# Goodwillie Towers of $\infty$-categories and Desuspension

Daniel Fuentes-Keuthan  Department of Mathematics, Johns Hopkins University,
Baltimore<EMAIL_ADDRESS>
###### Abstract.

We reconceptualize the process of forming $n$-excisive approximations to
$\infty$-categories, in the sense of Heuts, as inverting the suspension
functor lifted to $A_{n}$-cogroup objects. We characterize $n$-excisive
$\infty$-categories as those $\infty$-categories in which $A_{n}$-cogroup
objects admit desuspensions. Applying this result to pointed spaces we reprove
a theorem of Klein-Schwänzl-Vogt: every 2-connected cogroup-like
$A_{\infty}$-space admits a desuspension.

###### Contents

  1. 1 Punctured Cubes as coGroups
  2. 2 coGroups in Goodwillie Calculus
  3. 3 Desuspensions in Spaces

## Introduction

Given a suitably nice $\infty$-category $\mathcal{C}$ one may form the
stablization $Sp(\mathcal{C})$ by formally inverting the suspension functor
$\Sigma:\mathcal{C}\to\mathcal{C}$, as given by the filtered colimit

$\mathcal{C}\xrightarrow{\Sigma}\mathcal{C}\xrightarrow{\Sigma}\dots\mathcal{C}\xrightarrow{\Sigma}\dots\to
Sp(\mathcal{C})$

This new category $Sp(\mathcal{C})$ is easier to work with: it is the
homotopic analogue of the category of abelian group objects in $\mathcal{C}$
and satisfies formal properties which make it into a sort of linearization of
$\mathcal{C}$. This stabilization procedure however loses a lot of information
about $\mathcal{C}$, since, for instance, in $Sp(\mathcal{C})$ we can recover
an object from only its suspension. One might remedy this harsh truncation by
requiring additional structure on the suspension in order to recover the
object. In particular the suspension functor lifts to a functor landing in
$A_{\infty}$-cogroup objects in $\mathcal{C}$.

$\Sigma:\mathcal{C}\to coGrp_{\infty}(\mathcal{C})$

By forgetting down to the $A_{n}$-cogroup structures for each $n$ one obtains
a collection of functors

$\Sigma:\mathcal{C}\to coGrp_{n}(\mathcal{C})$

which can be formally inverted by taking the filtered colimit of

$\mathcal{C}\xrightarrow{\Sigma}coGrp_{n}(\mathcal{C})\xrightarrow{\Sigma}coGrp_{n}(coGrp_{n}(\mathcal{C}))\xrightarrow{\Sigma}\cdots$

to give a tower of approximations to $\mathcal{C}$ which capture more
information as $n$ grows larger. In particular
$coGrp_{1}(\mathcal{C})\simeq\mathcal{C}$ so the first stage of this process
is just the stabilization $Sp(\mathcal{C})$.

In [Heu15] Heuts introduces another tower of $\infty$-categories which
interpolates between $\mathcal{C}$ and $Sp(\mathcal{C})$

$\dots\xrightarrow{\Sigma_{n+2,n+1}}P_{n+1}\mathcal{C}\xrightarrow{\Sigma_{n+1,n}}P_{n}\mathcal{C}\xrightarrow{\Sigma_{n,n-1}}\dots
P_{1}\mathcal{C}\simeq Sp(\mathcal{C})$

via the $n$-excisive approximations $P_{n}(\mathcal{C})$ to $\mathcal{C}$. We
refer to this tower as the Goodwillie tower of $\mathcal{C}$. The
$\infty$-category $P_{n}(\mathcal{C})$ is built by formally inverting the
suspension functor viewed as landing in an $\infty$-category
$T_{n}(\mathcal{C})$ of certain “special” punctured $(n+1)$-cubes.

The main goal of this paper is to prove that the two constructions above are
the same by showing that the $\infty$-category $T_{n}(\mathcal{C})$ gives a
new model for $A_{n}$-cogroup objects in $\mathcal{C}$. Namely we have Theorem
1.22:

######  Theorem.

Let $\mathcal{C}$ be a pointed $\infty$-category with finite colimits, then
there is an equivalence

$coGrp_{n}(\mathcal{C})\overset{\sim}{\longrightarrow}T_{n}(\mathcal{C})$

As a corollary of this result we obtain a characterization of $n$-excsisive
$\infty$-categories as those $\infty$-categories in which one can desuspend
$A_{n}$-cogroup objects. This is Theorem 2.5:

######  Theorem.

Let $\mathcal{C}$ be a pointed, compactly generated $\infty$-category. Then
$\mathcal{C}$ is $n$-excisive if and only if the suspension functor induces an
equivalence

$\Sigma:\mathcal{C}\overset{\sim}{\longrightarrow}coGrp_{n}(\mathcal{C})$

Passing to the limit as $n$ goes to infinity of the Goodwillie tower, one
obtains an $\infty$-category $P_{\infty}\mathcal{C}$ which best approximates
$\mathcal{C}$ from the point of view of Goodwillie calculus. Theorem 2.5
extends to this $\infty$-category to obtain an $A_{\infty}$ desuspension
theorem, stated as Theorem 2.14:

######  Theorem.

Let $\mathcal{C}$ be a pointed, compactly generated $\infty$-category. The
suspension functor induces an equivalence

$\Sigma:P_{\infty}\mathcal{C}\overset{\sim}{\longrightarrow}coGrp_{\infty}(P_{\infty}\mathcal{C})$

These results together motivate a philosophical claim that Goodwillie calculus
attempts to reconstruct $\infty$-categories and functors via the $A_{\infty}$
cogroup structure on the suspension functor.

As an application, in this new light we may translate certain desuspension
problems into problems of the convergence of the Goodwillie tower. For
example, from knowing that the Goodwillie tower convergences on simply
connected spaces, we are able to recover the following theorem of Klein-
Schwänzl-Vogt, which we state as Theorem 3.2

######  Theorem.

The suspension functor induces an equivalence of $\infty$-categories

$\Sigma:\mathcal{S}_{*}^{\geq
2}\overset{\sim}{\longrightarrow}coGrp_{\infty}(\mathcal{S}_{*}^{\geq 3})$

In particular every 2-connected $A_{\infty}$-comonoid admits a desuspension to
a 1-connected space.

A brief outline of this article is as follows:

  1. (1)

In Section 1 we develop the punctured cube model of $A_{n}$-cogroup objects
and prove Theorem 1.22 which states that our model is equivalent to the
existing model developed by Lurie in [Lur20, Section 4.1].

  2. (2)

In Section 2 we prove Theorem 2.5 and Theorem 2.14 which establishes the
connection between Goodwillie calculus and cogroup objects.

  3. (3)

In Section 3 we apply Theorem 2.14 to the special case of pointed spaces in
order to recover Theorem 3.2: that all 2-connected $A_{\infty}$-cogroup
objects admit a desuspension.

Acknowledgements. The author is grateful to his advisor Emily Riehl for all of
her support during the completion of this project. This work was supported by
the National Science Foundation grant DMS-1652600 and was completed during a
visit to Utrecht University whom the author thanks for their generous
hospitality. The idea that special punctured cubical diagrams could give a
model for cogroup objects originated from Jacob Lurie. Finally, the author is
immensely grateful to Gijs Heuts for suggesting this project as well as for
providing considerable feedback, mentorship, and guidance towards it
completion.

##  1\. Punctured Cubes as coGroups

In this section we develop a new model for $A_{n}$-cogroup objects in an
$\infty$-category in terms of certain punctured cubical diagrams, and prove
that this model is equivalent to the model dual to the $A_{n}$-group model
developed in [Lur20, Section 4.1]. The dual story holds for group objects as
well, but we will not need this. We begin by recalling the established
formalism of $A_{n}$-comonoid objects.

######  Definition 1.1.

Let $\Delta^{inj}$ be the wide subcategory of the category $\Delta$ containing
only the injections. Let $\Delta_{n}\hookrightarrow\Delta^{inj}$ be the full
subcategory on the objects $[0],[1],...,[n]$.

######  Remark 1.2.

Throughout this paper when we discuss maps in $\Delta$ we only ever consider
the injections.

######  Definition 1.3.

Let $\mathcal{C}$ be a pointed $\infty$-category with finite colimits. A
(unital) $A_{n}$-comonoid object in $C$ is a functor
$A:\Delta_{n}\rightarrow\mathcal{C}$ satisfying the conditions

  1. (1)

$A[0]$ is a zero object.

  2. (2)

For each $t\leq n$ the maps $p_{k}:[1]\rightarrow[t]$ sending 0 to 0 and 1 to
$k$ induce an equivalence

$\coprod_{i=1}^{n}A[1]\xrightarrow{\sim}A[t]$

  3. (3)

The following composites are the identity on $A[1]$

$A[1]\xrightarrow{\partial_{1}}A[2]\simeq A[1]\amalg A[1]\rightarrow
A[1]\amalg 0\simeq A[1]$ $A[1]\xrightarrow{\partial_{1}}A[2]\simeq A[1]\amalg
A[1]\rightarrow 0\amalg A[1]\simeq A[1]$

where the right hand arrow is the coproduct of the identity of $A[1]$ and the
zero map.

Let $coMon_{n}(\mathcal{C})\hookrightarrow Fun(\Delta_{n},\mathcal{C})$ denote
the full subcategory spanned by the $A_{n}$-comonoid objects in $\mathcal{C}$.

######  Remark 1.4.

For $n=0,1$ conditions $(2)$ and $(3)$ do not make sense. By convention an
$A_{0}$-comonoid object is a zero object, and an $A_{1}$-comonoid object is
simply an object of $\mathcal{C}$. The second condition is dual to the usual
Segal condition for a monoid, the third condition says that the
comultiplication induced by the Segal conditions is counital. Because we are
in a pointed setting we have only one choice of counit, greatly simplifying
the data needed to specify counitality.

######  Definition 1.5.

An $A_{n}$-comonoid object $A$ in $\mathcal{C}$ is cogroup-like if it
satisfies conditions 1 and 3 above, and the stronger condition $(2^{\prime})$.

  * (2’)

For each $t\leq n$ and partition of $[n]$ into subsets $S,S^{\prime}$ agreeing
on at most one element the diagram

${A[S\cap S^{\prime}]}$${A[S]}$${A[S^{\prime}]}$${A[n]}$

is a pushout square in $\mathcal{C}$.

In particular condition $(2^{\prime})$ implies the usual Segal condition
$(2)$. We will denote by $coGrp_{n}(\mathcal{C})$ the $\infty$-category of
cogroup-like $A_{n}$-comonoids in $\mathcal{C}$. We will refer to these
objects as $A_{n}$-cogroup objects.

It is worth mentioning that the cogroup-like condition (2’) can be replaced
with a more natural condition involving all pushout squares in $\Delta_{n}$.

######  Lemma 1.6.

A functor $A:\Delta_{n}\rightarrow\mathcal{C}$ satisfies condition
$(2^{\prime})$ of Definition 1.5 if and only if it satisfies the following
condition

  * (2”)

For each $t\leq n$ and any two subsets $S,S^{\prime}$ of $[t]$ the diagram

${A[S]\cap A[S^{\prime}]}$${A[S]}$${A[S^{\prime}]}$${A[S\cup S^{\prime}]}$

is a pushout square in $\mathcal{C}$.

###### Proof.

This new condition is clearly stronger than $(2^{\prime})$, so we will show
that condition $(2^{\prime})$ suffices. Consider a pushout square in
$\Delta_{n}$

${S\cap S^{\prime}}$${S}$${S^{\prime}}$${S\cup S^{\prime}}$

If $S$ and $S^{\prime}$ intersect in one element then we are done by condition
$(2^{\prime})$. Otherwise they intersect in at least two objects, call one of
them $i$ so that $S\cap S^{\prime}-\\{i\\}$ is nonempty. Consider the diagram

${A[\\{i\\}]}$${A[S\cap S^{\prime}]}$${A[S]}$${A[(S^{\prime}-\\{S\cap
S^{\prime}-\\{i\\}\\})]}$${A[S^{\prime}]}$${A[S\cup S^{\prime}]}$

By condition $(2^{\prime})$ the outside and left hand squares are pushouts, so
that the right hand square is also a pushout. ∎

######  Remark 1.7.

The property of being cogroup like is detected already at the level of the
homotopy category, see for example [GGN13, Proposition 2.3] for a similar
discussion, and does not require us to specify any additional structure, only
properties.

Now we will give our alternative model of $A_{n}$-cogroup objects in terms of
punctured cubes.

######  Definition 1.8.

Let $\square[n+1]=P(\\{0,\dots,n\\})$ denote the powerset on $n+1$ elements.
An $(n+1)$-cube in an $\infty$-category $\mathcal{C}$ is a functor

$\mathcal{X}:\square[n+1]\rightarrow\mathcal{C}$

######  Definition 1.9.

Let $\square[n+1]_{0}=P_{0}(\\{0,\dots,n\\})$ denote the nerve of the powerset
on $n+1$ elements except that we remove the empty set. A punctured
$(n+1)$-cube in $\mathcal{C}$ is a functor

$\mathcal{X}:\square[n+1]_{0}\rightarrow\mathcal{C}$

In words it is an $(n+1)$-cube missing the initial vertex.

######  Definition 1.10.

A punctured $(n+1)$-cube
$\mathcal{X}:P_{0}(\\{0,\dots,n\\})\rightarrow\mathcal{C}$ is called special
if

  1. (1)

For every one element subset $s\in P(\\{0,\dots,n\\})$ there is an equivalence
$\mathcal{X}(s)\simeq 0$.

  2. (2)

The cube is strongly cocartesian: every face which exists in the cube is a
pushout.

Following [Heu15] we let $T_{n}\mathcal{C}$ denote the $\infty$-category of
special punctured $(n+1)$-cubes (note the change in index).

We will find it convenient to occasionally work with a different description
of $P_{0}(\\{1,\dots,n\\})$

######  Lemma 1.11.

There is an isomorphism of categories

$\square[n+1]_{0}\simeq\Delta^{inj}/[n]$

between the punctured $(n+1)$-cube and the slice category of $\Delta^{inj}$
over the object $[n]$.

###### Proof.

The idea is that an injection $[k]\to[n]$ designates a unique subset of $[n]$,
and this allows us to pass between the two categories. ∎

Under the above isomorphism there is a natural functor

$\pi_{n}:Fun(\Delta_{n},\mathcal{C})\to Fun(\square[n+1]_{0},\mathcal{C})$

given by restricting along the projection
$\pi_{n}:\Delta^{inj}/[n]\rightarrow\Delta_{n}$, which by Lemma 1.6 restricts
to a functor

$\pi_{n}:coGrp_{n}(\mathcal{C})\rightarrow T_{n}(\mathcal{C})$

######  Example 1.12.

The functor $Fun(\Delta_{2},\mathcal{C})\to Fun(\square[3]_{0},\mathcal{C})$
blows up a truncated cosimplicial object

${A[0]}$${{A[1]}}$${{A[2]}}$$\scriptstyle{\partial_{0}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\partial_{0}}$

into the punctured 3-cube

${{A[0]}}$${{A[0]}}$${{A[1]}}$${{A[0]}}$${{A[1]}}$${{A[1]}}$${{A[2]}}$$\scriptstyle{\partial_{0}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{0}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{0}}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\partial_{0}}$

For the remainder of this section we will establish our first main result,
Theorem 2.5, which states that the functor $\pi_{n}$ provides an equivalence
of models of $A_{n}$-cogroups. Because the proof is not exactly enlightening,
we give an explicit description of the $n=2$ case which contains our key
intuition.

######  Example 1.13 (Theorem 1.22 for $n=2$).

The data of a special punctured 3-cube is a diagram

${0}$${0}$${A}$${0}$${A^{\prime\prime}}$${A^{\prime}}$${A^{\prime\prime\prime}}$

with every face a pushout. Consider the diagram formed by the bottom and right
square faces.

${0}$${A^{\prime\prime}}$${0}$${A^{\prime}}$${A^{\prime\prime\prime}}$${A}$${A^{\prime}}$${Cof(a)}$${A}$$\scriptstyle{a}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

Because the squares are pushout, taking cofibers yields the equivalences as
displayed, giving an equivalence $A\simeq A^{\prime}$. By symmetry we have
also $A\simeq A^{\prime\prime}$. We can identify each non zero vertex with $A$
and $A\vee A$, as well as the bottom face with the coproduct inclusions.

${0}$${0}$${A}$${0}$${A}$${A}$${A\vee
A}$$\scriptstyle{m}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$

Note that once we substitute these equivalences, we are left with the
interesting right hand vertical map $m:A\to A\vee A$. Being strong cocartesian
tells us that $m$ satisfies counital properties as demonstrated below

${0}$${A}$${0}$${A}$${A}$${A\vee A}$${A}$${A\vee
A}$${A}$${A}$$\scriptstyle{m}$$\scriptstyle{m}$$\scriptstyle{i_{2}}$$\scriptstyle{0}$$\scriptstyle{i_{1}}$$\scriptstyle{0}$

Where the unique induced maps on the above left and right are respectively
$A\vee A\xrightarrow{A\vee\eta}A$ and $A\vee A\xrightarrow{\eta\vee A}A$.

By Definition 1.3 this gives the object ${0}$${A}$${A\vee A}$, obtained by
collapsing the cube, the structure of an $A_{2}$-comonoid. Furthermore,
because every face is a pushout, this comonoid object is cogroup like.

Now suppose we have a morphism in $T_{2}\mathcal{C}$ between the two punctured
cubes in which we’ve made the above identifications

${0}$${0}$${0}$${A}$${0}$${B}$${0}$${A}$${0}$${B}$${A}$${A\vee A}$${B}$${B\vee
B}$$\scriptstyle{m_{A}}$$\scriptstyle{m_{B}}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{i_{2}}$

The data of this morphism consists of three maps $f_{1},f_{2},f_{3}:A\to B$ as
well as a map $f_{4}:A\vee A\to B\vee B$ which makes the diagram formed by
these maps and the two cubes commute. Because the bottom squares of each cube
are pushouts we have $f_{4}\simeq f_{1}\vee f_{2}$

${0}$${A}$${A}$${A\vee A}$${B\vee
B}$$\scriptstyle{i_{2}}$$\scriptstyle{i_{2}f_{1}}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{1}f_{2}}$$\scriptstyle{f_{4}}$

By one half of counitality we obtain an equivalence $f_{3}\simeq f_{1}$ as
shown below

${A}$${B}$${A}$${B}$${A\vee A}$${B\vee
B}$${=}$${A}$${B}$${A}$${B}$$\scriptstyle{f_{3}}$$\scriptstyle{m_{A}}$$\scriptstyle{m_{B}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{1}\vee
f_{3}}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{1}}$

Likewise we have $f_{3}\simeq f_{2}$. Let $f$ denote any map equivalent to one
of $f_{1},f_{2},f_{3}$. The morphism of punctured cubes gives a morphism of
$A_{2}$-cogroup objects

${0}$${A}$${A\vee A}$${0}$${B}$${B\vee B}$$\scriptstyle{f}$$\scriptstyle{f\vee
f}$

Similarily we can show in general that a $k$-simplex in $T_{n}\mathcal{C}$
gives rise to a $k$-simplex in $coGrp_{2}(\mathcal{C})$ in an essentially
unique way, so that there is an equivalence $T_{2}\mathcal{C}\simeq
coGrp_{2}(\mathcal{C})$.

We will prove Theorem 1.22 using induction by considering the process of
building an $A_{n}$-cogroup object out of an $A_{n-1}$-cogroup object. Lurie
tackles this question in [Lur20, Section 4.1] by introducing an intermediate
object, an $A_{n}$-precogroup object, which adds no additional information to
an $A_{n-1}$-cogroup object.

######  Definition 1.14.

Let $\Delta_{n}^{0}$ be the largest simplicial subset of $\Delta_{n}$ which
does not include any simplex which includes the edge $e:[1]\to[n]$ as a face.

######  Definition 1.15.

An $A_{n}$-precogroup object of $\mathcal{C}$ is a functor
$A^{0}:\Delta_{n}^{0}\to\mathcal{C}$ which satisfies the conditions $(1)$ and
$(2^{\prime})$ of Definition 1.5, when they make sense, and additionally
satisfies condition $(3)$ if $n>2$. We denote the $\infty$-category of
$A_{n}$-precogroup objects in $\mathcal{C}$ by $coGrp_{n}^{pre}(\mathcal{C})$.

######  Theorem.

[Lur20, Theorem 4.1.5.8] The inclusion
$\Delta_{n-1}\hookrightarrow\Delta^{0}_{n}$ induces a trivial fibration

$coGrp_{n}^{pre}(\mathcal{C})\overset{\sim}{\longrightarrow}coGrp_{n-1}(\mathcal{C})$

We will utilize similar objects defined as diagrams on a cubical version of
$\Delta^{0}_{n}$.

######  Definition 1.16.

Let $\square[n+1]_{0}^{0}$ be defined by the pullback of simplicial sets

${\square[n+1]_{0}^{0}}$${\Delta^{0}_{n}}$${\square[n+1]_{0}}$${\Delta_{n}}$

######  Lemma 1.17.

The pullback square

${\square[n+1]_{0}^{0}}$${\Delta^{0}_{n}}$${\square[n+1]_{0}}$${\Delta_{n}}$

is additionally a pushout square.

###### Proof.

Consider a chain of inclusions
$[m_{0}]\xrightarrow{f_{1}}[m_{1}]\xrightarrow{f_{2}}\dots\xrightarrow{f_{k}}[m_{k}]$
in $\Delta_{n}$ which does not belong to $\Delta_{n}^{0}$. We want to show
that such a $k$-simplex lifts to one in $\square[n+1]_{0}$. Necessarily
$[m_{k}]=[n]$. The lift is given by the image of the chain of inclusions
$J_{0}\subseteq J_{1}\subseteq\dots\subseteq[n]$ where
$J_{i}=im(f_{k}\circ\cdots\circ f_{i+1})$. This is similar to the proof of
[Lur20, Proposition 4.1.5.7]. ∎

There is an inclusion

$P(\\{0,\dots,n-1\\})\cup P_{0}(\\{1,\dots,n\\})\hookrightarrow
P_{0}(\\{0,\dots,n\\})$

by partitioning $P_{0}(\\{0,\dots,n\\})$ into those subsets which respectively
contain and do not contain 0. By pulling back this gives a functor

$p:Fun(\square[n+1]_{0},\mathcal{C})\to Fun(\square[n],\mathcal{C})\times
Fun(\square[n]_{0},\mathcal{C})$

######  Definition 1.18.

Let $Fun^{sco}(\square[n]_{0},\mathcal{C})$ denote the full subcategory of
$Fun(\square[n]_{0},\mathcal{C})$ spanned by the $(n)$-cubes which are
strongly cocartesian and which have initial vertex a 0 object. We call these
objects special cocartesian $(n)$-cubes.

The functor $p$ restricts to a functor on special punctured cubes

$T_{n}(\mathcal{C})\to Fun^{sco}(\square[n],\mathcal{C})\times
T_{n-1}(\mathcal{C})$

In this setup we view a special punctured $(n+1)$-cube is containing the data
of a special punctured $n$-cube, a special cocartesian $n$-cube, and a
collection of “connecting structure” between the two.

The functor $p$ factors through a functor

$Fun(\square[n+1]_{0}^{0},\mathcal{C})\to Fun(\square[n],\mathcal{C})\times
Fun(\square[n]_{0},\mathcal{C})$

which motivates a definition of special $\square[n+1]_{0}^{0}$ diagrams.

######  Definition 1.19.

We define a category $T_{n}^{0}(\mathcal{C})$ of special
$\square[n+1]_{0}^{0}$ diagrams in an $\infty$-category $\mathcal{C}$ as the
pullback

${T_{n}^{0}(\mathcal{C})}$${Fun(\square[n]_{0}^{0},\mathcal{C})}$${Fun^{sco}(\square[n],\mathcal{C})\times
T_{n-1}(\mathcal{C})}$${Fun(\square[n],\mathcal{C})\times
Fun(\square[n]_{0},\mathcal{C})}$

Note that the forgetful functor

$Fun(\square[n+1]_{0},\mathcal{C})\to Fun(\square[n+1]_{0}^{0},\mathcal{C})$

restricts to a functor $T_{n}(\mathcal{C})\to T_{n}^{0}(\mathcal{C})$.

Again we can view an object of $T_{n}^{0}(\mathcal{C})$ as special punctured
$n$-cube, a special cocartesian $n$-cube, and a smaller collection of
connecting structure between the two. A key idea will be that the special
cocartesian $n$-cubes coming from a special punctured $(n+1)$-cube are already
determined by the rest of the structure. Already the $\infty$-category
$Fun^{sco}(\square[n],\mathcal{C})$ contains very little information.

######  Lemma 1.20.

The functor
$Fun^{sco}(\square[n],\mathcal{C})\xrightarrow{\sim}\mathcal{C}^{n}$ given by
pulling back along the inclusion of cardinality one subsets of $[n]$ is an
equivalence.

###### Proof.

There is a fully faithful left Kan extension functor

$Fun(P_{\leq 1}(\\{0,\dots,n-1\\},\mathcal{C})\hookrightarrow
Fun(\square[n],\mathcal{C})$

from diagrams defined on subsets of $[n-1]$ of cardinality less than two to
diagrams defined on the entire powerset of $[n-1]$.
$Fun^{sco}(\square[n],\mathcal{C})$ can be identified with the image of this
functor on the subcategory of $Fun(P_{\leq 1}(\\{0,\dots,n-1\\},\mathcal{C})$
of diagrams with initial vertex a 0 object. This category is equivalent to
$\mathcal{C}^{n}$. ∎

In Example 1.13 we saw by taking cofibers that a special punctured 3-cube
takes all initial non-zero vertices to equivalent objects. We will need a
generalization of this to special punctured $n$-cubes.

######  Lemma 1.21.

A special punctured $(n+1)$-cube $\mathcal{X}:\square[n+1]_{0}\to\mathcal{C}$
takes all cardinality two subsets of $[n]$ to equivalent objects in
$\mathcal{C}$.

###### Proof.

The case for $n=2$ was proved already in Example 1.13. If $n>2$ we can choose
four distinct elements $\\{a,b,c,d\\}$. Consider the subsets
$\\{a,b\\},\\{b,c\\}$ and form the special punctured 3-cube

${\\{b\\}}$${\\{c\\}}$${{\\{b,c\\}}}$${\\{a\\}}$${{\\{a,b\\}}}$${{\\{a,c\\}}}$${{\\{a,b,c\\}}}$

Any special punctured $n$-cube $\mathcal{X}$ takes this subdiagram to a
special punctured 3-cube, providing an equivalence
$\mathcal{X}\\{a,c\\}\simeq\mathcal{X}\\{b,c\\}$ via the argument of Example
1.13. Likewise we can form a punctured 3-cube amongst $\\{b,c,d\\}$ and obtain
an equivalence $\mathcal{X}\\{b,c\\}\simeq\mathcal{X}\\{b,d\\}$, giving an
equivalence $\mathcal{X}\\{a,c\\}\simeq\mathcal{X}\\{b,d\\}$. In this way we
can connect any two cardinality two subsets of $[n]$ via an equivalence. ∎

We are now ready to prove our first main result.

######  Theorem 1.22.

Let $\mathcal{C}$ be a pointed $\infty$-category with finite colimits. The
functor $\pi_{n}^{*}:coGrp_{n}(\mathcal{C})\rightarrow T_{n}(\mathcal{C})$ is
an equivalence of $\infty$-categories

###### Proof.

This is clearly the case for $n=1$. We will show that the square

${coGrp_{n}(\mathcal{C})}$${T_{n}(\mathcal{C})}$${coGrp_{n-1}(\mathcal{C})}$${T_{n-1}(\mathcal{C})}$$\scriptstyle{\pi_{n}}$$\scriptstyle{\pi_{n-1}}$

is a pullback, proving the claim by induction. This will be accomplished by
showing that for any $A_{n-1}$-cogroup object $A$ in $\mathcal{C}$ there is an
equivalence of fibers

$coGrp_{n}(\mathcal{C})_{A}\simeq T_{n}(\mathcal{C})_{\pi_{n-1}A}$

By [Lur20, Theorem 4.1.5.8] there is an essentially unique extension of $A$ to
an $A_{n}$-precogroup object $A^{0}:\Delta^{0}_{n}\to\mathcal{C}$ and so an
equivalence of fibers $coGrp_{n}(\mathcal{C})_{A}\simeq
coGrp_{n}(\mathcal{C})_{A^{0}}$. From the pushout square of Lemma 1.17 we
obtain a pullback square

${coGrp_{n}(\mathcal{C})}$${T_{n}(\mathcal{C})}$${coGrp_{n}^{pre}(\mathcal{C})}$${T^{0}_{n}(\mathcal{C})\par}$$\scriptstyle{\pi_{n}}$$\scriptstyle{\pi^{0}_{n}}$

and hence an equivalence of fibers $coGrp_{n}(\mathcal{C})_{A^{0}}\simeq
T_{n}(\mathcal{C})_{\pi^{0}_{n}A^{0}}$. It suffices to show now that the
natural map of fibers $T_{n}(\mathcal{C})_{\pi^{0}_{n}A^{0}}\to
T_{n}(\mathcal{C})_{\pi_{n-1}A}$ is an equivalence.

By abuse of notation, let $A$ denote the object $A[1]$, The functor
$\pi_{n}^{0}A^{0}$ factors via the pullback in the diagram below since the
cardinality two vertices are all equal to $A$. Note also that the pullback in
the bottom left vertex of the diagram below is a homotopy pullback, so that
the square is an actual pullback.

${{[0]}}$${\bullet}$${{Fun(\square[n+1]_{0}^{0},\mathcal{C})}}$${{(Fun^{sco}(\square[n],\mathcal{C})\times
T_{n-1}(\mathcal{C}))\times_{C^{2n}}\\{A\\}}}$${{Fun(\square[n],\mathcal{C})\times
Fun(\square[n]_{0},\mathcal{C})}}$$\scriptstyle{\pi_{n}^{0}A^{0}}$

The weak equivalence
$Fun^{sco}(\square[n],\mathcal{C})\xrightarrow{\sim}\mathcal{C}^{n}$ of Lemma
1.20 gives a weak equivalence

$(Fun^{sco}(\square[n],\mathcal{C})\times
T_{n-1}(\mathcal{C}))\times_{\mathcal{C}^{2n}}\\{A\\}\xrightarrow{\sim}T_{n-1}(\mathcal{C})\times_{\mathcal{C}^{n}}\\{A\\}$

which allows us to greatly simplify the bottom left vertex of the above
diagram.

Note now that $\pi_{n}^{0}A^{0}$ lands in
$T_{n}^{0}(\mathcal{C})\hookrightarrow Fun(\square[n+1]_{0}^{0},\mathcal{C})$
and that under the above equivalence
$T_{n-1}(\mathcal{C})\times_{\mathcal{C}^{n}}\\{A\\}$ maps into

$T_{n-1}(\mathcal{C})\times\mathcal{C}^{n}\simeq
Fun^{sco}(\square[n],\mathcal{C})\times T_{n-1}(\mathcal{C})\ \hookrightarrow
Fun(\square[n],\mathcal{C})\times Fun(\square[n]_{0},\mathcal{C})$

So that the above square factors and we can compute the fiber
$T_{n}(\mathcal{C})_{\pi_{n}^{0}A^{0}}$ equally as the pullback in the left
hand square below, which is seen to be a pullback using two applications of
the pullback pasting lemma.

${T_{n}(\mathcal{C})_{\pi_{n}^{0}A^{0}}}$${P^{0}}$${T_{n}(\mathcal{C})}$${{[0]}}$${\bullet}$${T_{n}^{0}(\mathcal{C})}$${T_{n-1}(\mathcal{C})\times_{\mathcal{C}^{n}}\\{A\\}}$${{T_{n-1}(\mathcal{C})\times\mathcal{C}^{n}}}$

Likewise $\pi_{n-1}A$ factors via
$T_{n-1}(\mathcal{C})\times_{\mathcal{C}^{n}}\\{A\\}$ and we can compute the
fiber $T_{n}(\mathcal{C})_{\pi_{n-1}A}$ as the left hand pullback in

${T_{n}(\mathcal{C})_{\pi_{n-1}A}}$${P}$${T_{n}(\mathcal{C})}$${{[0]}}$${T_{n-1}(\mathcal{C})\times_{\mathcal{C}^{n}}\\{A\\}}$${T_{n-1}(\mathcal{C})}$

Now compare the above right square with the square in the definition of
$P^{0}$.

${P^{0}}$${T_{n}(\mathcal{C})}$${T_{n-1}(\mathcal{C})\times_{\mathcal{C}^{n}}\\{A\\}}$${T_{n-1}(\mathcal{C})\times\mathcal{C}^{n}}$

We consider the difference in these two squares. By construction we demand
that the special punctured $(n+1)$-cubes in $P^{0}$ and $P$ both are equal to
$A$ on every initial vertex in the subset
$\square[n]_{0}\hookrightarrow\square[n+1]_{0}$. In addition, in $P^{0}$ we
demand that the vertices of cardinality one in
$\square[n]\hookrightarrow\square[n+1]_{0}$ are also equal to $A$. By Lemma
1.21 this later condition is automatic in a special punctured $(n+1)$-cube
already: the special $n$-cube in a special punctured $(n+1)$-cube provides no
additional data. Because of this we have an equivalence $P\simeq P^{0}$, and
so finally we obtain an equivalence of fibers
$T_{n}(\mathcal{C})_{\pi_{n}^{0}A^{0}}\simeq T_{n}(\mathcal{C})_{\pi_{n-1}A}$
as desired. ∎

##  2\. coGroups in Goodwillie Calculus

We begin by recalling the parts of the theory of Goodwillie approximation to
$\infty$-categories to which we will apply Theorem 1.22. We work always in the
category $\mathcal{C}\\!\mathit{at}^{\omega}$ of pointed, compactly generated
$\infty$-categories with morphisms the left adjoint functors which preserve
compact objects. Note that by the adjoint functor theorem such a functor
necessarily admits a right adjoint. For more details on this category see
[Heu15, Appendix A].

######  Definition 2.1.

(Heuts) An adjunction $F:\mathcal{C}\rightleftarrows\mathcal{D}:G$ is a weak
$n$-excisive approximation to $\mathcal{C}$ if

  * •

The identity $1_{\mathcal{D}}$ is $n$-exsisive.

  * •

The induced maps $P_{n}(1_{\mathcal{C}})\xrightarrow{\sim}GF$ and
$P_{n}(FG)\xrightarrow{\sim}1_{\mathcal{D}}$ are equivalences.

######  Definition 2.2.

(Heuts) An $\infty$-category $\mathcal{D}$ is $n$-excisive if every weak
$n$-excisive approximation to $\mathcal{D}$ is an equivalence. An adjunction
$F:\mathcal{C}\rightleftarrows\mathcal{D}:G$ is an $n$-excisive approximation
to $\mathcal{C}$ if it is a weak $n$-excisive approximation and the
$\infty$-category $\mathcal{D}$ is $n$-excisive.

######  Example 2.3.

An $\infty$-category is $1$-excisive if and only if it is stable. The
$1$-excisive approximation to $\mathcal{S}_{\star}$, the $\infty$-category of
pointed spaces, is the usual stabilization adjunction
$\Sigma^{\infty}:\mathcal{S}_{\star}\rightleftarrows Sp:\Omega^{\infty}$.

Just as one can construct $n$-excisive approximations to functors, there is a
natural construction of an $n$-excisive approximation to an $\infty$-category
using the category $T_{n}\mathcal{C}$.

Define the $\infty$-category $\overline{T_{n}}\mathcal{C}$ of (nonpunctured)
$(n+1)$-cubes in $\mathcal{C}$, whose objects consist of those cubes with a
zero object at the one element subsets and such that every face is a pushout.
Such a cube is completely determined by its initial vertex: it is a right Kan
extension for the initial vertex to the cardinality one subsets followed by a
left Kan extension to the entire cube. There is a natural span of functors

${\overline{T_{n}}\mathcal{C}}$${\mathcal{C}}$${T_{n}\mathcal{C}}$$\scriptstyle{\sim}$

with left leg, evaluation at the initial vertex, a trivial Kan fibration by
[Lur09, Thm 4.3.2.15], and with right leg the functor that forgets the initial
vertex. Choosing a section to the trivial fibration gives a functor
$L_{n}:\mathcal{C}\rightarrow T_{n}\mathcal{C}$ which sends an object of
$\mathcal{C}$ to the punctured $(n+1)$-cube which successively takes pushouts
of suspensions of the object, and then forgets the initial vertex. Note that
the functor $L_{n}$ admits a right adjoint
$R_{n}:T_{n}\mathcal{C}\rightarrow\mathcal{C}$ which takes the limit of the
punctured cube.

######  Example 2.4.

Let $A\in\mathcal{C}$. The functor $L_{2}:\mathcal{C}\rightarrow
T_{n}\mathcal{C}$ takes A to a cube with vertices equivalent to the objects
below left, and then forgets the initial vertex giving the cube below right

${A\ }$${0}$${0}$${\Sigma A}$${0}$${\Sigma A}$${\Sigma A}$${\Sigma A\vee\Sigma
A\par}$${0}$${0}$${\Sigma A}$${0}$${\Sigma A}$${\Sigma A}$${\Sigma A\vee\Sigma
A}$

Note that identifying the bottom square with the coproduct inclusion of
$\Sigma A$ into $\Sigma A\vee\Sigma A$ we recover the $A_{2}$-cogroup
structure on $\Sigma A$, as in Example 1.13.

Using the functor $L_{n}:\mathcal{C}\rightarrow T_{n}\mathcal{C}$ one may
build a chain

$\mathcal{C}\rightarrow T_{n}\mathcal{C}\rightarrow
T_{n}(T_{n}\mathcal{C})\rightarrow
T_{n}(T_{n}(T_{n}\mathcal{C}))\rightarrow\dots
P_{n}\mathcal{C}=\operatorname*{colim}_{k}T_{n}^{k}\mathcal{C}$

and an adjunction $\Sigma^{\infty}_{n}\mathcal{C}\leftrightarrows
P_{n}\mathcal{C}:\Omega^{\infty}_{n}$.

######  Proposition.

[Heu15, Proposition 3.15] Let $\mathcal{C}$ be a pointed, compactly generated
$\infty$-category. Then the adjunction
$\Sigma^{\infty}_{n}\mathcal{C}\rightleftarrows
P_{n}\mathcal{C}:\Omega^{\infty}_{n}$ resulting from the above construction is
an $n$-excisive approximation to $\mathcal{C}$.

Using the forgetful functors $T_{n}(\mathcal{C})\to T_{n-1}(\mathcal{C})$ we
can construct a tower of compact object preserving left adjoint functors

$\dots\xrightarrow{\Sigma_{n+2,n+1}}P_{n+1}\mathcal{C}\xrightarrow{\Sigma_{n+1,n+}}P_{n}\mathcal{C}\xrightarrow{\Sigma_{n,n-1}}\dots
P_{1}\mathcal{C}\simeq Sp(\mathcal{C})$

In fact $n$-excisiveness is detected already at the level of
$T_{n}\mathcal{C}$.

######  Proposition.

[Heu15, Proposition 3.16] Let $\mathcal{C}$ be a pointed, compactly generated
$\infty$-category. Then $\mathcal{C}$ is $n$-excisive if and only if the
functor $L_{n}:\mathcal{C}\rightarrow T_{n}\mathcal{C}$ is an equivalence.

This characterization of $n$-excisive $\infty$-categories [Heu15, Proposition
3.16] combined with the equivalence $coGrp_{n}(\mathcal{C})\simeq
T_{n}(\mathcal{C})$ immediately give our second main result.

######  Theorem 2.5.

Let $\mathcal{C}$ be a pointed, compactly generated $\infty$-category. Then
$\mathcal{C}$ is $n$-excisive if and only if the suspension functor
$\Sigma_{\mathcal{C}}$ induces an equivalence

$\Sigma:\mathcal{C}\xrightarrow{\sim}coGrp_{n}(\mathcal{C})$

between $\mathcal{C}$ and the $\infty$-category of $A_{n}$-cogroup objects in
$\mathcal{C}$.

######  Corollary 2.6.

Let $\mathcal{C}$ be a pointed, compactly generated $\infty$-category, then
the suspension functor on $P_{n}\mathcal{C}$ induces an equivalence

$P_{n}\mathcal{C}\xrightarrow{\sim}coGrp_{n}(P_{n}\mathcal{C})$

Theorem 2.5 has a nice conceptual interpretation. Of course in a stable
$\infty$-category all objects may be desuspended, hence desuspension follows
from the trivial $A_{1}$-cogroup structure on an object. In the unstable
setting, dual to the recognition theorem for loop spaces, one would assume a
full $A_{\infty}$-cogroup structure is needed for a desuspension111We take on
this idea in the case of spaces in Section 3.. What we have proved here is
that in the intermediate $n$-excisive setting an $A_{n}$-cogroup structure
suffices, and in fact this characterizes exactly the $n$-excisive setting.

It is interesting to reflect for a moment on the process of stabilization in a
new light. When one stabilizes an $\infty$-category $\mathcal{C}$ they invert
the suspension functor $\mathcal{C}\xrightarrow{\Sigma}\mathcal{C}$ by forming
the colimit

$\mathcal{C}\xrightarrow{\Sigma}\mathcal{C}\xrightarrow{\Sigma}\mathcal{C}\xrightarrow{\Sigma}\cdots\rightarrow
P_{1}\mathcal{C}$

This process forgets a lot of information about the values of the suspension
functor, in particular the $A_{\infty}$-cogroup structure that they carry. We
can carry out a less harshly forgetful inversion by remembering a part of this
structure, say the $A_{n}$-cogroup structure, and then inverting the
suspension functor by forming the colimit

$\mathcal{C}\xrightarrow{\Sigma}coGrp_{n}(\mathcal{C})\xrightarrow{\Sigma}coGrp_{n}(coGrp_{n}(\mathcal{C}))\xrightarrow{\Sigma}\cdots\rightarrow
P_{n}\mathcal{C}$

Theorem 2.5 says that forming the $n$-excisive approximation
$P_{n}\mathcal{C}$ is exactly this procedure. We can also give the following
slight weakening

######  Proposition 2.7.

Let $\mathcal{C}$ be a pointed, compactly generated $\infty$-category, then
$1_{\mathcal{C}}$ is $n$-excisive if and only if the suspension functor
$\Sigma:\mathcal{C}\rightarrow coGrp_{n}(\mathcal{C})$ is fully faithful.

###### Proof.

This follows from Theorem 2.5 since $P_{n}\mathcal{C}$ is a filtered colimit
of fully faithful functors $\Sigma$ starting from $\mathcal{C}$. ∎

Our goal now is to push Theorem 2.5 to the limit and study
$A_{\infty}$-cogroup objects in the limit of the tower of excisive
approximations. To begin we will state some technical results in the following
common setup.

######  Notation 2.8.

Consider a chain of left adjoint functors of compactly generated
$\infty$-categories

$\dots\xrightarrow{f_{n+2}}\mathcal{C}_{n+1}\xrightarrow{f_{n+1}}\mathcal{C}_{n}\xrightarrow{f_{n}}\dots\rightarrow
C_{1}$

Let $\mathcal{C}$ denote the limit, and
$\pi_{n}:\mathcal{C}\rightarrow\mathcal{C}_{n}$ the $n^{th}$ projection
functor.

######  Definition 2.9.

We say a family of functors $f_{\alpha}:C\to C_{\alpha}$ jointly creates
colimits (or equivalences) if the induced functor
$f:C\to\prod_{\alpha}C_{\alpha}$ creates colimits (resp. equivalences).

######  Lemma 2.10.

In the setup of Notation 2.8 the functors
$\mathcal{C}\xrightarrow{\pi_{n}}\mathcal{C}_{n}$ jointly create equivalences
and colimits.

###### Proof.

This follows from [RV14, Theorem 1.1] or as explained in the (not static!)
discussion at the end of the proof of [RV20, Proposition 8.2.11]. ∎

######  Proposition 2.11.

In the setup of Notation 2.14, for any $n$ there are equivalences of
$\infty$-categories

$coMon_{n}(C)\xrightarrow{\sim}lim_{k}coMon_{n}(C_{k})$

###### Proof.

There is an equivalence

$Fun(\Delta_{n},\mathcal{C})\xrightarrow{\sim}lim_{k}Fun(\Delta_{n},\mathcal{C}_{k})$

which we must show lifts to an equivalence

$coMon_{n}(\mathcal{C})\xrightarrow{\sim}lim_{k}coMon_{n}(C_{k})$

By Lemma 2.10 the legs of the limit cone
$\mathcal{C}\rightarrow\mathcal{C}_{k}$ create colimits and equivalences, so
they create the Segal condition (2’) of Definition 1.5. ∎

Now that we understand how to form $A_{n}$-cogroup objects in the limit of a
tower of $\infty$-categories each fixed $n$, we deal with the limit as $n$
goes to infinity.

######  Lemma 2.12.

Let $\mathcal{C}$ be an $\infty$-category. The sequence of forgetful functors

$coMon_{\infty}(\mathcal{C})\rightarrow\dots\rightarrow
coMon_{2}(\mathcal{C})\rightarrow coMon_{1}(\mathcal{C})$

induces an equivalence

$coMon_{\infty}(\mathcal{C})\overset{\sim}{\longrightarrow}lim_{k}coMon_{k}(\mathcal{C})$

###### Proof.

This is dual to the unital version of [Lur20, Prop 4.1.4.9]. ∎

Because being cogroup-like can be tested already at the level of the homotopy
category, combining Proposition 2.11 and Lemma 2.12 we obtain:

######  Proposition 2.13.

In the setup of Notation 2.8 there is an equivalence of $\infty$-categories

$coMon_{\infty}(\mathcal{C})\xrightarrow{\sim}lim_{k,n}coMon_{k}(\mathcal{C}_{n})$

which decends to an equivalence

$coGrp_{\infty}(\mathcal{C})\xrightarrow{\sim}lim_{k,n}coGrp_{k}(\mathcal{C}_{n})$

on the level of cogroup objects.

We will now apply Proposition 2.13 to the special case of Goodwillie calculus.
Recall that there is a sequence of compact object preserving left adjoint
functors

$\dots\xrightarrow{\Sigma_{n+2,n+1}}P_{n+1}\mathcal{C}\xrightarrow{\Sigma_{n+1,n+}}P_{n}\mathcal{C}\xrightarrow{\Sigma_{n,n-1}}\dots
P_{1}\mathcal{C}\simeq Sp(\mathcal{C})$

We let $P_{\infty}\mathcal{C}$ denote the limit of this tower in the category
$\mathcal{C}\\!\mathit{at}^{\omega}$, which is complete by [Heu15, Appendix
A]. In fact one can compute this limit by passing to the sequence

$\dots\xrightarrow{\Sigma_{n+2,n+1}}(P_{n+1}\mathcal{C})^{c}\xrightarrow{\Sigma_{n+1,n+}}(P_{n}\mathcal{C})^{c}\xrightarrow{\Sigma_{n,n-1}}\dots(P_{1}\mathcal{C})^{c}\simeq
Sp(\mathcal{C}^{c})$

computing the limit in $\mathcal{C}\\!\mathit{at}$, and taking Ind-objects of
the resulting category.

######  Theorem 2.14.

Let $\mathcal{C}$ be a pointed, compactly generated $\infty$-category. The
suspension functor

$\Sigma:P_{\infty}\mathcal{C}\overset{\sim}{\longrightarrow}coGrp_{\infty}(P_{\infty}\mathcal{C})$

is an equivalence of $\infty$-categories.

###### Proof.

The equivalences
$P_{n}(\mathcal{C})\xrightarrow{\Sigma}coGrp_{n}(P_{n}\mathcal{C})$ assemble
into a tower of equivalences

${P_{\infty}\mathcal{C}}$${lim_{n}coGrp_{n}(P_{n}\mathcal{C})}$${P_{n+1}\mathcal{C}}$${coGrp_{n+1}(P_{n+1}\mathcal{C})}$${P_{n}\mathcal{C}}$${coGrp_{n}(P_{n}\mathcal{C})}$${P_{1}\mathcal{C}}$${coGrp_{1}(P_{1}\mathcal{C})}$$\scriptstyle{\sim}$$\scriptstyle{\Sigma}$$\scriptstyle{\sim}$$\scriptstyle{\Sigma}$$\scriptstyle{\sim}$$\scriptstyle{\Sigma}$

Providing an equivalence
$P_{\infty}\xrightarrow{\sim}lim_{n}coGrp_{n}(P_{n}\mathcal{C})$. Because the
functors $P_{\infty}\mathcal{C}\rightarrow P_{n}\mathcal{C}$ jointly create
colimits we may identify this functor on underlying objects with the
suspension $\Sigma:P_{\infty}\mathcal{C}\rightarrow P_{\infty}\mathcal{C}$.

On the other hand, the family of functors
$P_{\infty}\mathcal{C}\xrightarrow{\pi_{n}}P_{n}\mathcal{C}$ satisfy the setup
of Notation 2.8, and so Proposition 2.13 provides an equivalence

$coGrp_{\infty}(P_{\infty}\mathcal{C})\xrightarrow{\sim}lim_{k,n}coGrp_{k}(P_{n}\mathcal{C})$

However, the diagonal functor
$\Delta:\mathbb{N}\rightarrow\mathbb{N}\times\mathbb{N}$ is final, inducing an
equivalence

$lim_{n}coGrp_{n}(P_{n}\mathcal{C})\xrightarrow{\sim}lim_{k,n}coGrp_{k}(P_{n}\mathcal{C})$

Providing the desired equivalence
$\Sigma:P_{\infty}\mathcal{C}\xrightarrow{\sim}coGrp_{\infty}(P_{\infty}\mathcal{C})$
which finishes the proof of Theorem 2.14. ∎

As in the discussion following Theorem 2.5, one could attempt to invert the
suspension with its full structure

$\mathcal{C}\xrightarrow{\Sigma}coGrp_{\infty}(\mathcal{C})\xrightarrow{\Sigma}coGrp_{\infty}(coGrp_{\infty}(\mathcal{C}))\xrightarrow{\Sigma}\cdots\rightarrow\hat{P}_{\infty}\mathcal{C}$

defining an $\infty$-category $\hat{P}_{\infty}\mathcal{C}$ as the colimit.
This begs the question question: are there equivalences
$\mathcal{C}\simeq\hat{P}_{\infty}\mathcal{C}$ or
$\hat{P}_{\infty}\mathcal{C}\simeq P_{\infty}\mathcal{C}$? Unfortunately
neither of these seem to be true. There is a natural map
$\hat{P}_{\infty}\mathcal{C}\rightarrow P_{\infty}\mathcal{C}$, which can be
obtained for example by contemplating the following
$\mathbb{N}\times\mathbb{N}^{op}$ indexed diagram
$\mathcal{J}_{\bullet,\bullet}$.

${\vdots}$${\vdots}$${\vdots}$${C}$${coGrp_{3}(\mathcal{C})}$${coGrp_{3}(coGrp_{3}(\mathcal{C}))}$${\cdots}$${C}$${coGrp_{2}(\mathcal{C})}$${coGrp_{2}(coGrp_{2}(\mathcal{C}))}$${\cdots}$${C}$${coGrp_{1}(\mathcal{C})}$${coGrp_{1}(coGrp_{1}(\mathcal{C}))}$${\cdots}$

######  Lemma 2.15.

In the diagram $\mathcal{J}_{\bullet,\bullet}$ above there are natural
equivalences

$\hat{P}_{\infty}\mathcal{C}\simeq colim_{n}lim_{m}\mathcal{J}_{n,m}\text{ and
}P_{\infty}\mathcal{C}\simeq lim_{n}colim_{m}\mathcal{J}_{n,m}$

In particular the natural functor
$colim_{n}lim_{m}\mathcal{J}_{n,m}\rightarrow
lim_{n}colim_{m}\mathcal{J}_{n,m}$ induces a functor
$\hat{P}_{\infty}\mathcal{C}\rightarrow P_{\infty}\mathcal{C}$ which factors
the comparison functor $\mathcal{C}\rightarrow P_{\infty}\mathcal{C}$.

###### Proof.

This follows from Lemma 2.12 and the proof of Theorem 2.14. ∎

Because filtered colimits and filtered limits do not commute in general in
$\mathcal{C}\\!\mathit{at}^{\omega}$ it seems unlikely that this map is an
equivalence. However one does have this result in the case that the Goodwillie
tower converges.

######  Lemma 2.16.

Suppose the comparison functor $\mathcal{C}\rightarrow
P_{\infty}(\mathcal{C})$ is an equivalence, then the functor
$\hat{P}_{\infty}\mathcal{C}\rightarrow P_{\infty}\mathcal{C}$ is an
equivalence.

###### Proof.

If $\mathcal{C}\xrightarrow{\sim}P_{\infty}(\mathcal{C})$ is an equivalence
then $\Sigma:\mathcal{C}\xrightarrow{\sim}coGrp_{\infty}(\mathcal{C})$ is an
equivalence by Theorem 2.14 and so
$\mathcal{C}\xrightarrow{\sim}\hat{P}_{\infty}\mathcal{C}$ is an equivalence.
The result then follows by the two out of three property for equivalences. ∎

It does not however appear that
$\mathcal{C}\rightarrow\hat{P}_{\infty}\mathcal{C}$ being an equivalence
implies that $\mathcal{C}\rightarrow P_{\infty}\mathcal{C}$ is an equivalence.
Convergence of the Goodwillie tower remains a stronger result.

##  3\. Desuspensions in Spaces

In this section we study how our results interact with the convergence of the
Goodwillie tower of spaces. Heuts shows that the layers of the tower are
equivalent to $\infty$-categories of truncated “Tate coalgebras” in spectra.
For a precise development see [Heu15]. Let $P_{\infty}\mathcal{S}_{*}^{\geq
n}$ denote the full sub $\infty$-category of $P_{\infty}\mathcal{S}_{*}$ whose
associated Tate coalgebra spectra are n-connected. Heuts proves the following
convergence result of the Goodwillie tower of spaces.

######  Theorem.

[Heu15, Theorem 1.3] There is an equivalence of categories for $n>1$

$\mathcal{S}_{*}^{\geq
n}\overset{\sim}{\longrightarrow}P_{\infty}\mathcal{S}_{*}^{\geq n}$

Theorem 2.14 in this case gives us the following result.

######  Corollary 3.1.

For any $n>0$ the suspension functor induces an equivalence of
$\infty$-categories

$\Sigma:P_{\infty}\mathcal{S}_{*}^{\geq
n}\overset{\sim}{\longrightarrow}coGrp_{\infty}(P_{\infty}\mathcal{S}_{*}^{\geq
n+1})$

Note that a 2-connected $A_{\infty}$ comonoid is automatically cogroup-like.
Combining this observation, Corollary 3.1, and [Heu15, Theorem 1.3] we recover
a theorem of Klein-Schwänzl-Vogt, [KSV97, Theorem 1.4] in the form of an
equivalence of $\infty$-categories.

######  Theorem 3.2.

The suspension functor induces an equivalence of $\infty$-categories

$\Sigma:\mathcal{S}_{*}^{\geq
2}\overset{\sim}{\longrightarrow}coMon_{\infty}(\mathcal{S}_{*}^{\geq 3})$

In particular every 2-connected $A_{\infty}$-comonoid admits a desuspension to
a 1-connected space.

###### Proof.

Consider the square

${S_{*}^{\geq 2}}$${P_{\infty}S_{\star}^{\geq
2}}$${coGrp_{\infty}(S_{\star}^{\geq
3})}$${coGrp_{\infty}(P_{\infty}S_{\star}^{\geq
3})}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

By [Heu15, Theorem 1.3] the horizontal functors are equivalences and by
Corollary 3.1 the right vertical map is an equivalence, hence also the left
hand vertical functor is an equivalence. ∎

The benefit to our approach has a distinct advantage to that of [KSV97], where
the authors prove their claims through a variety of arguments involving
connectivity. Our approach isolates these technicalities to a question purely
about the convergence of the Goodwillie tower. This general framework also
allows us to attack the question of desuspension in a general pointed,
compactly generated $\infty$-category.

In the 1-connected case, and in fact the Goddwillie convergent case which
includes nilpotent spaces, we can only make a weaker claim of fully
faithfulness. Let $\mathcal{S}_{*}^{conv}$ denote the $\infty$-category of
connected spaces for which the Goodwillie tower of the identity converges.

######  Proposition 3.3.

The suspension functor induces a fully faithful functor of $\infty$-categories

$\Sigma:\mathcal{S}_{*}^{conv}\hookrightarrow
coGrp_{\infty}(\mathcal{S}_{*}^{\geq 2})$

###### Proof.

In the corresponding square for the case of $S_{*}^{conv}$

${S_{*}^{conv}}$${P_{\infty}\mathcal{S}_{\star}^{\geq
1}}$${coGrp_{\infty}(S_{\star}^{\geq
2})}$${coGrp_{\infty}(P_{\infty}S_{\star}^{\geq
2})}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

we have again that the bottom right cospan is an equivalence, but now the top
horizontal functor is only fully faithful, so that the left vertical functor
is fully faithful. ∎

######  Remark 3.4.

Hopkins first claimed in [Hop84, Theorem 1’] that the suspension functor
$\Sigma:\mathcal{S}_{*}\to coGrp_{\infty}(\mathcal{S}_{*}^{\geq 2})$ is
essentially surjective. However this result does not follow immediately from
our work without further exploring the convergence of the Goodwillie tower of
pointed spaces. See also the discussion after Theorem 1.4 in [KSV97].

We end with a discussion on $n$-excisive approximations to functors. A reduced
functor $F:\mathcal{C}\rightarrow\mathcal{D}$ is 1-excisive if it takes a
suspension diagram to a pullback. This tells us that the value of $F$ on $X$
can be recovered in $\mathcal{D}$ from the suspension $F(X)\simeq\Omega
F(\Sigma X)$. In light of Theorem 2.5, we see that a reduced functor is
$n$-excisive when it can recover its values not from the suspension alone, but
from the suspension equipped with its $A_{n}$-cogroup structure. Examining the
construction of the excisive approximation in Goodwillie’s original work
[Goo03] we see this is exactly the property being forced. A functor can be
recovered from its tower when the value at any object can be recovered from
the functors value on the entire $A_{\infty}$-cogroup structure on the
suspension.

## References

  * [GGN13] D. Gepner, M. Groth, and T. Nikolaus.  Universality of multiplicative infinite loop space machines.  Algebraic & Geometric Topology, 15, 2013.
  * [Goo03] T. Goodwillie.  Calculus iii: Taylor series.  Geometry & Topology, 7, 2003.
  * [Heu15] G. Heuts.  Goodwillie approximations to higher categories, 2015.
  * [Hop84] M. Hopkins.  Formulations of cocategory and the iterated suspension.  In Homotopie algébrique et algèbre locale, number 113-114 in Astérisque. 1984.
  * [KSV97] J. Klein, R Schwänzl, and R. Vogt.  Comultiplication and suspension.  Topology and Its Applications, 77, 1997.
  * [Lur09] J. Lurie.  Higher Topos Theory (AM-170), volume 170.  Princeton University Press, 2009.
  * [Lur20] J. Lurie.  Higher Algebra.  2020\.  http://www.math.harvard.edu/$\sim$lurie/papers/HA.pdf.
  * [RV14] E. Riehl and D. Verity.  Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions, 2014.
  * [RV20] E. Riehl and D. Verity.  Elements of $\infty$-category Theory.  2020\.  http://www.math.jhu.edu/$\sim$eriehl/more-elements.pdf.


11institutetext: INAF Istituto di Radioastronomia, via Gobetti 101, 40129
Bologna, Italy. 22institutetext: Instituto de Astrofísica de Andalucía-CSIC,
Glorieta de la Astronomía s/n, 18008 Granada, Spain. 33institutetext: INAF
Osservatorio Astronomico di Padova, 36012 Asiago, VI, Italy. 44institutetext:
Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy,
University of Manchester, Manchester M13 9PL,UK. 55institutetext: Joint
Institute for VLBI ERIC, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The
Netherlands. 66institutetext: ASTRON, Oude Hoogeveensedijk 4, 7991 PD
Dwingeloo, The Netherlands. 77institutetext: Dept. of Space, Earth and
Environment, Chalmers University of Technology, Onsala Space Observatory,
43992 Onsala, Sweden. 88institutetext: Center for Space Plasma and Aeronomic
Research (CSPAR), University of Alabama in Huntsville, Huntsville, AL 35899,
USA. 99institutetext: Department of Astronomy University of Cape Town, Private
Bag X3, Rondebosch 7701, South Africa.

# High-resolution imaging of the evolving bipolar outflows in symbiotic novae:
the case of RS Ophiuchi 2021 nova outburst

R. Lico, Email<EMAIL_ADDRESS>M. Giroletti 11 U. Munari 33 T. J.
O’Brien 44 B. Marcote 5566 D. R. A. Williams 44 J. Yang 77 P. Veres 88 and P.
Woudt 99

(Received February xx, 2024; accepted xx)

###### Abstract

Context. The recurrent and symbiotic nova RS Ophiuchi (RS Oph) underwent a new
outburst phase during August 2021, about 15 years after the last event
occurred in 2006. This represents the first nova event ever detected at very-
high energies (VHE, E¿100 GeV), and a whole set of coordinated multi-
wavelength observations were triggered.

Aims. The main goals of this work are to characterize the evolving morphology
of the expanding bipolar ejecta with high accuracy and determine the physical
conditions of the surrounding medium in which they propagate.

Methods. By means of high-resolution very long baseline interferometry (VLBI)
radio observations, we monitored RS Oph with the European VLBI Network (EVN)
and e-MERLIN at 1.6 and 5 GHz during multiple epochs from 14 to 65 days after
the explosion.

Results. We reveal an evolving source structure, consisting of a central and
compact core and two elongated bipolar outflows, expanding on opposite sides
from the core in east-west direction. The ejecta angular separation with time
is consistent with a linear expansion with an average speed of $\sim 7000$ km
s-1. We find clear evidence of a radial dependence of the density along the
density enhancement on the orbital plane (DEOP), going from 1.1$\times$107
cm-3 close to the central binary to 3.8$\times$105 cm-3 at $\sim 175$ AU.

Conclusions. Thanks to the accurate source astrometric position provided by
Gaia DR3, in this work we draw a detailed scenario of the geometry and physics
of the RS Oph evolving source structure after the most recent nova event. We
conclude that most of the mass lost by the red giant companion goes in the
DEOP, for which we estimate a total mass of $4.3\times
10^{-6}\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{M_{\odot}}$, and in
the circumstellar region, while only a small fraction (about one-tenth) is
accreted by the white dwarf.

###### Key Words.:

novae – cataclysmic variables – stars: winds, outflows

##  1 Introduction

RS Ophiuchi (hereafter RS Oph) is a well-known recurrent and symbiotic nova,
located at 2.68$\pm$0.17 kpc distance according to Gaia DR3 parallax (Gaia
Collaboration et al. 2016, 2023). It consists of a massive white dwarf (WD)
and a M0III red giant (RG) in a close binary system with an orbital period of
$\sim$$454$ days (Brandi et al. 2009). Before accumulating on its surface, the
material accreted from the RG by the WD flows through a massive accretion
disk. The near-Chandrasekhar mass of the WD coupled with its large accretion-
rate fit with theoretical expectations of a nova outburst every $\sim$20 yrs
(cf. Nelson et al. 2011), and at least seven eruptions have been already
recorded from RS Oph, in 1898, 1933, 1958, 1967, 1985, 2006, and 2021. All
these events have evolved nearly identically at optical wavelengths. During an
outburst of RS Oph, most of the accreted material is ejected
($\sim$$10^{-6}M_{\odot}$; Das et al. 2006; Eyres et al. 2009; Pandey et al.
2022) at high velocity ($V_{\rm ej}$$\geq$7500 km s-1; Munari et al. 2022),
and the amount of accreted mass retained on the surface of the WD is
sufficient to support stable nuclear burning for about three months (Osborne
et al. 2011; Page et al. 2022; Munari & Valisa 2022). The accretion disk is
destroyed away by the nova ejecta, and it takes $\sim$260 days for it to
reform and return to pre-outburst brightness (Zamanov et al. 2022; Munari &
Tabacco 2022), marking the start of a new accretion cycle leading to the next
outburst.

Table 1: Details of the observations.

Epoch | Obs. date | $T-T_{0}$ | Freq. | Restoring beam | Map peak |  $1\sigma$ rms |  Stationsaa$a$EVN stations: Badary (BD), Effelsberg (EF), Hartebeesthoek (HH), Irbene (IR), Jodrell Bank - Lovell (JB1), Jodrell Bank - Mark2 (JB2), Medicina (MC), Noto (NT), Onsala (O8), Shanghai - 25 m (SH), Tianma - 65 m (T6), Svetloe (SV), Toruń (TR), Yebes (YS), Westerbork (WB), Zelenchukskaya (ZC). e-MERLIN outstations: Cambridge, Darnhall, Defford, Knockin, and Pickmere.
---|---|---|---|---|---|---|---
| (2021) | (days) | (GHz) |  (mas $\times$ mas, ∘) | mJy/beam | mJy/beam |
I | Aug. 22 | 14 | 5 |  $1.5\times 4.2$, 75.9 | 15.66 | 0.08 | JB1,WB,EF,MC,IR,YS,TR,HH,SH
| Aug. 23 | 15 | 1.6 |  $4.4\times 18.1$, 76.3 | 45.37 | 0.19 | JB1,WB,EF,MC,IR,TR,HH
II | Sep. 1 | 24 | 5 |  $6.9\times 11.5$, -5.7 | 9.58 | 0.01 | JB1,WB,EF,MC,IR,YS,O8,T6
| Sep. 2 | 15 | 1.6 |  $10.4\times 43.4$, 10.7 | 18.32 | 0.07 | JB1,WB,EF,MC,IR,TR,HH,O8,T6
III | Sep. 11 | 34 | 5 |  $1.3\times 2.8$, 75.4 | 3.25 | 0.03 | JB2,WB,EF,MC,IR,YS,TR,HH,SH,O8,SV,ZC,BD,e-MERLIN
| Sep. 12 | 35 | 1.6 |  $9.0\times 11.9$, 26.3 | 6.85 | 0.06 | JB2,WB,EF,MC,IR,TR,HH,O8,T6,SV,ZC,BD,e-MERLIN
IV | Sep. 26 | 49 | 5 | — | — | — | —
| Sep. 27 | 50 | 1.6 | — | — | — | —
V | Oct. 11 | 64 | 5 |  $1.3\times 2.6$, 71.3 | 4.50 | 0.04 | JB2,WB,EF,MC,IR,YS,TR,HH,SH,O8,SV,ZC,BD,NT,e-MERLIN
| Oct. 12 | 65 | 1.6 |  $9.9\times 45.6$, -1.3 | 4.56 | 0.05 | JB2,WB,EF,MC,IR,TR,HH,O8,NT,SV,ZC,BD,e-MERLIN
111

During August 2021, RS Oph underwent a new eruption that has been the first
nova outburst ever detected at the very-high energies (E¿100 GeV) mapped by
Cherenkov telescopes (H. E. S. S. Collaboration et al. 2022; Acciari et al.
2022). Prompt observations as well as monitoring campaigns were triggered
throughout electromagnetic spectrum, including radio (Sokolovsky et al. 2021;
Williams et al. 2021; de Ruiter et al. 2023; Nayana et al. 2024), infrared
(Woodward et al. 2021), optical (Munari & Valisa 2021, 2022; Azzollini et al.
2023; Molaro et al. 2023; Tomov et al. 2023), X-ray (Shidatsu et al. 2021;
Page et al. 2022; Orio et al. 2023; Ness et al. 2023; Islam et al. 2024), and
$\gamma$-rays (Cheung et al. 2022). The interpretation of the data thus
gathered has particularly benefited from the fact that similar instrumentation
and observing strategies as used for the 2006 eruption were also adopted in
2021, especially in monitoring the evolution in X-rays (eg. Osborne et al.
2011; Page et al. 2022) and imaging in the radio the expansion of the ejecta
(eg. O’Brien et al. 2006; Rupen et al. 2008; Munari et al. 2022; Giroletti et
al. 2023). Of special benefit has been the extremely accurate astrometric
position and proper motions of RS Oph provided by Gaia DR3, as in the case of
the recent work on the nova V407 Cygni by Giroletti et al. (2020).

Within this context, we have conducted a multi-frequency and five-epoch very
long baseline interferometry (VLBI) monitoring campaign with the European VLBI
network (EVN), at 1.6 and 5 GHz, which covered RS Oph from about two weeks
(late August) up to two months (early October) after the 2021 explosion. The 5
GHz data for the central epoch (day +34) have already been presented and 3D
modeled by Munari et al. (2022) in conjunction with results from high-
resolution optical spectroscopy, while a preliminary overlook to the whole set
of EVN observations has been provided by Giroletti et al. (2023). In this
paper we publish the complete set of results from the full EVN monitoring,
that allows us to characterize, with an unprecedented level of detail, the
evolving source structure and the physical conditions of the surrounding
medium in which the nova ejecta propagated.

Assuming the Gaia distance of 2.7 kpc, 1 mas corresponds to 2.7 AU. Throughout
the paper the radio spectral index $\alpha$ is defined as
$S_{\nu}\propto\nu^{\alpha}$, with $S_{\nu}$ representing the flux density at
frequency $\nu$, and all angles are measured from North to East. Finally, we
use as the reference epoch $T_{0}$ the start of the eruption derived to be
2021 August 08.50 ($\pm$0.01) UT by Munari & Valisa (2021).

##  2 Observations and data calibration

We monitored RS Oph from 22 August 2021 ($T_{0}$ \+ 14 days) to 12 October
2021 ($T_{0}$ \+ 65 days) with the EVN and e-MERLIN at 1.6 and 5 GHz. The
observations were organized in 10 observing sessions, grouped in five epochs,
with each epoch consisting of two 8-hour runs on consecutive days at 5 and 1.6
GHz, respectively (see Table1).

The EVN participating stations were: Badary, Effelsberg, Hartebeesthoek,
Irbene, Jodrell Bank, Medicina, Noto, Onsala, Shanghai, Svetloe, Torun, Yebes,
Westerbork, Zelenchukskaya. The last six observing runs also feature the
e-MERLIN outstations in Cambridge, Darnhall, Defford, Knockin, and Pickmere.
Observations were performed in dual polarization mode with a bandwidth of 128
MHz (divided into $4\times 32$ MHz-wide sub-bands) at 1.6 GHz and 256 MHz
(divided into $8\times 32$ MHz-wide sub-bands) at 5 GHz. We used a standard
phase-reference observing scheme, with 4 min on the target and 40 seconds on
the calibrator J1745-0753.

The data correlation was performed in real time with the software FX-kind
correlator (SFXC) at the Joint Institute for VLBI ERIC (JIVE, The Netherlands)
(Keimpema et al. 2015). The a-priori data reduction was carried out with the
JIVE EVN pipeline based on ParselTongue (Kettenis et al. 2006) and AIPS
(Greisen 2003). We used the DIFMAP software package (Shepherd 1997) to inspect
and edit the calibrated data downloaded from the EVN Data
Archive222http://archive.jive.nl/scripts/portal.php, as well as for the
imaging and self-calibration procedures. In more detail, for the imaging
process we used the classical inverse modeling CLEAN algorithm, implemented in
the DIFMAP, that represents the source structure as a collection of point
sources, whose location and flux densities are iteratively determined (Högbom
1974).

We note that the two observing runs of Epoch IV (26 and 27 September 2021)
were strongly affected by an hardware problem with e-MERLIN and were excluded
from the analysis presented in this paper. Additional details about the
observations can be found in Munari et al. (2022) and Giroletti et al. (2023).

##  3 Results

Figure 1: Natural-weighted total intensity images at 5 GHz (left frame) and
1.6 GHz (right frame), centered at the Gaia DR3 astrometric position for RS
Oph (RA $17$h$50$m$13.1610$s, Dec $-06$d$42^{\prime}28.61026^{\prime\prime}$),
corrected for proper motion (indicated by a black cross whose size represents
the position uncertainty magnified by a factor of 40 for visualization
purpose). The observing epoch, in units of days after $T_{0}$, is indicated in
the top-left corner of each image. All 5 GHz images (left panel) are displayed
with a Gaussian taper of 0.5 at 10 M$\lambda$ and convolved with a beam of 9.8
mas $\times$ 12.8 mas at 0∘. All 1.6 GHz images (right panel) are convolved
with a beam of 8.5 mas $\times$ 30.0 mas at 0∘. This restoring beam is
displayed in the bottom-left corner in each image. The color scale and the
overlaid contours represent the total intensity emission, with the lowest
contour at 2% of the map peak (see Table 1) and the following contours a
factor of two higher. The red cross indicates the average core position (see
Sect. 3.2).

###  3.1 Images and source structure

In the left panel of Fig. 1 we show the 5 GHz images, obtained for the epochs
I, II, III and V (14, 24, 34 and 64 days after $T_{0}$, respectively). All
images have been convolved with a beam of 9.8 mas $\times$ 12.8 mas at 0 deg,
resulting from the average beam sizes across all the epochs, and displayed
with natural weights and a Gaussian taper of 0.5 at 10 M$\lambda$. The 1.6 GHz
images, obtained for the epochs I, II, III and V (15, 25, 35 and 65 days after
$T_{0}$, respectively), are reported in the right panel of Fig. 1. All 1.6 GHz
images have been convolved with a beam of 8.5 mas $\times$ 30.0 mas at 0 deg,
resulting from the average beam sizes across all the epochs, and displayed
with natural weights. All images are centered at the Gaia DR3 astrometric
position for RS Oph, corrected for proper motion (highlighted with a black
cross at the center of each image).

Figure 2: Full resolution 5 GHz total intensity image during epoch I (14 days
after $T_{0}$), with relative coordinates referring to the Gaia DR3
astrometric position of RS Oph, corrected for proper motion. The convolving
beam, displayed to the bottom-left corner of the image, is 1.0 mas $\times$
2.0 mas at $90^{\circ}$. The color scale and the overlaid contours represent
the total intensity emission, with the lowest contour at 5% of the map peak
(see Table 1) and the following contours a factor of two higher. The gray dots
represent the best-fitting ellipse obtained by O’Brien et al. (2008) at the
same frequency and epoch after the 2006 outburst.

The images at both frequencies reveal an evolving source morphology expanding
in the east-west direction, that during the last observing epoch (64-65 days
after $T_{0}$) reaches a total extension of $\sim 200$ mas ($\sim 540$ AU). As
it is apparent since epoch II (24-25 days after $T_{0}$), at both frequencies,
the images reveal a triple-component source structure, with a compact central
component (core) and two elongated lobe structures propagating on opposite
sides from the core. We note that the compact core is located within a few mas
from the Gaia DR3 position (see Sect. 3.2 for more details). The flux density
depression between the eastern lobe and the rest of the source is interpreted
by Munari et al. (2022) as due to absorption from ionized gas in the so-called
density enhancement on the orbital plane (DEOP).

The western lobe is brighter than the eastern one and has a circular shape
that is consistent with what was found in the images at similar observing
frequencies from the previous 2006 nova event (O’Brien et al. 2006; Rupen et
al. 2008). On the other hand, the arc-like structure that we see on the
eastern side is assumed to represent the outer edge of the eastern lobe that
emerges from behind the DEOP. Within this scenario, we interpret the compact
structure that we see during the first epoch (14-15 days from $T_{0}$), at
both frequencies, as the western lobe, with the core in the background and the
eastern lobe behind the DEOP.

####  3.1.1 First epoch 5 GHz full-resolution image

To better investigate the structure at the early stages of evolution, we show
in Fig. 2 the total intensity image of epoch I (14 days after $T_{0}$) at 5
GHz in full resolution, centered at the Gaia position. The image is obtained
adopting uniform weights and a convolving beam of 1.0 mas $\times$ 2.0 mas at
$90^{\circ}$. The radio emission, which could hardly be resolved in the image
obtained with natural weights, is now found to be distributed in a complex and
elongated morphology. The total extension is over $\sim 30$ mas in the north-
south direction and $\sim 20$ mas in the east-west direction. Several spots of
enhanced brightness are distributed mostly along the outer rim of this roughly
elliptical region. These spots may mark locations of harder collision between
the fast ejecta and the pre-existing slow circumstellar material, a situation
reminiscent of what Chomiuk et al. (2014) first observed in Nova Mon 2012.

A similar elongated source structure, at the same observing frequency, was
reported by O’Brien et al. (2008) with the Very Long Baseline Array (VLBA), 14
days after the 2006 outburst. For comparison, in Fig. 2 we report the best-fit
ellipse (gray dots) obtained by O’Brien et al. (2008), with a semi-major axis
of $8.9$ mas and axial ratio of $1.14$ mas.

###  3.2 Lobe expansion speed

To represent and quantify the emission from the central core, that emerges
from the second epoch on, we model the brightness distribution with a 2-D
circular Gaussian component in the visibility plane with the modelfit
procedure in DIFMAP. We define a reference core position by averaging the core
Gaussian component position during the epochs II and III, when the core region
is compact, bright and clearly recognizable at both frequencies. The average
core position is indicated in all images in Fig. 1 with a red cross at a
distance from the Gaia DR3 position of $\sim 4$ and $\sim 10$ mas at 5 and 1.6
GHz, respectively. The size of the expanding lobes is determined as the
distance of the outermost lowest total intensity contour from the average core
position (see Table 2). We note that the orbital separation of RS Oph amounts
to $\sim 1.5$ AU or $4$ mas at the GAIA distance.

By considering the extension of the two lobes from the core region during the
different epochs (Fig. 3), and assuming zero separation at $T_{0}$, we obtain
their projected expansion speeds by means of a least-squares regression
(column 6 in table 2). We note that for the uncertainties on the distance from
the core we assume a conservative value equal to the beam minor axis. Since
during epoch I (14-15 days after $T_{0}$) the eastern lobe is not yet visible
and we cannot easily disentangle the western lobe from the core region, we do
not consider this epoch when estimating the lobe expansion rate.

For the western lobe we obtain an average projected expansion speed of
$6840\pm 370$ km s-1 at 5 GHz and $6660\pm 100$ km s-1 at 1.6 GHz. For the
eastern lobe we obtain an average projected expansion speed of $6830\pm 340$
km s-1 at 5 GHz and $7100\pm 540$ km s-1 at 1.6 GHz. The projected expansion
velocity of the two lobes is therefore the same well within their combined
errors.

###  3.3 Light-curves and spectral index

To quantify the emission from the different source components we use the
following approach. We determine the source total flux density by summing up
the flux density over all the clean components in DIFMAP, while the core flux
density is provided by the 2-D circular Gaussian component that represents the
core region (see Sect.3.1). The eastern lobe can be clearly isolated from the
overall source structure and its flux density is obtained by summing up the
flux density over all the clean components in this region. Lastly, the western
lobe flux density is determined by subtracting the emission of the eastern
lobe and the core from the total flux density.

In Fig. 4 we report the resultant light-curves. The flux density values are
reported in table 3, with the uncertainties calculated by considering a
calibration error of 10% of the flux density and a statistical error equal to
the map rms noise multiplied for number of beams in the area over which the
flux density was determined.

The total source emission (first frame in Fig. 4) shows an overall decreasing
trend with time, with a $\sim 50\%$ drop from epoch I ($\sim 80$ mJy and $\sim
40$ mJy at 1.6 GHz and 5 GHz, respectively) to Epoch V ($\sim 40$ mJy and
$\sim 20$ mJy at 1.6 GHz and 5 GHz, respectively). A similar decreasing trend
in the flux density with time is also found, at both frequencies, for the core
and the western lobe (second and third frame in Fig. 4, respectively). We also
note that the western lobe is brighter than both the core and the eastern
lobe. On the contrary, for the eastern lobe the flux density tends to increase
with time, as the outer part of the lobe itself emerges behind the DEOP
(fourth frame in Fig. 4). During the last observing epoch at 1.6 GHz the
eastern lobe flux density decreases with respect to the previous two epochs,
as the source becomes more diffuse.

The spectral index for the different source components is reported in Fig. 5
and Table 4. The western lobe (green squares) has slightly negative spectral
index, with an average value of $-0.38\pm 0.12$ across the entire observing
period covering about two months after $T_{0}$. The core region (black
circles) has an average value of $-0.50\pm 0.12$, until epoch III (34-35 days
after $T_{0}$), and then becomes steeper ($-1.13\pm 0.13$) during the last
epoch (64-65 days from $T_{0}$). On the other hand, the eastern lobe during
epoch II (24-25 days after $T_{0}$) shows a slightly positive spectral index
(blue triangles in Fig. 5) with a value of $0.10\pm 0.13$. In the following
epochs, the eastern lobe spectral index follows a similar trend as for the
western lobe: it becomes slightly negative during epoch III (34-35 days after
$T_{0}$) to a value of $-0.39\pm 0.13$ and later steepens to a value of
$-1.17\pm 0.14$ during the last epoch (64-65 days from $T_{0}$). The
uncertainties on the spectral index were calculated from the error propagation
theory (see Lico et al. 2012).

###  3.4 Core region brightness temperature

For the core region, that we represent by means of 2-D circular Gaussian
components across the different epochs (see Sect. 3.2), we estimate the
observed brightness temperature ($T_{B}^{obs}$), by using the following
equation (e.g. Tingay et al. 1998):

$T_{B}^{obs}=1.22\times
10^{12}\frac{S_{core}}{\theta_{maj}\theta_{min}\nu^{2}}$ (1)

with $S_{core}$ being the fitted core flux density in Jy, $\theta_{maj}$ and
$\theta_{min}$ the FWHM of the major and minor axes of the 2-D Gaussian
component in mas, and $\nu$ the observing frequency in GHz. The FWHMs of the
Gaussian components during epochs II, III and V are $1.6$, $4.2$, $12.7$ mas
at 5 GHz and $4.4$, $5.9$, $15.5$ mas at 1.6 GHz, respectively.

During epoch II (24-25 days after $T_{0}$), III (34-55 days after $T_{0}$) and
V (64-65 days after $T_{0}$) we find a brightness temperature of the order of
$7.0\times 10^{7}$ K, $7.8\times 10^{6}$ K and $3.0\times 10^{5}$ K at 5 GHz
and $4.0\times 10^{8}$ K, $6.7\times 10^{7}$ K and $4.6\times 10^{6}$ at 1.6
GHz, respectively, clearly supporting a non-thermal origin of the radio
emission, consistent with spectral index $<-0.1$.

Figure 3: Angular separation evolution with time of the eastern (blue) and the western (green) lobes, at 1.6 GHz (empty squares) and 5 GHz (filled circles). The solid and dashed lines represent the best least-square regression fit for the 5 and 1.6 GHz data, respectively.  Table 2: Lobe extension during the different observing epochs (columns 3 - 5) and expansion speed (column 6). | Lobe | Epoch II | Epoch III | Epoch V | Projected
---|---|---|---|---|---
|  |  |  |  | expansion speed
|  | mas (AU) | mas (AU) | mas (AU) |  km s-1
5 GHz | West | 36.3 (98) | 45.3 (122) | 95.8 (259) |  6840 $\pm$ 370
| East | 42.8 (116) | 54.0 (146) | 95.8 (259) |  6830 $\pm$ 340
1.6 GHz | West | 36.1 (97) | 48.8 (132) | 91.6 (247) |  6660 $\pm$ 100
| East | 44.1 (119) | 59.7 (161) | 97.9 (264) |  7100 $\pm$ 540
Figure 4: Light-curves at 1.6 GHz (blue circles) and 5 GHz (black triangles)
for the whole source (first frame), the core (second frame), the western lobe
(third frame) and the eastern lobe (fourth frame). All values are reported in
Table 3.

##  4 Discussion

With each new eruption, our understanding of the outburst mechanism of RS Oph
steadly improves, thanks to ever more accurate data and comparison to previous
events. Thanks to the radio imaging of the 2021 eruption presented in this
paper we are now able to characterize in great detail the evolving morphology
of the expanding bipolar ejecta, and determine the physical conditions of the
surrounding medium in which they propagate.

###  4.1 The evolving source morphology

From the high-resolution radio observations presented in this work we find an
expanding source structure consisting of three main components: a central and
compact core region, located within a few mas from the Gaia DR3 position (Gaia
Collaboration et al. 2016, 2023), and two expanding lobes in east-west
direction, on opposite sides from the central binary. This evolving source
structure is notably consistent with the images at similar frequencies for the
past RS Oph nova event in 2006 reported by O’Brien et al. (2006); Rupen et al.
(2008); Sokoloski et al. (2008). However, thanks to the accurate astrometric
position provided by the Gaia DR3 position, the key improvement from the 2021
data is the realisation that the ring-like structure at early times is
actually the western lobe, as the binary system is located just to the east of
this. This provides firm support for the bipolar outflow expansion scenario.
Furthermore, as we argued in Munari et al. (2022), the images obtained from
this observing campaign clearly reveal that the gravitational pull of the WD
channels most of the mass lost by the RG towards the orbital plane and forms
the so-called DEOP. The confinement of the ejected material into the two
observed bipolar outflow is produced by the combined effect of the DEOP and
the accretion disk around the WD (e.g. Orlando & Drake 2012). Within this
scenario, the western side represents the approaching lobe that is located in
the foreground to the DEOP, while the eastern side represents the receding
lobe expanding behind the DEOP.

Table 3: Flux densities of the different source regions. |  | Epoch I | Epoch II | Epoch III | Epoch V
---|---|---|---|---|---
|  | mJy | mJy | mJy | mJy
5 GHz | Total |  41.5 $\pm$ 4.2 |  43.8 $\pm$ 4.4 |  29.5 $\pm$ 3.0 |  18.0 $\pm$ 1.8
| Core | - |  8.6 $\pm$ 0.9 |  2.8 $\pm$ 0.3 |  1.0 $\pm$ 0.1
| East | - |  6.1 $\pm$ 0.6 |  7.0 $\pm$ 0.7 |  3.2 $\pm$ 0.4
| West | - |  29.1 $\pm$ 2.9 |  19.8 $\pm$ 2.0 |  13.8 $\pm$ 1.4
1.6 GHz | Total |  79.3 $\pm$ 7.9 |  64.8 $\pm$ 6.5 |  48.3 $\pm$ 4.8 |  36.7 $\pm$ 3.7
| Core | - |  16.3 $\pm$ 1.6 |  4.9 $\pm$ 0.5 |  3.5 $\pm$ 0.4
| East | - |  5.4 $\pm$ 0.6 |  10.9 $\pm$ 1.1 |  12.3 $\pm$ 1.3
| West | - |  43.1 $\pm$ 4.3 |  32.6 $\pm$ 3.3 |  20.9 $\pm$ 2.1

In more detail, this is the evolution process that we can infer from the
images presented in Fig.1. During the first epoch (14-15 days after $T_{0}$),
at both 1.6 and 5 GHz, we only see the approaching western lobe, projected on
the plane of the sky, with emission from the central region absorbed by the
dense and ionized gas and the western lobe laying in front. Similarly, the
eastern lobe is completely hidden from view by absorption from the densest and
ionized inner regions of DEOP. As the western lobe further propagates with a
circular shape, during epoch II (24-25 days from $T_{0}$) the central core
emerges with a bright and compact structure, and a brightness temperature of
the order of $7\times 10^{8}$ K at 5 GHz and $4\times 10^{8}$ K at 1.6 GHz,
indicating the non-thermal nature for the emission. A non-thermal origin for
the whole radio emission on these scales333The analysis of the source
structure evolution on larger scales, by means of multi-frequency e-MERLIN and
Jansky Very Large Array observations, will be presented in a separate work
(Williams at al. in prep.). is also indicated by the spectral properties: the
integrated spectral index in the core region and the western lobe are
negative, with average values of $-0.50\pm 0.12$ and $-0.40\pm 0.12$,
respectively; the eastern lobe has a slightly positive spectral index
($0.10\pm 0.13$), explained by the fact that as the eastern lobe leading edge
emerges behind the DEOP, it is clearly visible at 5 GHz while at 1.6 GHz the
free-free absorption effect is still significant (the optical depth going like
$\tau_{\nu}\propto\nu^{-2.1}$) and its emission is still opaque. We note that
the source extension and morphology during epoch II (24-25 days from $T_{0}$)
are in good agreement with the results about the 2006 outburst reported by
Rupen et al. (2008) (20.8 and 26.8 days after the outburst) and O’Brien et al.
(2006) (21.5 and 28.7 days after the outburst), obtained with the VLBA at 1.7
and 5 GHz.

During the following two epochs we can track the expanding source structure,
that reaches an overall east-west extension of $\sim 200$ mas ($\sim 540$ AU)
during the last observing epoch (64-65 days after $T_{0}$). As the source
structure expands and gets more diffuse it also becomes fainter, showing an
overall decreasing flux density trend with time, both for the core region and
the western lobe. On the other hand, the flux density of the eastern lobe
shows an increasing trend as it gradually emerges behind the DEOP (see Fig.
4). We note that in this last epoch the core flux density has dropped by $\sim
80\%$, and the brightest source region is found in the western lobe edge, that
maintains a less steep spectral index of $-0.36\pm 0.13$. On the other hand,
both the core region and the eastern lobe become more transparent with
spectral indexes of $-1.13\pm 0.13$ and $-1.17\pm 0.12$, respectively. We note
that during the last observing epoch the core brightness temperature decreases
as well, down to values of $3.0\times 10^{5}$ K at 5 GHz and $5.0\times
10^{6}$ K at at 1.6 GHz, indicating how the deceleration of the ejecta close
to the equatorial plane and therefore the densest regions of DEOP is almost
completed. The 1.6 GHz images during epochs III (35 days after $T_{0}$) and V
(65 days after $T_{0}$) are consistent with the images reported by Sokoloski
et al. (2008) obtained from 34 to 51 days after the 2006 outburst.

As it emerges from the analysis of the angular separation with time of the two
lobes (see Fig. 3) during the $14\leq(T-T_{0})\leq 65$ day interval probed by
our observations, the overall evolving source structure is consistent with a
linear expansion, with no hints of significant deceleration of eastern and
western lobes expanding perpendicular to the orbital plane. This is an
indication that the density gradient perpendicular to the orbital plane is
very steep, with no mass higher than 50 AU above the equatorial plane and
probably confined much closer to the plane than this upper limit.

###  4.2 Density profile and total mass of DEOP

If we assume intrinsic similarity and symmetry at any of the observing epochs
between the two expanding lobes, the flux ratio between the western lobe seen
in foreground and the eastern lobe seen through the absorbing DEOP allows to
estimate the radial dependence of density along the DEOP.

To this aim we adopt the following expression for the free-free optical
thickness of ionized gas (Mezger & Henderson 1967; Condon & Ransom 2016):

$\tau_{\nu}\approx 3.28\times
10^{-7}\,\,\left(\frac{T}{10^{4}}\right)^{-1.35}\nu^{-2.1}\,\,n_{e}^{2}\,\,s$
(2)

with the temperature $T$ in Kelvin, the frequency $\nu$ in GHz, the electron
density $n_{e}$ in cm-3, the geometrical thickness $s$ in pc, and assuming the
same number of electrons and ions $n_{e}n_{i}\approx n_{e}^{2}$.

Being interested in the order of magnitude of the derived density, we consider
the DEOP as being approximated by a plane-parallel slab of total $s$ thickness
constant through the radial extent, with an electron density $n_{e}$ dependent
only from the radial distance and otherwise constant in the height direction.
Considering that the nuclear burning at the surface of the WD continued
uninterrupted to day +82, i.e. well past our latest radio observation (Munari
& Valisa 2022; Page et al. 2022), we assume full ionization within DEOP over
its radial extent explored by our radio observations. In the previous section
we estimated that by an height of 50 AU above the equatorial plane, the
density within the DEOP has dropped to negligible values; assuming the drop to
be exponential with height (ie. $\rho$(z)$\propto$$\rho$(0)e-z/z0), the
accumulated column density is equivalent to the case of constant density over
an height $\leq$12 AU. Accordingly, we adopt $s$=20 AU in Eq.(2). Finally, for
de-projection purposes, we adopt the orbital inclination of $i$=54∘ derived by
Munari et al. (2022).

From Fig. 1 and its comparison with equivalent mapping of the 2006 eruption by
Rupen et al. (2008), the earliest appearance of the extincted eastern lobe
occurred at 5 GHz on day +20 at an angular distance of $\sim$25mas from the
position of RS Oph, corresponding to a de-projected distance of $\sim$50 AU.
Taking a brightness ratio 100 at this epoch for the the western lobe with
respect to the eastern one, the electron density results in ne(50
AU)=2.0$\times$106 cm-3. At the time of the latest radio epoch on day +65, the
brightness ratio at 5GHz has reduced to 4 for the apex of the lobes located at
$\sim$87mas from RS Oph, corresponding to a de-projected distance of $\sim$175
AU from central binary, resulting in an electron density of ne(175
AU)=3.8$\times$105 cm-3.

An estimate of the electron density at the DEOP inner regions can be obtained
from the recombination time following the initial UV-flash at the very start
of the outburst, for which Munari & Valisa, (2021) derived
$n_{e}$=1.1$\times$107 cm-3.

There is therefore a clear evidence of a radial dependence of the density
going from 1.1$\times$107 cm-3 close to the central binary, to 2.0$\times$106
at 50 AU distance, and to 3.8$\times$105 cm-3 at 175 AU. The behavior is
summarized in Fig. 6 where the fitting curve represents the relation $\rho$(r)
$\propto$ r-1.25. Integrating the density along the DEOP volume between
10$\leq$r$\leq$175 AU leads to a total DEOP mass of
$\mathrm{M_{DEOP}}=4.3\times 10^{-6}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{M_{\odot}}$.

Supposing the RG loses mass via a wind at a typical speed of 20 km s-1 and
that the vast majority of the mass lost ends up in the DEOP, the required RG
mass-loss rate is $\mathrm{\dot{M}_{\rm RG}=1\times
10^{-7}\leavevmode\nobreak\ \leavevmode\nobreak\ M_{\odot}/yr}$, which
compares well with standard estimates (eg. Livio & Warner 1984), arguing in
favor of the general soundness of the above reasoning.

To power an outburst every $\sim$20 yrs, the WD in RS Oph has to be massive
and accrete at a high rate. Theoretical models by Yaron et al. (2005) require
accretion rates $\dot{M}_{\rm WD}$$>$$10^{-8}$ M⊙ yr-1 for a 1.4 M⊙ WD, and
$\dot{M}_{\rm WD}$$>$$10^{-7}$ M⊙ yr-1 if the mass of the WD reduces to 1.25
M⊙. These estimates are in good agreement with the results of X-ray
observations of RS Oph in quiescence reported by Nelson et al. (2011), that
have placed an upper limit of $\dot{M}_{\rm WD}$$<$$2\times 10^{-8}$ M⊙ yr-1
for RS Oph. Comparing with our estimate for the RG mass loss of
$\mathrm{\dot{M}_{\rm RG}=1\times 10^{-7}\leavevmode\nobreak\
\leavevmode\nobreak\ M_{\odot}/yr}$, it is interesting to conclude that the WD
in RS Oph is likely accreting around one tenth of the mass lost by the RG and
that the great majority therefore goes to feed the circumstellar region, and
the DEOP in particular. This also has implications for the mass-reservoir
constituted by the RG to keep powering new eruptions on the WD and, pending
the mass-gain efficiency of eruption, the ultimate fate of the WD itself.

Figure 5: 1.6 - 5 GHz spectral index for the core region (black circles), western lobe (green squares) and eastern lobe (blue triangles). All values are reported in Table 4. Table 4: Spectral index values for the different source regions. Spectral index | Epoch II | Epoch III | Epoch V
---|---|---|---
Core |  $-0.56$ $\pm$ 0.12 |  $-0.49$ $\pm$ 0.12 |  $-1.13$ $\pm$ 0.13
East |  0.10 $\pm$ 0.13 |  $-0.39$ $\pm$ 0.13 |  $-1.17$ $\pm$ 0.14
West |  $-0.35$ $\pm$ 0.12 |  $-0.44$ $\pm$ 0.12 |  $-0.36$ $\pm$ 0.13

###  4.3 High-resolution source morphology

The full resolution image at 5 GHz for epoch I (14 days after $T_{0}$) reveals
a complex, granular-like, source morphology (see Fig. 2). Considering that
during this epoch the total extension in the north-south direction is $\sim
30$ mas, we obtain a transverse expansion speed of $\sim 0.02$c.

A similar elongated source morphology was found with the VLBA, at 5 GHz, 14
days after the 2006 outburst by O’Brien et al. (2008). The position, shape and
size of the brightness distribution best-fit ellipse reported in O’Brien et
al. (2008), are in good agreement with the total intensity source structure
that we find 14 days after the 2021 outburst (gray dots in Fig. 2). This
supports the idea that the way in which the RG wind refills the circumstellar
space - which has been emptied by the expanding ejecta of the previous nova
eruption - replicates self-similar at each new event, as the evolution at
optical wavelengths of each outburst is identical to all others.

A marked granular structure, although at a much closer distance to the WD, is
supported by the X-ray observations at the onset of the supersoft phase during
the 2006 outburst of RS Oph (Osborne et al. 2011), and to a lower degree also
during the 2011 event (Page et al. 2022). Our EVN observations suggest that
such granularity is present also to scales much larger than implied by the
X-ray observations. It is beyond the goals of this paper to investigate the
origin of such a clumpy structuring of the ejecta and of the circumstellar
medium around RS Oph, but various mechanisms may be at work at different
distances from the central binary, involving inhomogeneities in the wind
blowing off the WD during the burning phase as well as the turbulent nature of
shocks propagating through the pre-existing (and probably itself clumpy) slow
circumstellar material.

Figure 6: Dependence on radial distance from the central binary of the
electron density within DEOP.

##  5 Conclusions

In this work we present the results of a high-resolution VLBI radio monitoring
of the recurrent and symbiotic nova RS Oph after the most recent outburst
occurred during august 2021. By means of multi-epoch EVN + e-MERLIN
observations at 1.6 and 5 GHz, from 14 to 65 days after the explosion, we
characterize in great detail the evolving source morphology and the physical
conditions of the surrounding medium in which the ejected material propagates.

At both frequencies we identify a triple-component source morphology, with a
compact central core region and two elongated bipolar outflow expanding
linearly in east-west direction, with an average speed of $\sim 7000$ km s-1.
We estimate a total mass of $4.3\times 10^{-6}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{M_{\odot}}$ for the DEOP, representing the
density enhancement above the orbital plane that we reveal in our images as a
flux density depression between the eastern lobe and the rest of the source.
We find a clear radial dependence of the density within the DEOP, that varies
from 1.1$\times$107 cm-3 in the near vicinity of the central binary to
8$\times$105 cm-3 at $\sim 175$ AU. Based on our results, we can firmly
conclude that the majority of the mass lost by the RG companion ends up in the
DEOP and the circumstellar region, with only about one-tenth of it being
accreted by the WD.

###### Acknowledgements.

The European VLBI Network is a joint facility of independent European,
African, Asian, and North American radio astronomy institutes. Scientific
results from data presented in this publication are derived from the following
EVN project code: RG012. The research leading to these results has received
funding from the European Union’s Horizon 2020 Research and Innovation
Programme under grant agreement No. 101004719 (OPTICON RadioNet Pilot).
e-MERLIN is a National Facility operated by the University of Manchester at
Jodrell Bank Observatory on behalf of STFC. This work has made use of data
from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. RL, MG, UM acknowledge
financial support from INAF 2022 fundamental research programme ob. fun.
1.05.12.05.16. BM acknowledges financial support from the State Agency for
Research of the Spanish Ministry of Science and Innovation under grant
PID2019-105510GB-C31/AEI/10.13039/501100011033 and through the Unit of
Excellence María de Maeztu 2020–2023 award to the Institute of Cosmos Sciences
(CEX2019- 000918-M).

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