# Extreme HBL-like behavior of Markarian 421 and its two-zone photohadronic
interpretation

Sarira Sahu<EMAIL_ADDRESS>Instituto de Ciencias Nucleares,
Universidad Nacional Autónoma de México,
Circuito Exterior, C.U., A. Postal 70-543, 04510 Mexico DF, Mexico  Carlos E.
López Fortín<EMAIL_ADDRESS>Instituto de Ciencias
Nucleares, Universidad Nacional Autónoma de México,
Circuito Exterior, C.U., A. Postal 70-543, 04510 Mexico DF, Mexico  Isabel
Abigail Valadez Polanco<EMAIL_ADDRESS>Facultad de Ingeniería,
Universidad Autónoma de Yucatán,
Industrias No Contaminantes S/N, Sin Nombre de Col 27, Mérida, Yucatán, México
Subhash Rajpoot<EMAIL_ADDRESS>Department of Physics and
Astronomy, California State University,
1250 Bellflower Boulevard, Long Beach, CA 90840, USA

###### Abstract

Markarian 421 is the nearest high-energy peaked blazar and also is the first
extragalactic source to be detected in multi-TeV $\gamma$-rays. It has been
observed in multiwavelength for exceptionally long period of time with dense
monitoring and several major outbursts have been detected from this source. In
March 2010, the source was in a high state of activity and was observed in
multiwavelength by various telescopes for 13 consecutive days. During this
period the position of the synchrotron peak was found to be above $10^{17}$ Hz
and also the position of the second peak was shifted towards higher energy, a
signature of extreme HBL-like behavior. We observed that the standard
photohadronic model is inadequate to explain the observed spectra. However, a
recently proposed two-zone photohadronic model explains very well the GeV-TeV
flaring events observed by both MAGIC and VERITAS telescopes. From the
observation of the highest energy $\gamma$-ray event on MJD 55266 we also
estimated the minimum bulk Lorentz factor.

High energy astrophysics (739), Blazars (164), Gamma-rays (637), Relativistic
jets (1390), BL Lacertae objects (158)

mainReferences appendixReferences (Appendix)

##  1 Introduction

The spectral energy distributions (SEDs) of blazars are characterized by two
non-thermal peaks (Dermer & Schlickeiser, 1993). The first peak, in the
infrared to X-ray energy range, is produced by the synchrotron emission from
relativistic electrons in the jet. The second peak, in between X-ray to
$\gamma$-ray energy, is believed to be produced either from the synchrotron
Self-Compton (SSC) scattering of high-energy electrons with the low-energy
self-produced synchrotron photons in the jet (Maraschi et al., 1992; Murase et
al., 2012; Gao et al., 2013) or from the external Compton (EC) scattering with
external sources such as photons from the accretion disk or broad-line regions
(Sikora et al., 1994; Blazejowski et al., 2000). The model which explains
these two peaks is in general referred to as the leptonic model. Depending on
the position of the synchrotron peak, the blazars are classified into four
categories: low-energy peaked blazars (LBLs, $\nu_{peak}<10^{14}\,Hz$),
intermediate-energy peaked blazars (IBLs, $\nu_{peak}$ between $10^{14}\,Hz$
and $10^{15}\,Hz$), high energy-peaked blazars (HBLs, $\nu_{peak}$ between
$10^{15}\,Hz$ and $10^{17}\,Hz$) (Abdo et al., 2010) and extreme energy peaked
blazar (EHBL, $>10^{17}$ Hz) (Costamante et al., 2001). In particular, the
EHBLs are characterized by low luminosity and have limited variabilities which
make these objects difficult to detect (Costamante et al., 2018; Acciari et
al., 2020b).

So far, only a few EHBLs have been detected in the very high energy (VHE,
$>\,100$ GeV) regime and among them some well known objects are 1ES 0229+200,
1ES 0347-232, RGB J0710+591 and 1ES 1101-232 (Costamante et al., 2018). Among
all the EHBLs observed so far, the highest energy peak frequencies correspond
to 1ES 0229+200 with the synchrotron peak at $\nu^{peak}_{syn}\simeq 3.5\times
10^{19}$ Hz and the SSC peak at $\nu^{peak}_{IC}\simeq 1.5\times 10^{27}$ Hz
(Kaufmann et al., 2011) respectively. The nearest and the extensively studied
HBLs Markarian 421 (Mrk 421), Markarian 501 (Mrk 501) and 1ES 1959+650 have
also shown EHBL-like behavior with harder TeV spectra in which the synchrotron
and SSC peaks are shifted towards higher energy during some flaring episodes
(Ahnen et al., 2018; Acciari et al., 2020a; Aleksić et al., 2015). The EHBL-
like behavior of Mrk 501 was observed during two episodes of very-high energy
flaring events of May - July 2005 and June 2012. Also, during a
multiwavelength (MW) campaign of 1ES 1959+650 between 29th April to 21st
November 2016, the MAGIC telescopes observed multi-TeV flaring events during
the nights of 13th, 14th June and 1st July 2016 when the position of the
synchrotron peak was in EHBL range and exhibiting extreme HBL-like behavior.

It is very difficult to explain the shift of the second peak in the one-zone
leptonic model. However, it has been shown that by using unrealistic model
parameters, such as the large values of minimum electron Lorentz factor and
the bulk Lorentz factor (Ahnen et al., 2018; Acciari et al., 2020a) as well as
a low magnetic field, the shift can be explained. Alternative solutions have
also been proposed, such as: a two-zone leptonic model, IC scattering of the
electron with the cosmic microwave background, a spine-layer structured jet
model and variants of the hadronic model (Böttcher et al., 2008; Acciari et
al., 2019; Ahnen et al., 2018).

In previous studies, Sahu et al. (2019, 2020b) have used the photohadronic
model to explain the multi-TeV flaring from many HBLs very well. In all these
studies it is shown that the Fermi accelerated high energy protons interacting
with the seed SSC photons in the jet can explain the VHE spectrum very well.
The SSC photon flux $\Phi_{SSC}$ in this region is a perfect power-law given
as $\Phi_{SSC}\propto\epsilon_{\gamma}^{\beta}\propto E_{\gamma}^{-\beta}$ (in
units of $\mathrm{erg}\ \mathrm{cm^{-2}}\ \mathrm{s^{-1}}$), where
$\epsilon_{\gamma}$ is the seed photon energy, $E_{\gamma}$ is the energy of
the observed VHE gamma-ray and $\beta$ is the seed photon spectral index. The
value of $\beta$ lies in the range $0<\beta\leq 1$. However, recently, Sahu et
al. (2020a, 2021) have shown that this standard photohadronic scenario is
inadequate to explain the EHBL-like behavior of the flaring events of Mrk 501
and 1ES 1959+650. It was also shown that the seed SSC photon flux in the
kinematically allowed region for the $\Delta$-resonance production from
$p\gamma$ interaction is no more a single power-law due to the EHBL-like
behavior of the flaring events. So, the standard photohadronic model was
extended to include two different zones for the SSC flux. The zone-1 (low
energy region) has the conventional behavior having $0<\beta\leq 1$. But in
zone-2 (high energy region) the value of $\beta$ is different and in the range
$1<\beta\leq 1.5$. However, the spectral index $\alpha$ of Fermi accelerated
high energy protons in both the zones remain the same.

During the EHBL-like flaring events of Mrk 501 and 1ES 1959+650, the positions
of the synchrotron peak and the SSC peak were shifted towards the higher
energies. So, Sahu et al. (2020a, 2021) assumed that the transition from HBL
to EHBL-like behavior is due to the shift of the synchrotron peak and the SSC
peak in the SED towards higher energies. As a result of this shift, jet
parameters such as bulk Lorentz factor, blob size, and magnetic field may be
different from HBL, but the particle acceleration and emission mechanisms
remain the same. Therefore, the photohadronic process should still be the
dominant process in this changed environment. It is also argued that these
EHBL-like events are transient in nature and may repeat in the future.

During a MW campaign in 2010, flaring in GeV-TeV energy was observed in Mrk
421. The VHE activity started decreasing from a high flux to a normal flux
during the period from March 10 to March 22, i.e. for 13 consecutive days
(Aleksić et al., 2015). It was observed that the positions of both the
synchrotron peak and the SSC peak of Mrk 421 were shifted towards higher
energy values consistent with the EHBL classification of the object, were
similar to the flaring of Mrk 501 and 1ES1959+650 discussed above. This shift
in the peak positions towards higher energies is explained using electrostatic
acceleration mechanism (Zheng et al., 2021). Also recently it has been shown
that these VHE spectra of Mrk 421 can be explained better in the photohadronic
model than the one and the two-zone leptonic models (de León et al., 2021). As
the two-zone photohadronic model has successfully explained the EHBL-like
behavior of Mrk 501 and 1ES1959+650, we use this model once again to explain
the EHBL-like behavior of Mrk 421 observed during March 2010.

##  2 Flaring of Mrk 421

Mrk 421 is the nearest ($z=0.031$) well known HBL with a central black hole
mass $M_{H}\approx(2-9)\times 10^{8}M_{\odot}$ and is the first extragalactic
source to be detected in multi-TeV $\gamma$-rays (Punch et al., 1992). It is
also one of the fastest varying $\gamma$-ray sources. Since its discovery, the
object has been studied intensively through dedicated MW observations and
several major flares have been observed (Amenomori et al., 2003; Fossati et
al., 2008; Cui et al., 2005; Abeysekara et al., 2017; Mastichiadis et al.,
2013; Beck et al., 2020). Its average MW SED has been modelled with both
leptonic (Abdo et al., 2011; Banerjee et al., 2019) and hadronic (Cerruti et
al., 2015; Zech et al., 2017) models. During April-May 1994, a TeV/X-ray flare
was reported and on May 14/15 a correlation between the TeV emission and X-ray
flare was observed. Also, during a multiwavelength campaign in 2004, large
flares in X-ray by Rossi X-ray Timing Explorer (RXTE) and TeV flare by Whipple
10 m telescope were observed (Blazejowski et al., 2005). It is important to
note that during this period, the TeV flare had no coincident low energy
counterparts, and most significantly, the X-ray flux reached its peak 1.5 days
before the TeV flux during this outburst. A remarkable similarity between this
VHE flare and the orphan TeV flare in 1ES 1959+650 of 2002 (Krawczynski et
al., 2004) and a similar variation pattern in their X-ray emission was
established. Exceptionally long and dense monitoring in MW of Mrk 421 has been
undertaken since 2009 to understand the temporal evolution of the SED. Mrk 421
was in active state during a MW campaign in 2010 and an extraordinary flare
$\sim 27$ Crab Unit (c.u.) above 1 TeV was first detected by VERITAS
telescopes on February 16 and a follow-up observation was undertaken by the
HESS telescopes (Tluczykont, 2010) for four subsequent nights. So far, this is
the highest flare ever observed from Mrk 421\. Also, flaring in VHE gamma-ray
was observed for 13 consecutive days from March 10 (MJD 55265) to March 22
(MJD55277). Again, during a 6-month long multi-instrument campaign (Banerjee
et al., 2019), a large VHE flare about 16 times larger than the usual one, was
observed by MAGIC telescopes on 25th of April 2014 and soon it was followed-up
by XMM-Newton and VERITAS. Several of the above multi-TeV flaring events from
Mrk 421 have been explained extremely well by the standard photohadronic
scenario (Sahu et al., 2016, 2018, 2019).

##  3 Two-zone photohadronic model

The photohadronic model is very successful in explaining the multi-TeV flaring
events from many HBLs (Sahu, 2019; Sahu et al., 2019). The epoch of VHE
flaring is explained by assuming the formation of a double jet structure along
the same axis: an inner jet of size $R^{\prime}_{f}$ and photon density
$n^{\prime}_{\gamma,f}$ buried under an outer jet of size $R^{\prime}_{b}$ and
photon density $n^{\prime}_{\gamma}$ where $R^{\prime}_{f}<R^{\prime}_{b}$ and
$n^{\prime}_{\gamma,f}>n^{\prime}_{\gamma}$. Here the notation ′ implies
comoving frame. The internal and the external jets are moving with almost the
same bulk Lorentz factor $\Gamma_{in}\simeq\Gamma_{ext}\simeq\Gamma$. Also,
their common Doppler factor is $\mathcal{D}$ (for blazars
$\Gamma\simeq\mathcal{D}$). The geometrical structure of the jet and the
detailed description of the photohadronic model is given in Fig. 1 of Sahu
(2019).

In the inner jet region the Fermi accelerated protons with a differential
power-law spectrum $dN/dE_{p}\propto E^{-\alpha}_{p}$ ($\alpha\geq 2$)
interact with the background SSC seed photons to produce the
$\Delta$-resonance and its subsequent decay gives $\gamma$-rays and neutrinos
via the decay of neutral pion and charged pion respectively. As the inner
region is hidden, there is no way to directly estimate the photon density
there. Thus we assume a simple scaling behavior of the photon densities in the
inner and the outer jet regions satisfying the criteria

$\frac{n^{\prime}_{\gamma,f}(\epsilon_{\gamma,1})}{n^{\prime}_{\gamma,f}(\epsilon_{\gamma,2})}\simeq\frac{n^{\prime}_{\gamma}(\epsilon_{\gamma,1})}{n^{\prime}_{\gamma}(\epsilon_{\gamma,2})}.$
(1)

The above equation implies that the ratio of photon densities at two different
background energies $\epsilon_{\gamma,1}$ and $\epsilon_{\gamma,2}$ in the
inner and the outer jet regions are almost the same. By using the relation in
Eq. (1), we can express the inner photon density in terms of the observed
flux. The kinematical condition (Sahu, 2019) to produce the $\Delta$-resonance
is,

$E_{p}\epsilon_{\gamma}=\frac{0.32\ \Gamma{\mathcal{D}}}{(1+z)^{2}}\
\mathrm{GeV^{2}},$ (2)

where $z$ is the redshift of the object. The observed VHE $\gamma$-ray energy
$E_{\gamma}$ and the proton energy $E_{p}$ are related through
$E_{\gamma}\simeq 0.1\,E_{p}$. In previous works by Sahu (2019) it is
explicitly shown that for the production of $\Delta$-resonance, the value of
$\epsilon_{\gamma}$ always lies in the lower tail region of the SSC band and
for HBLs the flux in this region is a perfect power-law.

The VHE $\gamma$-rays en route to Earth undergo energy dependent attenuation
by the extragalactic background light (EBL) through electron-positron pair
production and this pair-production process not only attenuates the absolute
flux but also significantly changes the shape of the VHE spectra. The
measurement of the EBL is very difficult due to the uncertainties in the
contribution of zodiacal light (Hauser & Dwek, 2001; Chary & Pope, 2010) and
consequently galaxy counts result in a lower limit since the number of
unresolved sources are unknown (Pozzetti & Madau, 2001). Several approaches
with different degrees of complexity have been developed to calculate the EBL
density as a function of energy for different redshifts (Dominguez et al.,
2011; Inoue et al., 2013; Franceschini et al., 2008; Stecker et al., 2006,
2016). For the present analysis we use the EBL model of Franceschini et al.
(2008), which is consistent with a EBL peak flux of $\sim 15\
\mathrm{nW}\mathrm{n^{-2}}\ \mathrm{sr^{-1}}$ at $1.4\,\mathrm{\mu m}$ and is
also widely used in other leptonic and hybrid models (Aleksić et al., 2015;
Ahnen et al., 2017). Taking into account the EBL contribution the observed VHE
flux from the sources can be given as (Sahu, 2019)

$F_{\gamma,obs}(E_{\gamma})=A_{\gamma}\Phi_{SSC}(\epsilon_{\gamma})\left(\frac{E_{\gamma}}{TeV}\right)^{-\alpha+3}\,e^{-\tau_{\gamma\gamma}(E_{\gamma},z)},$
(3)

where $A_{\gamma}$ is a dimensionless constant and $\tau_{\gamma\gamma}$ is
the optical depth for the $\gamma\gamma\rightarrow e^{+}e^{-}$ process which
depends on $E_{\gamma}$ and the redshift. In the standard photohadronic model,
$\Phi_{SSC}\propto E^{-\beta}_{\gamma}$ and putting this in Eq. (3) the flux
can be written as

$F_{\gamma,obs}(E_{\gamma})=F_{0}\left(\frac{E_{\gamma}}{TeV}\right)^{-\delta+3}\,e^{-\tau_{\gamma\gamma}(E_{\gamma},z)}=F_{\gamma,in}(E_{\gamma})\,e^{-\tau_{\gamma\gamma}(E_{\gamma},z)}.$
(4)

The normalization constant $F_{0}$ is deduced from the observed VHE spectrum.
Thus the spectral index $\delta=\alpha+\beta$ is the only free parameter in
this model and $F_{\gamma,in}$ is the intrinsic VHE $\gamma$-ray flux. In
previous studies (Sahu et al., 2020a, 2021) it was shown that the EHBL-like
behavior of the flaring epoch can not be explained by the canonical
photohadronic model. In order to remedy this defect the tail region of the SSC
band which is responsible for the production of the $\Delta$-resonance is
assumed to have two different power-laws with spectral indices $\beta_{1}$
(lower part of the tail region) and $\beta_{2}$ (upper part of the tail
region). Thus the photon flux is expressed as

$\Phi_{SSC}\propto\left\\{\begin{array}[]{cr}E^{-\beta_{1}}_{\gamma},&\quad\mathrm{100\,GeV\,\lesssim
E_{\gamma}\lesssim E^{intd}_{\gamma}}\\\
E^{-\beta_{2}}_{\gamma},&\quad\mathrm{E_{\gamma}\gtrsim E^{intd}_{\gamma}}\\\
\end{array}\right..$ (5)

Here $\beta_{1}\neq\beta_{2}$ and $E^{intd}_{\gamma}$ is an energy scale
around which the transition takes place and its value can be fixed from the
individual flaring spectrum. We suppose that the change in the value of the
spectral index from $\beta_{1}$ to $\beta_{2}$ around $E^{intd}_{\gamma}$
could be due to the effect of magnetic field on the energy loss process of the
high energy electrons to the SSC photons or it could be some new physics.
However, we do not dwell on the transition between these two zones as the
exact physical mechanism behind it is unclear at this stage. The value of
$E^{intd}_{\gamma}$ corresponds to a value of $\epsilon_{\gamma}$ in the SSC
band. This separates $\Phi_{SSC}$ into two distinct regions. By inserting Eq.
(5) into Eq. (3), the observed VHE spectrum can be expressed as

$F_{\gamma,obs}=e^{-\tau_{\gamma\gamma}}\times\begin{cases}F_{1}\,\left(\frac{E_{\gamma}}{TeV}\right)^{-\delta_{1}+3},&\quad\mathrm{100\,GeV\,\lesssim
E_{\gamma}\lesssim E^{intd}_{\gamma}}\,\,\,\,\,(\text{zone-1})\\\
F_{2}\,\left(\frac{E_{\gamma}}{TeV}\right)^{-\delta_{2}+3},&\quad\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{E_{\gamma}\gtrsim
E^{intd}_{\gamma}}\,\,\,\,\,(\text{zone-2})\end{cases}.$ (6)

In this two-zone photohadronic model $F_{1}$ and $F_{2}$ are the normalization
constants and the spectral indices $\delta_{i}=\alpha+\beta_{i}$ ($i=1,2$) are
the free parameters to be adjusted by fitting the model to the observed VHE
spectrum of the EHBL. The proton spectral index is $\alpha\geq 2.0$, and we
use the generally accepted value of $\alpha=2.0$ (Dermer & Schlickeiser,
1993). This automatically constrains the value of $\beta_{i}$ for a given
$\delta_{i}$. However, it is important to mention that the value of
$\delta_{1}$ (zone-1) allows one to characterize the flaring state for a given
observation without depending explicitly on the modeling of the SSC region. In
other words, in the photohadronic scenario, the low energy region of the VHE
spectrum decides the flaring state of the outburst as the value of
$\delta_{1}$ always lies in the range $2.5\leq\ \delta\leq 3.0$ (Sahu et al.,
2019).

To fit the observed data for each VHE spectrum, we use a standard optimization
of the parameter space through the mean squared error (MSE) with respect to
the central values of the data points, and choose the best pair of
($F_{0},\delta$) by maximizing the Pearson chi-squared ($\chi^{2}$) statistic
and calculating the corresponding statistical significance.

The observed VHE spectrum of each day is fitted in the following manner. For
the low energy regime of the spectrum we take $2.5\leq\delta_{1}\leq 3.0$ and
fit it with different values of $F_{1}$. Similarly in the high energy regime
we take $3.1\leq\delta_{2}\leq 3.5$ and fit the spectrum for different values
of $F_{2}$. From both the zones we select the best fits by calculating their
statistical significance. The crossing point of these two curves gives the
value of $E^{intd}_{\gamma}$. Also, we have to make sure that these two curves
intersect only once.

Figure 1: The VHE flaring of MJD 55265 observed by MAGIC and VERITAS
telescopes are shown in (a). Also the VHE event of MJD 55266 observed by MAGIC
is shown in (b). Along with the two-zone photohadronic fit (blue for zone-1
and red for zone-2) the one-zone SSC fit for both days are shown for
comparison. In all the figures the normalization constants $F_{i}$ ($i=1,2$)
are defined in units of $\mathrm{erg\,cm^{-2}\,s^{-1}}$. The fitted
photohadronic model curves in zone-1 and zone-2 are extended by dashed line of
their respective color beyond the transition energy just to show their likely
behavior.

Figure 2: The VHE flaring data of MJD 55267 observed by VERITAS (a) and of MJD
55268 by both MAGIC and VERITAS (b) are fitted with two-zone photohadronc
mdoel which are shown along with the one-zone SSC fit for comparison.

Figure 3: The VHE spectra of MJD 55269 (a) and of MJD 55270 (b) observed by
both MAGIC and VERITAS telescopes are fitted with two-zone photohadronic
model. We have also shown the one-zone SSC fit for comparison.

Figure 4: The VHE spectra of MJD 55271 (a) and of MJD 55272 (b) observed by
VERITAS telescopes are fitted with two-zone photohadronic model and shown
along with their respective one-zone SSC fit for comparison.

Figure 5: The VHE flaring data of MJD 55273 (a), MJD 55274 (b), MJD 55276 (c),
and MJD 55277 (d) observed by the Cherenkov telescopes MAGIC and VERITAS are
fitted with two-zone photohadronic model. For comparison, we have also plotted
the one-zone SSC fits.

##  4 Results

During the MW campaign period of 2010, flaring in VHE gamma-rays was observed
for 13 consecutive days from March 10 (MJD 55265) to March 22 (MJD55277).
Initially the VHE flux was high ($\sim 2$ C.U.) and slowly the flux decreased
to a typical value of $\sim 0.5$ C.U. towards the end of the flaring epoch. In
the VHE energy band, MAGIC, VERITAS and Whipple telescope systems observed the
flaring events. The MAGIC telescopes made 11 observations and the exposure
time for each period was about 10 to 80 min, amounting to a total of 4.7 h of
good quality data. However, due to bad weather, data recorded on MJD 55272 and
MJD 55275 were removed from the MW observations.

The VERITAS telescopes observed the flaring of Mrk 421 on MJD 55260, 55265,
55267-55274 with a 10 min run time each day. The observations were performed
at zenith angles $18^{\circ}-23^{\circ}$ to benefit from the lowest possible
energy threshold of the events. Also, the Whipple 10 m telescope performed ten
observations in ON/OFF and TRK (tracking) modes (Pichel, 2009), lasting from
one to six hours each on MJD 55267-55271 and MJD 55273-55277.

The most important aspect of these 13 days observation is that the synchrotron
peak and the SSC peaks were displaced towards higher energies. Particularly,
the synchrotron peak frequency was above $10^{17}$Hz and according to the
classification scheme of the BL Lac objects, it belongs to EHBL category even
though Mrk 421 is well known as an HBL. As discussed in the introduction,
similar behavior has also been observed from Mrk 501 and 1ES 1959+650 which
are well known HBLs, and there spectra are explained very well with the two-
zone photohadronic scenario (Sahu et al., 2020a, 2021). Thus using the two-
zone photohadronic model, the 13 days VHE flaring events observed by MAGIC and
VERITAS are analysed here. For comparison, along with our results the one-zone
SSC fits are also shown (Aleksić et al., 2015). In this analysis, we ignore
the statistical significance of a fit when only three or less observed data
points are available.

The VHE spectrum of MJD 55265 (10 March 2010) observed by both MAGIC and
VERITAS is in the energy range $0.1\,TeV\leq E_{\gamma}\leq 4.3\,TeV$ and the
synchrotron peak is at $\epsilon_{\gamma}\sim 4.6\times 10^{17}$ Hz which
clearly shows its EHBL behavior. Using the two-zone photohadronic model, we
fit the low-energy part of the spectrum (zone-1) (for
$E^{intd}_{\gamma}\lesssim 1$ TeV) very well with $\delta_{1}=2.7$ and its
corresponding flux normalization is $F_{1}=2.1\times 10^{-10}\
\mathrm{erg\,cm^{-2}\,s^{-1}}$, with a statistical significance of 98%
($\chi^{2}=0.0008$). Also the best fit to the high energy part of the VHE
spectrum (zone-2) is obtained for $F_{2}=2.0\times 10^{-10}\
\mathrm{erg\,cm^{-2}\,s^{-1}}$ and $\delta_{2}=3.2$, with a statistical
significance of 96% ($\chi^{2}=0.0366$). These are shown in Fig. 1(a) along
with the one-zone SSC model of (Aleksić et al., 2015) for comparison. It can
be seen that our model fits better the data than the one-zone SSC model. The
zone-1 and zone-2 have their respective seed SSC flux spectral indices
$\beta_{1}=0.7$ and $\beta_{2}=1.2$ respectively. The VERITAS data behave
differently in its highest energy when the flux has suddenly gone up and is
non compatible with the two-zone model. However, previously it was shown that
this can be fitted well with conventional photohadronic model with
$F_{0}=2.1\times 10^{-10}\ \mathrm{erg\,cm^{-2}\,s^{-1}}$ and $\delta=2.6$
corresponding to a very high emission state (Sahu et al., 2019). As the MAGIC
and VERITAS observation windows are seven hours apart, it is possible that
intra-night variability might be responsible for this difference.

On MJD 55266 the flaring event was observed by MAGIC telescopes only and the
spectrum is in the energy range $0.1\,TeV\leq E_{\gamma}\leq 4.4$ TeV. The bow
shape spectrum on this day (MJD 55266) can not be fitted by a single power-
law. The zone-1 ($E^{intd}_{\gamma}\lesssim 1$ TeV) is fitted with
$\delta_{1}=2.7$ and $F_{1}=2.5\times 10^{-10}\ \mathrm{erg\,cm^{-2}\,s^{-1}}$
having a statistical significance of 99% ($\chi^{2}=0.0189$). Similarly, the
best fit to zone-2 is obtained for $F_{2}=2.45\times 10^{-10}\
\mathrm{erg\,cm^{-2}\,s^{-1}}$ and $\delta_{2}=3.5$, with a statistical
significance of 82% ($\chi^{2}=0.0008$) and on this day the synchrotron peak
is at $\epsilon_{\gamma}\sim 5.0\times 10^{17}$ Hz. Our fit and the one-zone
SSC results are compared in Fig. 1(b). It can be seen that the leptonic model
is unable to fit the data in the low energy regime.

The only data available on MJD 55267 are from VERITAS and the spectrum is very
well fitted with the two-zone photohadronic scenario with the transition
energy $E^{intd}_{\gamma}\simeq 1$ TeV, which is shown in Fig. 2(a). The best
fitted values for zone-1 are $F_{1}=2.5\times 10^{-10}\
\mathrm{erg\,cm^{-2}\,s^{-1}}$ and $\delta_{1}=2.7$ (statistical significance
of 83%, $\chi^{2}=0.0538$) while for zone-2 we have $F_{2}=2.5\times 10^{-10}\
\mathrm{erg\,cm^{-2}\,s^{-1}}$, and $\delta_{2}=3.3$. The SSC model of Aleksić
et al. (2015) predicts a VHE spectrum which falls significantly faster than
the observed spectrum.

Both MAGIC and VERITAS observed the flaring event of MJD 55268 and the
observed spectra are in the energy range $0.18\,TeV\leq E_{\gamma}\leq 2.1$
TeV. The MAGIC spectrum is well fitted in zone-1 with the spectral index
$\delta_{1}=2.7$ and the normalization factor $F_{1}=1.3\times 10^{-10}\
\mathrm{erg\,cm^{-2}\,s^{-1}}$ and here $E^{intd}_{\gamma}\lesssim 0.8$ TeV .
There are not many data points in zone-2, so we fitted this region with
$\delta_{2}=3.4,3.5$ and both values fit very well to the observed data.
However, above 2 TeV there is a minor difference between these fits. Also our
fit to the spectrum differs substantially from the one-zone SSC fit as shown
in Fig. 2(b).

The flaring event of MJD 55269 is very similar to the one observed on MJD
55268 and a good fit to the spectrum is achieved with $F_{1}=2.0\times
10^{-10}\ \mathrm{erg\,cm^{-2}\,s^{-1}}$ and $\delta_{1}=2.7$ for zone-1
(statistical significance of 96%, $\chi^{2}=0.0790$), while $F_{2}=1.8\times
10^{-10}\ \mathrm{erg\,cm^{-2}\,s^{-1}}$, $\delta_{2}=3.2$ are for zone-2 as
shown in Fig. 3(a).

On MJD 55270 the spectrum in the low energy regime i.e. zone-1 is flatter than
previous days which implies that the flaring event is moving towards low
emission state. However, because of less number of data points we fitted this
region with $\delta_{1}=2.7,2.8$ and both fit very well with very similar
normalization constants ($F_{1}=1.32\times 10^{-10}\
\mathrm{erg\,cm^{-2}\,s^{-1}}$ and $1.16\times 10^{-10}\
\mathrm{erg\,cm^{-2}\,s^{-1}}$). On the other hand, zone-2 is fitted well by
$F_{2}=8.7\times 10^{-11}\ \mathrm{erg\,cm^{-2}\,s^{-1}}$, $\delta_{2}=3.3$
(statistical significance of 87%, $\chi^{2}=0.0339$) as shown in Fig. 3(b).
One-zone SSC model also fits well to the spectrum. In the high energy regime,
however, it falls faster than our fit.

The flaring event of MJD 55271 was only observed by VERITAS in the energy
range $0.25\,TeV\leq E_{\gamma}\leq 2.6$ TeV. Below the transition energy
$E^{intd}_{\gamma}\lesssim 0.9$ TeV, the zone-1, we can fit well the spectrum
with $F_{1}=(1.7-1.62)\times 10^{-10}\ \mathrm{erg\,cm^{-2}\,s^{-1}}$ and
$\delta_{1}=(2.8-2.9)$. The zone-2 is fitted with $F_{2}\simeq 1.55\times
10^{-10}\ \mathrm{erg\,cm^{-2}\,s^{-1}}$ and $\delta_{2}=3.3$. The one-zone
SSC model fit is also shown in Fig. 4(a) to compare with our two-zone
photohadronic model fit.

For the observations from MJD 55272 to MJD 55277, with the exception MJD 55275
when there was no observation, the value of $E^{intd}_{\gamma}$ is very low
and in a narrow range of $0.25\,TeV\lesssim E^{intd}_{\gamma}\lesssim 0.4$
TeV. Also the spectrum for each individual day is flat which is best fitted
with either $\delta_{1}=2.9$ or $3.0$ and the statistical significance for
these fits ranges from 85% to 95%. This value of $\delta_{1}$ signifies that
the flaring of Mrk 421 has reached the low emission state and at the same time
the seed photon flux in the SSC band has reached the maximum value of
$\beta_{1}\simeq 1.0$. Also the maximum photon flux in zone-1 has
substantially decreased compared to the first day of the observation. The
zone-2 for these five days are also fitted very well with
$3.2\lesssim\delta_{2}\lesssim 3.5$ which correspond to the seed photons in
the low energy tail region of the SSC band and their fluxes having spectral
indices in the range $1.2\lesssim\beta_{2}\lesssim 1.5$. The flaring events of
all these days along with the one-zone SSC fits are shown in Fig. 4(b) and
Figs. 5(a), (b), (c) and (d) respectively.

Initially the VHE flaring was in high emission state and slowly it went down
to low emission state by the end of the flaring period. This transition was
accompanied by a gradual decrease in the transition energy $E^{intd}_{\gamma}$
from $\sim 1$ TeV to $\sim 0.25$ TeV and consequently zone-2 became wider and
spread into zone-1 making the latter region narrower. We recall that the
transition in the VHE spectrum took place due to the change in the seed SSC
photon flux. So, physically what it means is that initially the photon flux in
the tail region of the SSC band which is responsible for the
$\Delta$-resonance production had two distinct energy dependencies as shown in
Eq. (5) (lower part with $\epsilon^{\beta_{1}}_{\gamma}$ and upper part with
$\epsilon^{\beta_{2}}_{\gamma}$). Slowly the lower part drifted into the upper
part by squeezing the former region into a narrow strip.

On MJD 55266, the maximum observed $\gamma$-ray energy was $E_{\gamma}\simeq
4.4$ TeV which is produced from the interaction of Fermi accelerated proton of
energy $E_{p}\simeq 44$ TeV in the jet with the photons in the lower part of
the SSC spectrum. In the two-zone SSC model of Aleksić et al. (2015) for this
day the SSC spectrum starts around $\epsilon_{\gamma}\sim 7\times 10^{21}$ Hz
(Figure 8(b) of Aleksić et al. (2015)). For this value of $\epsilon_{\gamma}$,
the minimum value of the bulk Lorentz factor estimated in our model is
$\Gamma\simeq 21$. Similar or smaller values for the bulk Lorentz factor are
obtained for other days. On the other hand, in the one-zone SSC model (Figure
8(a) of Aleksić et al. (2015)), the SSC spectrum starts above $2\times
10^{21}$ Hz and for this value of $\epsilon_{\gamma}$ we get minimum
$\Gamma\simeq 35$ which is too high.

##  5 Discussion

Mrk 421 is the nearest HBL and is the first extragalactic source to be
observed in VHE. Since then it has undergone several episodes of multi-TeV
flaring. Mrk 421 is also the first source observed in multiwavelength for
exceptionally long period of time with dense monitoring. The VHE flaring
epochs of this source during all its previous observations were consistently
found to be in the HBL category which corresponds to the synchrotron peak in
the frequency range $10^{15}-10^{16}$ Hz. However, during the 13 consecutive
days observations undertaken during March 2010 the source was in the state of
high activity with a changed spectral behavior and the synchrotron as well as
the second peak shifted towards higher energies. The synchrotron peak shifted
above $10^{17}$ Hz, implying that the blazar has undergone a shift from HBL to
EHBL. Similar situations are observed in HBLs Mrk 501 and 1ES 1959+650, which
were explained very well by the two-zone photohadronic scenario. For Mrk 421,
we have used the same two-zone photohadronic model to interpret the 13
consecutive days of flaring in VHE and we find that the individual flaring
events are explained very well by our model. This shows once more the success
of two-zone photohadronic model in explaining the extreme HBL nature of the
VHE flaring events. We note that when the flaring in VHE was first observed on
March 10, 2010 (MJD56265), the source was in high emission state and it
continued to be in high emission state for about a week. Then it started
degrading to low emission state when the flux was low and also the spectrum in
zone-1 was almost flat. During the first week of the observation, the
transition energy $E^{intd}_{\gamma}\simeq 1$ TeV when the flaring in the
zone-1 was in high emission state. However, from MJD55272, the transition
energy $E^{intd}_{\gamma}$ started decreasing and reached a minimum value of
$0.25$ TeV on MJD55274 and for rest of the days the zone-1 was in low emission
state. The shift in the $E^{intd}_{\gamma}$ from $\sim 1$ TeV to $\lesssim
0.25$ TeV shows that towards the end of the VHE flaring, the zone-2 had
dominated almost the whole spectrum, implying that the seed photon flux in SSC
region has changed from a flatter behavior ($\beta_{1}\simeq 0.7$) to a
steeper ($\beta_{2}\simeq 1.5$) one. This change in spectral behavior, in
principle, can be detected by simultaneous observation of the SED in the low
energy tail region of the SSC band and in the VHE band during the EHBL-like
outburst of the source. We have also estimated the minimum bulk Lorentz
factor, obtaining a reasonable value of $\Gamma\approx 21$. Further analysis
of EHBL and EHBL-like transient behaviors will help solidify the evidence for
the two-zone photohadronic origin of these phenomena.

We are thankful to Luis H. Castañeda Hernandez for many useful discussions.
The work of S.S. is partially supported by DGAPA-UNAM (Mexico) Project No.
IN103019. Partial support from CSU-Long Beach is gratefully acknowledged.

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a discussion on Grothendieck topologies and sheaves in the $\infty$-setting
see [Lur09, Section 6.2.2.].

######  Definition 6.1.4.

Sheaves of anima
Let $\mathcal{C}$ be a site with finite limits in the usual categorical
setting viewed as $\infty$-category. A sheaf (of anima) is a presheaf $F$ of
anima such that for all coverings $(f_{i}\colon X_{i}\rightarrow X)_{i}$ the
map of anima

$\displaystyle
F(X)\rightarrow\lim(\prod_{i}F(X_{i})\rightrightarrows\prod_{i,j}F(X_{i}\times_{X}X_{j})\begin{subarray}{c}\rightarrow\\\\[-10.00002pt]
\rightarrow\\\\[-10.00002pt]
\rightarrow\end{subarray}\prod_{i,j,k}F(X_{i}\times_{X}X_{j}\times_{X}X_{k})\dots),$

where the limit is taken over an infinite diagram, is an equivalence. We
denote the $\infty$-category of all sheaves of anima on $\mathcal{C}$ by
$\mathcal{S}\mathrm{h}(\mathcal{C})$.

######  Definition 6.1.5.

Condensed anima
The $\infty$-category
$\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$ of condensed anima
is defined as the $\infty$-category of functors

$\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc}^{\mathrm{op}}\rightarrow\mathcal{A}\mathrm{ni}$

in $\mathcal{PS}\mathrm{h}(\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc})$
taking finite disjoint unions to finite products. As for condensed sets this
is obtained by restricting sheaves on the site of profinite sets to
$\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc}$.

######  Remark 6.1.6.

Condensed objects
More generally, one can define for some $\infty$-category $\mathcal{D}$
admitting certain limits and colimits the category of condensed objects of
$\mathcal{D}$ denoted by $\mathcal{C}\mathrm{ond}(\mathcal{D})$. Its objects
are, according to the definitions above, given by contravariant functors from
extremally disconnected profinite sets to $\mathcal{D}$ that take finite
disjoint unions to finite products. As for condensed sets, this definition has
some cardinality issues. See [Sch19a, Definition 11.7] for a precise
definition avoiding these issues.

######  Lemma 6.1.7.

Every presheaf
$F\colon(\mathcal{S}\mathrm{et}^{\mathrm{fin}})^{\mathrm{op}}\rightarrow\mathcal{A}\mathrm{ni}$
sending finite disjoint unions to finite products can be extended to a unique
condensed anima preserving cofiltered limits.

###### Proof.

The $\infty$-category of finite sets satisfies the conditions of Proposition
6.1.2. Thus, we obtain a unique presheaf
$\widetilde{F}\in\mathcal{PS}\mathrm{h}(\mathcal{P}\mathrm{ro}(\mathcal{C}))$
by the formula

$X=\\{X_{i}\\}_{i\in I}\rightarrow\mathrm{colim}_{i\in
I^{\mathrm{op}}}F(X_{i})$

preserving cofiltered limits. One has $X\amalg Y=\\{X_{i}\amalg
Y_{j}\\}_{(i,j)\in I\times J}$ for a disjoint union of profinite sets by
Corollary 5.1.11. Combining this with the assumption of $F$ sending finite
disjoint unions to finite products and the fact that filtered colimits commute
with finite products in $\mathcal{A}\mathrm{ni}$ by Remark 4.2.4, one can see
that $\widetilde{F}$ sends finite disjoint unions to finite products as well.
Hence, its restriction to
$\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc}^{\mathrm{op}}$ defines a
condensed anima. ∎

###  6.2 Discrete condensed anima

In Definition 4.2.8 we already defined
$\mathcal{A}\mathrm{ni(}\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}\mathrm{)}$
as the full-$\infty$-subcategory of

$\mathrm{Fun}(\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc}^{\mathrm{op}},\mathcal{A}\mathrm{ni}),$

generated under sifted colimits by the Yoneda image of
$\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc}$. The results of this section
give rise to a more precise description of these functors. Furthermore,
Theorem 6.3.1 shows how
$\mathcal{A}\mathrm{ni(}\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}\mathrm{)}$
unites topological and homotopical structures at the same time.

######  Lemma 6.2.1.

Let $\mathcal{C}$ be an $\infty$-category and
$v_{\mathcal{C}}\colon\mathcal{S}\mathrm{et}^{\mathrm{fin}}\rightarrow\mathcal{C}$
a fully faithful functor. To every object $A\in\mathcal{C}$, one can assign a
presheaf of anima

$A^{\mathrm{disc}}\colon\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et}^{\mathrm{fin}})^{\mathrm{op}}\rightarrow\mathcal{A}\mathrm{ni}$

preserving cofiltered limits in
$\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et^{fin}})$ and given by the
formula

$\displaystyle X=\\{X_{i}\\}_{i\in I}\mapsto\mathrm{colim}_{i\in
I^{\mathrm{op}}}\mathrm{Hom}_{\mathcal{C}}(v_{\mathcal{C}}(X_{i}),A),$ (6.1)

where the colimit is filtered.

###### Proof.

The presheaf is the extension of the presheaf of anima

$\mathrm{Hom}_{\mathcal{C}}(v_{\mathcal{C}}(\cdot),A)\colon(\mathcal{S}\mathrm{et}^{\mathrm{fin}})^{\mathrm{op}}\rightarrow\mathcal{A}\mathrm{ni}$

to $\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et^{fin}})$ as given in Lemma
6.1.3. With regards to Proposition 6.1.2, it preserves cofiltered limits in
$\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et^{fin}})$. ∎

######  Proposition 6.2.2.

If additionally $\mathcal{C}$ admits finite coproducts and $v_{\mathcal{C}}$
preserves this kind of coproducts, the restriction of $A^{\mathrm{disc}}$ to
the subcategory $\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc}^{\mathrm{op}}$
defines a condensed anima.

###### Proof.

As $v_{\mathcal{C}}$ preserves finite coproducts by assumption and one has the
functorial equivalence

$\mathrm{Hom}_{\mathcal{C}}(\mathrm{colim}_{i\in I}B_{i},A)=\mathrm{lim}_{i\in
I}\mathrm{Hom}_{\mathcal{C}}(B_{i},A)$

of hom-spaces, the presheaf of anima
$\mathrm{Hom}_{\mathcal{C}}(v_{\mathcal{C}}(\cdot),A)\colon(\mathcal{S}\mathrm{et}^{\mathrm{fin}})^{\mathrm{op}}\rightarrow\mathcal{A}\mathrm{ni}$
sends finite disjoint unions to finite products. By restriction we just obtain
the construction of Lemma 6.1.7. ∎

We will use the results above to define a condensed anima associated to an
anima. In Proposition 7.2.1 we will obtain by the same formula a condensed
anima associated to a pro-anima.

######  Definition 6.2.3.

Discrete condensed anima
Let $\mathcal{C}\coloneqq\mathcal{A}\mathrm{ni}$ be the category of animated
sets. For an anima $A$ we call the condensed anima obtained by Proposition
6.2.2 the discrete condensed anima attached to $A$ and denote it by
$\underline{A}^{\mathrm{disc}}$. The functor

$\underline{(\cdot)}^{\mathrm{disc}}\colon\mathcal{A}\mathrm{ni}\rightarrow\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)},\
A\mapsto\underline{A}^{\mathrm{disc}}$

is called discrete functor or constant sheaf functor.

We have another functor involving the categories $\mathcal{A}\mathrm{ni}$ and
$\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$ which is nicely
connected to the discrete functor as we will see.

######  Definition 6.2.4.

Global sections functor
The global sections functor on condensed anima is defined as the functor

$\displaystyle\Gamma\colon\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}\rightarrow\mathcal{A}\mathrm{ni},\
F\mapsto F(\ast)$ (6.2)

sending a condensed anima to its evaluation on the final object $\ast$ of
$\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et^{fin}})$.
(See Definition 7.1.1 for a precise definition of final objects in the
$\infty$-setting).

Recall that every discrete condensed anima is by definition and Proposition
6.1.2 (a restriction of) a left Kan extension along
$\iota_{\mathcal{S}\mathrm{et^{fin}}}\colon(\mathcal{S}\mathrm{et^{fin}})^{\mathrm{op}}\rightarrow\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et}^{\mathrm{fin}})^{\mathrm{op}}$.
The following proposition can be proven in an abstract way by applying the
fact that forming the left Kan extension along some morphism $G$ is precisely
the left adjoint $\infty$-functor of precomposition with this $G$. Note the
subsequent lemma for a reference of this fact. Nevertheless, in the proof of
the proposition we want to explicitly define how the claimed adjunction of the
global sections functor and the discrete functor can be obtained.

######  Proposition 6.2.5.

The global sections functor provides a right adjoint to the discrete functor,
i.e. one has a functorial equivalence of hom-spaces

$\displaystyle\mathrm{Hom}_{\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}}(\underline{A}^{\mathrm{disc}},F)\simeq\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(A,F(\ast))$
(6.3)

for all $A\in\mathcal{A}\mathrm{ni}$ and
$F\in\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$.

###### Proof.

First, we want to show the equivalence

$\displaystyle\mathrm{Hom}_{\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}}(\underline{A}^{\mathrm{disc}},F)\simeq\mathrm{Hom}_{\mathcal{PS}\mathrm{h}(\mathcal{S}\mathrm{et^{fin}})}(\overline{A},F|_{\mathcal{S}\mathrm{et^{fin}}})$
(6.4)

where
$\overline{A}\coloneqq\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(v_{\mathcal{S}\mathrm{et^{fin}}}(\cdot),A)$
is the representable functor of $A\in\mathcal{A}\mathrm{ni}$ restricted to
$\mathcal{S}\mathrm{et^{fin}}$.
In order to do so, we want to show that for every
$F\in\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$, any morphism
$h\colon\overline{A}\rightarrow F|_{\mathcal{S}\mathrm{et^{fin}}}$ corresponds
to a unique morphism $\underline{A}^{\mathrm{disc}}\rightarrow F$ of condensed
anima, s.t. restriction to $\mathcal{S}\mathrm{et^{fin}}$ yields the morphism
we started with. Note that
$\underline{A}^{\mathrm{disc}}|_{\mathcal{S}\mathrm{et^{fin}}}$ is just given
by the representable functor
$\overline{A}=\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(v_{\mathcal{S}\mathrm{et^{fin}}}(\cdot),A)$
on $\mathcal{S}\mathrm{et^{fin}}$. For every extremally disconnected profinite
set $X=\\{X_{i}\\}_{i\in I}$, one has maps $X\rightarrow X_{i}$ and thus maps
$F(X_{i})\rightarrow F(X)$ and $\overline{A}(X_{i})\rightarrow F(X_{i})$ being
compatible with all arrows in the diagram $\\{X_{i}\\}_{i\in I}$. Hence, one
obtains for every such $X$ a unique map

$\mathrm{colim}_{i\in I^{\mathrm{op}}}\overline{A}(X_{i})\rightarrow F(X)$

by the universal property of colimits. The resulting map
$N\colon\underline{A}^{\mathrm{disc}}\rightarrow F$ is a natural
transformation extending $h$ in the required way. As every other morphism
$\underline{A}^{\mathrm{disc}}\rightarrow F$ extending $h$ has to satisfy the
same conditions as we used for our construction, it has to be equivalent to
$N$. It remains to show the equivalence

$\displaystyle\mathrm{Hom}_{\mathcal{PS}\mathrm{h}(\mathcal{S}\mathrm{et^{fin}})}(\overline{A},F|_{\mathcal{S}\mathrm{et^{fin}}})\simeq\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(A,F(\ast)).$
(6.5)

Every finite set can be expressed as a finite disjoint union of one point
sets. As $\overline{A}$ and $F|_{\mathcal{S}\mathrm{et^{fin}}}$ define
presheaves on $\mathcal{S}\mathrm{et^{fin}}$ which send finite disjoint unions
to finite products, every morphism $\overline{A}\rightarrow
F|_{\mathcal{S}\mathrm{et^{fin}}}$ is fully determined by its evaluation on
the finite set $\\{\ast\\}$ which is the final object $\ast$ in
$\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et^{fin}})$. Hence, the morphism
$\overline{A}\rightarrow F|_{\mathcal{S}\mathrm{et^{fin}}}$ can be uniquely
identified with a morphism $\overline{A}(\ast)\rightarrow F(\ast)$ of anima
where
$\overline{A}(\ast)=\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(\\{\ast\\},A)=A$.
This proves (6.5) which combined with (6.4) gives the adjunction (6.3). ∎

As mentioned, the adjunction (6.4) can be formulated more generally for left
Kan extensions.

######  Lemma 6.2.6.

Let $\mathcal{C}$, $\widetilde{\mathcal{C}}$ and $\mathcal{D}$ be
$\infty$-categories. For every left Kan extension
$LS\colon\widetilde{\mathcal{C}}\rightarrow\mathcal{D}$ of a morphism
$S\colon\mathcal{C}\rightarrow\mathcal{D}$ along some
$G\colon\mathcal{C}\rightarrow\widetilde{\mathcal{C}}$, one has the
equivalence

$\displaystyle\mathrm{Hom}_{\mathrm{Fun}(\widetilde{\mathcal{C}},\mathcal{D})}(LS,F)\simeq\mathrm{Hom}_{\mathrm{Fun}(\mathcal{C},\mathcal{D})}(S,F\circ
G),$

natural in $F\in\mathrm{Fun}(\widetilde{\mathcal{C}},\mathcal{D})$.

###### Proof.

This follows by [Lur09, Proposition 4.3.3.7.]. ∎

###  6.3 Topological spaces as condensed anima

Recall the fully faithful embedding

$\mathcal{CG}\hookrightarrow\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)},\
X\mapsto\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(\cdot,X)$

as mentioned in Proposition 2.2.3. Using this and part (c) of the following
theorem, we can view compactly generated topological spaces as condensed
anima. Moreover, the theorem summarizes important relations of the
$\infty$-categories of anima, condensed anima and animated condensed sets.

######  Theorem 6.3.1.

With the definitions above, the following are true:

  1. (a)

There is a natural equivalence of the $\infty$-categories

$\mathrm{Cond(}\mathcal{A}\mathrm{ni(}\mathcal{S}\mathrm{et}))\cong\mathcal{A}\mathrm{ni(}\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}\mathrm{)}$

of condensed anima and animated condensed sets.

  2. (b)

The $\infty$-category $\mathcal{A}\mathrm{ni}$ embeds fully faithfully into
$\mathcal{A}\mathrm{ni(}\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}\mathrm{)}$
by the discrete functor
$\underline{(\cdot)}^{\mathrm{disc}}\colon\mathcal{A}\mathrm{ni}\hookrightarrow\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$
and

  3. (c)

the fully faithful embedding of the category
$\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}$ into
$\mathcal{A}\mathrm{ni(}\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}\mathrm{)}$
of Proposition 4.2.7 is given by composition of a condensed set with the fully
faithful embedding
$v\colon\mathcal{S}\mathrm{et}\hookrightarrow\mathcal{A}\mathrm{ni}$.

###### Proof.

  1. (a)

This is Lemma 11.8 in [Sch19a].

  2. (b)

We can conclude the statement directly as a corollary of Proposition 6.2.5 by
choosing $F\coloneqq\underline{B}^{\mathrm{disc}}$ for some anima
$B\in\mathcal{A}\mathrm{ni}$ and noting that
$\underline{B}^{\mathrm{disc}}(\ast)=B$.

  3. (c)

First of all, we note that for every
$F\in\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}$ the composition $v\circ
F$ is a condensed anima by $(a)$. Indeed, $F$ sends finite disjoint unions to
finite products by definition and the embedding $v$ preserves limits as it is
a right adjoint by Proposition 4.1.6. By Proposition 4.2.7, we already know
that there is a fully faithful embedding
$\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}\hookrightarrow\mathcal{A}\mathrm{ni(}\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}\mathrm{)}$
that identifies $\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}$ with the
$0$-truncated objects and is right adjoint to the composition with
$\pi_{0}\colon\mathcal{A}\mathrm{ni}\rightarrow\mathcal{S}\mathrm{et}$. It is
enough to prove that the composition with the embedding
$v\colon\mathcal{S}\mathrm{et}\hookrightarrow\mathcal{A}\mathrm{ni}$ is also
right adjoint to the composition with $\pi_{0}$. Then, the fully faithfulness
follows by uniqueness of adjoint functors. By the adjointness in Proposition
4.1.6 one has

$\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(v\circ
F(X),G(X))\cong\mathrm{Hom}_{\mathcal{S}\mathrm{et}}(F(X),\pi_{0}\circ G(X))$

for all $F\in\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}$,
$G\in\mathcal{A}\mathrm{ni(}\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}\mathrm{)}$
and $X\in\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc}$. This immediately
implies

$\mathrm{Hom}_{\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}}(v\circ
F,G)\cong\mathrm{Hom}_{\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}}(F,\pi_{0}\circ
G).$

∎

######  Definition 6.3.2.

For a compactly generated topological space $X$, we call the condensed anima
$\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(\cdot,X)\colon\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc}^{\mathrm{op}}\rightarrow\mathcal{A}\mathrm{ni}$
obtained by the fully faithful embedding of Theorem 6.3.1 (c) the condensed
anima corresponding to X or represented by X.

######  Proposition 6.3.3.

Regarding $\mathcal{S}\mathrm{et^{fin}}$ as a sub-$\infty$-category of
$\mathcal{A}\mathrm{ni}$ via the fully faithful embedding
$v\colon\mathcal{S}\mathrm{et}\hookrightarrow\mathcal{A}\mathrm{ni}$ one can
deduce the following:

  1. 1.

For any finite set $S$, the discrete condensed anima
$\underline{S}^{\mathrm{disc}}$ is the condensed anima
$\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(\cdot,S)$ represented by $S$ considered
as a finite discrete space.

  2. 2.

If $\\{X_{i}\\}_{i\in I}$ is a cofiltered system of finite sets, the limit
$\mathrm{lim}_{i\in I}\underline{X_{i}}^{\mathrm{disc}}$ is the condensed
anima $\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(\cdot,\mathrm{lim}_{i\in
I}X_{i})$ represented by the profinite set $\mathrm{lim}_{i\in I}X_{i}$.

###### Proof.

  1. 1.

Let $S$ be a finite set and
$v\colon\mathcal{S}\mathrm{et}\hookrightarrow\mathcal{A}\mathrm{ni}$ the
embedding of sets to animated sets. Then, the discrete condensed anima
associated to $S$ is given by the formula

$X=\\{X_{i}\\}_{i\in I}\mapsto\mathrm{colim}_{i\in
I^{\mathrm{op}}}\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(v(X_{i}),v(S))$

with $X$ extremally disconnected profinite set. As $v$ and the Yoneda
embedding
$\mathcal{S}\mathrm{et}^{\mathrm{fin}}\hookrightarrow\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et^{fin}})$
of finite sets into profinite sets are fully faithful, we can deduce

$\displaystyle\mathrm{colim}_{i\in
I^{\mathrm{op}}}\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(v(X_{i}),v(S))$
$\displaystyle=\mathrm{colim}_{i\in
I^{\mathrm{op}}}\mathrm{Hom}_{\mathcal{S}\mathrm{et}}(X_{i},S)$
$\displaystyle=\mathrm{colim}_{i\in
I^{\mathrm{op}}}\mathrm{Hom}_{\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et^{fin}})}(X_{i},S),$

using that $S$ and $X_{i}$ are finite sets for all $i\in I$. Moreover, every
finite set is cocompact in
$\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et^{fin}})$ as stated in Remark
5.1.4. Therefore, the filtered colimit can be computed as cofiltered limit
inside the hom-space. It follows that the discrete condensed anima can be
stated by the formula

$X=\\{X_{i}\\}_{i\in
I}\mapsto\mathrm{Hom}_{\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et^{fin}})}(\mathrm{lim}_{i\in
I}X_{i},S)=\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(X,S),$

as we can view $\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et^{fin}})$ as a
full sub-$\infty$-category of (the nerve of) $\mathcal{T}\mathrm{op}$ (since
this is true in the ordinary categorical setting). This is what we wanted to
prove.

  2. 2.

Let $\\{X_{i}\\}_{i\in I}$ be a cofiltered system of finite sets. As the limit
$\mathrm{lim}_{i\in I}\underline{X_{i}}^{\mathrm{disc}}$ can be written as

$\mathrm{lim}_{i\in
I}\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(\cdot,X_{i})=\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(\cdot,\mathrm{lim}_{i\in
I}X_{i}),$

the statement directly follows from (1.).

∎

######  Remark 6.3.4.

By Theorem 6.3.1 one obtains two natural functors from CW-spaces to condensed
anima. On the one hand, there is a fully faithful functor via the full
subcategory of condensed sets. On the other hand, there is a non-faithful
functor factoring over the full-$\infty$-subcategory of anima as
$\mathcal{CW}\rightarrow\mathcal{A}\mathrm{ni}$ is non-faithful. By
Proposition 6.3.3, the images under the two functors coincide if the CW-space
is a discrete finite set.

##  Chapter 7 Homotopy theory for condensed sets

The aim of this chapter is to develop an option of doing homotopy theory with
condensed sets in a meaningful way.
In his post [Cla20] at April 5, 2020 in the $n$-Category café, Clausen claims
that one can regard
$\mathcal{A}\mathrm{ni(}\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}\mathrm{)}\simeq\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$
as an "$\infty$-topos modulo cardinality issues". If we believe this statement
for a moment without demanding a proof, we can think about categorical
homotopy groups as they are defined for $\infty$-topoi [Lur09, Definition
6.5.1.1.]. Then by [Lur09, Proposition 6.5.1.7.], for all $0$-truncated
objects the non-zero homotopy groups vanish. Hence, by considering
$\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}$ as the full
sub-$\infty$-category of $0$-truncated objects of
$\mathcal{A}\mathrm{ni(}\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}\mathrm{)}$
by Proposition 4.2.7, this approach of defining a kind of homotopy groups for
condensed sets is not very meaningful.
Another, more promising, approach is to investigate to what extent the
embedding
$\mathcal{A}\mathrm{ni}\hookrightarrow\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$
of Theorem 6.3.1 has a left adjoint. By Lemma 11.9 in [Sch19a], there exists
at least a left adjoint if we restrict the functor to the full subcategory of
CW-spaces regarded as condensed anima. The functor is given by sending the
condensed anima corresponding to a CW-space to the singular simplicial complex
of the CW-space as we will see in Proposition 7.2.9. In fact, by Section 7.3
we cannot hope that this partially left adjoint can be extended to the whole
category $\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$.
Nevertheless, the construction of pro-$\infty$-categories allows us to define
a left adjoint if we replace $\mathcal{A}\mathrm{ni}$ by its
pro-$\infty$-category $\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$. Then,
as we will see, one can define homotopy pro-groups.

###  7.1 Pointed $\infty$-categories

In order to define homotopy groups, we have to define a notion of pointed
objects of an $\infty$-category. We follow the definition in [Lur09, Section
7.2.2].

######  Definition 7.1.1.

The $\infty$-category of pointed objects
Let $\mathcal{C}$ be an $\infty$-category.

  * 1.

We call an object $X\in\mathcal{C}$ final if $\mathrm{Hom}_{\mathcal{C}}(Y,X)$
is a contractible simplicial set for all $Y\in\mathcal{C}$, i.e. it is weakly
equivalent to the point. We denote a final object by $\ast_{\mathcal{C}}$.

  * 2.

A pointed object of $\mathcal{C}$ is a morphism
$y\colon\ast_{\mathcal{C}}\rightarrow Y$ in $\mathcal{C}$. The full
subcategory of $\mathcal{C}_{1}=\mathrm{Fun}(\triangle^{1},\mathcal{C})$
spanned by pointed objects is denoted by $\mathcal{C}_{*}$ and called pointed
$\infty$-category of $\mathcal{C}$.

######  Remark 7.1.2.

For simplicity, we write $x$ for an object of $\mathcal{C}_{*}$ meaning the
morphism $x\colon\ast_{\mathcal{C}}\rightarrow X$ and call $x$ an element of
$X$ denoted by $x\in X$.

######  Proposition 7.1.3.

Let $\mathcal{C}$ be an $\infty$-category and let $\mathcal{C}^{\prime}$ be
the full subcategory of final objects of $\mathcal{C}$. Then
$\mathcal{C}^{\prime}$ is either empty or a contractible Kan complex.

###### Proof.

See [Lur09, Proposition 1.2.12.9.] and notice the equivalence of our
definition of a final object with the definition provided in [Lur09,
Definition 1.2.12.1.] by [Lur09, Proposition 1.2.12.4., Corollary 1.2.12.5.
and Corollary 4.2.1.8.]. ∎

By the proposition, we can assume the final object of an $\infty$-category to
be unique (up to a contractible choice). In the ordinary categorical setting,
there is a similar statement for final objects. By the next remark, the
notions are connected in a natural way just as one wishes for an appropriate
definition.

######  Remark 7.1.4.

For an ordinary category $\mathcal{C}$ an object is final in the nerve
$N(\mathcal{C})$ if and only if it is final in $\mathcal{C}$ in the usual
sense by [Lur09, Example 1.2.12.7.] .

######  Example 7.1.5.

Final anima
The (up to a contractible choice unique) final object
$\ast_{\mathcal{A}\mathrm{ni}}$ in the $\infty$-category
$\mathcal{A}\mathrm{ni}$ is given by the final Kan complex which itself
corresponds to the final CW complex, i.e. it is given by the $0$-standard
simplex $\triangle^{0}=\\{\ast\\}$. An element $a$ of an anima $A$ can be
identified with a map $a\colon\triangle^{0}\rightarrow A$ via the Yoneda
embedding. This justifies our simplification of Remark 7.1.2.

######  Proposition 7.1.6.

Let $\mathcal{C}$ be an $\infty$-category and $\mathcal{C}_{*}$ its
$\infty$-category of pointed objects. Then, the following is true:

  * 1.

If $\mathcal{C}$ is accessible, $\mathcal{C}_{*}$ is accessible as well.

  * 2.

The limit over a diagram $\\{\ast_{\mathcal{C}}\rightarrow C_{i}\\}_{i\in I}$
exists in $\mathcal{C}_{*}$ whenever the limit over the induced diagram
$\\{C_{i}\\}_{i\in I}$ in $\mathcal{C}$ exists and it coincides with the
unique morphism $\ast_{\mathcal{C}}\rightarrow\lim_{i\in I}C_{i}$ induced by
the universal property of the limit.

###### Proof.

By [Lur09, Lemma 7.2.2.8., Proposition 4.2.1.5.], the category
$\mathcal{C}_{*}$ can be identified with the under-$\infty$-category
$\mathcal{C}_{*/}$. Thus, the first statement on accessibility follows by
[Lur09, Corollary 5.4.5.16.]. The second statement follows by the general fact
[Lur09, Corollary 5.1.2.3.] that limits in functor-$\infty$-categories can be
computed objectwise . ∎

######  Definition 7.1.7.

Morphisms of pointed $\infty$-categories
Let $F\colon\mathcal{C}\rightarrow\mathcal{D}$ be a morphism of
$\infty$-categories that preserves final objects, i.e. the image
$F(\ast_{\mathcal{C}})$ can be identified with the final object
$\ast_{\mathcal{D}}\in\mathcal{D}_{*}$. Then, one can define a morphism
$F_{\ast}\colon\mathcal{C}_{\ast}\rightarrow\mathcal{D}_{\ast}$ of pointed
$\infty$-categories which sends an object $\ast_{\mathcal{C}}\rightarrow
X\in\mathcal{C}_{*}$ to an object $\ast_{\mathcal{D}}\rightarrow
F(X)\in\mathcal{D}_{\ast}$.

######  Proposition 7.1.8.

Final objects in functor categories
Let $\ast_{\mathcal{C}}$ be the final object in an $\infty$-category
$\mathcal{C}$. Then, one has the following:

  * 1.

If the $\infty$-category $\mathcal{P}\mathrm{ro}(\mathcal{C})$ is defined, it
has a final object given by the corepresentable functor
$\mathrm{Hom}_{\mathcal{C}}(\ast_{\mathcal{C}},\cdot)$.

  * 2.

For every $\infty$-category $\mathcal{D}$, the $\infty$-category
$\mathrm{Fun}(\mathcal{D},\mathcal{C})$ has a final object given by the
functor with constant image $\ast_{\mathcal{C}}$.

###### Proof.

  * 1.

By the formula for morphisms in a pro-$\infty$-category, one has for every
$Y=\\{Y_{i}\\}_{i\in I}\in\mathcal{P}\mathrm{ro}(\mathcal{C})$ the following
identifications

$\displaystyle\mathrm{Hom}_{\mathcal{P}\mathrm{ro}(\mathcal{C})}(Y,\mathrm{Hom}_{\mathcal{C}}(\ast_{\mathcal{C}},\cdot))$
$\displaystyle=\mathrm{colim}_{i\in
I^{\mathrm{op}}}\mathrm{Hom}_{\mathcal{C}}(Y_{i},\ast_{\mathcal{C}})$
$\displaystyle=\mathrm{colim}_{i\in
I^{\mathrm{op}}}\ast_{\mathcal{A}\mathrm{ni}}=\ast_{\mathcal{A}\mathrm{ni}}$

where we used for the second equality the property of $\ast_{\mathcal{C}}$ to
be final in $\mathcal{C}$.

  * 2.

Let $c_{\ast}$ denote the functor with constant image $\ast_{\mathcal{C}}$ and
$F$ another functor in $\mathrm{Fun}(\mathcal{D},\mathcal{C})$. For all
$X\in\mathcal{D}$ the hom-spaces

$\mathrm{Hom}_{\mathcal{C}}(F(X),\ast_{\mathcal{C}})$

are contractible by definition of $\ast_{\mathcal{C}}$. Thus, the same holds
for

$\mathrm{Hom}_{\mathrm{Fun}(\mathcal{D},\mathcal{C})}(F,c_{\ast})$

and $c_{*}$ is final in $\mathrm{Fun}(\mathcal{D},\mathcal{C})$.

∎

###  7.2 Homotopy pro-groups for
$\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$

As stated in Chapter $4$, one can define homotopy groups for animated sets by
the construction of simplicial homotopy groups (cf. Def. 4.1.3). In order to
define homotopy groups for condensed sets viewed as condensed anima, we can
use the results of the preceding section to extend the construction of
simplicial homotopy groups first to pro-anima and then to condensed anima as
we will see in the following.

######  Lemma 7.2.1.

There exists a unique functor

$R_{\mathcal{A}\mathrm{ni}}\colon\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})\rightarrow\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$

preserving small cofiltered limits in
$\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$, which sends a pro-anima $A$
to a condensed anima given by the assignment

$\displaystyle X=\\{X_{i}\\}_{i\in I}\mapsto\mathrm{colim}_{i\in
I^{\mathrm{op}}}\mathrm{Hom}_{\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})}(v_{\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})}(X_{i}),A),$
(7.1)

where the colimit is filtered and the composition of the canonical embeddings

$v_{\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})}\coloneqq
j_{\mathcal{A}\mathrm{ni}}\circ
v_{\mathcal{A}\mathrm{ni}}\colon\mathcal{S}\mathrm{et^{fin}}\hookrightarrow\mathcal{A}\mathrm{ni}\hookrightarrow\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$

is fully faithful. The condensed anima defined by the formula (7.1) preserves
cofiltered limits in $\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et^{fin}})$
and the restriction of $R_{\mathcal{A}\mathrm{ni}}$ to
$\mathcal{A}\mathrm{ni}$ coincides with the discrete functor
$\underline{(\cdot)}^{\mathrm{disc}}$.

###### Proof.

Consider the $\infty$-category
$\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}\simeq\mathcal{A}\mathrm{ni(}\mathcal{C}\mathrm{ond(}\mathcal{S}\mathrm{et)}\mathrm{)}$
of condensed anima and the $\infty$-category $\mathcal{A}\mathrm{ni}$ of
animated sets. Both categories are presentable and complete by 4.2.3. In
particular, they are accessible $\infty$-categories that admit finite limits.
Hence, the hypotheses of Proposition 5.1.7 are satisfied.
Let
$\underline{(\cdot)}^{\mathrm{disc}}\colon\mathcal{A}\mathrm{ni}\rightarrow\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$
be the discrete functor that assigns to every animated set $A$ the discrete
condensed anima $\underline{A}^{\mathrm{disc}}$ as in 6.2.3. Then, by the
universal property of pro-$\infty$-objects, it extends to a unique functor

$R_{\mathcal{A}\mathrm{ni}}\colon\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})\rightarrow\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$

preserving small cofiltered limits. The mapping
$\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})\rightarrow\mathcal{PS}\mathrm{h}(\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc})$
given by sending a pro-anima $A$ to the presheaf defined by (7.1) preserves
cofiltered limits as the right hand side given by
$\mathrm{Hom}_{\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})}(v_{\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})}(X_{i}),A)$
commutes with limits in $A$. By fully faithfulness of the Yoneda embedding
$j_{\mathcal{A}\mathrm{ni}}$, it clearly extends the discrete functor. If we
can argue that (7.1) defines a condensed anima, the claim follows from the
uniqueness of the extension $R_{\mathcal{A}\mathrm{ni}}$. As
$v_{\mathcal{A}\mathrm{ni}}$ and $j_{\mathcal{A}\mathrm{ni}}$ preserve finite
coproducts (cf. Cor. 5.1.11), the same is true for their composition
$v_{\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})}$. Then, we can conclude by
the same argument as in Proposition 6.2.2. ∎

Note that the condensed anima corresponding to some pro-anima under the
functor $R_{\mathcal{A}\mathrm{ni}}$ is the same as one obtains by the
construction of Lemma 6.2.1 and Proposition 6.2.2 which we also used to define
discrete condensed anima.

######  Proposition 7.2.2.

There exists a left adjoint functor to $R_{\mathcal{A}\mathrm{ni}}$ denoted by

$L_{\mathcal{A}\mathrm{ni}}\colon\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}\rightarrow\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$

which preserves colimits. For every discrete condensed anima
$\underline{A}^{\mathrm{disc}}$ the corresponding pro-anima coincides with the
corepresentable pro-anima
$\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(A,\cdot)\colon\mathcal{A}\mathrm{ni}\rightarrow\mathcal{A}\mathrm{ni}$
obtained by the Yoneda embedding
$j_{\mathcal{A}\mathrm{ni}}\colon\mathcal{A}\mathrm{ni}\rightarrow\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$.

###### Proof.

With the same argument as in the preceding lemma, the preliminaries of
Proposition 5.1.14 are satisfied. Thus, there is a pro-existent left adjoint
to $R_{\mathcal{A}\mathrm{ni}}$ given by the functor

$\displaystyle
L_{\mathcal{A}\mathrm{ni}}\colon\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}\rightarrow\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$

sending a condensed anima $C$ to the assignment

$B\mapsto\mathrm{Hom}_{\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}}(C,\underline{B}^{\mathrm{disc}})$

with $B\in\mathcal{A}\mathrm{ni}$. The functor $L_{\mathcal{A}\mathrm{ni}}$
preserves colimits as it is a left adjoint. If $C$ is a discrete condensed
anima given by $C=\underline{A}^{\mathrm{disc}}$ for some anima $A$, this
reduces to the assignment

$B\mapsto\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(A,B)$

by fully faithfulness of the discrete functor. ∎

######  Corollary 7.2.3.

The final object in $\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$
can be represented by the discrete condensed anima

$\displaystyle\\{X_{i}\\}_{i\in I}\mapsto\mathrm{colim}_{i\in
I^{\mathrm{op}}}\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(v_{\mathcal{A}\mathrm{ni}}(X_{i}),\ast_{\mathcal{A}\mathrm{ni}})\cong\ast_{\mathcal{A}\mathrm{ni}}$
(7.2)

corresponding to $\ast_{\mathcal{A}\mathrm{ni}}$ and the functor
$R_{\mathcal{A}\mathrm{ni}}\colon\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})\rightarrow\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$
preserves final objects.

###### Proof.

As $\ast_{\mathcal{A}\mathrm{ni}}$ is the final object in
$\mathcal{A}\mathrm{ni}$, for every $X=\\{X_{i}\\}_{i\in
I}\in\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc}$ the assignment (7.2)
reduces to the functor
$c_{\ast}\in\mathrm{Fun}(\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc}^{\mathrm{op}},\mathcal{A}\mathrm{ni})$
with constant image $\ast_{\mathcal{A}\mathrm{ni}}$. By the Proposition 7.1.8
$(2.)$ and as $\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$ is a
full subcategory of
$\mathrm{Fun}(\mathcal{E}\mathrm{xtr}\mathcal{D}\mathrm{isc}^{\mathrm{op}},\mathcal{A}\mathrm{ni})$,
the functor
$c_{\ast}\in\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$ is also
final in $\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$ and can be
represented by the discrete condensed anima given by (7.2). As the final
object in $\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$ is simply given as
the image of the final object in $\mathcal{A}\mathrm{ni}$ under
$j_{\mathcal{A}\mathrm{ni}}$, its image under $R_{\mathcal{A}\mathrm{ni}}$ is
by Lemma 7.2.1 given by the discrete condensed anima corresponding to
$\ast_{\mathcal{A}\mathrm{ni}}$ and the claim follows. ∎

######  Corollary 7.2.4.

The functor
$L_{\mathcal{A}\mathrm{ni}}\colon\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}\rightarrow\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$
preserves final objects. Thus, we obtain a functor

$(L_{\mathcal{A}\mathrm{ni}})_{\ast}\colon\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}_{\ast}\rightarrow\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})_{*}$

of pointed $\infty$-categories given by the assignment

$c\mapsto L_{\mathcal{A}\mathrm{ni}}(c),$

where $L_{\mathcal{A}\mathrm{ni}}(c)$ denotes the image of the morphism
$c\colon\ast_{\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}}\rightarrow
C$ under $L_{\mathcal{A}\mathrm{ni}}$.

###### Proof.

By the preceding corollary and Proposition 7.2.2, the image of the final
object in $\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$ under the
left adjoint $L_{\mathcal{A}\mathrm{ni}}$ is given by the copresentable
functor
$\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(\ast,\cdot)\in\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$,
which is exactly the final object in the pro-$\infty$-category by Proposition
7.1.8. Thus, the functor $(L_{\mathcal{A}\mathrm{ni}})_{\ast}$ can be defined
as described in Definition 7.1.7. ∎

######  Remark 7.2.5.

The preceding results can be summarized as follows: One obtains a commutative
diagram

${\mathcal{A}\mathrm{ni}}$${\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}}$${\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})}$$\scriptstyle{\underline{(\cdot)}^{\mathrm{disc}}}$$\scriptstyle{j_{\mathcal{A}\mathrm{ni}}}$$\scriptstyle{L_{\mathcal{A}\mathrm{ni}}}$$\scriptstyle{R_{Ani}}$

which can be extended to a commutative diagram on the corresponding pointed
$\infty$-categories as all involved functors preserve final objects.

######  Proposition 7.2.6.

If the pro-$\infty$-category $\mathcal{P}\mathrm{ro}(\mathcal{C})$ is defined,
then the $\infty$-category $\mathcal{P}\mathrm{ro}(\mathcal{C}_{*})$ is
defined as well and one has an equivalence of $\infty$-categories

$\mathcal{P}\mathrm{ro}(\mathcal{C}_{*})=\mathcal{P}\mathrm{ro}(\mathcal{C})_{*}$

induced by the map
$f\colon\mathrm{Fun}(\triangle^{1},\mathcal{C})\rightarrow\mathrm{Fun}(\triangle^{1},\mathcal{P}\mathrm{ro}(\mathcal{C}))$,
which is itself induced by the Yoneda embedding
$j_{\mathcal{C}}\colon\mathcal{C}\rightarrow\mathcal{P}\mathrm{ro}(\mathcal{C})$.

###### Proof.

The $\infty$-category $\mathcal{D}_{*}$ admits finite limits and is accessible
whenever this is true for $\mathcal{D}$ by Proposition 7.1.6. This proves the
first claim. As $j_{\mathcal{C}}$ preserves final objects, the same is true
for the functor $f$. Thus, the restriction $f^{\prime}$ of $f$ to
$\mathcal{C}_{*}$ lands in $\mathcal{P}\mathrm{ro}(\mathcal{C})_{*}$. This
restriction $f^{\prime}$ can by Proposition 5.1.7 be extended to a unique
functor

$\mathcal{P}\mathrm{ro}(f^{\prime})\colon\mathcal{P}\mathrm{ro}(\mathcal{C}_{*})\rightarrow\mathcal{P}\mathrm{ro}(\mathcal{C})_{*},$

preserving cofiltered limits as $\mathcal{P}\mathrm{ro}(\mathcal{C})_{*}$
admits cofiltered limits by Proposition 7.1.6. We want to show that this
indeed defines an equivalence of $\infty$-categories. First, we note that an
object in $\mathcal{P}\mathrm{ro}(\mathcal{C})_{*}$ is given as a morphism

$\ast\rightarrow\mathrm{lim}_{i\in I}j_{\mathcal{C}}(X_{i}),$

where $\mathrm{lim}_{i\in I}j_{\mathcal{C}}(X_{i})$ is a pro-object in
$\mathcal{C}$ represented as a cofiltered limit as usual. This coincides with
$\mathrm{lim}_{i\in I}(\ast\rightarrow j_{\mathcal{C}}(X_{i}))$ by Proposition
7.1.6. Thus, every object in $\mathcal{P}\mathrm{ro}(\mathcal{C})_{*}$ is a
cofiltered limit of objects $\ast\rightarrow j_{\mathcal{C}}(X_{i})$. As we
know that objects in $\mathcal{P}\mathrm{ro}(\mathcal{C}_{*})$ can be written
as cofiltered limits of objects $j_{\mathcal{C}_{*}}(\ast^{\prime}\rightarrow
X_{i})$, where $\ast^{\prime}$ is the final object in $\mathcal{C}_{*}$, it is
enough to show the equivalence of the full subcategories given by the objects
$\ast\rightarrow j_{\mathcal{C}}(X_{i})$ in
$\mathcal{P}\mathrm{ro}(\mathcal{C})_{*}$ and the objects
$j_{\mathcal{C}_{*}}(\ast^{\prime}\rightarrow X_{i})$ in
$\mathcal{P}\mathrm{ro}(\mathcal{C}_{*})$. But this is clear, as the
subcategory of objects $j_{\mathcal{C}_{*}}(\ast^{\prime}\rightarrow X_{i})$
in $\mathcal{P}\mathrm{ro}(\mathcal{C}_{*})$ is equivalent to
$\mathcal{C}_{*}$ by fully faithfulness of $j_{\mathcal{C}_{*}}$ and
$\mathcal{C}_{*}$ itself is equivalent to the subcategory of objects
$\ast\rightarrow j_{\mathcal{C}}(X_{i})$ by fully faithfulness of the functor
$j_{\mathcal{C}}$. ∎

Now, we have collected all the necessary components to extend the notion of
homotopy groups to pro-$\infty$-categories and to use this to define homotopy
pro-groups for condensed anima.

######  Proposition 7.2.7.

There exist unique functors

$\displaystyle\mathcal{P}\mathrm{ro}(\pi_{0})\colon\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})\
\rightarrow\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et})$
$\displaystyle\mathcal{P}\mathrm{ro}(\pi_{1})\colon\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})_{*}\rightarrow\mathcal{P}\mathrm{ro}(\mathcal{G}\mathrm{rp})$
$\displaystyle\mathcal{P}\mathrm{ro}(\pi_{n})\colon\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})_{*}\rightarrow\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{b}\mathcal{G}\mathrm{rp})$

with $n\geq 2$ preserving small cofiltered limits, s.t. for every anima $A$
regarded as an object in $\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$ and
$a\in A$ the image coincides with the corresponding simplicial homotopy group
$\pi_{n}(A,a)$ or the set of path components $\pi_{0}(A)$, respectively.

###### Proof.

One has functors

$\displaystyle\pi_{0}\colon\mathcal{A}\mathrm{ni}\
\rightarrow\mathcal{S}\mathrm{et}$
$\displaystyle\pi_{1}\colon\mathcal{A}\mathrm{ni}_{*}\rightarrow\mathcal{G}\mathrm{rp}$
$\displaystyle\pi_{n}\colon\mathcal{A}\mathrm{ni}_{*}\rightarrow\mathcal{A}\mathrm{b}\mathcal{G}\mathrm{rp}$

with $n\geq 2$ by Definition 4.1.3 and Remark 4.1.4. As all included
$\infty$-categories are accessible and admit finite limits by Proposition
7.1.6, we can apply Corollary 5.1.13 to obtain the corresponding pro-functors
with the claimed properties by using the identification
$\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni}_{\ast})=\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})_{\ast}$
which is true by the preceding proposition. ∎

######  Definition 7.2.8.

Homotopy pro-groups for condensed anima
For every condensed anima
$C\in\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$ we define the
pro-set of path components by

$\widetilde{\pi}_{0}(C)\coloneqq\pi_{0}(L_{\mathcal{A}\mathrm{ni}}(C))$

and for some $c\in C$ the $n$th homotopy pro-group for $n\geq 1$ by

$\widetilde{\pi}_{n}(C,c)\coloneqq\pi_{n}((L_{\mathcal{A}\mathrm{ni}})_{\ast}(c)).$

We end this section by showing that our definition of homotopy pro-groups is a
proper extension of the notion of homotopy groups on topological spaces in the
sense that it deals with the class of CW-spaces properly.

######  Proposition 7.2.9.

For any CW-space $X$, the homotopy pro-groups of the condensed anima
$C\coloneqq\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(\cdot,X)$ represented by $X$
coincide with the usual topological homotopy groups.

###### Proof.

Recall the adjunction

${\mathrm{s}\mathcal{S}\mathrm{et}}$${\mathcal{T}\mathrm{op},}$$\scriptstyle{|\cdot|}$$\scriptstyle{\mathrm{Sing}(\cdot)}$

which induces an equivalence on the (homotopy) categories of of CW-spaces and
of Kan complexes. Let $A$ denote the anima corresponding to a CW-space $X$
through the (non fully faithful) map
$\mathcal{CW}\rightarrow\mathcal{A}\mathrm{ni}$ induced by
$\mathrm{Sing}(\cdot)$. We want to conclude that our left adjoint
$L_{\mathcal{A}\mathrm{ni}}$ restricted to CW-spaces coincides with the map
sending the condensed anima
$C\coloneqq\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(\cdot,X)$ corresponding to X
to the anima $A$. By Lemma 11.9. in [Sch19a], there is a natural map
$C\rightarrow\underline{A}^{\mathrm{disc}}$ of condensed anima which is
universal in this sense. Thus, one can deduce the following chain of
equivalences

$\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(A,B)\cong\mathrm{Hom}_{\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}}(\underline{A}^{\mathrm{disc}},\underline{B}^{\mathrm{disc}})\cong\mathrm{Hom}_{\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}}(C,\underline{B}^{\mathrm{disc}})$

by fully faithfulness of the discrete functor
$\underline{(\cdot)}^{\mathrm{disc}}\colon\mathcal{A}\mathrm{ni}\rightarrow\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$
and universality of the map $C\rightarrow\underline{A}^{\mathrm{disc}}$. This
directly implies left adjointness of the functor
$\widetilde{L}\colon\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}\supset\mathcal{CW}\rightarrow\mathcal{A}\mathrm{ni},\
C\mapsto A$ to the discrete functor. By uniqueness of adjoint functors, the
functor $\widetilde{L}$ has to coincide with the restriction of our left
adjoint $L_{\mathcal{A}\mathrm{ni}}$ to CW-spaces. Hence, as stated in
Proposition 7.2.7, the homotopy pro-groups of $C$ defined in Definition 7.2.8
coincide with the simplicial homotopy groups of $A$. But, by Remark 4.1.4
these groups coincide with the topological homotopy groups of $X$. ∎

###  7.3 Profinite sets as an example

In the proof of Proposition 7.2.9, we have seen that there is a partially left
adjoint to
$\mathcal{A}\mathrm{ni}\hookrightarrow\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$
defined on those condensed anima which are represented by CW-spaces. This
partially left adjoint induced the equivalence of the different notions of
homotopy (pro-)groups on CW-spaces. In this last section, we want to determine
the homotopy pro-groups of all condensed anima that can be represented by a
profinite set. These illustrate a class of condensed anima justifying our
approach of introducing pro-anima. Indeed, we will see that
$\mathcal{A}\mathrm{ni}\hookrightarrow\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$
does not preserve limits and hence cannot admit a left adjoint. This fact is
stated in more detail in Remark 7.3.2.

######  Proposition 7.3.1.

For every condensed anima represented by a profinite set, the corresponding
pro-anima under the functor $L_{\mathcal{A}\mathrm{ni}}$ of Proposition 7.2.2
is given by the profinite set itself and thus corresponds to an anima if and
only if the profinite set is finite.

###### Proof.

The pro-anima corresponding to a profinite set $S=\mathrm{lim}_{i\in I}S_{i}$
written as small cofiltered limit of finite (discrete) sets is given as a
functor

$\displaystyle\mathcal{A}\mathrm{ni}\rightarrow\mathcal{A}\mathrm{ni},\
A\mapsto\mathrm{lim}_{i\in
I}\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(v(S_{i}),A)$ (7.3)

in $\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$ by Proposition 5.2.3.
On the other hand, the image of the condensed anima
$S^{*}\coloneqq\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(\cdot,S)\in\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$
represented by the profinite set $S$ under the pro-existent left adjoint
$L_{\mathcal{A}\mathrm{ni}}\colon\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}\rightarrow\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$
of Proposition 7.2.2 is given by the pro-anima

$\displaystyle\mathcal{A}\mathrm{ni}\rightarrow\mathcal{A}\mathrm{ni},\
A\mapsto\mathrm{Hom}_{\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}}(S^{*},\underline{A}^{\mathrm{disc}}).$
(7.4)

In the following, it is our goal to show that the functors (7.4) and (7.3)
coincide.
We can view $S^{*}$ and $\underline{A}^{\mathrm{disc}}$ as presheaves from all
profinite sets to anima. This is possible as both condensed anima are given as
a restriction of the contravariant functor that was defined on all profinite
sets by the same formula $\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(\cdot,S)$ (cf.
Cor. 2.2.2) or $\mathrm{colim}_{i\in
I^{\mathrm{op}}}\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(v(\cdot),A)$ (cf. Prop.
6.2.2), respectively. For $\underline{A}^{\mathrm{disc}}$ we denoted this
functor by $A^{\mathrm{disc}}$ in Lemma 6.2.1. Then, one can conclude by the
$\infty$-categorical Yoneda Lemma

$\displaystyle\mathrm{Hom}_{\mathcal{PS}\mathrm{h}(\mathcal{P}\mathrm{ro}(\mathcal{S}\mathrm{et}^{\mathrm{fin}}))}(\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(\cdot,S),A^{\mathrm{disc}})\simeq$
$\displaystyle A^{\mathrm{disc}}(S)$

and moreover

$\displaystyle A^{\mathrm{disc}}(S)=\mathrm{colim}_{i\in
I^{\mathrm{op}}}\mathrm{Hom}_{\mathcal{A}\mathrm{ni}}(v(S_{i}),A)$

as in Lemma 6.2.1. We obtain a functor

$\displaystyle\mathcal{A}\mathrm{ni}\rightarrow\mathcal{A}\mathrm{ni},\
A\mapsto A^{\mathrm{disc}}(S)$

in $\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})^{\mathrm{op}}$. Thus, by
regarding it as a functor in $\mathcal{P}\mathrm{ro}(\mathcal{A}\mathrm{ni})$,
i.e. turning the filtered colimit into a cofiltered limit, one can see that
the functor (7.4) is the same as (7.3). ∎

######  Remark 7.3.2.

In Proposition 6.3.3, we have seen that the limit $\mathrm{lim}_{i\in
I}\underline{X_{i}}^{\mathrm{disc}}$ of discrete condensed anima defined by
finite sets $X_{i}$ is given by
$\mathrm{Hom}_{\mathcal{T}\mathrm{op}}(\cdot,\mathrm{lim}_{i\in I}X_{i})$,
i.e. coincides with the condensed anima represented by the profinite set
$\mathrm{lim}_{i\in I}X_{i}$. By Proposition 7.3.1, this condensed anima is
not discrete if the profinite set is not a finite set itself. Hence, the fully
faithful embedding
$\mathcal{A}\mathrm{ni}\hookrightarrow\mathcal{C}\mathrm{ond(}\mathcal{A}\mathrm{ni}\mathrm{)}$
does not preserve cofiltered limits. As every right adjoint preserves limits,
one cannot construct a left adjoint to the embedding. This justifies our
approach of passing to the category of pro-anima.

######  Corollary 7.3.3.

For every condensed anima represented by a profinite set the pro-set of path
components is given by the profinite set itself and all homotopy pro-groups
are trivial.

###### Proof.

Let $X=\\{X_{i}\\}_{i\in I}$ be the profinite set representing the condensed
anima. By Proposition 7.3.1 and Definition 7.2.8 the statement reduces to
computing the images of the (pointed) profinite set regarded as (pointed) pro-
anima $S=\\{S_{i}\\}_{i\in I}$ under the functors of Proposition 7.2.7, i.e.
the anima $S_{i}$ referring to the finite sets $X_{i}$. Since the functors
$\mathcal{P}\mathrm{ro}(\pi_{n})$ preserve small cofiltered limits, one can
identify the image of the profinite set $S$, respectively of the pointed
profinite set $(S,s)$, under $\mathcal{P}\mathrm{ro}(\pi_{0})$, respectively
some $\mathcal{P}\mathrm{ro}(\pi_{n})$ for $n\geq 1$, with a small cofiltered
limit of the images $\mathcal{P}\mathrm{ro}(\pi_{0}(S_{i}))$, respectively
$\mathcal{P}\mathrm{ro}(\pi_{n}(S_{i},s_{i}))$, where $s_{i}$ denotes the
image of $s$ under the natural map $S\rightarrow S_{i}$. Moreover, Proposition
7.2.7 states that for every anima $S_{i}$ these images coincide with the
simplicial homotopy groups $\pi_{0}(S_{i})$, $\pi_{n}(S_{i},s_{i})$ which in
turn coincide with the topological homotopy groups of the corresponding finite
set $X_{i}$ considered as finite discrete space. Thus, the set
$\pi_{0}(S_{i})$ of path components is the finite set $X_{i}$ and every
homotopy group $\pi_{n}(S_{i},s_{i})$ is trivial. Altogether, the set of path
components of $S$ is the cofiltered limit of the finite sets $X_{i}$ and thus
the profinite set $X$ itself. The homotopy pro-groups are trivial, as limits
of trivial groups are trivial. ∎

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