# Optical Studies of Seven Bright Southern Cataclysmic Variable Stars

John R. Thorstensen Chase K. Alvarado-Anderson Abigail D. Burrows Rowan M.
Goebel-Bain David C. Katz  Department of Physics and Astronomy, 6127 Wilder
Laboratory, Dartmouth College, Hanover, NH 03755-3528

###### Abstract

We report spectroscopic observations of seven bright southern cataclysmic
variable stars, collected on a single two-week observing run using the 1.9-m
Radcliffe telescope at the South African Astronomical Observatory. We used
radial velocity time series, in some cases in combination with other data, to
determine or clarify orbital periods for five of them, namely ATO
J061.1478$-$31.0634, BMAM-V547, MGAB-V202, NSV 4202, and V1147 Cen. For
BMAM-V547, we use data from the Transiting Exoplanet Survey Satellite (TESS)
to corroborate and sharpen the orbital period; the TESS data also show a
photometric period near 3.93 d, likely indicating precession of the accretion
disk. Also, we find a periodic modulation in the radial velocities of the SU
UMa-type dwarf nova Var Ret2005, but are unable to specify a unique cycle
count. Finally, we show a spectrum of ASASSN-V J061528.41$-$412007.3 that
appears typical of a luminous novalike variable.

keywords: stars

##  1 INTRODUCTION

Cataclysmic variables (CVs) are a subclass of mass-exchange binary stars, in
which a white dwarf (WD; the primary) accretes matter from a more extended
companion (or secondary) that fills its Roche critical lobe. Most commonly,
the companion resembles a main-sequence star, but with differences in detail
caused by the complicated history of mass transfer (Knigge, 2006; Knigge et
al., 2011). Material transferred from the secondary through the inner
Lagrangian point usually settles into an accretion disk around the white dwarf
primary, and in most CVs the disk dominates the optical luminosity.

The class is diverse; Warner (1995) gives a comprehensive review. If the WD
has a strong magnetic field, it can disrupt the formation of the disk;
material instead threads onto the field and falls down the field lines onto
the polar caps, leading to an AM Herculis star, or ‘polar’, so-called due to
the polarization of their optical light. If the accretion is very slow, the
disk can be faint enough for the WD to make a strong spectral contribution,
especially in the ultraviolet; such systems tend to have very short orbital
periods $P_{\rm orb}$. If $P_{\rm orb}$ is 4-6 hours or longer, the
secondary’s contribution to the combined spectrum often becomes visible. CVs
with $P_{\rm orb}>$ 1 d are rare.

As their name implies, all CVs are variable stars. Accretion disks evidently
are subject to a limit-cycle instability, leading to dramatic brightening of
typically a few magnitudes, developing over hours and lasting for several
days, during which enough mass is dropped onto the white dwarf to re-establish
the low-density state of the disk. Systems that do this are called dwarf novae
(DNae). Most known CVs are DNae, and there is an elaborate taxonomy that
describes their outbursts. Other disk CVs persist in their high-accretion
states; these are the novalike variables (NLs).

Spectroscopically, DNe at minimum light show strong Balmer and He I lines,
greatly broadened by motions in the disk, while NLs show strong continua. Some
novalikes show almost no emission, while others show complex line profiles
that vary with orbital phase; most of these are SW Sextantis stars
(Thorstensen et al., 1991; Rodríguez-Gil et al., 2007).

Population. All CVs vary, and most call attention to themselves through their
optical variation, though many have been discovered because of their unusually
blue or ultraviolet color, or through X-ray emission. The data have become
rich and complete enough that Pala et al. (2020) created a sample of CVs
within 110 pc that they claim is essentially complete. However, the pace of
discovery remains extremely high due to the proliferation of high-cadence
surveys of sufficient depth such as ZTF (Bellm et al., 2019), ASAS-SN (Shappee
et al., 2014), and ATLAS (Tonry et al., 2018). With increasingly-complete
samples, it should be possible to extend Pala’s project to much greater depth.
Complete samples are key observables for CV population synthesis models such
as those of Goliasch & Nelson (2015) and Kalomeni et al. (2016).

We undertook this study to elucidate the nature of several CVs and candidate
CVs in the south celestial hemisphere, which remains somewhat less explored
than the north. We selected our targets from a master CV list maintained by
the lead author. Because of time and aperture constraints, we targeted CVs
that remained little-studied, in particular objects with unknown or uncertain
$P_{\rm orb}$.

##  2 OBSERVATIONS

All our observations are from the 1.9 m Radcliffe telescope operated by the
South African Astronomical Observatory. We used the SpUpNIC spectrograph
(Crause et al., 2019) with Grating 6, which covered from 4220 to 6860 Å. The
1.′′1 slit yielded a FWHM resolution of $\sim 4.5$ Å. Most of our individual
exposures were between 8 and 20 min. We took spectra of a CuAr arc at each new
setting of the telescope, and about once an hour as the telescope tracked. For
our final calibration we used the night-sky airglow lines, especially the
strong [OI] lines at $\lambda\lambda$ 5577 and 6300, to adjust the calibration
slightly, typically by $\sim 20$ km s-1. The night-sky adjustment failed for a
few spectra; for those, we reverted to the arc calibration. When the weather
was clear, we observed flux standards in twilight. From the scatter in the
standard star normalizations – most likely caused by seeing variations and the
narrow slit – we estimate that the absolute calibration is accurate to $\sim
20$ per cent, but the relative flux scale should be better than that.

To reduce the data we used a combination of IRAF routines called from pyraf,
and python scripts that made extensive use of astropy routines. In particular,
we extracted 1-dimensional spectra from the images using our own
implementation of the variance-weighted extraction algorithm of Horne (1986),
as well as the modified wavelength calibration described earlier.

We measured radial velocities of the H$\alpha$ emission line – the strongest
emission feature in all these objects – using convolution techniques described
by Schneider & Young (1980) and Shafter (1983). When a contribution from a
late-type star was present, we also measured its radial velocity using the
fxcor task in IRAF, which implements a cross-correlation technique similar to
Tonry & Davis (1979). For the correlation template, we used the sum of 76
spectra of IAU velocity standards, mostly early K stars, which were
individually shifted to zero velocity before summing.

To search for periods we created an oversampled grid of test frequencies
$\omega$, and at each $\omega$ fit the velocities $v(t)$ with a general least-
squares sinusoid

$v(t)=A\sin(\omega t)+B\cos(\omega t)+C,$

and transformed this to

$v(t)=\gamma+K\sin(\omega(t-T_{0})).$

The periodograms we present are derived from these fits; the quantity plotted
as a function of $\omega$ is

${1\over\chi^{2}}=\left[{1\over
N-3}\sum_{i=0}^{i=N}\left(v(t)-v_{i}\over\sigma_{i}\right)^{2}\right]^{-1},$

where the $v_{i}$ are the $N$ measured velocities, and $\sigma_{i}$ are their
estimated uncertainties. The $N-3$ term in the denominator arises because at
each $\omega$, the three parameters $K$, $T_{0}$, and $\gamma$ are adjusted.

When a late-type star was present, we estimated its spectral type and
contribution to the total spectrum using the procedure described by Peters &
Thorstensen (2005).

Table 2 lists the stars we observed. The first column gives the primary name
used in the American Association of Variable Star Observer’s International
Variable Star Index (VSX) 111 at https://www.aavso.org/vsx/. Some of these
objects have multiple designations, generally because they have appeared in
multiple surveys, and VSX lists these designations. All the objects here are
variable, so the $G$ magnitude is only illustrative. In the discussion below,
we shorten the lengthier coordinate-based names.

Table 1: Stars Discussed Here Table 2: List of Objects

VSX Name | $\alpha_{\rm ICRS}$ | $\delta_{\rm ICRS}$ | $G$ | $1/\pi_{\rm DR2}$ | SIMBAD name
---|---|---|---|---|---
| [h:m:s] | [d:m:s] | [mag] | [pc] |
ATO J061.1478$-$31.0634 | 4 04 35.483 |  $-$31 03 48.38 | 14.4 | 481(4) | Gaia 19fes
Var Ret2005 | 4 11 09.288 |  $-$59 11 16.27 | 16.1 | 329(4) | EC 04102-5918
ASASSN-V J061528.41$-$412007.3 | 6 15 28.406 |  $-$41 20 07.24 | 13.2 | 636(6) | UCAC4 244-008602
BMAM-V547 | 6 57 33.663 |  $-$53 34 22.03 | 14.1 | 1072(17) | Gaia DR3 …
MGAB-V202 | 8 18 08.715 |  $-$42 34 16.91 | 14.1 | 783(11) | Gaia DR3 …
NSV 4202 | 8 39 18.497 |  $-$70 32 41.64 | 16.6 | 730(24) | OGLE MC-DN-32
V1147 Cen | 13 00 57.58 |  $-$49 12 12.46 | 12.6 | 351(3) | V* V1147 Cen

Note. — The celestial coordinates and distance estimates are from the Gaia
Data Release 2 (Gaia Collaboration et al., 2016, 2018). Distances are inverses
of the parallax, without further adjustment. The SIMBAD designations in the
final column omit the Gaia numbers for the sake of space. SIMBAD entries for
these objects can be found using coordinates.

We list all our radial velocities in Table 3, and give parameters of the best-
fitting sinusoids in Table 4. The next section discusses the individual stars
in greater detail.

Table 3: Radial Velocities

Object | TimeaaTime of mid-exposure in Barycentric Julian Days, minus 2,400,000, referred to UTC. | $v_{\rm abs}$ | $\sigma$ | $v_{\rm emn}$ | $\sigma$
---|---|---|---|---|---
| d | km s-1 | km s-1 | km s-1 | km s-1
ATO J061$-$31 | 59990.2968 | $0$ | $14$ | $48$ | $11$
ATO J061$-$31 | 59990.3024 | $35$ | $11$ | $39$ | $9$
ATO J061$-$31 | 59990.3096 | $53$ | $10$ | $34$ | $10$
ATO J061$-$31 | 59990.3180 | $73$ | $11$ | $-22$ | $10$
ATO J061$-$31 | 59991.2852 | $47$ | $10$ | $25$ | $9$
ATO J061$-$31 | 59991.2956 | $74$ | $11$ | $-15$ | $10$
ATO J061$-$31 | 59991.3061 | $108$ | $14$ | $-21$ | $10$

Note. — Radial velocities used in this study. The time argument is referred to
UTC (not TAI) and is the barycentric julian date of mid-exposure minus
2,400,000., which differs from MJD by 0.5 d. The full table is published as a
machine-readable table, and the first few lines are shown here to indicate its
form and content.

Table 4: Parameters of Sinusoidal Velocity Fits Data set | $T\\_0$ | $P$ | $K$ | $\gamma$ | $N$ | $\sigma$
---|---|---|---|---|---|---
| BJD | [d] | km s-1 | km s-1 |  | km s-1
ATO J061-31 abs. | 59994.2295(13) | 0.245282aaPeriod held fixed at twice Monard’s value. | 132(5) | $39(3)$ | 25 | 11
ATO J061-31 emn. | 59994.363(4) | 0.245282aaPeriod held fixed at twice Monard’s value. | 93(10) | $4(7)$ | 25 | 23
BMAM-V547 | 59986.376(9) | 0.15536bbPeriod chosen corresponds to photometric modulation in TESS data. | 19(7) | $12(5)$ | 28 | 17
MGAB-V202 wings | 59988.401(2) | 0.15612(10) | 188(19) | $17(12)$ | 76 | 57
NSV-4202 | 59989.553(4) | 0.2839(6) | 80(7) | $-21(5)$ | 31 | 16
V1147 Cen abs. | 59997.3812(13) | 0.4190(5)bbPeriod chosen corresponds to photometric modulation in TESS data. | 152(3) | $-34(2)$ | 24 | 8
V1148 Cen emn. | 59997.613(2) | 0.4190 | 134(5) | $-7(4)$ | 24 | 12

##  3 THE INDIVIDUAL STARS

###  3.1 ATO J061$-$31

The Catalina light curve (Drake et al., 2009) of this bright dwarf nova shows
a relatively steady minimum near $14.4<V<14.7$, and a single outburst to
$V=12.3$. It has been followed for some years, but there are evidently no
spectra in the literature. $P_{\rm orb}$ is not definitively determined; VSX
lists 0.122641(4) d ($\approx$ 2.94 h) from a photometric modulation at
minimum light, attributed to B. Monard222 The vsnet-alert site maintained at
Kyoto University retains an archive of messages about variable stars, mainly
CVs. Monthly digests of the messages can be downloaded from their website,
http://ooruri.kusastro.kyoto-u.ac.jp/mailman3/postorius/lists/vsnet-
alert.ooruri.kusastro.kyoto-u.ac.jp/. Monard’s period is relayed by T. Kato in
vsnet-alert 23816 (from 2019 December).  . The 0.1226-d period would be
unusual for a dwarf nova; CVs near this period, near the long edge of the
roughly 2- to 3-h ‘gap’ in the CV period distribution, tend to be NLs rather
than DNae. On the other hand, DNae with $P_{\rm orb}$ twice as long($\sim
5.88$ h) often have prominent secondary stars and display two ‘humps’ per
orbit due to the changing aspect of the tidally-elongated secondary.

During our observing run, the target was west of the meridian at evening
twilight, so we could not determine a definitive $P_{\rm orb}$ from our
velocities alone. Our aim instead was to distinguish between candidate periods
of 2.94 h and 5.88 h. We obtained 25 exposures totaling 5.9 h, spread over
three nights, spanning somewhat over 3 h of hour angle.

Figure 1:  Top: Mean spectrum. Middle: Periodogram of the absorption
velocities. Although the sampling was not ideal, the highest peak corresponds
to the correct period. Lower: Absorption (red triangles) and emission (blue
squares) velocities folded on the orbital period; the data are repeated over a
second cycle to preserve continuity. The zero of phase is chosen to correspond
to the absorption velocity increasing through its mean, which should be the
inferior conjunction of the late-type companion.

Fig. 1 shows the results. The mean spectrum (top panel) shows multiple
absorption features from a late-type star in addition to the broadened Balmer
and HeI emission typical of dwarf novae. The middle panel shows the
periodogram of the absorption velocities, which despite the limited sampling,
clearly indicates a 5.88-h period, double the VSX value. There is no
significant modulation at half this period ($P$ = 2.94 h). Allowing the period
to vary we find $P=0.24521(12)$ d, consistent within the uncertainties with
(double) the more precise Monard period; we therefore adopt $P_{\rm
orb}=0.245282(8)$ d. The lower panel shows the folded radial velocities. As
expected, the emission velocities move approximately in antiphase to the
absorption.

The upper trace of Fig. 2 shows the mean spectrum, and the lower shows it
after subtraction of a scaled spectrum of the K0.5V-type star, HD124752. The
scaling factor was chosen interactively to best cancel the late-type features
in the difference spectrum. Our best estimate of the spectral type is K0-1,
with a plausible range from G6 to K4. Knigge (2006) compiled numerous
spectral-type estimates for CV donor stars with known period, and finds that
around $P_{\rm orb}$ = 6 hr, the typical spectral type is near M0 (see his
Fig. 7). The secondary in ATO 061-31 therefore appears to be significantly
warmer than typical. This might indicate that some nuclear evolution has taken
place in the secondary. Evolved donor stars can be much hotter than expected
at a given $P_{\rm orb}$ (Thorstensen & Armstrong, 2005; Thorstensen, 2013,
2015; Wakamatsu et al., 2021).

Figure 2:  The top trace shows the mean spectrum of ATO J061$-$31, averaged
after shifting the individual spectra to the secondary star’s rest frame. The
lower trace shows the same spectrum after subtraction of a scaled late-type
spectrum (see text).

ATO J061$-$31 was observed by the Transiting Exoplanet Survey Satellite (TESS)
in Sectors 4 and 5, with 1800 sec cadence. We downloaded the TESS ‘PSDCSAP’
data using the lightkurve python module, edited out obvious artifacts, and
folded the remaining data on the 0.245282-d period. The result (Fig. 3) shows
double-humped modulation due to the changing aspect of the tidally-distorted
secondary star. Note that one maximum appears slightly fainter than the other,
corroborating once again that the period is 5.88 hours and not half that. The
TESS data were taken in late 2018, about 4.5 years before our spectra, and our
nominal period (based on doubling the Monard value) is not quite precise
enough to specify an unambiguous cycle count across this gap.

Figure 3:  TESS data folded on our adopted period. The period is not known
accurately enough to extend the cycle count back to this epoch, so the epoch
chosen for the fold corresponds approximately to one of the two minima.

###  3.2 Var Ret2005

The Gaia alerts-index lists this object as Gaia20cdc, and the Gaia light curve
shows typical quiescent $G$ magnitude between 16 and 17, but with outbursts to
$G\sim 13$ at intervals ranging from several months to over a year. Outbursts
were also noted on the vsnet-alert message board; T. Kato, in vsnet-alert
23341 (2019 July) classified some of these as superoutbursts, and concluded
that the object is “almost certainly an SU UMa star.” However, we were unable
to find mention of any candidate superhump period, nor any spectroscopic
studies in the literature.

The average of our 20 spectra (Fig. 4, top) is typical of quiescent short-
period dwarf novae (see, e.g., Thorstensen 2020b), with strong Balmer and HeI
emission lines, and no hint of a late-type companion.

On the nights we observed we could not obtain a range of hour angles
sufficient to determine an unambiguous radial velocity period, and weather
constrained our two visits to be two nights apart. Consequently, the
periodogram (Fig. 4, middle) shows strong aliases spaced by $\Delta f=1/(2{\rm
d})$. Nonetheless, we constrain the period to the values shown in Table 5. The
candidate periods are shorter than 2 h. This corroborates Kato’s suggested SU
UMa classification, since nearly all dwarf novae in this range are SU UMa
stars. Assuming the classification is correct, more complete observations of a
superoutburst should reveal a superhump period, which in turn would resolve
the orbital period ambiguity, since $P_{\rm sh}$ is generally a few per cent
longer than the orbital period in SU UMa stars (see, e.g. Kato et al. 2020 and
references therein).

Table 5: Candidate Periods in Var Ret2005

Rank | $P$ | $1/P$ | $\sigma$
---|---|---|---
| (d) | (d-1) | (km s-1)
1 | 0.06355 | 15.736 | 11.9
2 | 0.06557 | 15.252 | 12.1
3 | 0.06165 | 16.220 | 12.5
4 | 0.06771 | 14.769 | 13.0
5 | 0.05986 | 16.706 | 13.6
6 | 0.07000 | 14.286 | 14.4
7 | 0.05816 | 17.193 | 15.3
8 | 0.07244 | 13.804 | 16.3
9 | 0.05656 | 17.682 | 17.2

The uncertainties in the individual periods are of order $5\times 10^{-5}$ d.

Note. — Ranked list of alias periods and corresponding frequencies from the
Var Ret2005 velocities. The last column gives the scatter around the best fit
at each period.

The lower panel of Fig. 4 shows our H$\alpha$ radial velocities folded at our
best period, but readers are cautioned that the period chosen is not
unambiguous.

Figure 4:  Top: Mean spectrum of Var Ret2005. Middle: Periodogram of the
H$\alpha$ emission velocities. Lower: Emission line velocities folded on the
best-fitting period, which is not determined uniquely.

###  3.3 ASASSN J0615$-$41

In a 2018 July message on vsnet-chat333http://ooruri.kusastro.kyoto-u.ac.jp
/mailarchive/vsnet-chat/8036, T. Kato suggested this object is a novalike CV,
based on its absolute magnitude. The Catalina Real Time Survey (CRTS; Drake et
al. 2009) collected 284 magnitudes from 2005 to 2013, which show irregular
variation between 12.8 and 13.8, similar to many NLs. The object is listed in
the SIMBAD database as a ‘star’. We observed the object because we were unable
to find any published spectra.

Fig. 5 shows the average of two 480-s exposures. H$\alpha$ emission is present
with an emission equivalent width of 3.3 Åand a FWHM around 14 Å. H$\beta$ is
in absorption, with an emission core, and the higher Balmer lines are entirely
in absorption. The absorption feature just shortward of $\lambda 5900$ appears
to be NaD absorption, likely interstellar, blended somewhat with weak HeI
$\lambda 5876$ absorption.

The spectrum is consistent with a thick-disk, or UX-UMa type, novalike
variable (Warner, 1995). The variability and spectrum bolster the case that
this is a bona fide novalike CV.

Figure 5:  Spectrum of ASASSN 0615$-$41\.

###  3.4 BMAM-V547

This object was first noted by Mariusz Bajer in archival data444This and
MGAB-V202 have apparently not been discussed in the literature indexed by
SIMBAD and ADS. Please refer to the VSX entries for details.. The ASAS-SN
light curve shows it varying irregularly around $V=14.2$, and more recently
fading to about $g=15.0$, still with irregular variations. It is not
classified as a CV in SIMBAD.

The mean spectrum (top panel of Fig. 6) shows a strong, blue continuum and
weak, narrow emission lines, typical of a novalike variable. The amplitude of
the emission radial velocity variations is small, and their periodogram
(middle panel) does not indicate a unique period. However, one of the possible
periods, marked with a vertical line in the figure, aligns with the
photometric period we derive from TESS observations (see below). The lower
panel shows the velocities folded on this period, which we identify as the
likely $P_{\rm orb}$.

Figure 6:  Top: Mean spectrum of BMAM-V547 from 4.8 h of cumulative exposure. The continuum shape is not precisely determined. Middle: Periodogram of H$\alpha$ emission velocities. The period corresponding to the TESS modulation is indicated. Lower: H$\alpha$ emission velocities folded on the indicated period. The modulation is weak, but apparently significant.  Table 6: TESS Observations of BMAM-V547 Sector | StartaaDates are the UT of the first and last points used, in year-month-day form. | End | Mean Flux
---|---|---|---
|  |  | (electrons s-1)
2 | 2018-08-23 | 2018-09-15 | 838
6 | 2018-12-15 | 2019-01-06 | 644
29 | 2020-08-26 | 2020-09-19 | 320
33 | 2020-12-18 | 2021-01-13 | 329
34 | 2021-01-14 | 2021-02-08 | 286
35 | 2021-02-10 | 2021-03-06 | 330
39 | 2021-05-27 | 2021-06-24 | 307

We also analyzed TESS observations of this star, which are summarized in Table
6. We downloaded the PSDCSAP files, edited out apparent artifacts (and some
possible flares, as well, since our aim was to find periodicities), and
computed periodograms using the LombScargle task from the astropy module
timeseries. All the sectors separately showed very strong modulation near
6.435 cycles d-1, equivalent to $P=0.1554$ d, or 3.730 h. To explore this, we
combined data from four sectors in which the mean brightness was consistent
and relatively low – sectors 29, 33, 35, and 39 – and searched this data set
for periods (see the top panel of Fig. 7). This refined the frequency to
6.4365(9) d-1, or $P=0.15536(2)$ d (near 3.73 hr), where the uncertainty was
estimated by examining light curves folded over a range of nearby periods. The
modulation apparently maintains coherence over the 301-day span of the data,
which amounts to 1940 cycles. The middle panel of Fig. 7 shows the low-state
TESS data set folded on this period.

All the TESS data sectors except Sector 2 (during which the source was
brightest) show a second, weaker modulation (also indicated in Fig. 7) near
0.254 d-1, or 3.93 d. The lower panel of Fig. 7 shows the low-state TESS data
folded on this much longer period.

The period of the $\sim 3.73$-hr modulation in the TESS data is typical of NL
orbital periods. This, together with its coherence and the evidence for radial
velocity modulation consistent with the same period, suggests that the
0.15536-d period is $P_{\rm orb}$, rather than being caused by some other
clock in the system, although given the relatively weak velocity modulation we
cannot be certain of this. The spectrum, photometric modulation, and velocity
modulation are all consistent with a novalike variable.

Periods comparable to the 3.93-d period, much longer than $P_{\rm orb}$ and
often called superorbital periods, are seen in other novalike CVs (see, e.g.,
Armstrong et al. 2013 and references therein). These are generally attributed
to the precession of a disk – either precession of the major axis of an
elliptical disk, or precession of the line of nodes of a tilted disk. Often,
systems with eccentric or tilted disks also show superhumps, periodic
modulations at frequencies close to the orbital frequency $f_{0}$. These
frequencies are thought to be beats between the precession and the orbit, and
appear at $f_{0}+f_{\rm p}$ or $f_{0}-f{\rm p}$, where $f_{\rm p}$ is a disk
precessional frequency. We do not find these frequencies in MGAB-V547. Our
favored $P_{\rm orb}$ is flanked by sidelobes, but these appear to artifacts
of the gaps between TESS sectors. In particular, we do not detect noticeable
power near $f_{0}\pm f_{\rm p}$.

Figure 7:  Top: Lomb-scargle periodogram of the low-state TESS data set
(described in the text). The vertical lines indicate the two periods discussed
in the text; our favored orbital period is near 6.43 d-1, and the longer
$\sim$3.93-di period, likely due to disk precession, is near 0.254 d-1.
Middle: Edited low-state TESS data folded on the likely orbital period. Each
point is plotted twice for continuity. Lower panel: The same data, folded on
the $\sim$3.93 d period.

###  3.5 MGAB-V202

This object was apparently first identified as a CV candidate by Gabriel
Murawski. The VSX listing includes a light curve from ASAS-SN showing
irregular variation $13.8\lesssim V\lesssim 14.4$. Again, SIMBAD does not
include a classification as a CV.

TESS observed the source in Sectors 34, 35 (2021 February and March, roughly)
and 61 (starting in 2023 January). Lomb-Scargle periodograms of the data from
Sectors 34 and 35 both show an apparently significant periodicity near 5.797
cycles d-1. In a simple fold of the data (Fig. 8, top), the modulation is
evidently masked by irregular flickering, but averaging in phase bins does
reveal a low-amplitude modulation (middle panel). The data from Sector 61,
shows a stronger periodicity near a different frequency, 6.037 d-1. This
modulation is discernible in the folded data (lower panel). In phase-binned
averages (not shown), its maximum and minimum are respectively near 305 and
325 TESS counts s-1.

Figure 8:  Top: Unbinned TESS 120-s cadence data from Sectors 34 and 35 folded
on the 0.1726 d period. No modulation is evident. Middle: TESS data from
Sectors 34 and 35 averaged in phase bins on the 0.1726-d period, to bring out
the small modulation. Lower: TESS data from Sector 61, folded on the 0.1656-d
period. A low-amplitude modulation is evident even though masked by other
variation. In all these, the zero-point of phase is chosen arbitrarily.

We obtained 76 spectra of MGAB-V202, a total of 13.8 h of exposure time
spanning hour angles from $-2.1$ h to $+5.6$ h. The mean spectrum (top panel
of Fig. 9) shows relatively strong Balmer and HeI emission lines on a blue
continuum. For the radial velocities, we obtained the clearest result using
the double-gaussian convolution with a separation of 42 Å, which isolated the
motion of the rather faint wings (or base) of the H$\alpha$ emission line.
This gave the periodogram shown in the middle panel. The prominent peak is at
6.405 cycles d-1, or 0.1561 d. This is, notably, not seen in any of the TESS
photometry, and it is not a daily alias of any of the TESS periods, either.
Thanks to the large span of hour angle, it is determined without significant
ambiguity in the cycle count; a 1000-trial Monte Carlo simulation of the
measurement (Thorstensen & Freed, 1985) returned the correct period every
time. The lower panel shows the folded line-wing velocities with the best-fit
sinusoid superposed.

Fig. 10 displays the spectra as a function of phase in a two-dimensional
image. The lower panel is ‘stretched’ to show the large-amplitude motion of
the H$\alpha$ line wings. Also, the HeI emission lines at $\lambda\lambda$
5876 and 6678 both show absorption over part of the phase that appears to
drift blueward, which is a classic symptom of the SW Sex phenomenon
(Thorstensen et al., 1991). Faint, large-amplitude Balmer line wings are seen
also in the novalikes V795 Her (Casares et al., 1996) and LAMOST
J204305.95+341340.6 (Thorstensen, 2020a). Based on its spectral appearance,
orbital period, and detailed spectral behavior, MGAB-V202 is clearly an SW Sex
star.

Both of the photometric periods seen in TESS data taken $\sim 2$ years apart
are distinct from $P_{\rm orb}$. The 0.1725-d period seen in 2021 is 10.5 per
cent longer than $P_{\rm orb}$, and the stronger 0.1656-d period in early 2023
is 6.1 per cent longer. As noted earlier, novalikes in this range of $P_{\rm
orb}$ frequently show superhumps, either called positive superhumps, with
$P_{\rm sh}$ somewhat longer than $P_{\rm orb}$, thought to be caused by
precession of an eccentric disk, or negative superhumps with periods shorter
than $P_{\rm orb}$, thought to be from apsidal precession of a tilted disk.
Bruch (2023) recently studied long-term TESS light curves of a large sample of
novalikes, and found many examples in which the the superhump modulations
disappear and/or change period, as seen here. The modulations in MGAB-V202
appear to be examples of positive superhumps.

Figure 9:  Top: Mean spectrum of MGAB-V202. Middle: Periodogram of H$\alpha$
line-wing emission velocities. Lower: H$\alpha$ line-wing emission velocities
folded on the adopted period.  Figure 10:  Spectra as a function of phase. The
image was constructed from continuum-divided spectra. Each line is a smoothed
average of spectra immediately adjacent to the corresponding phase; there are
100 lines per cycle. The ‘stretch’ in the upper is chosen to show the nearly
motionless line cores, while the lower panel brings out the large-amplitude
motion of the H$\alpha$ line wings.

###  3.6 NSV 4202

This object was apparently first noticed by Knigge & Bauernfeind (1967).
Sebastian Otero added it to the VSX catalog and classified it as a low-
amplitude dwarf nova based on its light curve from ASAS-3. It was also
detected by the OGLE-III survey (Mróz et al., 2015). The ASAS-SN light curve
shows shows a rather flat quiescence near $V=14.4$, and outbursts to $V\sim
12.8$ at irregular intervals of order 100 days, all typical of dwarf novae. We
were unable to find any candidate orbital period in the literature.

The mean spectrum (Fig. 11, top panel) shows strong Balmer and HeI emission
typical of a dwarf nova at minimum light. The H$\alpha$ radial velocities show
periodicity at 0.2839(6) d, or 6.81 h. This is 3.52 cycle d-1. A daily cycle-
count alias at 4.55 cycle d-1, or 5.3 h, is marginally possible but gives a
much poorer fit. The data have good alias discrimination because of the 6.2-h
range of hour angle covered, and because of the amplitude of the modulation
relative to the noise (i.e., $K/\sigma$); the Monte Carlo test (Thorstensen &
Freed, 1985) prefers the stronger period more than 97 per cent of the time.
The lower panel shows velocities folded on the 6.81-h period.

Figure 11:  Top: Mean spectrum of NSV 4202. Middle: Periodogram of H$\alpha$
emission velocities. Lower: H$\alpha$ emission velocities folded on the best
period.

The period is fairly long for a dwarf nova, but it is somewhat surprising that
the mean spectrum shows no contribution from a late-type secondary. A
secondary contribution is almost always seen in high signal-to-noise spectra
of quiescent dwarf novae with periods above six hours or so (see, e.g., the
spectra of ATO 061-31 that were discussed earlier), so we looked for other
evidence to corroborate the period. Unfortunately, none of the synoptic
surveys appear to have sampled this object densely enough to corroborate our
period, and although TESS has observed this deep southerly location many
times, NSV 4202 is $\sim 15$ arcsec from a significantly brighter star and
light curves are not available. We also prepared a phase-resolved image of the
rectified spectra, similar to that of MGAB-V202 shown in Fig. 10, but that
also showed no sign of a secondary star’s spectrum.

###  3.7 V1147 Cen

This object, first noted as a variable star by Luyten (1935), is apparently
the longest-known of those studied here. It was recognized as a likely U Gem
star by Pastukhova (2007), who presented an ASAS-3 light curve showing a
quiescent level near 13.5 mag and frequent outbursts to 11.0 mag. Kazarovets
et al. (2008) bestowed the designation V1147 Cen, and listed the type as
“UGSS:”. The ASAS-SN light curve is entirely typical of an active dwarf nova,
with outbursts typically $\sim 40$ days apart. No detailed study appears to
have been published, and the orbital period remains unknown.

TESS light curves are available from Sectors 11 and 37. Both show a strong
periodicity at 4.767 cycles d-1 ($P=5.035$ h), with less power at half that
frequency (10.07 h).

The mean spectrum (Fig. 12, top panel) shows typical dwarf nova emission lines
and also a contribution from a late-type star. We have only 4.0 hours of data
covering 4.8 hours of hour angle, so the velocities do not define the period
uniquely, but both the emission and absorption velocities show a strong,
consistent low-frequency modulation (Fig. 12, middle panel). One of the
aliases of this modulation is at $P=0.4190(5)$ d, or 2.38(3) cycle d-1,
consistent with half the dominant TESS frequency. The TESS modulation is
clearly due to ellipsoidal variation of the secondary, with two humps per
orbit. The lower panel shows both emission and absorption velocities folded on
the spectroscopic orbital period, which amounts to 10.06 h.

Figure 12:  Top: Mean spectrum of V1147 Cen. Middle: Periodogram of the
absorption (cross-correlation) velocities. The period is not uniquely
determined; the vertical tick indicates the periodicity that is consistent
with the TESS light curve. Lower: Absorption (red) and H$\alpha$ emission
(blue) velocities folded on the adopted period. The epoch is chosen to
correspond to inferior conjunction of the secondary star. The emission line
modulation is approximately, but not exactly, in antiphase to this.

The top trace in Fig. 13 is the average flux-calibrated spectrum of V1147 Cen,
resampled into the rest frame of the secondary prior to averaging. The lower
trace shows the difference between this average and a scaled spectrum of the
K2V star HD109111. The secondary features cancel very well; we found adequate
cancellation for types K0 to K3. Comparing to other CVs with $P_{\rm orb}\sim
10$ h in Fig. 7 of Knigge (2006), the secondary in V1147 is cooler than the
analytic fit, but similar to several other examples plotted in his figure and
listed in Knigge’s Table 2.

Figure 13:  The upper trace shows the average spectrum of V1147 Cen in the
rest frame of the secondary star, and the lower trace shows the result of
subtracting the K2 star HD109011 from this. The subtracted spectrum nominally
has $V=14.15.$ The apparent bump in the continuum shortward of 4800 Å may be a
calibration artifact.

In summary, we confirm that V1147 Cen is a UGSS star and show that it has a
relatively long $P_{\rm orb}$. Had it been more northerly, it would likely
have already been swept up by SDSS, LAMOST, and other surveys and attracted
more attention.

##  4 SUMMARY

We obtained spectra of selected bright CVs in the southern hemisphere, with
the aim of characterizing them more fully. The orbital period of a CV is its
most fundamental observable, and for most of our targets we succeeded in
measuring $P_{\rm orb}$. Our targets, while selected on the basis of
tractability, represent several different subclasses of CVs – three novalikes,
including an apparent SW Sextantis star, and four dwarf novae, including two
with visible secondary stars and one short-period SU UMa-type system.

None of the objects appears grossly atypical, but there are a few notable
findings:

  * •

MGAB-V202 is evidently a new SW Sextantis star. In TESS photometry it shows
two periods clearly different from $P_{\rm orb}$, neither of which is detected
consistently. These may be related to disk precession, and further monitoring
may be enlightening.

  * •

The TESS photometry of BMAM-V547 shows a clear, persistent modulation at a
period that agrees with one of our possible radial velocity periods. In
addition, the TESS photometry shows a superorbital period near 3.93 d.

  * •

The secondary star in the dwarf nova ATO J061$-$31 is slightly warmer than
expected at its orbital period.

  * •

The dwarf nova NSV 4202 does not show a secondary-star spectrum, despite its
relatively long $P_{\rm orb}$.

##  5 ACKNOWLEDGMENTS

This paper uses observations made at the South African Astronomical
Observatory (SAAO). We are deeply thankful to the SAAO staff for their warm
hospitality and expert assistance. Student travel to and from the observatory,
and accommodations at SAAO, were underwritten by a generous donation from
Heather and Jay Weed. The observations reported here were taken as part of the
Dartmouth College Foreign Study Program in astronomy; Professors Brian
Chaboyer and Ryan Hickox were essential in arranging, supporting, and carrying
out this program.

This paper includes data collected with the TESS mission, obtained from the
MAST data archive at the Space Telescope Science Institute (STScI). Funding
for the TESS mission is provided by the NASA Explorer Program. STScI is
operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS 5–26555

## References

  * Armstrong et al. (2013) Armstrong, E., Patterson, J., Michelsen, E., et al. 2013, MNRAS, 435, 707
  * Bellm et al. (2019) Bellm, E. C., Kulkarni, S. R., Graham, M. J., et al. 2019, PASP, 131, 018002
  * Bruch (2023) Bruch, A. 2023, MNRAS, 519, 352
  * Casares et al. (1996) Casares, J., Martinez-Pais, I. G., Marsh, T. R., Charles, P. A., & Lazaro, C. 1996, MNRAS, 278, 219
  * Crause et al. (2019) Crause, L. A., Gilbank, D., Gend, C. v., et al. 2019, Journal of Astronomical Telescopes, Instruments, and Systems, 5, 024007
  * Drake et al. (2009) Drake, A. J., Djorgovski, S. G., Mahabal, A., et al. 2009, ApJ, 696, 870
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
  * Goliasch & Nelson (2015) Goliasch, J., & Nelson, L. 2015, ApJ, 809, 80
  * Horne (1986) Horne, K. 1986, PASP, 98, 609
  * Kalomeni et al. (2016) Kalomeni, B., Nelson, L., Rappaport, S., et al. 2016, ApJ, 833, 83
  * Kato et al. (2020) Kato, T., Isogai, K., Wakamatsu, Y., et al. 2020, PASJ, 72, 14
  * Kazarovets et al. (2008) Kazarovets, E. V., Samus, N. N., Durlevich, O. V., Kireeva, N. N., & Pastukhova, E. N. 2008, Information Bulletin on Variable Stars, 5863, 1
  * Knigge (2006) Knigge, C. 2006, MNRAS, 373, 484
  * Knigge et al. (2011) Knigge, C., Baraffe, I., & Patterson, J. 2011, ApJS, 194, 28
  * Knigge & Bauernfeind (1967) Knigge, R., & Bauernfeind, H. 1967, Bamberg Veroeffentlichungen der Remeis-Sternwarte, 7, 51
  * Luyten (1935) Luyten, W. J. 1935, Astronomische Nachrichten, 256, 325
  * Mróz et al. (2015) Mróz, P., Udalski, A., Poleski, R., et al. 2015, Acta Astron., 65, 313
  * Pala et al. (2020) Pala, A. F., Gänsicke, B. T., Breedt, E., et al. 2020, MNRAS, arXiv:1907.13152 [astro-ph.SR]
  * Pastukhova (2007) Pastukhova, E. N. 2007, Peremennye Zvezdy Prilozhenie, 7, 8
  * Peters & Thorstensen (2005) Peters, C. S., & Thorstensen, J. R. 2005, PASP, 117, 1386
  * Rodríguez-Gil et al. (2007) Rodríguez-Gil, P., Gänsicke, B. T., Hagen, H. J., et al. 2007, MNRAS, 377, 1747
  * Schneider & Young (1980) Schneider, D. P., & Young, P. 1980, ApJ, 238, 946
  * Shafter (1983) Shafter, A. W. 1983, ApJ, 267, 222
  * Shappee et al. (2014) Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, ApJ, 788, 48
  * Thorstensen (2013) Thorstensen, J. R. 2013, PASP, 125, 506
  * Thorstensen (2015) —. 2015, PASP, 127, 351
  * Thorstensen (2020a) —. 2020a, AJ, 160, 151
  * Thorstensen (2020b) —. 2020b, arXiv e-prints, arXiv:2005.02150
  * Thorstensen & Armstrong (2005) Thorstensen, J. R., & Armstrong, E. 2005, AJ, 130, 759
  * Thorstensen & Freed (1985) Thorstensen, J. R., & Freed, I. W. 1985, AJ, 90, 2082
  * Thorstensen et al. (1991) Thorstensen, J. R., Ringwald, F. A., Wade, R. A., Schmidt, G. D., & Norsworthy, J. E. 1991, AJ, 102, 272
  * Tonry & Davis (1979) Tonry, J., & Davis, M. 1979, AJ, 84, 1511
  * Tonry et al. (2018) Tonry, J. L., Denneau, L., Heinze, A. N., et al. 2018, PASP, 130, 064505
  * Wakamatsu et al. (2021) Wakamatsu, Y., Thorstensen, J. R., Kojiguchi, N., et al. 2021, PASJ, 73, 1209
  * Warner (1995) Warner, B. 1995, Cambridge Astrophysics Series, 28


Sebastian Halbig
S.H., Philipps-Universität Marburg, Arbeitsgruppe Algebraische Lie-Theorie, Hans-Meerwein-Straße 6, 35043 Marburg

Tony Zorman
T.Z., Technische Universität Dresden, Institut für Geometrie, Zellescher Weg 12–14, 01062 Dresden

[2020]primary: 18M30; secondary: 18M05, 16T05, 18C15

Diagrammatics for Comodule Monads

We extend Willerton's <cit.> graphical calculus for bimonads to comodule monads,
a monadic interpretation of module categories over a monoidal category.
As an application, we prove a version of Tannaka–Krein duality for these structures.

§ INTRODUCTION

Given a monad \(B\) on a monoidal category \(\cat{C}\),
one might ask to which extend monoidal structures on the category \(\cat{C}^B\) of \(B\)-algebras
are controlled by additional structures on the monad itself.
This can be seen as an extension of the classical Tannaka–Krein duality,
and was proved by Moerdijk <cit.> and McCrudden <cit.>.

Let \(B\) be a monad on a monoidal category \(\cat{C}\).
Bimonad structures on \(B\) are
in one-to-one correspondence with
monoidal structures on \(\cat{C}^B\), such that the forgetful functor \(U^B\) is strict monoidal.

In <cit.>,
<ref> is proved in a graphical fashion.
The aim of this article is to generalise both the statement
and its graphical proof to comodule monads,
as developed by Aguiar and Chase <cit.>.
More precisely, we show the following.

Let \(B\) be a bimonad on the monoidal category \(\cat{C}\),
and \(K\) a monad on a right \(\cat{C}\)-module category \(\cat{M}\).
Coactions of \(B\) on \(K\) are
in bijection with
right actions of \(\cat{C}^B\) on \(\cat{M}^K\), such that \(U^{K}\) is a strict comodule functor over \(U^{B}\).

The article is structured as follows.
<Ref> serves to review basic concepts from category theory,
as well as their associated string diagrammatic counterparts.
<Ref> introduces Willerton's graphical calculus for monoidal categories,
and <ref> extends this construction to module categories and comodule monads,
allowing us to prove <ref>.

§ 2-CATEGORIES AND THE GRAPHICAL CALCULUS

We assume the reader's familiarity with basic concepts of the theory of monoidal and 2-categories as discussed for example in <cit.>.
Without loss of generality, we take “2-category” to mean “strict 2-category”.
Throughout this article, we work in a monoidal 2-category $(\mathbb{X}, \times, \1)$ which admits the construction of algebras[
Monoidal 2-categories are discussed in <cit.>.
The term `admitting the construction of algebras' is due to Street, see <cit.>.
For example, we may take $\mathbb{X}$ to be the 2-category of (small) categories, functors, and natural transformations  with the monoidal structure induced by the Cartesian product.
We use juxtaposition for the horizontal composition of $\mathbb{X}$, or, in case we want to emphasise the direction of composition,

\begin{equation*}
\blank \odot_{\mathcal{A},\mathcal{B},\mathcal{C}} \blank \from \mathbb{X}(\mathcal{B},\mathcal{C}) \times \mathbb{X}(\mathcal{A},\mathcal{B}) \to \mathbb{X}(\mathcal{A},\mathcal{C}), \qquad \qquad \mathcal{A},\mathcal{B},\mathcal{C} \in \Ob(\mathbb{X}).
\end{equation*}

In order to appeal to more standard nomenclature of monads and to emphasise that our main example for $\mathbb{X}$ is given by $\BiCat{Cat}$, we call its 0-cells categories, its 1-cells functors, and its 2-cells natural transformations.

String diagrams will serve as the main tool for doing computations.
We closely follow the presentation and conventions in <cit.>.
In the case of a 2-category \(\mathbb{X}\),
a string diagram comprises regions, labelled with categories, strings labelled with functors,
and vertices between the strings labelled with natural transformations.
If two string diagrams can be transformed into each other, the natural transformations they represent are equal.
A more detailed description is given in <cit.>.
Our convention is to read diagrams from top to bottom and left to right.
Horizontal and vertical composition are given by horizontal and vertical gluing of  diagrams, respectively.
Identity natural transformations are given by unlabelled vertices.
The identity functorof a category is represented by the empty edge.
If the involved categories are clear from the context, we omit writing them explicitly.

Recall from  <cit.> that a monad on a category \(\cat{C}\in \mathbb{X}\) is
a functor \(T \from \cat{C} \to \cat{C}\)
together with two natural transformations \(\mu \from T^2 \nt T\) and \(u \from \Id_{\cat{C}} \nt T\),
called the multiplication and unit of \(T\),
satisfying appropriate associativity and unitality axioms:

\[
\begin{mytikzcd}
{T^3} & { T^2} & {T^2} & T & T \\
{T^2} & T && T
\arrow["T\mu", Rightarrow, from=1-1, to=1-2]
\arrow["\mu", Rightarrow, from=1-2, to=2-2]
\arrow["{\mu T}"', Rightarrow, from=1-1, to=2-1]
\arrow["\mu"', Rightarrow, from=2-1, to=2-2]
\arrow[Rightarrow, no head, from=1-4, to=2-4]
\arrow["{u T}"', Rightarrow, from=1-4, to=1-3]
\arrow["\mu"', Rightarrow, from=1-3, to=2-4]
\arrow["{T u}", Rightarrow, from=1-4, to=1-5]
\arrow["\mu", Rightarrow, from=1-5, to=2-4]
\end{mytikzcd}
\]

We represent the multiplication and unit of a monad \(T\)
in terms of string diagrams:

\begin{equation} \label[diagram]{eq:mult-and-unit-of-monad}
\inputtikz{monads-multiplication-and-unit}
\end{equation}

Their associativity and unitality then equate to

\begin{equation} \label[diagram]{eq:assoc-and-unit-of-monad-string-diagram}
\inputtikz{monads-assoc-and-unit}
\end{equation}

As discussed in <cit.>, there is a $2$-category $\Mon(\mathbb{X})$ of monads on $\mathbb{X}$.

\begin{equation} \label{eq:inclusion-triv-monad}
\Inc \from \mathbb{X} \to \Mon(\mathbb{X}),\qquad \cat{C} \mapsto (1_{\cat{C}}, \id_{1_{\cat{C}}}, \id_{1_{\cat{C}}})
\end{equation}

be the inclusion 2-functor, which maps any category to its identity monad.
By assuption $\mathbb{X}$ admits the construction of algebras.
That is the above functor has  a right adjoint $\Alg \from \Mon(\mathbb{X}) \to \mathbb{X}$,
which maps any monad $(T, \mu, u) \from\cat{C} \to\cat{C}$ to its Eilenberg–Moore category $\cat{C}^T \in \Ob(\mathbb{X})$.
Using the previous 2-adjunction, one proves that to every monad $(T, \mu, u) \from\cat{C} \to\cat{C}$ one can associate adjoint functors

\begin{equation} \label{eq:Eilenberg--More-adjunction}
F^T \from \cat{C} \to \cat{C}^T \qquad \text{ and } \qquad
U^T \from \cat{C}^T \to \cat{C},
\end{equation}

whose unit and counit transformation we denote by
$\eta \from 1_{\cat{C}} \nt U^TF^T$ and $\varepsilon \from F^TU^T \nt 1_{\cat{C}^T}$,
such that
$T = U^TF^T$,
$\mu = F^T \varepsilon U^T$,
and $u = \eta$.
We call the adjunction $\adj{F_T}{U_T}{\cat{C}}{\cat{C}^T}$ the -adjunction of $(T, \mu, u)$,
and depict \(\eta\) and \(\varepsilon\) by

\[
\inputtikz{adjunction-unit-counit}
\]

Graphically, the defining equations of adjunctions translate to the snake equations

\[
\inputtikz{adjunction-snake-identities}.
\]

As noted in <cit.>, the category of  a monad \(T\) can be incorporated into the graphical calculus.
Defining the natural transformation

\[
\vartheta \eqdef T U^T = U^T F^T U^T \xrightarrow{\;U^T\varepsilon\;} U^T
\]

we may write
\begin{equation}\label[diagram]{eq:action-over-monad-string-diagram}
\inputtikz{action-over-monad}
\end{equation}

We think of $\vartheta \from T U^T \to U^T$ as an action of $T$ on $U^T$ due to the identities

\[
\inputtikz{axioms-of-action-over-monad}
\]

In fact, any adjunction $\stdadj$ defines a monad \((UF, F \varepsilon U, \eta)\).
A question one might ask is
how much the functors \(F\) and \(U\) “differ” from the free and forgetful functors
\(F^T \from \cat{C} \to \cat{C}^T\) and \(U^T \from \cat{C}^T \to \cat{C}\)  of \(T\), respectively.

Let \(T\) be the monad of the adjunction .
There exists a unique functor \(\Sigma \from \cat D \to \cat{C}^T\) satisfying \(\Sigma F = F^T\) and \(U^T \Sigma = U\).

Given an adjunction $\stdadj$, we call the unique functor \(\Sigma \from \cat D \to \cat{C}^T\) discussed in the previous lemma the comparison functor and refer to the adjunction $F \lad U$ as monadic if $\Sigma$ is an equivalence.

§ BIMONADS

Bimonads are a vast generalisation of bialgebras.
They naturally arise in the study of (rigid) monoidal categories and topological quantum field theories, see amongst others <cit.>.

Due to the lack of a braiding on the endofunctors \(\mathrm{End}(\cat{C})\) over \(\cat{C}\), the naive notion of bialgebra does not generalise to the monadic setting and needs to be adjusted.
One possible way of overcoming this problem was introduced and studied by Moerdijk under the name “Hopf monad”[As remarked in <cit.>, the concept of Hopf monads is strictly dual to that of monoidal comonads, which are studied for example in <cit.>.] in <cit.>; the idea being that the opmonoidal structure of a functor replaces the bialgebra's comultiplication and counit.
Following the conventions of <cit.>, we refer to such structures as bimonads.

An opmonoidal functor between (strict) monoidal categories \((\cat{C}, \otimes, 1)\) and \((\cat{D}, \otimes, 1)\)
is a functor \(F\from \cat{C} \to \cat{D}\)
together with two natural transformations[
We view the monoidal unit of $\cat{C}$ as a functor $1 \from \1 \to\cat{C}$, where $\1$ is the unit of $\mathbb{X}$. ]

\[
F_2 \from F(\blank \otimes \bblank) \nt F\blank \otimes F\bblank \qquad \text{ and } \qquad
F_0 \from F1 \to 1
\]

called the comultiplication and  counit,
satisfying coassociativity and counitality:

\begin{equation} \label{eq:opmon-fun->coassociativity-counitality}
\mathscale{0.95}{
\begin{mytikzcd}
{F(\blank \otimes \bblank \otimes \bbblank)} & {F\blank \otimes F(\bblank \otimes \bbblank)} & {F1 \otimes F\blank} & F\blank & {F\blank \otimes F1} \\
{F(\blank \otimes \bblank) \otimes F\bbblank} & {F\blank \otimes F\bblank \otimes F\bbblank} && F\blank
\arrow["{{F_{2;\blank, \bblank \otimes \bbblank}}}", from=1-1, to=1-2]
\arrow["{{F\blank \otimes F_{2;\bblank,\bbblank}}}", from=1-2, to=2-2]
\arrow["{{F_{2; \blank \otimes \bblank, \bbblank}}}"', from=1-1, to=2-1]
\arrow["{{F_{2;\blank,\bblank} \otimes F\bbblank}}"', from=2-1, to=2-2]
\arrow["{{F_{2; 1, \blank}}}"', from=1-4, to=1-3]
\arrow["{{F_0 \otimes F\blank}}"', from=1-3, to=2-4]
\arrow[Rightarrow, no head, from=1-4, to=2-4]
\arrow["{{F_{2;\blank,1}}}", from=1-4, to=1-5]
\arrow["{{F\blank \otimes F_0}}", from=1-5, to=2-4]
\end{mytikzcd}
\end{equation}

If \(F_2\) and \(F_0\) are isomorphisms or identities,
we call \(F\) strong monoidal or strict monoidal, respectively.

An opmonoidal natural transformation
between opmonoidal functors \(F, G \from \cat{C} \to \cat{D}\) is
a natural transformation \(\rho \from F \nt G\)
such that

\[
\begin{mytikzcd}
{F(\blank \otimes \bblank)} & {G(\blank \otimes \bblank)} & F1 & G1 \\
{F\blank \otimes F\bblank} & {G\blank \otimes G\bblank} && 1
\arrow["{{\rho_{\blank \otimes \bblank}}}", from=1-1, to=1-2]
\arrow["{{G_{2,\blank,\bblank}}}", from=1-2, to=2-2]
\arrow["{{F_{2,\blank,\bblank}}}"', from=1-1, to=2-1]
\arrow["{{\rho_\blank \otimes \rho_\bblank}}"', from=2-1, to=2-2]
\arrow["{{F_0}}"', from=1-3, to=2-4]
\arrow["{{\rho_1}}", from=1-3, to=1-4]
\arrow["{{G_0}}", from=1-4, to=2-4]
\end{mytikzcd}
\]

If \(\rho\) is additionally a natural isomorphism, we call it an opmonoidal natural isomorphism.

Willerton introduced a graphical calculus for opmonoidal functors in <cit.>.
The key idea is to incorporate the Cartesian product of categories, or, more generally, the tensor product of a monoidal 2-category into the diagramatic calculus.

As before, we consider strings and vertices between them.
These are labelled with functors and natural transformations, respectively.
The strings and vertices are embedded into bounded rectangles, which we will call sheets.
Each (connected) region of a sheet is decorated with a category.
The same mechanics as for string diagrams apply: horizontal and vertical gluing represents composition of functors and natural transformations.
On top of these operations, we add stacking sheets behind each other to depict the monoidal product of $\mathbb{X}$.
Our convention is to read diagrams from front to back, left to right, and top to bottom.

Two of the most vital building blocks in this new graphical language are  the tensor product and unit of a monoidal category \((\cat{C}, \otimes, 1)\):

\[
\inputtikz{tensor-product}
\]

On the left, there are two sheets equating to two copies of \(\cat{C}\) joined by a line: the tensor product of \(\cat{C}\).
On the right, we have the unit of \(\cat{C}\) considered as a functor \(1 \from \1 \to \cat{C}\), where \(\1\) is the terminal category.
We represent \(\1\) by the empty sheet, and the unit of \(\cat{C}\) by a dashed line.

Consider an opmonoidal functor \((F, F_2, F_0) \from \cat{C} \to \cat{D}\).
Graphically, one may depict \(F_2\) and \(F_0\) like the following:

\[
\inputtikz{comonoidal-functor}
\]

The coassociativity and counitality of <ref> become:

\begin{equation} \label[diagram]{eq:coass-of-opmonoidal-functor}
\inputtikz{coassoc-of-bimonad}
\end{equation}

\begin{equation} \label[diagram]{eq:counit-of-opmonoidal-functor}
\inputtikz{counitality-of-bimonad}
\end{equation}

A bimonad on a monoidal category \(\cat{C}\) comprises
an opmonoidal endofunctor \((B, B_2, B_0) \from \cat{C} \to \cat{C}\)
together with opmonoidal natural transformations \(\mu \from B^2 \nt B\) and \(\eta \from \Id_{\cat{C}} \nt B\)
implementing a monad structure on \(B\).

Graphically, what it means for \(\mu\) and \(\eta\) to be opmonoidal natural transformations is:

\begin{equation} \label[diagram]{eq:mu-opmonoidal-nt}
\inputtikz{comult-of-mult}
\end{equation}

\begin{equation} \label[diagram]{eq:eta-opmonoidal-nt}
\inputtikz{counit-of-unit}
\end{equation}

Let be an opmonoidal adjunction between \(\cat{C}\) and \(\cat{D}\).
The monad of the adjunction \(UF \from \cat{C} \to \cat{C}\) is a bimonad;
its comultiplication is defined  as the composition

\[
UF(\blank \otimes \bblank) \trightarrow{UF_{2;\blank,\bblank}} U(F\blank\otimes F\bblank) \trightarrow{U_{2; F\blank,F\bblank}} UF\blank \otimes UF\bblank,
\]

and its counit is given by

\[
UF1 \trightarrow{U F_0} U1 \trightarrow{\;U_0\;} 1.
\]

Adjunctions between monoidal categories are a broad topic with many facets, see <cit.>.
For our purposes, we can restrict ourselves to the following situation.

We call an adjunction \(\stdadj\) between monoidal categories \(\cat{C}\) and \(\cat{D}\) opmonoidal if
\(F\) and \(U\) are opmonoidal functors, and
the unit and counit of the adjunction are opmonoidal natural transformations.
If \(F\) and \(U\) are moreover strong monoidal,
we call \(\stdadj\) a (strong) monoidal adjunction.

Suppose \(\stdadj\) to be an opmonoidal adjunction.
In our graphical language, the conditions that the unit \(\eta\) and counit \(\varepsilon\) need to be opmonoidal natural transformations take the following form.

\begin{equation} \label[diagram]{eq:unit-adjunction-opmonoidal-nt}
\inputtikz{unit-monoidal-adjunction}
\end{equation}

\begin{equation} \label[diagram]{eq:counit-adjunction-opmonoidal-nt}
\inputtikz{counit-monoidal-adjunction}
\end{equation}

The next result is a slightly simplified version of <cit.>.

Let be a pair of adjoint functors between two monoidal categories.
The adjunction \(F \adjoint U\) is monoidal if and only if \(U\) is a strong monoidal functor.
That is, the coherence morphisms of \(U\) are invertible.

Let \(B \from \cat{C} \to \cat{C}\) be the bimonad arising from the monoidal adjunction .
In many cases, in particular $\mathbb{X}= \BiCat{Cat}$, one can show that $\cat{C}^B$ carries a monoidal structure such that the forgetful functor \(U^B \from \cat{C}^B \to \cat{C}\) is strict, see <cit.>.
By the above lemma, the adjunction \(F^B \adjoint U^B\) is monoidal.
This raises the question whether the comparison functor—
mediating between the two adjunctions—
is compatible with this additional structure.
Due to <cit.>, see also <cit.>, we have the following result for $\mathbb{X}= \BiCat{Cat}$.

Let be a monoidal adjunction and write \(B \from \cat{C} \to \cat{C}\) for its induced bimonad.
The comparison functor \(\Sigma \from \cat D \to \cat{C}^B\) is strong monoidal and
\(U^B \Sigma = U\) as well as \(\Sigma F = F^B\) as strong, respectively, opmonoidal functors.

For a more general version see <cit.>.

§ COMODULE MONADS

Monads with a `coaction' over a bimonad were defined and studied by Aguiar and Chase in <cit.>.
This concept is needed to obtain an adequate monadic interpretation of twisted centres.
We briefly summarise the aspects of the aforementioned article that are needed for our investigation.[
We slightly deviate from <cit.> in that we study right comodule monads as opposed to their left versions.
Throughout this section, we fix $\mathbb{X}= \BiCat{Cat}$.

Let \(\cat{C}\) be a monoidal category.
A right right \(\cat{C}\)-module category comprises
a category \(\cat{M}\)
together with an action functor \(\ract \from \cat{M} \times \cat{C} \to \cat{M}\),
satisfying appropriate associativity and unitality conditions,
see for example <cit.>.

To keep our notation concise,
for the rest of this article
we fix two monoidal categories \((\cat{C}, \otimes, 1)\) and \((\cat{D}, \otimes, 1)\)
and over each a right module category \((\cat{M}, \ract)\) and \((\cat{N}, \ract)\).

Let \(F \from \cat{C} \to \cat{D}\) be an opmonoidal functor.
A (right) comodule functor over \(F\) is a pair \((G, \delta)\) comprising
a functor \(G \from \cat{M} \to \cat{N}\) together with
a natural transformation \(\delta \from G(\blank \ract \bblank) \nt G\blank \ract F\bblank\),
called the coaction of \(G\), which is coassociative and counital.

A comodule functor is called strong if its coaction is an isomorphism.

The string diagrammatic calculus of <ref> can be adapted to this setting as follows:
in addition to \(\otimes\) “splitting” the sheets,
we now additionally allow \(\ract\) to do so as well.
In order to keep track of which splitting occurred,
functors between the module categories will be coloured blue.
For example, the coaction \(\delta \from G(\blank \ract \bblank) \nt G\blank \ract F\bblank\) from <ref> is represented by

\[
\inputtikz{coaction}
\]

The mentioned compatibility of the coaction with the comultiplication and counit of \(F\)
result in diagrams analogous
to (<ref>)
and (<ref>)
—except that one of the strings will be in a different colour.

Let \(G, K \from \cat{M} \to \cat{N}\) be comodule functors over  \(B, F \from \cat{C} \to \cat{D}\).
A comodule natural transformation from \(G\) to \(K\) comprises
a pair of natural transformations \(\phi \from G \nt K\) and \(\psi \from B \nt F\),
such that

\begin{equation} \label[diagram]{eq:comodule-natural-transformation}
\begin{mytikzcd}
{G(m \ract x)} & {K(m \ract x)} \\
{Gm \ract Bx} & {Km \ract Fx}
\arrow["{\phi_{m \ract x}}", from=1-1, to=1-2]
\arrow["{\delta^{(K)}_{m,x}}", from=1-2, to=2-2]
\arrow["{\delta_{m,x}^{(G)}}"', from=1-1, to=2-1]
\arrow["{\phi_m \ract \psi_x}"', from=2-1, to=2-2]
\end{mytikzcd}
\end{equation}

commutes, for all \(x \in \cat{C}\) and \(m \in \cat{M}\).

We call \((\phi, \psi)\) a morphism of comodule functors if \(B=F\) and \(\psi=\id_B\).

Given a comodule natural transformation $(\phi \from G \nt K,\, \psi \from B \nt F)$,
the graphical version of <ref> is displayed in our next picture,
where the blue dot indicates \(\phi\) and the black dot indicates $\psi$.

\[
\inputtikz{comodule-morphism}
\]

Suppose the pair \(\phi \from G \nt K\) and \(\psi \from B \nt F\) to constitute a comodule natural transformation.
We can view \(\phi \from G \nt K\) as a morphism of comodule functors over \(F\) if we equip \(G\) with a new coaction.
It is given for all \(x \in \cat{C}\) and \(m \in \cat{M}\) by

\[
G(m \ract x) \trightarrow{\;\delta^{(G)}_{m,x}\;} Gm \ract Bx \trightarrow{Gm\ract\psi_x} Gm \ract Fx.
\]

It follows that by altering the involved coactions suitably,
comodule natural transformations and morphisms of comodule functors can be identified with each other.

Let \((B, \mu, \eta, B_2, B_0)\) be a bimonad on \(\cat{C}\).
The unit \(\eta \from \Id_{\cat{C}} \to B\) implements a coaction on \(\Id_{\cat{M}} \from \cat{M} \to \cat{M}\) via

\[
\id_m \ract \eta_x \from \Id_{\cat{M}}(m \ract x) \to \Id_{\cat{M}}m \ract Bx, \qquad \text{ for all } x \in \cat{C}, m \in \cat{M}.
\]

Using the multiplication \(\mu \from B^2 \nt B\),
we can equip the composition \(GK\from \cat{M} \to \cat{M}\) of
two \(B\)-comodule functors \(G, K \from \cat{M} \to \cat{M}\) with a comodule structure:

\[
GK(\blank \ract \bblank) \xrightarrow{G\delta^{(K)}} G(K\blank \ract B\bblank) \xrightarrow{\delta^{(K)}} GK\blank \ract B^2\bblank \xrightarrow{\id \ract \mu} GK\blank \ract B\bblank.
\]

Due to the associativity and unitality of the multiplication of \(B\), the category \(\Com(B, \cat{M})\) of comodule endofunctors on \(\cat{M}\) over \(B\) is monoidal.
Studying its monoids will be a main focus of the rest of this article.

Consider a bimonad \(B\) on \(\cat{C}\).
A comodule monad over \(B\) on \(\cat{M}\) is a comodule endofunctor \((K, \delta) \from \cat{M} \to \cat{M}\)
together with morphisms of comodule functors \(\mu \from K^2 \nt K\)  and \(\eta \from \Id_{\cat{M}} \nt K\)
such that \((K,\mu,\eta)\) is a monad.

The conditions for the multiplication and unit of
a comodule monad \(K \from \cat{M} \to \cat{M}\)
over a bimonad \(B \from \cat{C} \to \cat{C}\)
to be morphisms of comodule functors amount to

\[
\inputtikz{mult-and-unit-comodule-monad}
\]

Notice how these conditions are analogous to those in

Let \(B \from \cat{C} \to \cat{C}\) be a bimonad and  \((K, \delta) \from \cat{M} \to \cat{M}\) a comodule monad over it.
The coaction of \(K\) allows us to define an action \(\ract \from \cat{M}^{K}\times \cat{C}^B \to \cat{M}^K\).
For any two modules \((m, \vartheta_{m}) \in \cat{M}^K\) and \((x, \vartheta_{x}) \in \cat{C}^B\), it is given by

\[
(m, \vartheta_{m}) \ract (x, \vartheta_{x}) \eqdef \big(m \ract x, (\vartheta_{m}\ract \vartheta_{x}) \circ \delta_{m,x} \big).
\]

The axioms of the coaction of \(B\) on \(K\) translate precisely to the compatibility of the action of \(\cat{C}^B\) on \(\cat{M}^{K}\) with the tensor product and unit of \(\cat{C}^B\).

We have already seen that monads and adjunctions are in close correspondence, and that additional structures on the monads have their counterparts expressed in terms of the units and counits of adjunctions.
In the case of comodule monads this is slightly more complicated as we have two adjunctions to consider: one corresponding to the bimonad and one to the comodule monad.

Consider two adjunctions \(\stdadj\) and \(\adj{G}{V}{\cat{M}}{\cat{N}}\) such that \(F \dashv U\) is monoidal and \(G,V\) are comodule functors over \(F,U\).
We call the pair \((G \dashv V, F \dashv U)\) a comodule adjunction if the following two identities hold:

\[
\begin{mytikzcd}
{m \ract x} & {VG(m \ract x)} & {GV(m \ract x)} & {G(Vm \ract Ux)} \\
{VGm \ract UFx} & {V(Gm \ract Fx)} & {m \ract x} & {GVm \ract FUx}
\arrow["{\eta^{(G \lad V)}_{m\ract x}}", from=1-1, to=1-2]
\arrow["{V\delta^{(G)}_{m,x}}", from=1-2, to=2-2]
\arrow["{\delta^{(V)}_{Gm \ract Fx}}", from=2-2, to=2-1]
\arrow["{\eta^{(G \lad V)}_m \ract \eta^{(F \lad U)}_x}"', from=1-1, to=2-1]
\arrow["{\varepsilon^{(G \lad V)}_{m \ract x}}"', from=1-3, to=2-3]
\arrow["{G\delta^{(V)}_{m,x}}", from=1-3, to=1-4]
\arrow["{\delta^{(G)}_{Vm \ract Ux}}", from=1-4, to=2-4]
\arrow["{\varepsilon^{(G \lad V)}_m \ract \varepsilon^{(F \lad U)}_x}", from=2-4, to=2-3]
\end{mytikzcd}
\]

From their string diagrammatic representations, one observes that the conditions required by <ref> are analogous to

\begin{equation} \label[diagram]{eq:comodule-adjunction-unit-and-counit}
\inputtikz{comodule-adjunction-unit-and-counit}
\end{equation}

The philosophy that monads and adjunctions are two sides of the same coin extends to the comodule setting.
Suppose that we have a monoidal adjunction and over it a comodule adjunction \(\adj{G}{V}{\cat{M}}{\cat{N}}\).
As stated in <cit.>, the bimonad \(B\eqdef UF\) admits a coaction on the monad \(K \eqdef VG\).
For any \(m \in \cat{M}\) and \(x \in \cat{C}\) it is given by

\[
VG(m \ract x) \trightarrow{V\delta^{(G)}_{m,x}} V(Gm\ract Fx) \trightarrow{\delta^{(V)}_{Gm,Fx}} Km\ract Bx.
\]

The next result clarifies the structure of comparison functors associated to comodule adjunctions.
Its proof is analogous to <cit.>.

Consider a comodule adjunction \(\adj{G}{V}{\cat{M}}{\cat{N}}\) over a monoidal adjunction \(\stdadj\),
and denote the associated comodule monad and bimonad by \(K \eqdef VG \from \cat{M} \to \cat{M}\) and \(B \eqdef UF \from \cat{C} \to \cat{C}\), respectively.
The comparison functor \(\Sigma^{K} \from \cat{N} \to \cat{M}^K\) is a strong comodule functor over \(\Sigma^{(B)} \from \cat{D} \to \cat{C}^B\);
we furthermore have that

\[
U^K \Sigma^{K} = V \qquad\text{and}\qquad \Sigma^{K} G = F^K
\]

as comodule functors.

For any \(n \in \cat{N}\) we have \(\Sigma^{K}n= (Vn, V\varepsilon_{n})\) and a direct computation shows that the coaction of \(V\) lifts to a coaction of \(\Sigma^{K}\).
That is,

\[
U^{K} \delta^{\Sigma^{K}}_{n, y} = \delta ^{V}_{n, y}, \qquad \qquad \text{ for all } n \in \cat{N} \text{ and } y \in \cat{D}.
\]

Using that \(U^{K} \from \cat{M}^{K} \to \cat{M}\) is a faithful and conservative functor,
one observes that \(\Sigma^{K}\) becomes a strong comodule functor in this manner.
Furthermore, as \(U^{K}\) is strict comodule, the coactions of \(U^{K}\Sigma^{K}\) and \(V\) coincide.
Lastly, we compute for any \(x \in \cat{C}\) and
\(m \in \cat{M}\)

\begin{align*}
\delta^{(\Sigma^{K} G)}_{m, x}
= \delta^{(U^K \Sigma^{K} G)}_{m, x}
= \delta^{(VG)}_{m, x}
= \delta^{(K)}_{m, x}
= \delta^{(U^K F^K)}_{m, x}
= \delta^{(F^K)}_{m, x}.
\end{align*}

Our main theorem extends <cit.> and <cit.>;
we prove it analogously to the latter.

Let \(\cat{C}\) and \(\cat{D}\) be monoidal categories,
and suppose that \(\cat{M}\) and \(\cat{N}\) are right \(\cat{C}\)- and \(\cat{D}\)-module categories, respectively.
Suppose that  is a monoidal adjunction,
and let \(\adj{G}{V}{\cat{M}}{\cat{N}}\) be any adjunction.
Lifts of \(G \dashv V\) to a comodule adjunction are in bijection with lifts of \(\,V \from \cat{N} \to \cat{M}\) to a strong comodule functor.

Let \(G \dashv V\) be a comodule adjunction over \(F \dashv U\),
and write \(\delta^{(V)}\) and \(\delta^{(G)}\) for the coactions of \(V\) and \(G\), respectively.
For all \(m \in \cat{M}\) and \(x \in \cat{C}\),
we define the inverse \(\delta^{-(V)}\from V\blank \ract U\bblank \nt V(\blank \ract \bblank)\) of \(\delta^{(V)}\) via

\[
Vm \ract Ux
\xrightarrow{\eta^{(G \adjoint V)}_{Vm \,\ract\, Ux}} VG(Vm \ract Ux)
\xrightarrow{V\delta^{(G)}_{Vm, Ux}} V(GVm \ract FUx)
\xrightarrow{V\left(\varepsilon^{(G \adjoint V)}_m \,\ract\, \varepsilon^{(F \adjoint U)}_x\right)} V(m \ract x).
\]

That is,

\begin{equation} \label[diagram]{eq:inverse-of-coaction}
\inputtikz{lifting-adjunctions-c2-inverse-def}
\end{equation}

Using that \(G\) and \(V\) are part of a comodule adjunction,
a straightforward computation proves \(\delta^{-(V)} \circ \delta^{(V)} = \Id_{V(\blank \ract \bblank)}\):

\begin{equation*}
\inputtikz{lifting-adjunctions-c2-post-inverse}
\end{equation*}

A similar strategy can be used to show that \(\delta^{(V)} \circ \delta^{-(V)} = \id_{V\blank \ract U\bblank}\).
Thus, \(\delta^{(V)}\) is a natural isomorphism and therefore \(V\) is a strong comodule functor.

Conversely, suppose that \((V, \delta^{(V)}) \from \cat{N} \to \cat{M}\) is a strong comodule functor over \(U\).
Define an arrow \(\delta^{(G)}\from G(\blank \ract \bblank) \nt G\blank \ract F\bblank\) by

\begin{equation} \label[diagram]{eq:coaction-via-counit-unit-and-inverse}
\inputtikz{lifting-adjunctions-c2-def}
\end{equation}

Suppose that \(\eta\) and \(\varepsilon\) are the unit and counit of the adjunction \(F \lad U\).
Due to <cit.>, the comultiplication and counit of \(F \from \cat{C} \to \cat{D}\) are for all \(x,y \in \cat{C}\) given by

\[
F_{2,x,y} \eqdef F(x \otimes y)
\xrightarrow{F(\eta_x \otimes \eta_y)} F(UFx \otimes UFy)
\xrightarrow{FU_{-2, Fx, Fy}} FU(Fx \otimes Fy)
\xrightarrow{\varepsilon_{Fx \otimes Fx}} Fx \otimes Fy,
\]

\[
F_0 \eqdef F1 \xrightarrow{FU_{-0}} FU1 \xrightarrow{\epsilon_1} 1.
\]

Note that, graphically, \(F_2\) looks just like <ref>,
with black strings taking the place of blue ones.
The following shows that \(\delta^{(G)}\from G(\blank \ract \bblank) \nt G\blank \ract F\bblank\) is a coaction;
satisfies the coassociativity and counitality conditions of <ref>:

\[
\inputtikz{lifting-adjunctions-c2-prop-1}
\]

\[
\inputtikz{lifting-adjunctions-c2-prop-2}
\]

A straightforward computation proves that the unit of the adjunction \(G \dashv V\) satisfies
the axioms displayed in  <ref>:

\begin{equation*}
\inputtikz{lifting-adjunctions-eta}
\end{equation*}

An analogous computation for the counit shows that \(G \dashv V\) is a comodule adjunction.

To see that these constructions are inverse to of each other, first suppose that we have a comodule adjunction \((G, \delta^{(G)}) \adjoint (V, \delta^{(V)})\).
By utilising \(\delta^{-(V)}\) as given in <ref>, we obtain another coaction \(\lambda^{(G)}\) on \(G\),
see <ref>.
A direct computation shows that \(\delta^{(G)} = \lambda^{(G)}\):

\begin{equation*}
\inputtikz{lifting-adjunctions-ops-are-inverses}
\end{equation*}

The converse direction is clear since the map which associated to any strong comodule structure on $V$ a comodule adjunction $G \lad V$ preserves the coaction of $V$.

The above result yields a description of a comodule monad's coaction in terms of its adjunction;
<ref>, our desired analogue of <ref>.

Suppose \(\cat{C}^{B}\) acts from the right on \(\cat{M}^{K}\) such that \(U^{K}\) is a strict comodule functor.
Due to <ref>, \(K = U^K F^K\) is a comodule monad via the coaction

\begin{equation}
\delta^{(K)} \eqdef \delta^{(U^K)} \circ U^K\delta^{(F^K)} = U^K\delta^{(F^K)}.
\end{equation}

Conversely, if \(K\) is a comodule monad, \(\cat{M}^{K}\) becomes a suitable right module over \(\cat{C}^{B}\) with the action as given in <ref>.

Since the coaction on \(K\) and the action of \(\cat{C}^B\) on \(\cat{M}^K\) determine the coactions of \(F^{K}\) uniquely, the above constructions are inverse to each other by <ref>.

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

18C15 (18A40 18D10 18D35)
titleGeneralized Hopf modules for bimonads
journaltitleTheory and Applications of Categories

American Mathematical Society, Providence, RI

18D10 (05A30 16T30)
noteWith forewords by Kenneth Brown and Stephen Chase and
André Joyal
seriesCRM Monograph Series
titleMonoidal functors, species and Hopf algebras

howpublishedBénabou, Jean et al., Reports of the Midwest
Category Seminar. Lect. Notes Math. 47, 1-77 (1967).
titleIntroduction to bicategories

18C20 (16T05 18D10 18D15)
titleHopf monads on monoidal categories
journaltitleAdvances in Mathematics

North-Holland, Amsterdam

55S25 (55N20)
booktitleHandbook of algebraic topology
titleStable operations in generalized cohomology

18C20 (16W30 18D10)
titleHopf monads
journaltitleAdvances in Mathematics

18C15, 18D05, 18D10
shortjournalTheory Appl. Categ.
titleThe Formal Theory of Multimonoidal Monads
journaltitleTheory and Applications of Categories

* Can't use 'journaltitle' + 'journal'
* Can't use 'date' +

American Mathematical Society, Providence, RI

seriesMathematical Surveys and Monographs
titleTensor categories

math.QA, math.CT, primary:18M15, secondary: 16T05, 18C20, 18M30
titlePivotality, twisted centres and the anti-double of a Hopf
journaltitlearXiv e-prints

titleThe geometry of tensor calculus. I
journaltitleAdv. Math.

Oxford University Press

Abstract. 2-Dimensional Categories provides an introduction to 2-categories
and bicategories, assuming only the most elementary aspects of category
theory. A re

title2-Dimensional Categories

* Can't use 'date' + 'year'

Springer Berlin Heidelberg

booktitleCategory Seminar
titleDoctrinal adjunction
Berlin, Heidelberg

Springer-Verlag, Berlin

57R56 (16W30 18D10 57M27 57N10 57N70)
seriesLecture Notes in Mathematics
titleNon-semisimple topological quantum field theories for
3-manifolds with corners

18D10 (18D05 18D25)
pagesNo. 19, 469485
titleOpmonoidal monads
journaltitleTheory and Applications of Categories

Springer-Verlag, New York

seriesGraduate Texts in Mathematics
titleCategories for the working mathematician

noteCategory theory 1999 (Coimbra)
titleMonads on tensor categories
journaltitleJ. Pure Appl. Algebra

Courier Dover Publications

titleCategory theory in context

Springer, Heidelberg

booktitleNew structures for physics
seriesLecture Notes in Phys.
titleA survey of graphical languages for monoidal categories

titleThe formal theory of monads
journaltitleJournal of Pure and Applied Algebra

Birkhäuser/Springer, Cham

57R56 (18-02 18C20 18D10 57-02 57M27)
isbn978-3-319-49833-1; 978-3-319-49834-8
seriesProgress in Mathematics
titleMonoidal categories and topological field theory

math.CT, math.QA
Given a Hopf algebra in a symmetric monoidal category with duals, the
category of modules inherits the structure of a monoidal category with duals.
If the notion of algebra is replaced with that of monad on a monoidal
category with duals then Bruguieres and Virelizier showed when the category
of modules inherits this structure of being monoidal with duals, and this
gave rise to what they called a Hopf monad. In this paper it is shown that
there are good diagrammatic descriptions of dinatural transformations which
allows the three-dimensional, object-free nature of their constructions to
become apparent.

titleA diagrammatic approach to Hopf monads
journaltitleArabian Journal of Science and Engineering C - Theme
Issue "Interactions of algebraic and coalgebraic structures (theory and
applications)" December 2008; Vol. 33, Number 2C, 561-585.

* Can't use 'journaltitle' + 'journal'
* Can't use 'date' +

18D10, 18C15, 18C20, 18D05
We give a 3-categorical, purely formal argument explaining why on the
category of Kleisli algebras for a lax monoidal monad, and dually on the
category of Eilenberg-Moore algebras for an oplax monoidal monad, we always
have a natural monoidal structure. The key observation is that the 2-category
of lax monoidal monads in any 2-category
\(\mathcal D\) with finite
products is isomorphic to the 2-category of monoidal objects with oplax
morphisms in the 2-category of monads with lax morphisms in
\(\mathcal D\). We explain at
the end of the paper that a similar phenomenon occurs in many other

shortjournalJ. Pure Appl. Algebra
titleThe Formal Theory of Monoidal Monads
journaltitleJournal of Pure and Applied Algebra

* Can't use 'journaltitle' + 'journal'
* Can't use 'date' +