdot I_{C}$ are 2-cells

such that the following diagrams commute:

$\otimes_{C^{\prime}}\cdot(F\times
F)\cdot(1_{C}\times\otimes_{C})$$\otimes_{C^{\prime}}\cdot(1_{C^{\prime}}\times\otimes_{C^{\prime}})\cdot(F\times
F\times
F)$$F\cdot\otimes_{C}\cdot(1_{C}\times\otimes_{C})$$\otimes_{C^{\prime}}\cdot(F\times
F)\cdot(\otimes_{C}\times
1_{C})$$\otimes_{C^{\prime}}\cdot(\otimes_{C^{\prime}}\times
1_{C^{\prime}})\cdot(F\times F\times
F)$$F\cdot\otimes_{C}\cdot(\otimes_{C}\times
1_{C})$$\otimes_{C^{\prime}}(1_{F}\times\phi)$$\alpha_{{C^{\prime}}{F,F,F}}$$\otimes_{C^{\prime}}(\phi\times
1_{F})$$\phi(1_{C}\times\otimes_{C})$$\phi(\otimes_{C}\times
1_{C})$$F\alpha_{C}$ (1)

$\otimes_{C^{\prime}}\cdot(F\times F)\cdot\langle
1_{C},I_{C}\rangle$$\otimes_{C^{\prime}}\cdot\langle
1_{C^{\prime}},I_{C^{\prime}}\rangle\cdot F$$F\cdot\otimes_{C}\cdot\langle
1_{C},I_{C}\rangle$$F$$\otimes_{C^{\prime}}(1_{F}\times\overline{\phi})$$\phi\langle
1_{C},I_{C}\rangle$$\rho_{C^{\prime}}F$$F\rho_{C}$ (2)

$\otimes_{C^{\prime}}\cdot(F\times F)\cdot\langle
I_{C},1_{C}\rangle$$\otimes_{C^{\prime}}\cdot\langle
I_{C^{\prime}},1_{C^{\prime}}\rangle\cdot F$$F\cdot\otimes_{C}\cdot\langle
I_{C},1_{C}\rangle$$F$$\otimes_{C^{\prime}}(\overline{\phi}\times
1_{F})$$\phi\langle I_{C},1_{C}\rangle$$\lambda_{C^{\prime}}F$$F\lambda_{C}$
(3)

An oplax monoidal morphism of monoidal objects consists of a triple
$\big{(}F,\phi,\overline{\phi}\big{)}:(C,\otimes_{C},I_{C},\alpha_{C},\lambda_{C},\rho_{C})\longrightarrow(C^{\prime},\otimes_{C^{\prime}},I_{C^{\prime}},\alpha_{C}{{}^{\prime}},\lambda_{C^{\prime}},\rho_{C^{\prime}})$
in which:

  * •

$F:C\longrightarrow C^{\prime}$ is a 1-cell;

  * •

$\phi:F\cdot\otimes_{C}\longrightarrow\otimes_{C^{\prime}}\cdot(F\times F)$
and $\overline{\phi}:F\cdot I_{C}\longrightarrow I_{C^{\prime}}$ are 2-cells

(note the opposite direction) making similar diagrams commute. A
transformation of lax monoidal morphisms
$\tau:(F,\phi,\overline{\phi})\longrightarrow(F^{\prime},\phi^{\prime},\overline{\phi^{\prime}})$
is a 2-cell $\tau:F\longrightarrow F^{\prime}$ such that the following
diagrams commute:

$F\cdot\otimes_{C}$$\otimes_{C^{\prime}}\cdot(F\times
F)$$F^{\prime}\cdot\otimes_{C}$$\otimes_{C^{\prime}}\cdot(F^{\prime}\times
F^{\prime})$$\phi$$\tau\otimes_{C}$$\otimes_{C^{\prime}}(\tau\times\tau)$$\phi^{\prime}$
(4)

$I_{C^{\prime}}$$F^{\prime}\cdot I_{C}$$F\cdot
I_{C}$$\overline{\phi^{\prime}}$$\tau I_{C}$$\overline{\phi}$ (5)

There are similar transformations of oplax morphisms.

Denoting $2\mathsf{Cat}_{\times}$ the sub-3-category of $2\mathsf{Cat}$ full
on 2-natural transformations and modifications whose 0-cells are 2-categories
with finite products of 0-cells, and 1-cells are 2-functors preserving finite
products of 0-cells, we have 3-functors:

$\displaystyle\mathsf{Mon}:2\mathsf{Cat}_{\times}\longrightarrow
2\mathsf{Cat}_{\times}$

and

$\displaystyle\mathsf{Mon}_{\text{op}}:2\mathsf{Cat}_{\times}\longrightarrow
2\mathsf{Cat}_{\times}$

given on objects by sending each 2-category $\mathcal{D}$ with finite products
of 0-cells, to the 2-category $\mathsf{Mon}(\mathcal{D})$ (resp.
$\mathsf{Mon}_{\text{op}}(\mathcal{D})$) of monoidal objects in $\mathcal{D}$,
lax (resp. oplax) monoidal morphisms, and transformations of lax (resp. oplax)
monoidal morphisms. There are also the obvious canonical forgetful 2-functors

$\displaystyle
U_{\mathcal{D}}:\mathsf{Mon}(\mathcal{D})\longrightarrow\mathcal{D}$

and

$\displaystyle
U_{\text{op},\mathcal{D}}:\mathsf{Mon}_{\text{op}}(\mathcal{D})\longrightarrow\mathcal{D}$

Furthermore, these 2-functors determine 3-natural transformations
$U:\mathsf{Mon}\longrightarrow\mathsf{Id}_{2\mathsf{Cat}_{\times}}$ and
$U_{\text{op}}:\mathsf{Mon}_{\text{op}}\longrightarrow\mathsf{Id}_{2\mathsf{Cat}_{\times}}$.

######  Definition 2.

For a monoidal category $\mathsf{M}$, there is a bicategory
$\Sigma(\mathsf{M})$ called the suspension of $\mathsf{M}$. The bicategory
$\Sigma(\mathsf{M})$ has one object $*$ and
$\Sigma(\mathsf{M})(*,*)=\mathsf{M}$, and with composition given by the
monoidal structure.

Let us next show that, for a monoidal category
$(\mathsf{M},\otimes_{M},I_{M})$, an $\mathsf{M}$-graded monad (over
$\mathsf{Cat}$) is nothing but a lax 2-functor
$\Sigma(\mathsf{M})\longrightarrow\mathsf{Cat}$. For, to give such a lax
2-functor (see, for example, [10, Ch. 7.5]) is to give:

  * •

A category $\mathbf{C}$;

  * •

A functor $F:\mathsf{M}\longrightarrow[\mathbf{C},\mathbf{C}]$, whose value at
some object $X\in\mathsf{M}$ we denote by $F_{X}$;

  * •

A natural transformation $\gamma:\circ\cdot(F\times F)\longrightarrow
F\cdot\otimes_{M}$, where, to distinguish compositions, $\circ$ is the
composition of functors in $[\mathbf{C},\mathbf{C}]$;

  * •

A natural transformation $\delta:1_{\mathbf{C}}\longrightarrow F_{I_{M}}$;

such that, for all objects $X,Y,Z$ in $\mathsf{M}$, the following diagrams
commute:

$F_{X\otimes_{M}Y}F_{Z}$$F_{X}F_{Y}F_{Z}$$F_{X\otimes_{M}Y\otimes_{M}Z}$$F_{X}F_{Y\otimes_{M}Z}$$F_{X}\gamma_{Y,Z}$$\gamma_{X,Y\otimes_{M}Z}$$\gamma_{X,Y}F_{Z}$$\gamma_{X\otimes_{M}Y,Z}$
(6)

$F_{X}$$F_{X}\cdot
1_{\mathbf{C}}$$F_{X\otimes_{M}I_{M}}$$F_{X}F_{I_{M}}$$F_{X}\delta$$\gamma_{X,I_{M}}$$F_{\rho_{X}}$
(7)

$F_{X}$$1_{\mathbf{C}}\cdot
F_{X}$$F_{I_{M}\otimes_{M}X}$$F_{I_{M}}F_{X}$$\delta
F_{X}$$\gamma_{I_{M},X}$$F_{\lambda_{X}}$ (8)

A simple comparison with Definition 1 above easily shows then that this is
nothing but a lax monoidal functor
$F:\mathsf{M}\longrightarrow[\mathbf{C},\mathbf{C}]$, where
$[\mathbf{C},\mathbf{C}]$ carries the obvious strict monoidal structure.
Inspired by this example, we have the following:

######  Definition 3.

For a monoidal category $(\mathsf{M},\otimes_{M},I_{M})$ and bicategory
$\mathcal{B}$, an $\mathsf{M}$-graded monad over $\mathcal{B}$ is a lax
2-functor $\Sigma(\mathsf{M})\longrightarrow\mathcal{B}$. To give such a lax
2-functor is equivalently to give a quadruple $(C,F,\gamma,\delta)$ in which:

  * •

$C$ is an object of $\mathcal{D}$;

  * •

$F:\mathsf{M}\longrightarrow\text{End}_{\mathcal{D}}(C)$ is a functor, whose
value at some object $X\in\mathsf{M}$ we denote by $F_{X}$;

  * •

A natural transformation $\gamma:\circ\cdot(F\times F)\longrightarrow
F\cdot\otimes_{M}$;

  * •

A 2-cell $\delta:1_{C}\longrightarrow F_{I_{M}}$;

such that, for all objects $X,Y,Z$ in $\mathsf{M}$, the similar diagrams of
(6), (7) and (8) commute.

We now give our main definition.

######  Definition 4.

For a monoidal category $(\mathsf{M},\otimes_{M},I_{M})$ and 2-category
$\mathcal{D}$ with finite products of 0-cells, a commutative
$\mathsf{M}$-graded monad over $\mathcal{D}$ is a lax 2-functor
$\Sigma(\mathsf{M})\longrightarrow\mathsf{Mon}(\mathcal{D})$. Equivalently,
this consists of an 11-tuple

$\displaystyle(C,\otimes_{C},I_{C},\alpha_{C},\lambda_{C},\rho_{C},F,\Phi,\overline{\Phi},\Gamma,\Delta)$

in which:

  * •

$(C,\otimes_{C},I_{C},\alpha_{C},\lambda_{C},\rho_{C})$ (which we will
abbreviate as $(C,\otimes_{C},I_{C})$ when the context is clear) is a monoidal
object of $\mathcal{D}$;

  * •

$(F,\Phi,\overline{\Phi})$ is a functor

$\displaystyle\mathsf{M}\longrightarrow\text{End}_{\mathsf{Mon}(\mathcal{D})}\big{(}(C,\otimes_{C},I_{C})\big{)}$

whose value at some object $X\in\mathsf{M}$ is the lax monoidal morphism
denoted by $(F,\Phi,\overline{\Phi})(X)=(F_{X},\Phi_{X},\overline{\Phi}_{X})$;

  * •

$\Gamma:\circ\cdot(F,\Phi,\overline{\Phi})\times(F,\Phi,\overline{\Phi})\longrightarrow(F,\Phi,\overline{\Phi})\cdot\otimes_{M}$
is a natural transformation;

  * •

$\Delta:(1_{C},1,1)\longrightarrow(F_{I_{M}},\Phi_{I_{M}},\overline{\Phi}_{I_{M}})$
is a transformation of lax monoidal morphisms

such that, for all objects $X,Y,Z$ in $\mathsf{M}$, the similar diagrams of
(6), (7) and (8) commute.

Unwinding this definition more, we have that, for each $X\in\mathsf{M}$,
$F_{X}:C\longrightarrow C$ is a 1-cell, while
$\Phi_{X}:F_{X}\cdot\otimes_{C}\longrightarrow\otimes_{C}\cdot(F_{X}\times
F_{X})$ and $\overline{\Phi}_{X}:F_{X}\cdot I_{C}\longrightarrow I_{C}$ are
2-cells such that
$(F_{X},\Phi_{X},\overline{\Phi}_{X}):(C,\otimes_{C},I_{C})\longrightarrow(C,\otimes_{C},I_{C})$
is a lax monoidal morphism. For each morphism $f:X\longrightarrow Y$ in
$\mathsf{M}$, $(F,\Phi,\overline{\Phi})(f)$ – which we denote by $Ff$ – is a
2-cell (in $\mathcal{D}$) $F_{X}\longrightarrow F_{Y}$ which is additionally a
transformation of lax monoidal morphisms
$(F_{X},\Phi_{X},\overline{\Phi}_{X})\longrightarrow(F_{Y},\Phi_{Y},\overline{\Phi}_{Y})$.
It is now clear that, for example, sending $X$ to $F_{X}$ and $f$ to $Ff$
determines a functor $\mathsf{M}\longrightarrow\text{End}_{\mathcal{D}}(C)$.

Furthermore, for each pair $X$, $Y$ of objects of $\mathsf{M}$, $\Gamma_{X,Y}$
is a 2-cell (in $\mathcal{D}$) $F_{X}F_{Y}\longrightarrow F_{X\otimes_{M}Y}$
which additionally is a transformation of lax monoidal morphisms

$\displaystyle\big{(}F_{X}F_{Y},F_{X}\Phi_{Y}\cdot\Phi_{X}(F_{Y}\times
F_{Y}),F_{X}\overline{\Phi}_{Y}\cdot\overline{\Phi}_{X}\big{)}\longrightarrow\big{(}F_{X\otimes_{M}Y},\Phi_{X\otimes_{M}Y},\overline{\Phi}_{X\otimes_{M}Y}\big{)}$

Again, clearly, $\Gamma$ is a natural transformation

$\displaystyle\circ\cdot(F\times F)\longrightarrow F\cdot\otimes_{M}$

Finally, $\Delta:1_{C}\longrightarrow F_{I_{M}}$ is a 2-cell which is
additionally a transformation of lax monoidal morphisms
$(1_{C},1,1)\longrightarrow(F_{I_{M}},\Phi_{I_{M}},\overline{\Phi}_{I_{M}})$.
Therefore we immediately conclude the obvious result:

######  Proposition 5.

Each commutative $\mathsf{M}$-graded monad over a 2-category $\mathcal{D}$ is
an $\mathsf{M}$-graded monad over $\mathcal{D}$.

######  Example 6.

Take $\mathsf{M}$ to be the terminal category $\mathbbm{1}$, and $\mathcal{D}$
to be the 2-category $\mathsf{Cat}$. Applying this to Definition 4 gives a
monoidal monad. That is, a commutative monad.

######  Definition 7.

Let $\mathsf{M}$ be a monoidal category and $\mathcal{D}$ a 2-category. Given
$\mathsf{M}$-graded monads $(C,F,\gamma,\delta)$ and $(D,G,\mu,\eta)$ over
$\mathcal{D}$, a lax morphism of $\mathsf{M}$-graded monads
$(C,F,\gamma,\delta)\longrightarrow(D,G,\mu,\eta)$ is a lax natural
transformation of the corresponding lax 2-functors. It consists of a pair
$(\Omega,\omega)$ in which:

  * •

$\Omega:C\longrightarrow D$ is a 1-cell in $\mathcal{D}$;

  * •

$\omega:\text{hom}_{\mathcal{D}}(C,\Omega)\cdot
F\longrightarrow\text{hom}_{\mathcal{D}}(\Omega,C)\cdot G$ is a natural
transformation – that is, a family of 2-cells $\big{(}\omega_{X}:\Omega
F_{X}\longrightarrow G_{X}\Omega\big{)}_{X\in\mathsf{M}}$

such that, for all objects $X,Y$ in $\mathsf{M}$, the following diagrams
commute:

$\Omega F_{X\otimes_{M}Y}$$\Omega
F_{X}F_{Y}$$G_{X\otimes_{M}Y}\Omega$$G_{X}G_{Y}\Omega$$G_{X}\Omega
F_{Y}$$\omega_{X}F_{Y}$$G_{X}\omega_{Y}$$\mu_{X,Y}\Omega$$\Omega\gamma_{X,Y}$$\omega_{X\otimes_{M}Y}$
(9)

$\Omega\cdot 1_{C}$$\Omega$$\Omega
F_{I_{M}}$$G_{I_{M}}\Omega$$1_{D}\cdot\Omega$$\eta\Omega$$\omega_{I_{M}}$$\Omega\delta$
(10)

One obtains oplax morphisms of graded monads as oplax natural transformations
of the corresponding lax 2-functors. Note in particular, that an oplax
morphism of graded monads $(C,F,\gamma,\delta)\longrightarrow(D,G,\mu,\eta)$
consists of a pair $(\Omega,\omega)$ in which:

  * •

$\Omega:C\longrightarrow D$ is a 1-cell in $\mathcal{D}$;

  * •

$\omega$ is a family of 2-cells
$\big{(}\omega_{X}:G_{X}\Omega\longrightarrow\Omega
F_{X}\big{)}_{X\in\mathsf{M}}$ (note the direction change!) natural in $X$

making similar diagrams commute.

######  Definition 8.

For the $\mathsf{M}$-graded monads $(C,F,\gamma,\delta)$ and $(D,G,\mu,\eta)$
over a 2-category $\mathcal{D}$ in Definition 7 above, and lax morphisms of
$\mathsf{M}$-graded monads
$(\Omega,\omega),(\Xi,\xi):(C,F,\gamma,\delta)\longrightarrow(D,G,\mu,\eta)$,
a transformation of lax morphisms of $\mathsf{M}$-graded monads
$\beta:(\Omega,\omega)\longrightarrow(\Xi,\xi)$ is a modification of the
corresponding lax natural transformations. That is, a 2-cell
$\beta:\Omega\longrightarrow\Xi$ such that, for every pair $X,Y$ of objects,
and every morphism $f:X\longrightarrow Y$ of $\mathsf{M}$, the following
diagram commutes:

$\Omega F_{X}$$G_{X}\Omega$$\Xi
F_{Y}$$G_{Y}\Xi$$\omega_{X}$$\beta(Ff)$$(Gf)\beta$$\xi_{Y}$ (11)

In a similar fashion, we obtain transformations of oplax morphisms. Once
again, we have 3-functors:

$\displaystyle\Sigma_{\mathsf{M}}\mathsf{Mnd}:2\mathsf{Cat}\longrightarrow
2\mathsf{Cat}$

and

$\displaystyle\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}:2\mathsf{Cat}\longrightarrow
2\mathsf{Cat}$

given on objects by sending each 2-category $\mathcal{D}$, to the 2-category
$\Sigma_{\mathsf{M}}\mathsf{Mnd}(\mathcal{D})$ (resp.
$\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})$) of
$\mathsf{M}$-graded monads over $\mathcal{D}$, lax morphisms (resp. oplax) of
graded monads, and transformations of lax (resp. oplax) morphisms of graded
monads.

We now turn to algebras for $\mathsf{M}$-graded monads. Our definition is
based on the fact that, given a (formal) monad $(C,t)$ in a 2-category
$\mathcal{D}$, the Eilenberg-Moore object for $(C,t)$ (if it exists) is
nothing but the lax-limit of the lax 2-functor
$\mathbf{1}\longrightarrow\mathcal{D}$ (as an object of the 2-category of lax
2-functors, lax natural transformations and modifications) corresponding to
the monad, while the Kleisli object for $(C,t)$ is the lax-colimit of the lax
2-functor (as an object of the 2-category of lax 2-functors, oplax natural
transformations and oplax modifications) corresponding to the monad.
Furthermore, in [11, Theorem 13], Street shows that the two constructions of
the paper (thought of as generalisations of the construction of Eilenberg-
Moore algebras and the construction of Kleisli algebras [12] for a formal
monad) capture the lax-limits (resp. lax-colimits) of so-called lax functors
(resp. oplax functors) $\mathbf{A}\longrightarrow\mathsf{Cat}$, for a category
$\mathbf{A}$ and $\mathsf{Cat}$ the 2-category of categories, functors and
natural transformations. We, too, think of these notions as synonymous.

######  Definition 9.

For an $\mathsf{M}$-graded monad $(C,F,\gamma,\delta)$ over a 2-category
$\mathcal{D}$, the Eilenberg-Moore object of the graded monad is the lax limit
(if it exists) of the corresponding lax 2-functor as an object in
$\Sigma_{\mathsf{M}}\mathsf{Mnd}(\mathcal{D})$. Similarly, the Kleisli object
of the graded monad is the lax colimit of the corresponding lax 2-functor as
an object in $\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})$.

When all such lax limits (resp. lax colimits) exists in a particular
2-category, we have the following terminology.

######  Definition 10.

For a monoidal category $\mathsf{M}$ and a 2-category $\mathcal{D}$, we say
that $\mathcal{D}$ admits the construction of Eilenberg-Moore algebras if the
canonical inclusion 2-functor
$\Delta:\mathcal{D}\longrightarrow\Sigma_{\mathsf{M}}\mathsf{Mnd}(\mathcal{D})$
has a right 2-adjoint

$\displaystyle\mathsf{EM}:\Sigma_{\mathsf{M}}\mathsf{Mnd}(\mathcal{D})\longrightarrow\mathcal{D}$

Dually, $\mathcal{D}$ admits the construction of Kleisli algebras if the
inclusion 2-functor
$\Delta_{\text{op}}:\mathcal{D}\longrightarrow\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})$
has a left 2-adjoint

$\displaystyle\mathsf{K}:\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})\longrightarrow\mathcal{D}$

##  2 Lifting EM and Kleisli objects under $\mathsf{Mon}$

Note that, for a monoidal category $\mathsf{M}$, 2-categories $\mathcal{D}$
and $\mathcal{C}$ and a 2-functor $G:\mathcal{D}\longrightarrow\mathcal{C}$,
the 2-functor

$\displaystyle\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(G):\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})\longrightarrow\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{C})$

is given by sending an $\mathsf{M}$-graded monad $\mathcal{F}$ over
$\mathcal{D}$ – considered as a lax 2-functor
$\Sigma(\mathsf{M})\longrightarrow\mathcal{D}$ – to the $\mathsf{M}$-graded
monad $G\cdot\mathcal{F}$ over $\mathcal{C}$.

######  Definition 11.

For a monoidal category $\mathsf{M}$, 2-categories $\mathcal{D}$ and
$\mathcal{C}$ which admit the construction of Kleisli algebras, and a fixed
2-functor $G:\mathcal{D}\longrightarrow\mathcal{C}$, $G$ preserves Kleisli
objects in its domain when the diagram below

$\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})$$\mathcal{D}$$\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{C})$$\mathcal{C}$$\mathsf{K}_{\mathcal{D}}$$\Delta_{\text{op},\mathcal{D}}$$\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(G)$$G$$\mathsf{K}_{\mathcal{C}}$$\Delta_{\text{op},\mathcal{C}}$

commutes up to the canonical isomorphism. If this is the case when
$\mathcal{D}$ is the 2-category $\mathsf{Mon}(\mathcal{C})$, and
$G=U_{\mathcal{C}}:\mathsf{Mon}(\mathcal{C})\longrightarrow\mathcal{C}$ is the
canonical forgetful 2-functor, we say that $\mathsf{Mon}(\mathcal{C})$ has
standard Kleisli objects with respect to $\mathcal{C}$. Dually, we may define
standard Eilenberg-Moore objects with respect to $\mathcal{C}$.

######  Proposition 12.

For a monoidal category $\mathsf{M}$ and 2-category $\mathcal{D}$, there is a
canonical isomorphism of 2-categories

$\displaystyle\xi_{\mathcal{D}}:\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}\big{(}\mathsf{Mon}(\mathcal{D})\big{)}\longrightarrow\mathsf{Mon}\big{(}\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})\big{)}$

which makes the following diagram commute

$\mathsf{Mon}\big{(}\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})\big{)}$$\mathsf{Mon}(\mathcal{D})$$\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})$$\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}\big{(}\mathsf{Mon}(\mathcal{D})\big{)}$$\mathsf{Mon}(\Delta_{\text{op},\mathcal{D}})$$U_{\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})}$$\Delta_{\text{op},\mathsf{Mon}(\mathcal{D})}$$\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(U_{\mathcal{D}})$$\xi_{\mathcal{D}}$
(12)

###### Proof.

A 0-cell of
$\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}\big{(}\mathsf{Mon}(\mathcal{D})\big{)}$
is precisely a commutative $\mathsf{M}$-graded monad over $\mathcal{D}$ as we
have described in Definition 4. On the other hand, a 0-cell of
$\mathsf{Mon}\big{(}\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})\big{)}$
is also an 11-tuple

$\displaystyle\big{(}(D,G,\mu,\eta),(\otimes_{D},\phi),(I_{D},\overline{\phi}),\alpha_{D},\lambda_{D},\rho_{D}\big{)}$
(13)

in which:

  * •

$(D,G,\mu,\eta)$ is an $\mathsf{M}$-graded monad over $\mathcal{D}$;

  * •

$(\otimes_{D},\phi):(D\times
D,(G,G),(\mu,\mu),(\eta,\eta))\longrightarrow(D,G,\mu,\eta)$ is an oplax
morphism of graded monads;

  * •

$(I_{D},\overline{\phi}):(1_{\mathcal{D}},1,1,1)\longrightarrow(D,G,\mu,\eta)$
is an oplax morphism of graded monads;

  * •

$\alpha_{D}$, $\lambda_{D}$, $\rho_{D}$ are transformations of oplax morphisms
of graded monads.

which altogether make (13) a monoidal object in
$\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})$. Now, routine but
long checks show that

$\displaystyle(D,\otimes_{D},I_{D},\alpha_{D},\lambda_{D},\rho_{D})$

is a monoidal object in $\mathcal{D}$ exactly because $(\otimes_{D},\phi)$,
$\alpha_{D}$, $\lambda_{D}$ and $\rho_{D}$ satisfy similar pentagonal and
triangular diagrams.

Furthermore, since $\alpha_{D}$, $\lambda_{D}$ and $\rho_{D}$ are
transformations of oplax morphisms of graded monads, they satisfy oplax
versions of the diagram (11), which is exactly to say that for each
$X\in\mathsf{M}$, the triple

$\displaystyle(G_{X},\phi_{X},\overline{\phi}_{X})$

satisfies the oplax versions diagrams of (1), (2) and (3). That is, for each
$X\in\mathsf{M}$,
$(G_{X},\phi_{X},\overline{\phi}_{X}):(D,\otimes_{D},I_{D})\longrightarrow(D,\otimes_{D},I_{D})$
is an oplax monoidal morphism. Also, since
$G:\mathsf{M}\longrightarrow\text{End}_{\mathcal{D}}(D)$ is a functor and
because both of

$\displaystyle\phi:\text{hom}_{\mathcal{D}}(\otimes_{D},D)\cdot
G\longrightarrow\text{hom}_{\mathcal{D}}(D\times D,\otimes_{D})\cdot(G,G)$

and

$\displaystyle\overline{\phi}:\text{hom}_{\mathcal{D}}(I_{D},D)\cdot
G\longrightarrow\text{hom}_{\mathcal{D}}(1_{\mathcal{D}},I_{D})$

are natural transformations, for each morphism $f:X\longrightarrow Y$ of
$\mathsf{M}$, $Gf$ satisfies the oplax versions of diagrams (4) and (5), and
so
$Gf:(G_{X},\phi_{X},\overline{\phi}_{X})\longrightarrow(G_{Y},\phi_{Y},\overline{\phi}_{Y})$
is a transformation of oplax morphisms. In total, this is to say that the
assignments

$\displaystyle X\longmapsto(G_{X},\phi_{X},\overline{\phi}_{X})\quad\text{ ,
}\quad f\longmapsto G_{f}$

define a functor
$(G,\phi,\overline{\phi}):\mathsf{M}\longrightarrow\text{End}_{\mathsf{Mon}(\mathcal{D})}\big{(}(D,\otimes_{D},I_{D})\big{)}$.

Next to say that $\mu:\circ\cdot(G\times G)\longrightarrow G\cdot\otimes_{M}$
is a natural transformation, and that $(\otimes_{D},\phi)$ and
$(I_{D},\overline{\phi})$ both satisfy oplax versions of (9) is the same as to
say that

$\displaystyle\mu:\circ\cdot(G,\phi,\overline{\phi})\times(G,\phi,\overline{\phi})\longrightarrow(G,\phi,\overline{\phi})\cdot\otimes_{M}$

is a natural transformation making a similar diagram to (6) commute.

Finally, to say that $\eta:1_{D}\longrightarrow G_{I_{M}}$ is a 2-cell such
that $(\otimes_{D},\phi)$ and $(I_{D},\overline{\phi})$ both satisfy oplax
versions of (10) is the same as to say that

$\displaystyle\eta:(1_{D},1,1)\longrightarrow(G_{I_{M}},\phi_{I_{M}},\overline{\phi}{I_{M}})$

is a transformation oplax morphisms, making similar diagrams to (7) and (8)
commute.

Similar verifications may also be made for the associated 1- and 2-cells in
each the above categories. ∎

######  Theorem 13.

Let $\mathcal{D}$ be a 2-category with finite products of 0-cells that admits
the construction of Kleisli algebras. Furthermore, suppose the left 2-adjoint
$\mathsf{K}_{\mathcal{D}}:\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})\longrightarrow\mathcal{D}$
preserves finite products of 0-cells in its domain. Then, the 2-category
$\mathsf{Mon}(\mathcal{D})$ admits the construction of Kleisli algebras and
they are standard with respect to $\mathcal{D}$.

###### Proof.

Since $\mathsf{Mon}$ is a 3-functor, it sends 2-adjunctions to 2-adjunctions.
Therefore, in the diagram below, because $\mathsf{K}_{\mathcal{D}}$ preserves
finite products of 0-cells we have that
$\mathsf{Mon}(\mathsf{K}_{\mathcal{D}})\dashv\mathsf{Mon}(\Delta_{\text{op},\mathcal{D}})$.

$\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})$$\mathcal{D}$$\mathsf{Mon}\big{(}\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})\big{)}$$\mathsf{Mon}(\mathcal{D})$$\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}\big{(}\mathsf{Mon}(\mathcal{D})\big{)}$$\mathsf{K}_{\mathcal{D}}$$\Delta_{\text{op},\mathcal{D}}$$U_{\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(\mathcal{D})}$$U_{\mathcal{D}}$$\mathsf{Mon}(\mathsf{K}_{\mathcal{D}})$$\mathsf{Mon}(\Delta_{\text{op},\mathcal{D}})$$\Delta_{\text{op},\mathsf{Mon}(\mathcal{D})}$$\Sigma_{\mathsf{M}}\mathsf{Mnd}_{\text{op}}(U_{\mathcal{D}})$$\xi_{\mathcal{D}}$

Next, the diagram is commutative because diagram (12) of Proposition 12 is
commutative and because $U:\mathsf{Mon}\longrightarrow\mathsf{Id}$ is a
3-natural transformation. Finally, since $\xi_{\mathcal{D}}$ is an isomorphism
of 2-categories, one has that

$\displaystyle\mathsf{Mon}(\mathsf{K}_{\mathcal{D}})\cdot\xi_{\mathcal{D}}\dashv\Delta_{\text{op},\mathsf{Mon}(\mathcal{D})}$

and so $\mathsf{Mon}(\mathcal{D})$ admits the construction of Kleisli algebras
which are standard with respect to $\mathcal{D}$. ∎

Since the 2-functor
$\mathsf{EM}_{\mathcal{D}}:\Sigma_{\mathsf{M}}\mathsf{Mnd}(\mathcal{D})\longrightarrow\mathcal{D}$
is a right 2-adjoint, it automatically preserves finite products of 0-cells in
its domain. Therefore, we have a slightly stronger result in the dual case.

######  Theorem 14.

Let $\mathcal{D}$ be a 2-category with finite products of 0-cells that admits
the construction of Eilenberg-Moore algebras. Then, the 2-category
$\mathsf{Mon}_{\text{op}}(\mathcal{D})$ admits the construction of Eilenberg-
Moore algebras and they are standard with respect to $\mathcal{D}$.

Since the 2-category $\mathsf{Cat}$ has all lax limits, we have the following.

######  Corollary 15.

$\mathsf{Mon}_{\text{op}}(\mathsf{Cat})$ admits the construction of Eilenberg-
Moore algebras and they are standard with respect to $\mathsf{Cat}$.

##  3 Commutative graded monads and localisable monads

In [13, Lemma 27], it is shown that for a stiff monoidal category
$\mathbf{C}$, $\mathsf{ZI}(\mathbf{C})$-graded monads correspond bijectively
to formal monads in the 2-category
$[\mathsf{ZI}(\mathbf{C})^{\text{op}},\mathsf{Cat}]$. In fact, this may be
upgraded to an isomorphism of 2-categories.

######  Proposition 16.

For a stiff monoidal category $\mathbf{C}$, there is an isomorphism of
2-categories between the 2-category of (formal) monads, morphism of monads,
and monad morphism transformations in
$[\mathsf{ZI}(\mathbf{C})^{\text{op}},\mathsf{Cat}]$, and the 2-category
$\Sigma_{\mathsf{ZI}(\mathbf{C})}\mathsf{Mnd}_{\text{op}}(\mathsf{Cat})$.

###### Proof.

Suppose
$(\overline{\mathbf{C}},\overline{T},\overline{\mu}^{T},\overline{\eta}^{T})$
and
$(\overline{\mathbf{C}},\overline{S},\overline{\mu}^{S},\overline{\eta}^{S})$
are formal monads in $[\mathsf{ZI}(\mathbf{C})^{\text{op}},\mathsf{Cat}]$. By
[13, Lemma 27], these correspond bijectively to
$\mathsf{ZI}(\mathbf{C})$-graded monads $(\mathbf{C},T,\mu^{T},\eta^{T})$ and
$(\mathbf{C},S,\mu^{S},\eta^{S})$. For a morphism of monads

$\displaystyle(1_{\overline{\mathbf{C}}},\overline{\phi}):(\overline{\mathbf{C}},\overline{T},\overline{\mu}^{T},\overline{\eta}^{T})\longrightarrow(\overline{\mathbf{C}},\overline{S},\overline{\mu}^{S},\overline{\eta}^{S})$

we define an oplax morphism of $\mathsf{ZI}(\mathbf{C})$-graded monads
$(1_{\mathbf{C}},\phi):(\mathbf{C},T,\mu^{T},\eta^{T})\longrightarrow(\mathbf{C},S,\mu^{S},\eta^{S})$
as follows:

  * •

$1_{\mathbf{C}}$ is the usual identity functor;

  * •

$\phi$ is the family of natural transformations
$(\overline{\phi}_{u}:S_{u}\longrightarrow
T_{u})_{u\in\mathsf{ZI}(\mathbf{C})}$

The pair $(1_{\mathbf{C}},\phi)$ indeed satisfies the oplax version of diagram
(9) because the following diagram commutes:

$FT_{u}T_{v}(A)$$FT_{u\otimes 0}T_{0\otimes v}(A)$$FT^{2}_{u\otimes
v}(A)$$S_{u}T_{v}(A)$$S_{u\otimes 0}T_{0\otimes v}(A)$$S_{u\otimes
v}T_{u\otimes v}(A)$$S_{u}S_{v}F(A)$$S_{u\otimes 0}S_{0\otimes
v}F(A)$$S^{2}_{u\otimes v}F(A)$$S_{u\otimes v}F(A)$$FT_{u\otimes
v}(A)$$T_{\rho^{-1}}T_{\lambda^{-1}}(A)$$T_{u\otimes!}T_{!\otimes
v}(A)$$S_{\rho^{-1}}T_{\lambda^{-1}}(A)$$S_{u\otimes!}T_{!\otimes
v}(A)$$S_{\rho^{-1}}S_{\lambda^{-1}}(A)$$S_{u\otimes!}S_{!\otimes
v}(A)$$\overline{\phi}_{u,T_{v}(A)}$$\overline{\phi}_{u\otimes 0,T_{0\otimes
v}(A)}$$S_{u}\overline{\phi}_{v,A}$$S_{u\otimes 0}\overline{\phi}_{0\otimes
v,A}$$S_{u\otimes v}\overline{\phi}_{u\otimes v,A}$$\overline{\phi}_{u\otimes
v,T_{u\otimes v}(A)}$$\overline{\mu}^{S}_{u\otimes
v,A}$$\overline{\mu}^{T}_{u\otimes v,A}$$\overline{\phi}_{u\otimes v,A}$

A similar diagram will show the commutativity of the oplax version of diagram
(10).

On the other hand, given an oplax morphism of $\mathsf{ZI}(\mathbf{C})$-graded
monads
$(1_{\mathbf{C}},\phi):(\mathbf{C},T,\mu^{T},\eta^{T})\longrightarrow(\mathbf{C},S,\mu^{S},\eta^{S})$,
we get a morphism of monads
$(\mathbf{C},T,\mu^{T},\eta^{T})\longrightarrow(\mathbf{C},S,\mu^{S},\eta^{S})$
because

$S_{u}(A)$$S_{u}T_{u}(A)$$T_{u}(A)$$S_{u\otimes u}(A)$$T_{u\otimes
u}$$S^{2}_{u}(A)$$T^{2}_{u}(A)$$S_{u}\phi_{u,A}$$\phi_{u\otimes
u,A}$$\phi_{u,T_{u}(A)}$$\phi_{u,A}$$\mu^{S}_{u,u,A}$$S_{\nabla
u,A}$$\mu^{T}_{u,u,A}$$T_{\nabla u,A}$

commutes, where the top box commutes because $(1_{\mathbf{C}},\phi)$ satisfies
the oplax version of diagram (9).

Finally, it is trivial that transformations of such oplax morphisms of graded
monads are exactly transformations of the corresponding morphisms of monads. ∎

Under the above isomorphism of 2-categories, one can see that the Eilenberg-
Moore objects for formal monads in
$[\mathsf{ZI}(\mathbf{C})^{\text{op}},\mathsf{Cat}]$ correspond bijectively to
the Eilenberg-Moore objects for the corresponding
$\mathsf{ZI}(\mathbf{C})$-graded monads. For, denoting by $\Theta$ the above
the above isomorphism of 2-categories, we have the following commutative
diagram

$\mathsf{Cat}$$\Sigma_{\mathsf{ZI}(\mathbf{C})}\mathsf{Mnd}(\mathsf{Cat})$$\mathsf{Mnd}\big{(}[\mathsf{ZI}(\mathbf{C})^{\text{op}},\mathsf{Cat}]\big{)}$$\Delta_{\mathsf{Cat}}$$\Theta$incl

in which the inclusion 2-functor
$\text{incl}:\mathsf{Cat}\longrightarrow\mathsf{Mnd}\big{(}[\mathsf{ZI}(\mathbf{C})^{\text{op}},\mathsf{Cat}]\big{)}$
is exactly the 2-functor of [11, Theorem 13]. Therefore, to give a right
2-adjoint to $\Delta_{\mathsf{Cat}}$ is to give a right 2-adjoint to incl.

##  4 Future work

Our main interest in this work is in studying commutative
$\mathsf{ZI}(\mathbf{C})$-graded monads for a stiff monoidal category
$\mathbf{C}$. Since we have the isomorphism $\Theta$ above, and a previously
described isomorphism extending the bijective correspondence of [13, Theorem
25], we can ask, what are those localisable monads on a stiff monoidal
category $\mathbf{C}$ which, as $\mathsf{ZI}(\mathbf{C})$-graded monads are in
fact commutative $\mathsf{ZI}(\mathbf{C})$-graded monads? We may call such
localisable monads commutative localisable monads.

For such commutative localisable monads, the Eilenberg-Moore objects for their
corresponding commutative $\mathsf{ZI}(\mathbf{C})$-graded monads will have,
by Theorem 14, a monoidal structure inherited from $\mathbf{C}$. In turn, this
should imply that the Eilenberg-Moore objects for their corresponding formal
monads in $[\mathsf{ZI}(\mathbf{C})^{\text{op}},\mathsf{Cat}]$ will have
monoidal structures. This solves a previous open question posed in [13].

We should be able to go further too. We should aim to establish a connection
(perhaps an equivalence) between our construction of Eilenberg-Moore algebras
in Definition 10 above and the Eilenberg-Moore objects of [4, Section 3.1]. In
particular, the latter have strong connections to graded theories in [5,
Theorem 20]. It is envisaged that we may be able to calculate free models for
commutative graded monads corresponding to relatively simple commutative
localisable monads. These models, having an additional commutative structure,
may themselves serve as insightful models for certain aspects of concurrency.

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