# Incidence bicomodules, Möbius inversion, and a Rota formula for infinity
adjunctions

Louis Carlier

###### Abstract

In the same way decomposition spaces, also known as unital 2-Segal spaces,
have incidence (co)algebras, and certain relative decomposition spaces have
incidence (co)modules, we identify the structures that have incidence
bi(co)modules: they are certain augmented double Segal spaces subject to some
exactness conditions. We establish a Möbius inversion principle for
(co)modules, and a Rota formula for certain more involved structures called
Möbius bicomodule configurations. The most important instance of the latter
notion arises as mapping cylinders of infinity adjunctions, or more generally
of adjunctions between Möbius decomposition spaces, in the spirit of Rota’s
original formula.

###### Contents

  1. 1 Preliminaries
  2. 2 Bicomodules
  3. 3 Correspondences, fibrations, and adjunctions
  4. 4 Möbius inversion for comodules and a Rota formula

## Introduction

The theory of Möbius categories, developed by Leroux [21] generalises the
theory for locally finite posets [25] and Cartier–Foata finite-decomposition
monoids [8], which admit incidence (co)algebras and a Möbius inversion
principle. It has recently been generalised to $\infty$-categories and
decomposition spaces by Gálvez-Carrillo, Kock, and Tonks [13, 14].
(Decomposition spaces are the same thing as the unital $2$-Segal spaces of
Dyckerhoff and Kapranov [10].)

An important tool for computing the Möbius function in the incidence algebra
of a locally finite poset is the classical formula of Rota [25, Theorem 1]
which compares the Möbius functions across a Galois connection. Rota’s
original work already gave many applications of this formula, and it also
features prominently in standard text books such as Aigner [3] and Stanley
[26]. The idea that one should not just work with posets individually, but
rather exploit relationships between posets is of course a very modern idea,
appealing to any mathematician with a categorical bias.

The original goal of the present work, thought to be a routine exercise, was
to generalise this formula to $\infty$-adjunctions. It turned out a lot of
machinery was required to do this in a satisfactory way, and developing this
machinery ended up as a substantial contribution, warranting the present
paper: they are general constructions in the theory of decomposition
spaces/2-Segal spaces, concerning bicomodules, which are of interest not just
in combinatorics but also in representation theory, in connection with Hall
algebras.

Before stating and proving this formula for categories, let us recall some
definitions.

### Incidence coalgebras

Given a small category $X$, write $X_{0}$ for its set of objects and $X_{1}$
for its set of arrows. Let $\mathbb{Q}_{X_{1}}$ be the free vector space on
$X_{1}$. We say a category $X$ is _locally finite_ if each morphism $f:x\to z$
in $X$ admits only finitely many two-step factorisations
$x\xrightarrow{g}y\xrightarrow{h}z$. This condition guarantees that the
comultiplication on $\mathbb{Q}_{X_{1}}$, given by

$\displaystyle\Delta:\mathbb{Q}_{X_{1}}$
$\displaystyle\to\mathbb{Q}_{X_{1}}\otimes\mathbb{Q}_{X_{1}}$ $\displaystyle
f$ $\displaystyle\mapsto\sum_{hg=f}g\otimes h$

is well defined. The counit $\delta:\mathbb{Q}_{X_{1}}\to\mathbb{Q}$ is given
by $\delta(\operatorname{id}_{x})=1$, and $\delta(f)=0$ else.

The _incidence algebra_ $\mathcal{I}_{X}$ is the linear dual,
($\textrm{Lin}(\mathbb{Q}_{X_{1}},\mathbb{Q}),\ast,\delta)$ with the
convolution product:

$(\alpha\ast\beta)(f)=\sum_{hg=f}\alpha(g)\,\beta(h),$

where $\alpha,\beta\in\mathcal{I}_{X}$ and $f\in\mathbb{Q}_{X_{1}}$.

The _zeta function_ $\zeta_{X}:\mathbb{Q}_{X_{1}}\to\mathbb{Q}$ is defined by
$\zeta_{X}(f)=1$ for all $f\in\mathbb{Q}_{X_{1}}$.

Define $\Phi_{\text{even}}:\mathbb{Q}_{X_{1}}\to\mathbb{Q}$ to be the number
of even-length factorisations of a morphism, without identities, and
$\Phi_{\text{odd}}:\mathbb{Q}_{X_{1}}\to\mathbb{Q}$ to be the number of odd-
length factorisations, without identities. A category is _Möbius_ [21] if it
is locally finite and $\Phi_{\text{even}}$ and $\Phi_{\text{odd}}$ are finite.

######  Theorem (Content, Lemay, Leroux [9]).

If $X$ is a Möbius category then the zeta function is invertible, and the
inverse, called the Möbius function, is given by
$\mu=\Phi_{\operatorname{even}}-\Phi_{\operatorname{odd}}$.

Examples of Möbius categories are locally finite posets and monoids with the
finite-decompositions property, and this theorem generalises similar theorems
for these more specialised settings.

### Rota formula for categories

A classical formula due to Rota [25] compares the Möbius functions of two
posets related by a _Galois connection_. The following generalisation of
Rota’s formula to Möbius categories is both natural and straightforward (but
seems not to have been made before).

######  Theorem (Rota formula for Möbius categories).

Let $X$ and $Y$ be Möbius categories, and let $F:X\rightleftarrows Y:G$ be an
adjunction, $F\dashv G$. Then for all $x\in X$, $y\in Y$,

$\sum_{\mathclap{\begin{subarray}{c}x^{\prime}\in X\\\ f:x\to x^{\prime}\\\
Fx^{\prime}=y\end{subarray}}}\mu_{X}(f)=\sum_{\mathclap{\begin{subarray}{c}y^{\prime}\in
Y\\\ g:y^{\prime}\to y\\\ Gy^{\prime}=x\end{subarray}}}\mu_{Y}(g).$

The reader is not expected to read the following elementary proof, but only
notice that it looks like an associativity formula for a convolution product,
except that the arrows live in different categories.

###### Proof of the Rota formula.

$\displaystyle\smashoperator[]{\sum_{\begin{subarray}{c}x^{\prime}\in X\\\
f:x\to x^{\prime}\\\
Fx^{\prime}=y\end{subarray}}^{}}\mu_{X}(x\xrightarrow{f}x^{\prime})$
$\displaystyle\stackrel{{\scriptstyle(1)}}{{=}}\sum_{\begin{subarray}{c}x^{\prime}\in
X\\\ f:x\to x^{\prime}\\\ h:Fx^{\prime}\to
y\end{subarray}}\mu_{X}(x\xrightarrow{f}x^{\prime})\delta_{Y}(Fx^{\prime}\xrightarrow{h}y)$
$\displaystyle\stackrel{{\scriptstyle(2)}}{{=}}\sum_{\begin{subarray}{c}x^{\prime}\in
X\\\ f:x\to x^{\prime}\\\ h:Fx^{\prime}\to
y\end{subarray}}\mu_{X}(x\xrightarrow{f}x^{\prime})\left(\sum_{\begin{subarray}{c}y^{\prime}\in
Y\\\ g:y^{\prime}\to y\\\ h^{\prime}:Fx^{\prime}\to y^{\prime}\\\ \textrm{s.t.
}h=gh^{\prime}\end{subarray}}\zeta_{Y}(Fx^{\prime}\xrightarrow{h^{\prime}}y^{\prime})\mu_{Y}(y^{\prime}\xrightarrow{g}y)\right)$
$\displaystyle\stackrel{{\scriptstyle(3)}}{{=}}\sum_{\begin{subarray}{c}x^{\prime}\in
X,y^{\prime}\in Y\\\ f:x\to x^{\prime}\\\ h:Fx^{\prime}\to y\\\
g:y^{\prime}\to y\\\ h^{\prime}:Fx^{\prime}\to y^{\prime}\\\ \textrm{s.t.
}h=gh^{\prime}\end{subarray}}\mu_{X}(x\xrightarrow{f}x^{\prime})\zeta_{Y}(Fx^{\prime}\xrightarrow{h^{\prime}}y^{\prime})\mu_{Y}(y^{\prime}\xrightarrow{g}y)$
$\displaystyle\stackrel{{\scriptstyle(4)}}{{=}}\sum_{\begin{subarray}{c}x^{\prime}\in
X,y^{\prime}\in Y\\\ g:y^{\prime}\to y\\\ k:x\to Gy^{\prime}\\\ f:x\to
x^{\prime}\\\ k^{\prime}:x^{\prime}\to Gy^{\prime}\\\ \textrm{s.t.
}k=k^{\prime}f\end{subarray}}\mu_{X}(x\xrightarrow{f}x^{\prime})\zeta_{X}(x^{\prime}\xrightarrow{k^{\prime}}Gy^{\prime})\mu_{Y}(y^{\prime}\xrightarrow{g}y)\qquad\text{
by adjunction}$
$\displaystyle\stackrel{{\scriptstyle(5)}}{{=}}\sum_{\begin{subarray}{c}y^{\prime}\in
Y\\\ g:y^{\prime}\to y\\\ k:x\to
Gy^{\prime}\end{subarray}}\left(\sum_{\begin{subarray}{c}x^{\prime}\in X\\\
f:x\to x^{\prime}\\\ k^{\prime}:x^{\prime}\to Gy^{\prime}\\\ \textrm{s.t.
}k=k^{\prime}f\end{subarray}}\mu_{X}(x\xrightarrow{f}x^{\prime})\zeta_{X}(x^{\prime}\xrightarrow{k^{\prime}}Gy^{\prime})\right)\mu_{Y}(y^{\prime}\xrightarrow{g}y)$
$\displaystyle\stackrel{{\scriptstyle(6)}}{{=}}\sum_{\begin{subarray}{c}y^{\prime}\in
Y\\\ g:y^{\prime}\to y\\\ k:x\to
Gy^{\prime}\end{subarray}}\delta_{X}(x\xrightarrow{k}Gy^{\prime})\mu_{Y}(y^{\prime}\xrightarrow{g}y)\stackrel{{\scriptstyle(7)}}{{=}}\sum_{\begin{subarray}{c}y^{\prime}\in
Y\\\ g:y^{\prime}\to y\\\
Gy^{\prime}=x\end{subarray}}\mu_{Y}(y^{\prime}\xrightarrow{g}y).$

∎

In the main result of the present paper, Theorem 4.5.2, we write this formula
as

$\mu_{X}\star_{l}\delta_{Y}=\delta_{X}\star_{r}\mu_{Y},$

with the following more conceptual proof

$\mu_{X}\star_{l}\delta_{Y}\stackrel{{\scriptstyle(2)}}{{=}}\mu_{X}\star_{l}(\zeta\star_{r}\mu_{Y})\stackrel{{\scriptstyle(3-4-5)}}{{=}}(\mu_{X}\star_{l}\zeta)\star_{r}\mu_{Y}\stackrel{{\scriptstyle(6)}}{{=}}\delta_{X}\star_{r}\mu_{Y},$

referring to certain left and right convolution actions. The important insight
here, which is due to Aguiar and Ferrer [4], is that the ‘mixed arrows’ which
appear in the middle factors (those of the form $Fx\to y$, which in the
crucial step of the proof are reinterpreted as $x\to Gy$ by adjunction) belong
to a _bimodule_ : they are acted upon from the left by arrows in the category
$X$ and from the right by arrows in $Y$ (see Example 3.2.3 for an explicit
description). The long complicated sums in the proof are thus condensed into
convolution actions from the left and right, denoted $\star_{l}$ and
$\star_{r}$. Aguiar and Ferrer [4] established a bimodule proof of the Rota
formula in the setting of posets.

Two ingredients are necessary to make sense of the pleasing convolution proof
above: one is to exhibit the data necessary to induce bicomodules and
establish that adjunctions constitute an example. This already accounts for
equalities (3-4-5) in the proof. The other is to establish a Möbius inversion
principle for (co)modules (a notion which has not previously been considered
in the literature, to the knowledge of the author), to account for the
equalities (2) and (6).

In fact, as the notions and arguments become increasingly abstract and
conceptual, it is natural to ask for further generalisation. In this work we
take three considerable abstraction steps (beyond passing from posets to
categories, which is already a fruitful step). First, we pass from categories
and adjunctions to $\infty$-categories and $\infty$-adjunctions. Any
$\infty$-adjunction defines a bicomodule in our sense. This step in itself is
not so easy to justify from the viewpoint of combinatorics, but the homotopy
content inherent in $\infty$-categories is important since already classical
combinatorial structures have symmetries, and these can be handled more
conveniently with groupoids than with sets, as advocated by Baez–Dolan [5],
Gálvez-Kock-Tonks [16] and others. (This aspect will not be of importance in
the present contribution, though.) Second, we pass from $\infty$-categories to
decomposition spaces (also called $2$-Segal spaces) and introduce a notion of
adjunction for them. This step has an important combinatorial motivation,
because many combinatorial coalgebras admit a natural realisation as incidence
coalgebras of decomposition spaces which are not posets or categories. An
important example is the coalgebra of all finite posets, which will serve as a
running example in this paper. The final abstraction step consists in noticing
that the abstract Rota formula works equally well for certain bicomodules
which do not come from adjunctions or $\infty$-adjunctions. In fact the
running example chosen to illustrate the theory is of this type: it is a
certain bicomodule interpolating between the decomposition space of finite
sets and the decomposition space of finite posets. The outcome is the formula
$\mu(P)=(-1)^{n}$ for the Möbius function of a poset with $n$ elements. This
formula is well known (see for example [1]) but its derivation via a Rota
formula is new and interesting, since the coalgebra of finite posets is not
the incidence coalgebra of a locally finite poset or Möbius category.

Finally, a word should be said about the objective approach, an important
aspect of the decomposition-space viewpoint on incidence algebras and Möbius
inversion. The point here is to lift combinatorial identities to bijections of
sets, and more generally equivalences of $\infty$-groupoids. Specifically, the
Möbius-inversion formula, which classically is an equation in vector spaces,
is lifted to an equivalence between $\infty$-groupoids defining spans, whose
homotopy cardinality are the linear maps in question (the introductions of
[13] and [16] contain further motivation). The present work inscribes itself
in this tradition, seeking to find objective structures for bicomodules and
the Rota formula. However, it must be admitted that the objective level is not
fully achieved in this paper. On one hand, the Möbius formula obtained for
comodules is not directly realised as the homotopy cardinality of an
equivalence: it has been found necessary here to take homotopy cardinality a
little bit earlier in the constructions, so that the final arguments take
place at the vector space level. This is due to the increased complexity
compared to the plain Möbius-inversion formula of [14], where an even-odd
splitting could be found for the single decomposition space involved. In the
present situation, _two_ decomposition spaces are involved, and the even-odd
splitting at the objective level could not be found. Furthermore, the
objective analogue of bicomodules given here is not fully satisfactory from
the homotopy viewpoint. While it is shown to induce bicomodules up to
homotopy, the _coherence_ of this up-to-homotopy structure is not established
in this work, and would seem to require considerable further efforts, in the
line of coherence proofs given by [13] and [24]. Further discussion is
included in the main text. The justification for not establishing coherence in
the present contribution is that it is not necessary for the sake of taking
cardinality, as required anyway in the final constructions for the Rota
formula established.

Although the motivation and the statement of the theorem belongs to
combinatorics, the setting for this work and the tools employed are from
simplicial homotopy theory, in the style of [13, 14], working with
$\infty$-groupoids, homotopy pullbacks, mapping spaces, fibrations and fibre
sequences. One technical novelty compared to [13] and [14] is that we exploit
general simplicial maps between decomposition spaces, not just culf ones, and
introduce the notion of adjunction between decomposition spaces. Another is
that the notion of mapping cylinder is exploited systematically: on one hand
locally to model the shapes needed to index the various configurations, and on
the other hand globally, as infinity mapping cylinders.

### Outline of the paper

We begin in Section 1 with a brief review of needed notions from the theory of
$\infty$-categories, with an emphasis on decomposition spaces.

In Section 2, following Walde [27] and Young [28], we first explain how to
obtain a comodule in the context of decomposition spaces.

Proposition 2.1.1. If $f:C\to X$ is a culf map between two simplicial
$\infty$-groupoids such that $C$ is Segal and $X$ is a decomposition space,
then the span

$C_{0}\xleftarrow{d_{1}}C_{1}\xrightarrow{(f_{1},d_{0})}X_{1}\times C_{0}$

induces on the slice $\infty$-category ${\mathcal{S}}_{/C_{0}}$ the structure
of a left ${\mathcal{S}}_{/X_{1}}$-comodule (at the $\pi_{0}$ level), and the
span

$C_{0}\xleftarrow{d_{0}}C_{1}\xrightarrow{(d_{1},f_{1})}C_{0}\times X_{1}.$

induces on ${\mathcal{S}}_{/C_{0}}$ the structure of a right
${\mathcal{S}}_{/X_{1}}$-comodule (at the $\pi_{0}$ level).

The data needed to obtain a comodule is called a _comodule configuration_. In
order to obtain a bicomodule structure, we first need an augmented
bisimplicial $\infty$-groupoid Segal in each direction. We furthermore require
this bisimplicial $\infty$-groupoid to be stable, see Section 2.3. This
stability condition is a pullback condition on certain squares, and is a
$\infty$-categorical reformulation of the notion of
Bergner–Osorno–Ozornova–Rovelli–Scheimbauer [6], suitable for
$\infty$-groupoids.

Theorem 2.4.1. Let $B$ be an augmented stable double Segal space, and such
that the augmentation maps are culf. Suppose moreover $X:=B_{\bullet,-1}$ and
$Y:=B_{-1,\bullet}$ are decomposition spaces. Then the spans

$B_{0,0}\xleftarrow{e_{1}}B_{1,0}\xrightarrow{(u,e_{0})}X_{1}\times B_{0,0}$

and

$B_{0,0}\xleftarrow{d_{0}}B_{0,1}\xrightarrow{(d_{1},v)}B_{0,0}\times Y_{1}$

induce on ${\mathcal{S}}_{/B_{0,0}}$ the structure of a bicomodule over
${\mathcal{S}}_{/X_{1}}$ and ${\mathcal{S}}_{/Y_{1}}$ (at the $\pi_{0}$
level).

An augmented bisimplicial $\infty$-groupoid satisfying the conditions of the
theorem is called a _bicomodule configuration_. We describe as an example the
bicomodule configuration of layered sets and posets, treated in more details
in [7].

In Section 3, we introduce the notion of correspondence of decomposition
spaces: it is a decomposition space $\mathcal{M}$ with a map
$\mathcal{M}\to\Delta^{1}$. We show that any correspondence of decomposition
spaces gives rise to a bicomodule configuration. We then introduce the notion
of _cartesian_ and _cocartesian fibration_ of decomposition spaces, adapting a
homotopy-invariant definition for $\infty$-categories which can be found in
[4]. They give rise to left and right pointed comodule configurations. We
define an _adjunction_ between decomposition spaces $X$ and $Y$ to be a
simplicial map between decomposition spaces $p:\mathcal{M}\to\Delta^{1}$ which
is both a cartesian and a cocartesian fibration, equipped with equivalences
$X\simeq\mathcal{M}_{\\{0\\}}$ and $Y\simeq\mathcal{M}_{\\{1\\}}$. Adjunctions
give rise to bicomodule configurations with two pointings.

In Section 4, we define left and right convolution actions $\star_{l}$ and
$\star_{r}$ dual to the comodule structures. The following is a consequence of
Theorem 2.4.1.

Corollary 4.3.1. Given a bicomodule configuration, the left and right
convolutions satisfy the associative law

$\alpha\star_{l}(\theta\star_{r}\beta)\simeq(\alpha\star_{l}\theta)\star_{r}\beta.$

We then establish in Section 4.4 a Möbius inversion principle for complete
comodules. Let $C\to Y$ be a right comodule configuration such that the
simplicial $\infty$-groupoid $C$ is augmented and with new bottom degeneracies
$s_{-1}:C_{n-1}\to C_{n}$ which are sections to $d_{0}$. We say it is
_complete_ (Section 4.2) if the sections $s_{-1}$ are monomorphisms.

For a complete decomposition space $Y$, let $\overrightarrow{Y_{n}}$ denote
the full subgroupoid of simplices with all principal edges nondegenerate. The
spans $Y_{1}\xleftarrow{d_{1}^{n-1}}\overrightarrow{Y_{n}}\rightarrow 1$
define linear functors, the _Phi functors_
$\Phi_{n}:{\mathcal{S}}_{/Y_{1}}\to{\mathcal{S}}$. We also put
$\displaystyle\Phi_{\text{even}}:=\sum_{n\text{ even}}\Phi_{n}$, and
$\displaystyle\Phi_{\text{odd}}:=\sum_{n\text{ odd}}\Phi_{n}$.

The zeta functor $\zeta^{C}:{\mathcal{S}}_{/C_{0}}\to{\mathcal{S}}$ is the
linear functor defined by the span $C_{0}\xleftarrow{=}C_{0}\xrightarrow{}1$,
and $\delta^{R}:{\mathcal{S}}_{/C_{0}}\to{\mathcal{S}}$ is the linear functor
given by the span $C_{0}\xleftarrow{s_{-1}}C_{-1}\xrightarrow{}1$. We define
$\delta^{L}$ similarly for left comodule configurations.

Theorem 4.4.4 and 4.4.5. Given $C\to Y$ a complete right comodule
configuration and $D\to X$ a complete left comodule configuration, then

$\zeta^{C}\star_{r}\Phi_{\text{even}}^{Y}\simeq\delta^{R}+\zeta^{C}\star_{r}\Phi_{\text{odd}}^{Y},$
$\Phi_{\text{even}}^{X}\star_{l}\zeta^{D}\simeq\delta^{L}+\Phi_{\text{odd}}^{X}\star_{l}\zeta^{D}.$

In Section 4.5, we establish a Möbius inversion principle at the algebraic
level. To this end, we need to impose some finiteness conditions in order to
take homotopy cardinality. Define the _Möbius functions_ as the homotopy
cardinalities $|\mu^{Y}|:=|\Phi^{Y}_{\text{even}}|-|\Phi^{Y}_{\text{odd}}|$
and $|\mu^{X}|:=|\Phi^{X}_{\text{even}}|-|\Phi^{X}_{\text{odd}}|$.

Theorem 4.5.1. Given $C\to Y$ a right Möbius comodule configuration and $D\to
X$ a left Möbius comodule configuration,

$|\zeta^{C}|\star_{r}|\mu^{Y}|=|\delta^{R}|,\qquad|\mu^{X}|\star_{l}|\zeta^{D}|=|\delta^{L}|.$

Finally we can extend the Rota formula to bicomodules with Möbius inversion
for both comodules, called _Möbius bicomodule configurations_. Combining
Proposition 4.3.1 and Theorem 4.5.1, we obtain the main theorem of the present
paper:

Theorem 4.5.2. Given a Möbius bicomodule configuration $B$ with
$X:=B_{\bullet,-1}$ and $Y:=B_{-1,\bullet}$, we have

$|\mu^{X}|\star_{l}|\delta^{R}|=|\delta^{L}|\star_{r}|\mu^{Y}|,$

where $\delta^{R}$ is the linear functor given by the span

$B_{0,0}\xleftarrow{}X_{0}\xrightarrow{}1,$

and $\delta^{L}$ is the linear functor given by the span

$B_{0,0}\xleftarrow{}Y_{0}\xrightarrow{}1.$

The motivating example, treated in Section 4.6, shows that any (co)cartesian
fibration $p:\mathcal{M}\to\Delta^{1}$ such that $\mathcal{M}$ is a complete
decomposition space gives rise to a complete left (or right) comodule
configuration:

Theorem 4.6.3. Given an adjunction of decomposition spaces in the form of a
bicartesian fibration $p:\mathcal{M}\to\Delta^{1}$, suppose moreover that
$\mathcal{M}$ is a Möbius decomposition space. Then the bicomodule
configuration extracted from this data is Möbius. In particular, we have the
Rota formula for the adjunction $p$:

$|\mu^{X}|\star_{l}|\delta^{R}|=|\delta^{L}|\star_{r}|\mu^{Y}|.$

When specialised to the case of a classical adjunction between 1-categories,
this is the classical Rota formula from page Theorem.

Finally, the bicomodule configuration of layered sets and posets defined in
Section 2 is Möbius, and we apply the generalised Rota formula to compute the
Möbius function of the incidence algebra of the decomposition space of finite
posets.

Theorem 4.7.3 ([7, Theorem 3.4]). The Möbius function of the incidence algebra
of the decomposition space $X$ of finite posets is

$\mu(P)=\begin{cases}(-1)^{n}&\text{ if $P\in X_{1}$ is a discrete poset with
$n$ elements,}\\\ 0&\text{ else.}\end{cases}$

It is shown in [7] how this result implies similar results for any directed
restriction species or free operad.

### Acknowledgements

The author would like to thank Joachim Kock not only for suggesting to
investigate this Rota formula but also for help and support all along the
project, and also Christian Sattler for useful remarks. The author is grateful
to Kurusch Ebrahimi-Fard and Yannic Vargas, who independently pointed out the
work of Aguiar and Ferrer. Finally the author wish to thank the anonymous
referee for a thorough and constructive report that led to many expository
improvements. The author was supported by PhD grant attached to
MTM2013-42293-P, and grant number MTM2016-80439-P of Spain.

##  1 Preliminaries

We work in the $\infty$-category of $\infty$-groupoids, denoted
${\mathcal{S}}$, following the notation of [13]. Our $\infty$-categories are
quasi-categories; the theory of quasi-categories has been substantially
developed by Joyal [18, 19] and Lurie [22]. An $\infty$-groupoid is an
$\infty$-category in which all morphisms are invertible. They are precisely
Kan complexes: simplicial sets in which every horn admits a filler (and not
only the inner ones).

###  1.1 Pullbacks and fibres, slices and linear functors

The main tool used throughout this paper are pullbacks. We use the following
standard lemma many times.

######  Lemma 1.1.1 ([22, Lemma 4.4.2.1]).

Given a prism diagram of $\infty$-groupoids

${X}$${X^{\prime}}$${X^{\prime\prime}}$${Y}$${Y^{\prime}}$${Y^{\prime\prime}}$${\lrcorner}$

in which the right-hand square is a pullback. Then the outer rectangle is a
pullback if and only if the left-hand square is.

Given a map of $\infty$-groupoids $p:X\to S$, an object $s\in S$, the _fibre_
$X_{s}$ of $p$ over $s$ is the pullback

${X_{s}}$${X}$${1}$${S.}$${\lrcorner}$$\scriptstyle{p}$$\scriptstyle{\ulcorner
s\urcorner}$

A map of $\infty$-groupoids is a _monomorphism_ when its fibres are
$(-1)$-groupoids, that is, are either empty or contractible. If $f:X\to Y$ is
a monomorphism, then there is a complement $Z:=Y\backslash X$ such that
$Y\simeq X+Z$; a monomorphism is essentially an equivalence from X onto some
connected components of Y.

Recall that the objects of the slice $\infty$-category ${\mathcal{S}}_{/I}$
are maps of $\infty$-groupoids with codomain $I$. Pullback along a morphism
$f:J\to I$, defines an functor
$f^{*}:{\mathcal{S}}_{/I}\to{\mathcal{S}}_{/J}$. This functor is right adjoint
to the functor $f_{!}:{\mathcal{S}}_{/J}\to{\mathcal{S}}_{/I}$ given by post-
composing with $f$. A span $I\xleftarrow{p}M\xrightarrow{q}J$ induces a
functor between the slices by pullback and postcomposition

${\mathcal{S}}_{/I}\xrightarrow{p^{*}}{\mathcal{S}}_{/M}\xrightarrow{q_{!}}{\mathcal{S}}_{/J}.$

A functor is _linear_ if it is homotopy equivalent to a functor induced by a
span. The following Beck-Chevalley rule holds for $\infty$-groupoids: for any
pullback square

${J}$${I}$${V}$${U,}$$\scriptstyle{f}$$\scriptstyle{p}$${\lrcorner}$$\scriptstyle{q}$$\scriptstyle{g}$

the functors $p_{!}f^{*},g^{*}q_{!}:{\mathcal{S}}_{/I}\to{\mathcal{S}}_{/V}$
are naturally homotopy equivalent (see [11] for the technical details
regarding coherence of these equivalences). By the Beck-Chevalley rule, the
composition of two linear functors is linear. For an extended treatment of
linear functors and homotopy linear algebra, we refer to [12].

###  1.2 Segal spaces and decomposition spaces

We consider the functor $\infty$-category

$\operatorname{Fun}{(\boldsymbol{\Delta}^{\text{{\rm{op}}}},{\mathcal{S}})}$

whose objects are _simplicial $\infty$-groupoids_, that is functors from the
$\infty$-category $\boldsymbol{\Delta}^{\text{{\rm{op}}}}$ to the
$\infty$-category ${\mathcal{S}}$.

A simplicial $\infty$-groupoid $X$ is called a _Segal space_ if the following
squares are pullbacks, for all $n>0$:

${X_{n+1}}$${X_{n}}$${X_{n}\
}$${X_{n-1}.}$${\lrcorner}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{n+1}}$$\scriptstyle{d_{n}}$$\scriptstyle{d_{0}}$

The simplex category $\boldsymbol{\Delta}$ has an active-inert factorisation
system. A morphism $[m]\to[n]$ is _active_ (also called _generic_) if it
preserves endpoints: $g(0)=0$, $g(m)=n$. A morphism is _inert_ (also called
_free_) if it is distance preserving: $f(i+1)=f(i)+1$, for $0\leq i\leq m-1$.
The active maps are generated by the codegeneracy maps and the inner coface
maps, and the inert maps are generated by the outer coface maps
$d^{\bot}:=d^{0}$ and $d^{\top}:=d^{n}$.

A _decomposition space_
$X:\boldsymbol{\Delta}^{\text{{\rm{op}}}}\to{\mathcal{S}}$ is a simplicial
$\infty$-groupoid such that the image of any pushout diagram in
$\boldsymbol{\Delta}$ of an active map $g$ along an inert map $f$ is a
pullback of $\infty$-groupoids. It is enough to check that the following
squares are pullbacks, where $0\leq k\leq n$:

${X_{n+1}}$${X_{n+2}}$${X_{n}}$${X_{n+1},}$$\scriptstyle{s_{k+1}}$$\scriptstyle{d_{\bot}}$${\lrcorner}$$\scriptstyle{d_{\bot}}$$\scriptstyle{s_{k}}$

${X_{n+2}}$${X_{n+3}}$${X_{n+1}}$${X_{n+2},}$$\scriptstyle{d_{\bot}}$$\scriptstyle{d_{k+2}}$$\scriptstyle{d_{\bot}}$${\llcorner}$$\scriptstyle{d_{k+1}}$

${X_{n+1}}$${X_{n+2}}$${X_{n}}$${X_{n+1},}$$\scriptstyle{s_{k}}$$\scriptstyle{d_{\top}}$${\lrcorner}$$\scriptstyle{d_{\top}}$$\scriptstyle{s_{k}}$

${X_{n+2}}$${X_{n+3}}$${X_{n+1}}$${X_{n+2}.}$$\scriptstyle{d_{\top}}$$\scriptstyle{d_{k+1}}$$\scriptstyle{d_{\top}}$${\llcorner}$$\scriptstyle{d_{k+1}}$

The notion of decomposition space was introduced by Gálvez-Carrillo, Kock, and
Tonks [13], and independently by Dyckerhoff and Kapranov [10] under the name
unital $2$-Segal space. It can be seen as an abstraction of posets. The
equivalence of the two notions follows from the pullback formulation of
2-Segal spaces given in Proposition 2.3.2 of [10]. It is precisely the
condition required to obtain a counital coassociative comultiplication on
${\mathcal{S}}_{/X_{1}}$, see also [24] for the exact role played by the
decomposition space condition. Since the motivation in the present paper comes
from combinatorics, we follow the terminology of [13]; for a survey motivated
by combinatorics, see [16].

######  Proposition 1.2.1 ([10, Proposition 2.3.3], [13, Proposition 3.5]).

Every Segal space is a decomposition space.

There are plenty of examples of decomposition spaces which are not Segal, e.g.
Schmitt’s Hopf algebra of graphs, which is a running example in [16].

###  1.3 Incidence coalgebras and culf functors

For any decomposition space $X$, we get an incidence coalgebra [10], [13]. The
span $X_{1}\xleftarrow{d_{1}}X_{2}\xrightarrow{(d_{2},d_{0})}X_{1}\times
X_{1}$ defines a linear functor, the _comultiplication_ :

$\displaystyle\Delta:{\mathcal{S}}_{/X_{1}}$
$\displaystyle\to{\mathcal{S}}_{/X_{1}\times X_{1}}$
$\displaystyle(T\xrightarrow{t}X_{1})$
$\displaystyle\mapsto(d_{2},d_{0})_{!}\circ d_{1}^{*}(t).$

The span $X_{1}\xleftarrow{s_{0}}X_{0}\xrightarrow{z}1$ defines a linear
functor, the _counit_ :

$\displaystyle\delta:{\mathcal{S}}_{/X_{1}}$ $\displaystyle\to{\mathcal{S}}$
$\displaystyle(T\xrightarrow{t}X_{1})$ $\displaystyle\mapsto z_{!}\circ
s_{0}^{*}(t).$

The up-to-coherent-homotopy coassociativity follows from the decomposition
space axioms, see [13, §5 and 7] or [24, §4.3] for a proof. We obtain a
coalgebra $({\mathcal{S}}_{/X_{1}},\Delta,\delta)$ called the _incidence
coalgebra_.

The $\infty$-category ${\mathcal{S}}_{/I}$ plays the role of the vector space
with basis $I$. The presheaf category ${\mathcal{S}}^{I}$ can be considered
the linear dual of the slice category ${\mathcal{S}}_{/I}$ (see [12] for the
precise statements and proofs). A span $I\xleftarrow{}M\xrightarrow{}J$
defines both a linear functor
${\mathcal{S}}_{/I}\xrightarrow{}{\mathcal{S}}_{/J}$ and the dual linear
functor ${\mathcal{S}}^{J}\to{\mathcal{S}}^{I}$. If X is a decomposition
space, the coalgebra structure on ${\mathcal{S}}_{/X_{1}}$ therefore induces
an algebra structure on ${\mathcal{S}}^{X_{1}}$. In details, the _convolution
product_ of two linear functors $F,G:{\mathcal{S}}_{/X_{1}}\to{\mathcal{S}},$
given by the spans $X_{1}\xleftarrow{}M\xrightarrow{}1$ and
$X_{1}\xleftarrow{}N\xrightarrow{}1$, is the composite of their tensor product
$F\otimes G$ with the comultiplication:

$F\ast
G:{\mathcal{S}}_{/X_{1}}\xrightarrow{\Delta}{\mathcal{S}}_{/X_{1}}\otimes{\mathcal{S}}_{/X_{1}}\xrightarrow{F\otimes
G}{\mathcal{S}}\otimes{\mathcal{S}}\simeq{\mathcal{S}},$

where the tensor product $F\otimes G$ is given by the span $X_{1}\times
X_{1}\xleftarrow{}M\times N\xrightarrow{}1$. The neutral element for
convolution is

$\delta:{\mathcal{S}}_{/X_{1}}\to{\mathcal{S}}$

defined by the span $X_{1}\xleftarrow{s_{0}}X_{0}\xrightarrow{}1.$

A map $f:X\to Y$ of simplicial spaces is _cartesian_ on an arrow $[n]\to[k]$
in $\boldsymbol{\Delta}$ if the naturality square for $F$ with respect to this
arrow is a pullback. It is called a _right fibration_ if it is cartesian on
$d_{\bot}$ and on all active maps, and is called a _left fibration_ if it is
cartesian on $d_{\top}$ and on all active maps.

A simplicial map $f:X\to Y$ is _conservative_ if it is cartesian with respect
to codegeneracy maps

${X_{n}}$${X_{n+1}}$${Y_{n}\
}$${Y_{n+1}}$${\lrcorner}$$\scriptstyle{s_{i}}$$\scriptstyle{f_{n}}$$\scriptstyle{f_{n+1}}$$\scriptstyle{s_{i}}$

, $\qquad 0\leq i\leq n$.

It is _ulf_ (unique lifting of factorisations) if it is cartesian with respect
to inner coface maps

${X_{n+1}}$${X_{n+2}}$${Y_{n+1}}$${Y_{n+2}}$${\lrcorner}$$\scriptstyle{f_{n+1}}$$\scriptstyle{d_{i+1}}$$\scriptstyle{f_{n+2}}$$\scriptstyle{d_{i+1}}$

, $\qquad 0\leq i\leq n$.

We write _culf_ for conservative and ulf, that is cartesian on all active
maps. The culf functors induce coalgebra homomorphisms between the incidence
algebras. They play an essential role in [13] and [14] as a natural notion of
morphism between decomposition spaces, but the present paper deals also with
general simplicial maps.

##### Layered finite posets and layered finite sets

We refer to [15] for the following material. An _$n$ -layering_ of a finite
poset $P$ is a monotone map $l:P\to\underline{n}$, where
$\underline{n}=\\{1,\dots,n\\}$ are the objects of the skeleton of the
category of finite ordered sets (possibly empty) and monotone maps. The fibres
$P_{i}=l^{-1}(i)$, $i\in\underline{n}$ are called layers, and can be empty.
The objects of the groupoid $\mathbf{C}_{n}$ of $n$-layered finite posets are
monotone maps $l:P\to\underline{n}$ and the morphisms are triangles

${P}$${P^{\prime}}$${\underline{n},}$$\scriptstyle{\simeq}$

where $P\to P^{\prime}$ is a monotone bijection. They assemble into a
simplicial groupoid $\mathbf{C}$. The face maps are given by joining layers,
or deleting an outer layer for the top and bottom face maps. The degeneracy
maps are given by inserting empty layers.

######  Proposition 1.3.1 ([15, Proposition 6.12, Lemma 6.13]).

The simplicial groupoid $\mathbf{C}$ of layered finite posets is a
decomposition space (but not a Segal space), and is complete, locally finite,
locally discrete, and of locally finite length.

The incidence coalgebra of $\mathbf{C}$ has comultiplication given by the span

$\mathbf{C}_{1}\xleftarrow{d_{1}}\mathbf{C}_{2}\xrightarrow{(d_{2},d_{0})}\mathbf{C}_{1}\times\mathbf{C}_{1},$

where $d_{1}$ joins the two layers, and $d_{2}$ and $d_{0}$ return the two
layers. The comultiplication of a poset is thus obtained by summing over
admissible cuts (a $2$-layering of the poset) and taking tensor product of the
two layers.

Similarly, let $\mathbf{I}_{n}$ denote the groupoid of all layerings of finite
sets. Again these groupoids assemble into a simplicial groupoid, denoted
$\mathbf{I}$.

######  Proposition 1.3.2 ([15, Proposition 4.3, Lemma 4.4]).

The simplicial groupoid $\mathbf{I}$ is a Segal space, and hence a
decomposition space, which is complete, locally finite, locally discrete, and
of locally finite length.

The simplicial groupoid $\mathbf{C}$ is the decomposition space corresponding
to the terminal directed restriction species, finite posets and convex maps,
while $\mathbf{I}$ is the decomposition space corresponding to the terminal
restriction species, finite sets and injections. The incidence coalgebra of
$\mathbf{I}$ is the binomial coalgebra [15, §2.4] with well-known Möbius
function $(-1)^{n}$ for a set with $n$ elements.

##  2 Bicomodules

###  2.1 Comodules

The theory of modules in the context of decomposition spaces has been
developed by Walde [27], and independently by Young [28], both in the context
of Hall algebras. They call them relative 2-Segal spaces. Here we give a
conceptual way to reformulate their definitions using linear functors.

Given a map between two simplicial $\infty$-groupoids $f:C\to X$, the span
$C_{0}\xleftarrow{d_{1}}C_{1}\xrightarrow{(f_{1},d_{0})}X_{1}\times C_{0}$
defines a linear functor
$\gamma_{l}:{\mathcal{S}}_{/{C_{0}}}\to{\mathcal{S}}_{/X_{1}}\otimes{\mathcal{S}}_{/{C_{0}}}$,
and the span
$C_{0}\xleftarrow{d_{0}}C_{1}\xrightarrow{(d_{1},f_{1})}C_{0}\times X_{1}$
defines a linear functor
$\gamma_{r}:{\mathcal{S}}_{/{C_{0}}}\to{\mathcal{S}}_{/C_{0}}\otimes{\mathcal{S}}_{/{X_{1}}}$.

######  Proposition 2.1.1.

Let $f:C\to X$ be a map between two simplicial $\infty$-groupoids. Suppose
moreover that $C$ is Segal, $X$ is a decomposition space and the map $f:C\to
X$ is culf, then the span

$C_{0}\xleftarrow{d_{1}}C_{1}\xrightarrow{(f_{1},d_{0})}X_{1}\times C_{0}$

induces on the slice $\infty$-category ${\mathcal{S}}_{/C_{0}}$ the structure
of a left ${\mathcal{S}}_{/X_{1}}$-comodule (at the $\pi_{0}$ level), and the
span

$C_{0}\xleftarrow{d_{0}}C_{1}\xrightarrow{(d_{1},f_{1})}C_{0}\times X_{1}.$

induces on ${\mathcal{S}}_{/C_{0}}$ the structure of a right
${\mathcal{S}}_{/X_{1}}$-comodule (at the $\pi_{0}$ level).

The data needed to obtain a comodule is called a _comodule configuration_ ,
that is a culf map from a Segal space to a decomposition space.

######  Remark 2.1.2.

The relevance of the Segal condition on $C$ and the culf condition on $f$ can
be explained individually as follows. It is standard that for a category $C$,
the coalgebra of arrows $C_{1}$ coacts on $C_{0}$: the coaction (from the
right) is given by $b\mapsto\sum_{f:a\to b}a\otimes f$. Coassociativity of
this coaction is equivalent to the Segal condition. Now a culf map $C\to X$
defines a coalgebra homomorphism, and in this way, also $X_{1}$ coacts on
$C_{0}$, by “corestriction of coscalars”.

######  Remark 2.1.3.

The proposition is stated only at the $\pi_{0}$-level. This means that we
establish only the comodule structure up to homotopy, but do not establish the
_coherence_ of this up-to-homotopy structure. A stronger result, a partial
coherence result, is given by [27] and [28], who establish the coherence at
the $1$-truncated level (rather than the $0$-truncated level established
here). It is most likely that full coherence can be established by exploiting
the techniques employed by [13] and [24]. While only a small bit of the axioms
are used to establish the proposition as stated, the full decomposition-space
axioms and the culf condition are expected to be required for the fully
coherent result, and this is why these conditions have been included in the
definition of comodule configuration.

###### Proof.

We want to prove that the map $\gamma_{l}$ is a left
${\mathcal{S}}_{/X_{1}}$-coaction. The desired diagram, commutative up to
homotopy

${{\mathcal{S}}_{/C_{0}}}$${{\mathcal{S}}_{/X_{1}\times
C_{0}}}$${{\mathcal{S}}_{/X_{1}\times C_{0}}}$${{\mathcal{S}}_{/X_{1}\times
X_{1}\times
C_{0}}}$$\scriptstyle{\gamma_{l}}$$\scriptstyle{\gamma_{l}}$$\scriptstyle{\operatorname{Id}\otimes\gamma_{l}}$$\scriptstyle{\Delta\otimes\operatorname{Id}}$

is induced by the solid spans in the diagram

${C_{0}}$${C_{1}}$${X_{1}\times C_{0}}$${C_{1}}$${C_{2}}$${X_{1}\times
C_{1}}$${X_{1}\times C_{0}}$${X_{2}\times C_{0}}$${X_{1}\times X_{1}\times
C_{0}.}$$\scriptstyle{d_{1}}$$\scriptstyle{(f_{1},d_{0})}$$\scriptstyle{d_{1}}$$\scriptstyle{(f_{1},d_{0})}$$\scriptstyle{d_{1}}$$\scriptstyle{(f_{1},d_{0}d_{0})}$$\scriptstyle{(d_{2}f_{1},d_{0})}$$\scriptstyle{d_{2}}$${\urcorner}$${\llcorner}$$\scriptstyle{\operatorname{Id}\otimes
d_{1}}$$\scriptstyle{\operatorname{Id}\otimes{(f_{1},d_{0})}}$$\scriptstyle{d_{1}\otimes\operatorname{Id}}$$\scriptstyle{(d_{2},d_{0})\otimes\operatorname{Id}}$

The coassociativity (at the $\pi_{0}$ level) will follow from Beck-Chevalley
equivalences if we have the two pullbacks indicated in the diagram. The upper
right-hand square is a pullback if and only if its composite with the second
projection is a pullback. This composite outer square is a pullback because
$C$ satisfies the Segal condition. Similarly, the lower left-hand square is a
pullback if its composite with the first projection is a pullback. This
composite outer square is a pullback because $f:C\to X$ is culf. ∎

######  Example 2.1.4 (Décalage [17]).

Given a simplicial space $X$, the _lower décalage_
$\operatorname{Dec}_{\bot}(X)$ is the simplicial space obtained by deleting
$X_{0}$, all $d_{0}$ face maps and $s_{0}$ degeneracy maps. The original
$d_{0}$ maps induce a simplicial map $d_{\bot}:\operatorname{Dec}_{\bot}(X)\to
X$, called the décalage map. Similarly, the _upper décalage_
$\operatorname{Dec}_{\top}(X)$ is the simplicial space obtained by deleting
$X_{0}$, all last face maps $d_{\top}$ and last degeneracy maps $s_{\top}$.
The original $d_{\top}$ maps induce a simplicial map
$d_{\top}:\operatorname{Dec}_{\top}X\to X$.

It is well known that $\operatorname{Dec}_{\bot}(X)$ is a Segal space and the
décalage map is culf (see [13, Proposition 4.9]). Hence we have a comodule
configuration. The resulting comodule is the incidence coalgebra of $X$ as a
(right) comodule over itself.

For categories, given a functor $f:C\to D$, define the _mapping cylinder_ (or
_collage_ in [20]) $M_{f}$ to be the category where objects are either objects
of $C$ or objects of $D$ and

$\operatorname{Hom}_{M_{f}}(x,y)=\begin{cases}\operatorname{Hom}_{C}(x,y)\,&\text{
if }x,y\in C,\\\ \operatorname{Hom}_{D}(x,y)\,&\text{ if }x,y\in D,\\\
\operatorname{Hom}_{D}(f(x),y)\,&\text{ if }x\in C,y\in D,\\\
\emptyset\,&\text{ else }.\\\ \end{cases}$

There exists a unique $p:M_{f}\to\Delta^{1}$ such that $p^{-1}(0)=C$ and
$p^{-1}(1)=D$. This is moreover a cocartesian fibration, the cocartesian lift
for $x\in C$ being given by
$\operatorname{Id}_{f(x)}\in\operatorname{Map}_{M_{f}}(x,f(x))$. The shape of
a comodule configuration is that of
$(M_{\operatorname{id}})^{\text{{\rm{op}}}}$, where $M_{\operatorname{id}}$ is
the mapping cylinder of the identity of $\boldsymbol{\Delta}$. In other words,
a comodule configuration is a functor from
$(M_{\operatorname{id}})^{\text{{\rm{op}}}}$ to ${\mathcal{S}}$ (satisfying
certain conditions).

Let $\boldsymbol{\Delta}_{\text{bot}}$ be the simplex category of finite
linear orders with a specified bottom element, and bottom-preserving monotone
maps. Consider the mapping cylinder $M_{j}$ of the functor
$j:\boldsymbol{\Delta}\to\boldsymbol{\Delta}_{\text{bot}}$ freely adding a
bottom element. Presheaves on $M_{j}$ are diagrams of the following shape.

${X_{0}}$${X_{1}}$${X_{2}}$${C_{-1}}$${C_{0}}$${C_{1}}$${C_{2}}$$\scriptstyle{s_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{s_{1}}$$\scriptstyle{s_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{2}}$${\dots}$$\scriptstyle{v}$$\scriptstyle{u}$$\scriptstyle{s_{0}}$$\scriptstyle{v}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{s_{1}}$$\scriptstyle{s_{0}}$$\scriptstyle{v}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{2}}$${\dots}$

This is the shape of what we call a _right pointed comodule configuration_ :
it is a comodule configuration $C\to X$ such that the Segal space $C$ is
augmented, and with new bottom sections $s_{-1}:C_{n-1}\to C_{n}$. The
importance of the pointing (the extra bottom degeneracy maps) is that it makes
possible to formulate the notion of completeness and the condition locally
finite length, see Section 4.5 below; it guarantees the existence of a
filtration on the associated comodule (see [14, §6] for a similar argument),
which is of independent interest.

######  Example 2.1.5.

The comodule configuration obtained from the lower décalage of a decomposition
space $X$ is also right pointed, the augmentation map is given by
$d_{1}:X_{1}\to X_{0}$, and the extra bottom sections by $s_{0}$.

###  2.2 Augmented bisimplicial infinity-groupoids

We shall establish conditions under which left and right comodule structures
define a bicomodule. The main objects of interest are augmented bisimplicial
$\infty$-groupoids subject to conditions, which are formulated in terms of
pullbacks. We consider the functor $\infty$-category

$\operatorname{Fun}{(\boldsymbol{\Delta}^{\text{{\rm{op}}}}\times\boldsymbol{\Delta}^{\text{{\rm{op}}}},{\mathcal{S}})}$

whose objects are _bisimplicial $\infty$-groupoids_, that is functors from the
$\infty$-category
$\boldsymbol{\Delta}^{\text{{\rm{op}}}}\times\boldsymbol{\Delta}^{\text{{\rm{op}}}}$
to the $\infty$-category ${\mathcal{S}}$.

A _double Segal space_ is a bisimplicial $\infty$-groupoid satisfying the
Segal condition for each restriction
$\boldsymbol{\Delta}^{\text{{\rm{op}}}}\times\\{[n]\\}\to{\mathcal{S}}$ (the
columns) and
$\\{[n]\\}\times\boldsymbol{\Delta}^{\text{{\rm{op}}}}\to{\mathcal{S}}$ (the
rows).

Let $\boldsymbol{\Delta}_{+}$ be the augmented simplex category of all finite
ordinals and order-preserving maps. An _augmented_ bisimplicial
$\infty$-groupoid $B$ has in addition $\infty$-groupoids $B_{i,-1}$ and
$B_{-1,i}$ of $(-1)$-simplices. We consider the functor $\infty$-category

$\operatorname{Fun}{(\boldsymbol{\Delta}^{\text{{\rm{op}}}}_{+}\times\boldsymbol{\Delta}^{\text{{\rm{op}}}}_{+}\backslash\\{(-1,-1)\\},{\mathcal{S}})}$

whose objects are _augmented bisimplicial $\infty$-groupoids_.

######  Remark 2.2.1.

The shape of an augmented bisimplicial $\infty$-groupoid is
$(\boldsymbol{\Delta}_{/\Delta^{1}})^{\text{{\rm{op}}}}$. We denote $[i,j]$
the object given by the map $\Delta^{i+1+j}\to\Delta^{1}$ sending the $i+1$
first vertices to $0$ and the others to $1$. We allow $i$ or $j$ to be equal
to $-1$ but not both. Maps $[i,j]\to[k,l]$ are given by the inclusions
respecting the horizontal map. For example, the object $[2,1]$ can be drawn as
follows

.....

where the horizontal maps lie over the map in $\Delta^{1}$.

We can draw $[i,j]$ as a column of $i{+}1$ black dots followed by $j{+}1$
white dots. Maps send black dots to black dots and white dots to white dots,
without crossing.

We use the following notation for an augmented bisimplicial $\infty$-groupoid.
We denote $d_{k}:B_{i,j}\to B_{i,j-1}$ and $e_{l}:B_{i,j}\to B_{i-1,j}$ the
face maps, and $s_{k}:B_{i,j-1}\to B_{i,j}$ and $t_{l}:B_{i-1,j}\to B_{i,j}$
the degeneracy maps; $u$ and $v$ are the augmentation maps.

${B_{-1,0}}$${B_{-1,1}}$${B_{-1,2}}$${B_{0,-1}}$${B_{0,0}}$${B_{0,1}}$${B_{0,2}}$${B_{1,-1}}$${B_{1,0}}$${B_{1,1}}$${B_{1,2}}$$\scriptstyle{s_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{s_{1}}$$\scriptstyle{s_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{2}}$${\dots}$$\scriptstyle{t_{0}}$$\scriptstyle{t_{0}}$$\scriptstyle{v}$$\scriptstyle{u}$$\scriptstyle{s_{0}}$$\scriptstyle{t_{0}}$$\scriptstyle{v}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{s_{1}}$$\scriptstyle{s_{0}}$$\scriptstyle{v}$$\scriptstyle{t_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{2}}$${\dots}$$\scriptstyle{e_{0}}$$\scriptstyle{e_{1}}$${\vdots}$$\scriptstyle{e_{0}}$$\scriptstyle{e_{1}}$$\scriptstyle{u}$$\scriptstyle{s_{0}}$${\vdots}$$\scriptstyle{e_{0}}$$\scriptstyle{e_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{s_{1}}$$\scriptstyle{s_{0}}$${\vdots}$$\scriptstyle{e_{0}}$$\scriptstyle{e_{1}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{2}}$${\dots}$${\vdots}$

An augmented double Segal space satisfies that rows and columns are Segal. If
we suppose that augmentations are culf and $B_{\bullet,-1}$ and
$B_{-1,\bullet}$ are decomposition spaces, we can apply Proposition 2.1.1 to
obtain comodules: the span

$B_{0,0}\xleftarrow{e_{1}}B_{1,0}\xrightarrow{(u,e_{0})}B_{1,-1}\times
B_{0,0}$

induces on ${\mathcal{S}}_{/B_{0,0}}$ the structure of a left comodule over
${\mathcal{S}}_{/B_{1,-1}}$, and the span

$B_{0,0}\xleftarrow{d_{0}}B_{0,1}\xrightarrow{(d_{1},v)}B_{0,0}\times
B_{-1,1}$

induces on ${\mathcal{S}}_{/B_{0,0}}$ the structure of a right comodule over
${\mathcal{S}}_{/B_{-1,1}}$.

###  2.3 Stability

We say a bisimplicial $\infty$-groupoid is _stable_ if the following squares
are pullbacks:

${B_{i-1,j-1}}$${B_{i-1,j}}$${B_{i,j-1}}$${B_{i,j},}$$\scriptstyle{d_{k}}$$\scriptstyle{e_{l}}$$\scriptstyle{d_{k}}$$\scriptstyle{e_{l}}$${\ulcorner}$${B_{i-1,j-1}}$${B_{i-1,j}}$${B_{i,j-1}}$${B_{i,j},}$$\scriptstyle{s_{k}}$$\scriptstyle{e_{l}}$$\scriptstyle{s_{k}}$${\urcorner}$$\scriptstyle{e_{l}}$

${B_{i-1,j-1}}$${B_{i-1,j}}$${B_{i,j-1}}$${B_{i,j},}$$\scriptstyle{t_{l}}$$\scriptstyle{t_{l}}$$\scriptstyle{d_{k}}$${\llcorner}$$\scriptstyle{d_{k}}$${B_{i-1,j-1}}$${B_{i-1,j}}$${B_{i,j-1}}$${B_{i,j},}$$\scriptstyle{s_{k}}$$\scriptstyle{t_{l}}$${\lrcorner}$$\scriptstyle{t_{l}}$$\scriptstyle{s_{k}}$

for all values of the indices except for $d_{\bot}$ along $e_{\top}$ and
$d_{\top}$ along $e_{\bot}$.

######  Remark 2.3.1.

A bisimplicial $\infty$-groupoid is stable if it satisfies all the following
properties:

  * •

$s_{k}:B_{i,j-1}\to B_{i,j}$ is a cartesian natural transformation, for all
$0\leq k\leq j-1$;

  * •

$d_{k}$, $k\neq\top,\bot$, is a cartesian natural transformation;

  * •

$d_{\top}$ is a left fibration;

  * •

$d_{\bot}$ is a right fibration.

######  Remark 2.3.2.

Bergner, Osorno, Ozornova, Rovelli, and Scheimbauer introduced the notion of
stable double category (bisimplicial set) in [6]: they define a double
category to be stable if every square is uniquely determined by its span of
source morphisms and, independently by its cospan of target morphisms. The
present definition is a categorical reformulation of their notion suitable for
$\infty$-groupoids. The motivation for the terminology is the following
example. Let $C$ be a stable $\infty$-category (see [23, Chapter 1]). Define a
double Segal space $B$ where $B_{0,0}$ is the $\infty$-groupoid of objects of
$C$, where $B_{0,1}$ is the $\infty$-groupoid of arrows of $C$ (as in the fat
nerve), and $B_{1,1}$ is the $\infty$-groupoid of pullback squares
(equivalently, pushout squares). More generally, $B_{m,n}$ is the
$\infty$-groupoid of ($\Delta^{m}{\times}\Delta^{n}$)-diagrams in $C$ for
which all the rectangles are pullbacks (and hence pushouts). This is a stable
bisimplicial $\infty$-groupoid (which of course is a double Segal space). This
is almost by definition: since we only took pullback and pushout squares, they
are determined by their sources by pushout or their targets by pullback, in
the sense of our definition.

######  Lemma 2.3.3.

Let $B$ be a double Segal space. Suppose we have the two following pullbacks:

${B_{0,0}}$${B_{0,1}}$${B_{1,0}}$${B_{1,1}}$$\scriptstyle{d_{0}}$$\scriptstyle{e_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{e_{0}}$${\ulcorner}$${B_{0,0}}$${B_{0,1}}$${B_{1,0}}$${B_{1,1},}$$\scriptstyle{d_{1}}$$\scriptstyle{e_{1}}$$\scriptstyle{d_{1}}$$\scriptstyle{e_{1}}$${\ulcorner}$

then the double Segal space is stable.

###### Proof.

First, the second pullback implies that every square with top face maps
$d_{\top}:B_{i,j+1}\to B_{i,j}$ is a pullback. Indeed, in the cube

${B_{0,0}}$${B_{0,1}}$${B_{0,1}}$${B_{0,2}}$${B_{1,0}}$${B_{1,1},}$${B_{1,1}}$${B_{1,2}}$$\scriptstyle{d_{\top}}$$\scriptstyle{d_{\bot}}$$\scriptstyle{d_{\bot}}$$\scriptstyle{e_{\top}}$$\scriptstyle{d_{\top}}$$\scriptstyle{e_{\top}}$$\scriptstyle{d_{\top}}$$\scriptstyle{e_{\top}}$$\scriptstyle{d_{\bot}}$$\scriptstyle{e_{\top}}$$\scriptstyle{d_{\top}}$$\scriptstyle{d_{\bot}}$

the top and bottom squares are pullbacks because every row is Segal, and the
back square is a pullback by hypothesis. Thus the rectangle consisting of
bottom and back is a pullback because bottom and back squares are; front is a
pullback because top and rectangle are. By induction, suppose the squares

${B_{i-1,j-1}}$${B_{i-1,j}}$${B_{i,j-1}}$${B_{i,j}}$$\scriptstyle{d_{\top}}$$\scriptstyle{e_{\top}}$$\scriptstyle{d_{\top}}$$\scriptstyle{e_{\top}}$${\ulcorner}$

are pullback, we can form cubes with the top and bottom faces pullbacks thanks
to the Segal condition, and the back square is a pullback by hypothesis. This
proves that every square involving top face maps are pullbacks. Starting with
the first pullback, we prove in the same way that every square involving
bottom face maps are pullbacks.

Now we want to prove that the following square is a pullback, for $0<k<i$,

${B_{i-1,j-1}}$${B_{i-1,j}}$${B_{i,j-1}}$${B_{i,j}.}$$\scriptstyle{d_{\top}}$$\scriptstyle{e_{k}}$$\scriptstyle{d_{\top}}$$\scriptstyle{e_{k}}$

If $k=i-r$, we postcompose vertically with $(e_{\top})^{r}$:

${B_{k-1,j-1}}$${B_{k-1,j}}$${B_{i-1,j-1}}$${B_{i-1,j}}$${B_{i,j-1}}$${B_{i,j}}$$\scriptstyle{d_{\top}}$$\scriptstyle{(e_{\top})^{r}}$$\scriptstyle{(e_{\top})^{r}}$$\scriptstyle{d_{\top}}$$\scriptstyle{e_{k}}$$\scriptstyle{e_{k}}$$\scriptstyle{d_{\top}}$

Then the vertical composite is equivalent to $e_{\top}\circ(e_{\top})^{r}$ (by
face map identities), so both the rectangle and the upper square are pullbacks
by assumption, and therefore by Lemma 1.1.1, the lower square is a pullback
too, as required.

We can do the same proof with bottom face maps. We can also replace $d_{\top}$
in the new previous pullback squares and obtain the remaining pullbacks
involving the face maps.

For squares with face and degeneracy maps, we use the following strategy: in
the diagram

${B_{00}}$${B_{01}}$${B_{00}}$${B_{10}}$${B_{11}}$${B_{10},}$$\scriptstyle{s_{0}}$$\scriptstyle{d_{\bot}}$$\scriptstyle{s_{0}}$$\scriptstyle{e_{\bot}}$$\scriptstyle{d_{\bot}}$$\scriptstyle{e_{\bot}}$$\scriptstyle{e_{\bot}}$

the map $s_{0}$ is a section of $d_{1}$, then the long edge is an identity.
The right-hand square is a pullback (it is one of the two pullback in the
hypothesis). Thus the left-hand square is a pullback. We can proceed in the
same way for the other degeneracy maps.

There remains the case of squares involving only degeneracy:

${B_{ij}}$${B_{i,j+1}}$${B_{i+1,j}}$${B_{i+1,j+1}.}$$\scriptstyle{s_{0}}$$\scriptstyle{t_{k}}$$\scriptstyle{t_{k}}$$\scriptstyle{s_{0}}$

We again glue on the right a square with face map such that the long edge is
an identity and use once again the Lemma 1.1.1. ∎

###  2.4 Bicomodules

######  Theorem 2.4.1.

Let $B$ be an augmented stable double Segal space, and such that the
augmentation maps are culf. Suppose moreover $X:=B_{\bullet,-1}$ and
$Y:=B_{-1,\bullet}$ are decomposition spaces. Then the spans

$B_{0,0}\xleftarrow{e_{1}}B_{1,0}\xrightarrow{(u,e_{0})}X_{1}\times B_{0,0}$

and

$B_{0,0}\xleftarrow{d_{0}}B_{0,1}\xrightarrow{(d_{1},v)}B_{0,0}\times Y_{1}$

induce on ${\mathcal{S}}_{/B_{0,0}}$ the structure of a bicomodule over
${\mathcal{S}}_{/X_{1}}$ and ${\mathcal{S}}_{/Y_{1}}$ (at the $\pi_{0}$
level).

A bisimplicial $\infty$-groupoid satisfying the conditions of the theorem is
called a _bicomodule configuration_.

######  Remark 2.4.2.

In analogy with Remark 2.1.2, the notion of bicomodule configuration can be
broken up into steps. First, for any double Segal space $B$, since the zeroth
column $B_{\bullet,0}$ is a Segal space, ${\mathcal{S}}_{/B_{0,0}}$ is a left
comodule over ${\mathcal{S}}_{/B_{1,0}}$, and similarly
${\mathcal{S}}_{/B_{0,0}}$ is a right comodule over
${\mathcal{S}}_{/B_{0,1}}$. It is now the stability of $B$ that expresses the
bicomodule condition. From here, a culf augmentation on the left to a
decomposition space $X$ induces a coalgebra homomorphism, and a culf
augmentation on the right to a decomposition space $Y$ induces another
coalgebra homomorphism, and coextension of coscalars along these coalgebra
homomorphisms makes ${\mathcal{S}}_{/B_{0,0}}$ an
${\mathcal{S}}_{/X_{1}}$-${\mathcal{S}}_{/Y_{1}}$ bicomodule. This viewpoint
might well be useful in the proof of full coherence.

######  Remark 2.4.3.

As for the Proposition 2.1.1, the theorem is stated at the $\pi_{0}$-level. It
is most likely that the full coherence can be established using the techniques
employed in [13] and [24]. It is expected that all the stability pullbacks are
required for the fully coherent result. For the present purposes, we are going
to take homotopy cardinality anyway, and for that, coherence is not essential.

###### Proof.

The left and right comodule structures were established in Proposition 2.1.1.
The desired homotopy coherent diagram

${{\mathcal{S}}_{/B_{0,0}}}$${{\mathcal{S}}_{/B_{1,-1}\times
B_{0,0}}}$${{\mathcal{S}}_{/B_{0,0}\times
B_{-1,1}}}$${{\mathcal{S}}_{/B_{1,-1}\times B_{0,0}\times
B_{-1,1}}}$$\scriptstyle{\gamma_{l}}$$\scriptstyle{\gamma_{r}}$$\scriptstyle{\operatorname{Id}\otimes\gamma_{r}}$$\scriptstyle{\gamma_{l}\otimes\operatorname{Id}}$

is induced by the solid spans in the diagram

${B_{0,0}}$${B_{1,0}}$${B_{1,-1}\times
B_{0,0}}$${B_{0,1}}$${B_{1,1}}$${B_{1,-1}\times B_{0,1}}$${B_{0,0}\times
B_{-1,1}}$${B_{1,0}\times B_{-1,1}}$${B_{1,-1}\times B_{0,0}\times
B_{-1,1}.}$$\scriptstyle{e_{1}}$$\scriptstyle{(u,e_{0})}$$\scriptstyle{d_{0}}$$\scriptstyle{(d_{1},v)}$$\scriptstyle{e_{1}}$$\scriptstyle{(d_{1},ve_{1})}$$\scriptstyle{(ud_{0},e_{0})}$$\scriptstyle{d_{0}}$${\urcorner}$${\llcorner}$$\scriptstyle{\operatorname{Id}\otimes
d_{0}}$$\scriptstyle{\operatorname{Id}\otimes{(d_{1},v)}}$$\scriptstyle{e_{1}\otimes\operatorname{Id}}$$\scriptstyle{{(u,e_{0})}\otimes\operatorname{Id}}$

The homotopy commutativity of the squares follows one again from the new
augmentation simplicial identities. The upper-right hand square is a pullback
if and only if its composite with the second projection is a pullback and,
similarly, the lower-left hand square is a pullback if and only if its
composite with the first projection is a pullback. These composite outer
squares are pullbacks due to the stability condition. ∎

######  Example 2.4.4.

In analogy with Example 2.1.4, given a decomposition space $X$, let $B$ be the
total decalage of $X$. (Its zeroth column is $\operatorname{Dec}_{\top}(X)$
and its zeroth row is $\operatorname{Dec}_{\bot}(X)$. With its natural
augmentation maps, this becomes a bicomodule configuration, realising the
coalgebra of $X$ as a bicomodule over itself.

###  2.5 Augmented double Segal space of layered sets and posets

The following example is developped in [7], where details of the general
construction and proofs can be found.

Suppose $\mathbf{I}$ is the decomposition space of layered finite sets and
$\mathbf{C}$ is the decomposition space of layered finite posets. We obtain a
bisimplicial groupoid $\mathbf{B}$ with the following explicit description:
the groupoid $\mathbf{B}_{i,j}$ consists of pairs of layerings
$(S\to\underline{i},P\to\underline{j{+}1})$ where $S$ is a finite set, and $P$
is a finite poset. For example, $\mathbf{B}_{0,0}$ is the groupoid of
$1$-layered finite posets. The horizontal face maps (taking place only on the
$(j{+}1)$-layered finite poset part) are given by:

  * •

$d_{k}:\mathbf{B}_{i,j}\to\mathbf{B}_{i,j-1}$ joins the layers $(k{+}1)$ and
$(k{+}2)$ of the poset, for all $j>0$ and $0\leq k\leq j-1$;

  * •

$d_{\top}=d_{j}:\mathbf{B}_{i,j}\to\mathbf{B}_{i,j-1}$ deletes the last layer.

Horizontal degeneracy maps are given by inserting empty layers:
$s_{k}:\mathbf{B}_{i,j}\to\mathbf{B}_{i,j+1}$ inserts an empty $(k{+}2)$nd
layer in the poset, for all $j\geq 0$ and $0\leq k\leq j$.

The vertical face maps are given by:

  * •

$e_{\bot}=e_{0}:\mathbf{B}_{i,j}\to\mathbf{B}_{i-1,j}$ deletes the first layer
of the set, for all $i>0$;

  * •

$e_{k}:\mathbf{B}_{i,j}\to\mathbf{B}_{i-1,j}$ joins the layers $k$ and $k{+}1$
of the set, for all $0<k<i$.

  * •

$e_{\top}=e_{i}:\mathbf{B}_{i,j}\to\mathbf{B}_{i-1,j}$ joins the last layer of
the set and the first layer of the poset into the first layer of the poset.

Vertical degeneracy maps are given by inserting empty layers:
$t_{k}:\mathbf{B}_{i,j}\to\mathbf{B}_{i+1,j}$ inserts an empty $(k{+}1)$st
layer to the set, for all $0\leq k\leq i$. The augmentation maps are
$u:\mathbf{B}_{i,0}\to\mathbf{I}_{i}$ deleting the whole $1$-layered poset and
$v:\mathbf{B}_{0,j}\to\mathbf{C}_{j}$ deleting the first layer of the poset.
It should be noted that the row $\mathbf{B}_{0,\bullet}$ is the lower décalage
of $\mathbf{C}$, that $v$ is the décalage map given by the original $d_{0}$,
and that $u$ is the augmentation map that décalage always have.

######  Proposition 2.5.1 ([7, Proposition 2.14]).

With augmentations maps $u$ and $v$, the bisimplicial groupoid $\mathbf{B}$ is
a bicomodule configuration.

##  3 Correspondences, fibrations, and adjunctions

###  3.1 Decomposition space correspondences

A _correspondence_ is by definition a decomposition space $\mathcal{M}$ with a
map to the $1$-simplex $\Delta^{1}$. We consider the slice $\infty$-category
$\text{{{Cat}}}_{\infty/\Delta^{1}}$. It contains in particular
$\boldsymbol{\Delta}_{/\Delta^{1}}$, whose objects are $[i,j]$, see Remark
2.2.1. There is now a natural notion of nerve in this context. Given a
correspondence $p:\mathcal{M}\to\Delta^{1}$, the relative nerve
$N_{\Delta^{1}}:\text{{{Cat}}}_{\infty/\Delta^{1}}\to\operatorname{Fun}((\boldsymbol{\Delta}_{/\Delta^{1}})^{\text{{\rm{op}}}},{\mathcal{S}})$
of $p$ is the augmented bisimplicial $\infty$-groupoid given by
$B_{i,j}:=N_{\Delta^{1}}(p)_{i,j}=\operatorname{Map}_{/\Delta^{1}}([i,j],p)$,
where $[i,j]$ is given by the map $\Delta^{i+1+j}\to\Delta^{1}$ sending the
$i+1$ first vertices to $0$ and the others to $1$. It is allowed for $i$ or
$j$ to be equal to $-1$ but not both.

From the nerve definition, the following square is a standard mapping-space
fibre sequence for slices:

${B_{i,j}}$${\operatorname{Map}(\Delta^{i+1+j},\mathcal{M})}$${1}$${\operatorname{Map}(\Delta^{i+1+j},\Delta^{1}).}$${\lrcorner}$$\scriptstyle{\text{post
}p}$$\scriptstyle{\ulcorner[i,j]\urcorner}$

######  Proposition 3.1.1.

Given a decomposition space correspondence $p:\mathcal{M}\to\Delta^{1}$, the
bisimplicial $\infty$-groupoid $B$ described above enjoys the following
properties:

  1. 1.

it is Segal in both directions;

  2. 2.

it is stable;

  3. 3.

it is augmented;

  4. 4.

these augmentations are culf.

To prove these properties, we will use the following lemmas.

######  Lemma 3.1.2.

Given a diagram such that top and bottom are two fibre sequences

${F}$${E}$${B}$${F^{\prime}}$${E^{\prime}}$${B^{\prime}.}$$\scriptstyle{q}$

If $q$ is an equivalence, then the left-hand square is a pullback.

###### Proof.

In the following cube, the front and back squares are pullbacks by assumption;
the bottom one is since $q$ is an equivalence.

${F}$${E}$${F^{\prime}}$${E^{\prime}}$${1}$${B}$${1}$${B^{\prime}.}$$\scriptstyle{q}$

The rectangle consisting of the back square and the bottom square is a
pullback by Lemma 1.1.1, since both squares are pullbacks. Thus the rectangle
consisting of the top square and the front one is. Applying one more time
Lemma 1.1.1, we conclude the top square is a pullback. ∎

######  Lemma 3.1.3.

Given a diagram such that horizontal maps form fibre sequences

${F_{4}}$${E_{4}}$${B_{4}}$${F_{3}}$${E_{3}}$${B_{3}}$${F_{2}}$${E_{2}}$${B_{2}}$${F_{1}}$${E_{1}}$${B_{1}.}$$\scriptstyle{u}$$\scriptstyle{v}$

Suppose the vertical middle square (involving $E_{i},1\leq i\leq 4$) is a
pullback, and suppose $u$ and $v$ are equivalences, then the left vertical
square is a pullback.

###### Proof.

By Lemma 3.1.2, since $u$ and $v$ are equivalences, the front and back squares
of the left cube of the diagram are pullbacks. We conclude by applying Lemma
1.1.1 twice, as in the proof of Lemma 3.1.2. ∎

###### Proof of Proposition 3.1.1.

(1) Segal in both directions means: for any $i$, the squares

${B_{n+1,i}}$${B_{n,i}}$${B_{n,i}}$${B_{n-1,i},}$${\lrcorner}$$\scriptstyle{e_{0}}$$\scriptstyle{e_{n+1}}$$\scriptstyle{e_{n}}$$\scriptstyle{e_{0}}$${B_{i,n+1}}$${B_{i,n}}$${B_{i,n}}$${B_{i,n-1}.}$${\lrcorner}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{n+1}}$$\scriptstyle{d_{n}}$$\scriptstyle{d_{0}}$

are pullbacks.

The $\infty$-groupoids $B_{n,i}$, for $n\geq 0$ are also given by the
following fibre sequences:

${B_{n,i}}$${\operatorname{Map}(\Delta^{n+1+i},\mathcal{M})}$${1}$${\operatorname{Map}(\Delta^{1},\Delta^{1}),}$${\lrcorner}$$\scriptstyle{R_{n+1+i}}$$\scriptstyle{\ulcorner\operatorname{id}\urcorner}$

the right-hand map $R_{n+1+j}$ sends
$\sigma\in\operatorname{Map}(\Delta^{n+1+j},\mathcal{M})$ to
$p\circ\sigma\circ\rho_{n+1}$ where $\rho_{n+1}:\Delta^{1}\to\Delta^{n+1+j}$
maps the arrow in $\Delta^{1}$ to the $(n+1)$st edge of $\Delta^{n+1+j}$, that
is $\rho_{n+1}=(d^{\bot})^{n}(d^{\top})^{j}$. Indeed, in the diagram

${B_{i,j}}$${\operatorname{Map}(\Delta^{i+1+j},\mathcal{M})}$${1}$${\operatorname{Map}(\Delta^{i+1+j},\Delta^{1})}$${1}$${\operatorname{Map}(\Delta^{1},\Delta^{1}),}$${\lrcorner}$$\scriptstyle{\text{post
}p}$$\scriptstyle{\ulcorner\beta_{i,j}\urcorner}$$\scriptstyle{\text{pre
}\rho_{i+1}}$$\scriptstyle{\ulcorner\operatorname{id}\urcorner}$

the bottom square is a pullback, because the fibre of the right bottom map is
contractible, thus the whole rectangle is a pullback by Lemma 1.1.1.

Using the Lemma 3.1.2, we only have to check that the front square in the cube

${B_{n+1,i}}$${B_{n+1,i}}$${\operatorname{Map}(\Delta^{n+1+1+i},\mathcal{M})}$${\operatorname{Map}(\Delta^{n+1+i},\mathcal{M})}$${B_{n,i}}$${B_{n-1,i}}$${\operatorname{Map}(\Delta^{n+1+i},\mathcal{M})}$${\operatorname{Map}(\Delta^{n+i},\mathcal{M})}$

is a pullback, and apply Lemma 1.1.1. But the squares

${\mathcal{M}_{n+1+1+i}}$${\mathcal{M}_{n+1+i}}$${\mathcal{M}_{n+1+i}}$${\mathcal{M}_{n+i}}$$\scriptstyle{d_{\bot}}$$\scriptstyle{d_{n+1}}$$\scriptstyle{d_{n}}$$\scriptstyle{d_{\bot}}$

are pullbacks because $\mathcal{M}$ is a decomposition space, $d_{\bot}$ is an
inert map and $d_{n+1}$ and $d_{n}$ are always inner coface maps thus active
maps. For the remaining squares, we use that the squares

${\mathcal{M}_{i+1+n+1}}$${\mathcal{M}_{i+1+n}}$${\mathcal{M}_{i+1+n}}$${\mathcal{M}_{i+1+n-1}}$$\scriptstyle{d_{i+1}}$$\scriptstyle{d_{\top}}$$\scriptstyle{d_{\top}}$$\scriptstyle{d_{i+1}}$

are also pullbacks because $\mathcal{M}$ is a decomposition space.

(2) To establish the stability condition, by Lemma 2.3.3 it is enough to prove
that the two following squares are pullbacks:

${B_{0,0}}$${B_{0,1}}$${B_{1,0}}$${B_{1,1},}$$\scriptstyle{d_{0}}$$\scriptstyle{e_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{e_{0}}$${\ulcorner}$${B_{0,0}}$${B_{0,1}}$${B_{1,0}}$${B_{1,1}.}$$\scriptstyle{d_{1}}$$\scriptstyle{e_{1}}$$\scriptstyle{d_{1}}$$\scriptstyle{e_{1}}$${\ulcorner}$

We can prove this with the same strategy used above. The decomposition space
axioms used here are that the following squares are pullbacks

${\mathcal{M}_{0}}$${\mathcal{M}_{1}}$${\mathcal{M}_{1}}$${\mathcal{M}_{2},}$$\scriptstyle{s_{0}}$$\scriptstyle{s_{1}}$$\scriptstyle{d_{\bot}}$${\urcorner}$$\scriptstyle{d_{\bot}}$${\mathcal{M}_{0}}$${\mathcal{M}_{1}}$${\mathcal{M}_{1}}$${\mathcal{M}_{2}.}$$\scriptstyle{s_{0}}$$\scriptstyle{s_{0}}$$\scriptstyle{d_{\top}}$${\urcorner}$$\scriptstyle{d_{\top}}$

(3) and (4) The augmentations $Y_{j}:=B_{-1,j}$ and $X_{i}:=B_{i,-1}$ are also
given by the following fibre sequences

${Y_{j}}$${\operatorname{Map}(\Delta^{j},\mathcal{M})}$${1}$${\operatorname{Map}(\Delta^{0},\Delta^{1}),}$${\lrcorner}$$\scriptstyle{S}$$\scriptstyle{\ulcorner
d_{0}\urcorner}$${X_{i}}$${\operatorname{Map}(\Delta^{i},\mathcal{M})}$${1}$${\operatorname{Map}(\Delta^{0},\Delta^{1}),}$${\lrcorner}$$\scriptstyle{T}$$\scriptstyle{\ulcorner
d_{1}\urcorner}$

where the map $S$ sends $\sigma$ to $p\circ\sigma\circ(d^{\top})^{j}$, and the
map $T$ sends $\sigma$ to $p\circ\sigma\circ(d^{\bot})^{i}$. Since the
following squares commute

${\operatorname{Map}(\Delta^{i+1+j},\mathcal{M})}$${\operatorname{Map}(\Delta^{j},\mathcal{M})}$${\operatorname{Map}(\Delta^{1},\Delta^{1})}$${\operatorname{Map}(\Delta^{0},\Delta^{1}),}$$\scriptstyle{(d_{\bot})^{i+1}}$$\scriptstyle{R_{i+1+j}}$$\scriptstyle{S}$$\scriptstyle{d_{\bot}}$${\operatorname{Map}(\Delta^{i+1+j},\mathcal{M})}$${\operatorname{Map}(\Delta^{i},\mathcal{M})}$${\operatorname{Map}(\Delta^{1},\Delta^{1})}$${\operatorname{Map}(\Delta^{0},\Delta^{1}),}$$\scriptstyle{(d_{\top})^{j+1}}$$\scriptstyle{R_{i+1+j}}$$\scriptstyle{T}$$\scriptstyle{d_{\top}}$

it is enough to define maps $B_{i,j}\to Y_{j}$ and $B_{i,j}\to X_{i}$.

The augmentation maps are culf: we need to prove that the back squares of the
following cubes are pullbacks:

${Y_{0}}$${Y_{1}}$${\operatorname{Map}(\Delta^{0},\mathcal{M})}$${\operatorname{Map}(\Delta^{1},\mathcal{M})}$${B_{0,0}}$${B_{0,1}}$${\operatorname{Map}(\Delta^{1},\mathcal{M})}$${\operatorname{Map}(\Delta^{2},\mathcal{M}),}$$\scriptstyle{s_{0}}$$\scriptstyle{s_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{s_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{s_{1}}$$\scriptstyle{d_{0}}$

${Y_{1}}$${Y_{2}}$${\operatorname{Map}(\Delta^{1},\mathcal{M})}$${\operatorname{Map}(\Delta^{2},\mathcal{M})}$${B_{0,1}}$${B_{0,2}}$${\operatorname{Map}(\Delta^{2},\mathcal{M})}$${\operatorname{Map}(\Delta^{3},\mathcal{M}).}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{0}}$

We can apply the Lemma 3.1.3 since the front square is a pullback because
$\mathcal{M}$ is a decomposition space. ∎

To summarise, given a decomposition space correspondence
$p:\mathcal{M}\to\Delta^{1}$, we get a bicomodule configuration and then
${\mathcal{S}}_{/B_{0,0}}$ is a bicomodule by Theorem 2.4.1.

###  3.2 Cocartesian and cartesian fibrations of decomposition spaces

Ayala and Francis [4] formulate a homotopy-invariant definition of cartesian
and cocartesian fibrations so it can be equally well formulated in any model
for $\infty$-categories. We adapt here those definitions to decomposition
spaces.

Let $p:X\to Y$ be a simplicial map between decomposition spaces. A morphism
$\Delta^{1}\xrightarrow{<s\xrightarrow{a}t>}X$ is _$p$ -cocartesian_ if the
diagram of coslices of decomposition spaces

${{}^{a/}X}$${{}^{s/}X}$${{}^{pa/}Y}$${{}^{ps/}Y}$

is a pullback, where the coslice ${}^{s/}X$ is given by pullback of lower
décalage $\operatorname{Dec}_{\bot}(X)$:

${(^{s/}X)_{n}}$${\operatorname{Dec}_{\bot}(X)_{n}}$${1}$${X_{0},}$${\lrcorner}$$\scriptstyle{(d_{\top})^{n+1}}$$\scriptstyle{\ulcorner
s\urcorner}$

similarly the coslice ${}^{a/}X$ is given by pullback of
$\operatorname{Dec}_{\bot}(\operatorname{Dec}_{\bot}(X))$

${(^{a/}X)_{n}}$${\operatorname{Dec}_{\bot}(\operatorname{Dec}_{\bot}(X))_{n}}$${1}$${X_{1},}$${\lrcorner}$$\scriptstyle{(d_{\top})^{n+1}}$$\scriptstyle{\ulcorner
a\urcorner}$

and the functor ${}^{a/}X\to{}^{s/}X$ is given by
$\operatorname{Dec}_{\bot}(d_{\bot})$, where the simplicial map
$d_{\bot}:\operatorname{Dec}_{\bot}(X)\to X$ is given by the original $d_{0}$.

The functor $p:X\to Y$ is a _cocartesian fibration_ if any diagram of solid
arrows

${\Delta^{0}}$${X}$${\Delta^{1}}$${Y}$$\scriptstyle{d_{1}}$$\scriptstyle{p}$

admits a $p$-cocartesian diagonal filler.

Similarly, a morphism $\Delta^{1}\xrightarrow{<s\xrightarrow{a}t>}X$ is _$p$
-cartesian_ if the diagram of slice decomposition spaces

${X_{/a}}$${X_{/t}}$${Y_{/pa}}$${Y_{/pt}}$

is a pullback, where the slice $X_{/t}$ is given by pullback of the upper
décalage $\operatorname{Dec}_{\top}(X)$, the slice $X_{/a}$ is given by
pullback of $\operatorname{Dec}_{\top}(\operatorname{Dec}_{\top}(X))$, and the
functor $X_{/a}\to X_{/t}$ is given by
$\operatorname{Dec}_{\top}(d_{\top})=d_{\top-1}$, where
$d_{\top}:\operatorname{Dec}_{\top}(X)\to X$ is given by the original
$d_{\top}$.

The functor $p:X\to Y$ is a _cartesian fibration_ if any diagram of solid
arrows

${\Delta^{0}}$${X}$${\Delta^{1}}$${Y}$$\scriptstyle{d_{0}}$$\scriptstyle{p}$

admits a $p$-cartesian diagonal filler.

#### Bisimplex category with diagonal maps

We define $\overline{[i,j]}:{M}_{\phi_{i,j}}\to\Delta^{1}$ to be the canonical
projection from the mapping cylinder of
$\phi_{i,j}:=(d^{\top})^{j}:\Delta^{i}\to\Delta^{i+j}$; it is a cocartesian
fibration. They assemble into a category, denoted
$\overline{\boldsymbol{\Delta}_{/\Delta^{1}}}$, of shape like
$\boldsymbol{\Delta}_{/\Delta^{1}}$, but with extra diagonal maps
$d:\overline{[i{-}1,j]}\to\overline{[i,j{-}1]}$ given by inclusion. These
satisfy new simplicial identities: $\sigma_{k}d=d\sigma_{k+1}$, $0\leq k\leq
j$, where $\sigma_{k}$ are degeneracy maps “on $j$” (horizontal) and $d$ are
diagonal maps, and with face maps: $d\delta_{k+1}=\delta_{k}d$, $0\leq k\leq
j$, where $\delta_{k}$ are horizontal face maps. Similarly for degeneracy maps
$\tau_{k}$ “on $i$” (vertical), $\tau_{k}d=d\tau_{k}$, $0\leq k<i$ and
$d\epsilon_{k}=\epsilon_{k}d$, $0\leq k<i$, where $\epsilon_{k}$ are vertical
face maps. For example, we can draw $\overline{[2,1]}$ as follows

.......

where the horizontal maps lie over the map in $\Delta^{1}$. It is like a
cocartesian version of the earlier drawing of Remark 2.2.1.

######  Remark 3.2.1.

We can draw $\overline{[i,j]}$ as a column of $i{+}1$ black dots followed by
$j{+}1$ white dots. Where arrows in $\boldsymbol{\Delta}_{/\Delta^{1}}$ send
black dots to black dots and white dots to white dots (without crossing), in
$\overline{\boldsymbol{\Delta}_{/\Delta^{1}}}$ we allow moreover to map white
dots to black dots.

There is a natural notion of nerve in the context of cocartesian fibrations
over $\Delta^{1}$: given a cocartesian fibration $p:\mathcal{M}\to\Delta^{1}$
between decomposition spaces, define the cocartesian nerve
$N_{\text{cocart}}:\text{{{Cat}}}_{\infty/\Delta^{1}}^{\text{cocart}}\to\operatorname{Fun}((\overline{\boldsymbol{\Delta}_{/\Delta^{1}}})^{\text{{\rm{op}}}},{\mathcal{S}})$
by
$N_{\text{cocart}}(p)_{\overline{i,j}}:=\operatorname{Map}_{/\Delta^{1}}^{\text{cocart}}(\overline{[i,j]},\mathcal{M})$,
the mapping space preserving cocartesian arrows.

Similarly to the previous Section 3.1, we get a bicomodule configuration and
${\mathcal{S}}_{/B_{0,0}}$ is a bicomodule over ${\mathcal{S}}_{/X_{1}}$ and
${\mathcal{S}}_{/Y_{1}}$. We have here moreover diagonal maps $B_{i,j-1}\to
B_{i-1,j}$ and new sections $s_{-1}:B_{i,j-1}\to B_{i,j}$, for $i\geq 0$ given
by the composition with a diagonal map. That is ${\mathcal{S}}_{/B_{0,0}}$ is
pointed as a right comodule over ${\mathcal{S}}_{/Y_{1}}$.

${B_{-1,0}}$${B_{-1,1}}$${B_{-1,2}}$${B_{0,-1}}$${B_{0,0}}$${B_{0,1}}$${B_{0,2}}$${B_{1,-1}}$${B_{1,0}}$${B_{1,1}}$${B_{1,2}}$
${\dots}$${\dots}$${\vdots}$${\vdots}$${\vdots}$${\dots}$${\vdots}$

We now adapt Lurie’s definition of adjunction of $\infty$-categories [22] to
decomposition spaces.

An _adjunction_ between decomposition spaces $X$ and $Y$ is a simplicial map
between decomposition spaces $p:\mathcal{M}\to\Delta^{1}$ which is both a
cartesian and a cocartesian fibration together with equivalences
$X\simeq\mathcal{M}_{\\{0\\}}$ and $Y\simeq\mathcal{M}_{\\{1\\}}$.

######  Proposition 3.2.2.

Given an adjunction $p:\mathcal{M}\to\Delta^{1}$, the bisimplicial
$\infty$-groupoid $B$ described above is a bicomodule configuration. Moreover
${\mathcal{S}}_{/B_{0,0}}$ is pointed as a right comodule over
${\mathcal{S}}_{/Y_{1}}$, and as a left comodule over
${\mathcal{S}}_{/X_{1}}$.

###### Proof.

By Proposition 3.1.1, the bisimplicial $\infty$-groupoid $B$ is a bicomodule
configuration. The pointings are given by the cartesian and cocartesian
conditions. ∎

Adjunctions between $\infty$-categories are examples of adjunctions between
decomposition spaces.

######  Example 3.2.3.

To illustrate these abstract concepts, let us spell out the bicomodule
configuration associated to an ordinary adjunction of $1$-categories $F\colon
X\rightleftarrows Y\colon G$. The $\infty$-groupoid $B_{0,0}$ is now just the
set of arrows $Fx\to y$, which by adjunction correspond to arrows $x\to Gy$,
and $B_{1,0}$ is the set of composable pair $Fx\to Fx^{\prime}\to y$ (which is
the same as $x\to x^{\prime}\to Gy$). In general $B_{i,j}$ is the set of
chains of composable arrows $Fx_{0}\to\dots\to Fx_{i}\to y_{j}\to\dots\to
y_{0}$. These chains can be drawn as in the picture page 2.2.1, where the
horizontal arrow is a ‘mixed’ arrow $Fx\to y$. This drawing can be filled as
in the picture page 3.2, and thus giving a right pointing by the following
rearrangement:

${Fx^{\prime}}$${y}$${Fx}$

$\quad\mapsto\quad$ ${y}$${Fx}$${Fx^{\prime}}$

Similarly the functor $G:Y\to X$ induces a left pointing on the equivalent (by
adjunction) augmented simplicial set of chains $x_{0}\to\dots\to x_{i}\to
Gy_{j}\to\dots\to Gy_{0}$.

##  4 Möbius inversion for comodules and a Rota formula

###  4.1 Finiteness and cardinality

An $\infty$-groupoid $X$ is _locally finite_ if at each base point $x$ the
homotopy groups $\pi_{i}(X,x)$ are finite for $i\geq 1$ and are trivial for
$i$ sufficiently large. It is called _finite_ if furthermore it has only
finitely many components. We denote ${\mathcal{F}}$ (following the notation of
[14]) the $\infty$-category of finite $\infty$-groupoids. A map is _finite_ if
each fibre is finite. A pullback of any homotopy finite map is again finite. A
span $I\xleftarrow{p}M\xrightarrow{q}J$ and the corresponding linear functor
${\mathcal{S}}_{/I}\xrightarrow{}{\mathcal{S}}_{/J}$ are _finite_ if the map
$p$ is finite.

A decomposition space $X$ is _locally finite_ if $X_{1}$ is locally finite and
both $s_{0}$ and $d_{1}$ are finite maps [14, §7.4].

######  Proposition 4.1.1 ([12, Proposition 4.3]).

Let $I$, $J$, $M$ be locally finite $\infty$-groupoids and
$I\xleftarrow{p}M\xrightarrow{q}J$ a finite span. Then the induced finite
linear functor ${\mathcal{S}}_{/I}\xrightarrow{}{\mathcal{S}}_{/J}$ restricts
to ${\mathcal{F}}_{/I}\xrightarrow{}{\mathcal{F}}_{/J}$.

The cardinality [5] of a finite $\infty$-groupoid $X$ is the alternating
product of cardinalities of the homotopy groups

$|X|=\sum_{x\in\pi_{0}(X)}\prod_{k=1}^{\infty}|\pi_{k}(X,x)|^{(-1)^{k}}.$

For a locally finite $\infty$-groupoid $S$, there is a notion of cardinality
$|-|:{\mathcal{F}}_{/S}\to\mathbb{Q}_{\pi_{0}S}$ sending a basis element
$\ulcorner s\urcorner$ to the basis element $\delta_{s}=|\ulcorner
s\urcorner|$.

For any locally finite decomposition space $X$, we can take the cardinality of
the linear functors $\delta:{\mathcal{F}}_{/X_{1}}\xrightarrow{}{\mathcal{F}}$
and $\Delta:{\mathcal{F}}_{/X_{1}}\xrightarrow{}{\mathcal{F}}_{/X_{1}\times
X_{1}}$ to obtain a coalgebra structure

$\displaystyle\mathbb{Q}_{\pi_{0}X_{1}}$
$\displaystyle\xrightarrow{|\delta|}\mathbb{Q}$
$\displaystyle\mathbb{Q}_{\pi_{0}X_{1}}$
$\displaystyle\xrightarrow{|\Delta|}\mathbb{Q}_{\pi_{0}X_{1}}\otimes\mathbb{Q}_{\pi_{0}X_{1}}$

called the _numerical incidence coalgebra_ of $X$, see [14, §7.7].

###  4.2 Completeness and Möbius condition

A decomposition space is called _complete_ if $s_{0}:X_{0}\to X_{1}$ is a
monomorphism [14, §2]. Since $s_{0}$ is a monomorphism, we can identify
$X_{0}$ with a $\infty$-subgroupoid of $X_{1}$. We denote $X_{a}$ its
complement: $X_{1}\simeq X_{0}+X_{a}$. More generally, recall the word
notation from [14]: consider the alphabet with three letters $\\{0,1,a\\}$;
$0$ indicates degenerate edges $s_{0}(x)\in X_{1}$, $a$ denotes edges
specified to be nondegenerate, and $1$ denotes unspecified edges. For $w$ a
word of length $n$ in this alphabet, define

$X^{w}=\prod_{i\in w}X_{i}\subset(X_{1})^{n}.$

This inclusion is full since $X_{a}\subset X_{1}$ is full by completeness.

Denote by $X_{w}$ the $\infty$-groupoid of $n$-simplices whose principal edges
have the types indicated in the word $w$, that is the full subgroupoid of
$X_{n}$ given by the following pullback

${X_{w}}$${X_{n}}$${X^{w}}$${(X_{1})^{n}.}$${\lrcorner}$

We define $\overrightarrow{X}_{n}=X_{a\dots a}\subset X_{n}$ to be the full
subgroupoid of simplices with all principal edges nondegenerate. It is the
complement of the union of the essential images of the degeneracy maps
$s_{i}\colon X_{n-1}\to X_{n}$, that is

$\overrightarrow{X}_{n}=X_{n}\setminus\bigcup_{i=0}^{n-1}\operatorname{Im}(s_{i}).$

By definition $\overrightarrow{X}_{0}=X_{0}$.

For a complete decomposition space, the spans
$X_{1}\xleftarrow{d_{1}^{n-1}}\overrightarrow{X}_{n}\rightarrow 1$ define
linear functors, the _Phi functors_

$\Phi_{n}:{\mathcal{S}}_{/X_{1}}\to{\mathcal{S}}.$

We also put $\displaystyle\Phi_{\text{even}}:=\sum_{n\text{ even}}\Phi_{n}$,
and $\displaystyle\Phi_{\text{odd}}:=\sum_{n\text{ odd}}\Phi_{n}$.

The incidence algebra of a decomposition space contains the _zeta functor_

$\zeta:{\mathcal{S}}_{/X_{1}}\to{\mathcal{S}}$

given by the span $X_{1}\xleftarrow{=}X_{1}\xrightarrow{}1$.

######  Theorem 4.2.1 ([14, Theorem 3.8]).

For a complete decomposition space, the following Möbius inversion holds:

$\displaystyle\zeta\ast\Phi_{\text{even}}$
$\displaystyle\simeq\delta+\zeta\ast\Phi_{\text{odd}}$
$\displaystyle\simeq\Phi_{\text{even}}\ast\zeta$
$\displaystyle\simeq\delta+\Phi_{\text{odd}}\ast\zeta.$

This is however not enough to allow the Möbius inversion formula to descend to
the vector space level. A complete decomposition space $X$ is of _locally
finite length_ [14] if every edge $f\in X_{1}$ has finite length, that is, the
fibres $(\overrightarrow{X}_{n})_{f}$ of
$d_{1}^{(n)}:\overrightarrow{X}_{n}\to X_{1}$ over $f$ are empty for $n$
sufficiently large.

A _Möbius decomposition space_ [14] is a decomposition space which is locally
finite and of locally finite length; the fibre $(\overrightarrow{X}_{n})_{f}$
is finite (eventually empty). It follows that the map

$\sum_{n}d_{1}^{n-1}:\sum_{n}\overrightarrow{X}_{n}\to X_{1}$

is finite; by Proposition 4.1.1, the Phi functors descend to

$\Phi_{n}:{\mathcal{F}}_{/X_{1}}\to{\mathcal{F}}$

and we can take cardinality to obtain functions
$|\Phi_{n}|:\mathbb{Q}_{\pi_{0}X_{1}}\to\mathbb{Q}$.

Finally, we can take cardinality of the abstract Möbius inversion formula of
4.2.1, see [14] for a complete exposition.

######  Theorem 4.2.2 ([14, Theorem 8.9]).

If $X$ is a Möbius decomposition space, then the cardinality of the zeta
functor, $|\zeta|:\mathbb{Q}_{\pi_{0}X_{1}}\to\mathbb{Q}$, is convolution
invertible with inverse $|\mu|:=|\Phi_{\text{even}}|-|\Phi_{\text{odd}}|$:

$|\zeta|\ast|\mu|=|\delta|=|\mu|\ast|\zeta|.$

###  4.3 Right and left convolutions

We introduce left and right convolution actions as dual to the comodule
structures. Explicitly, given a right comodule configuration $C\to Y$, we get
a right comodule ${\mathcal{S}}_{/C_{0}}$ over ${\mathcal{S}}_{/Y_{1}}$. The
_right convolution_ $\theta\star_{r}\beta$ of the two functors
$\theta:{\mathcal{S}}_{/C_{0}}\to{\mathcal{S}}$ and
$\beta:{\mathcal{S}}_{/Y_{1}}\to{\mathcal{S}}$, given by the spans
$C_{0}\xleftarrow{}M\xrightarrow{}1$ and $Y_{1}\xleftarrow{}N\xrightarrow{}1$,
is the composite of $\theta\otimes\beta$ with the right coaction $\gamma_{r}$:

$\theta\star_{r}\beta:{\mathcal{S}}_{/C_{0}}\xrightarrow{\gamma_{r}}{\mathcal{S}}_{/C_{0}}\otimes{\mathcal{S}}_{/Y_{1}}\xrightarrow{\theta\otimes\beta}{\mathcal{S}},$

where the tensor product $\theta\otimes\beta$ is given by the span
$C_{0}\times Y_{1}\xleftarrow{}M\times N\xrightarrow{}1$.

Similarly, given a left comodule configuration, we can define the _left
convolution_ $\alpha\star_{l}\theta$ of
$\alpha:{\mathcal{S}}_{/X_{1}}\to{\mathcal{S}}$ and
$\theta:{\mathcal{S}}_{/C_{0}}\to{\mathcal{S}}$:

$\alpha\star_{l}\theta:{\mathcal{S}}_{/C_{0}}\xrightarrow{\gamma_{l}}{\mathcal{S}}_{/X_{1}}\otimes{\mathcal{S}}_{/C_{0}}\xrightarrow{\alpha\otimes\theta}{\mathcal{S}}.$

If we have a bicomodule configuration, then the following associativity
formula expresses the compatibility of coactions from Theorem 2.4.1.

######  Corollary 4.3.1.

Given a bicomodule configuration, the convolutions defined above satisfy

$\alpha\star_{l}(\theta\star_{r}\beta)\simeq(\alpha\star_{l}\theta)\star_{r}\beta.$

###  4.4 Möbius inversion for (co)modules

Let $C\to Y$ be a comodule configuration. The zeta functor

$\zeta^{C}:{\mathcal{S}}_{/C_{0}}\to{\mathcal{S}}$

is the linear functor defined by the span

$C_{0}\xleftarrow{=}C_{0}\xrightarrow{}1.$

Let $C\to Y$ be a right pointed comodule configuration. The augmented
simplicial $\infty$-groupoid $C$ is an object of the functor $\infty$-category

$\operatorname{Fun}{(\boldsymbol{\Delta}^{\text{{\rm{op}}}}_{\text{bot}},{\mathcal{S}})}$

where $\boldsymbol{\Delta}_{\text{bot}}$ is the simplex category of finite
linear orders with a specified bottom element, and with monotone maps
preserving the bottom element. The forgetful functor
$\boldsymbol{\Delta}_{\text{bot}}\to\boldsymbol{\Delta}$ is right adjoint to
the functor $j:\boldsymbol{\Delta}\to\boldsymbol{\Delta}_{\text{bot}}$ adding
a bottom element.

######  Remark 4.4.1.

In the situation where $Y$ is Segal and $C=\operatorname{Dec}_{\bot}Y$, we can
take $C_{-1}$ to be $Y_{0}$, with $d_{0}$ as augmentation map. By [22, Lemma
6.1.3.16], this is a colimit diagram.

A right pointed comodule configuration $f:C\to Y$ is _complete_ if the new
degeneracies $s_{-1}:C_{n-1}\to C_{n}$ are monomorphisms. Since $s_{-1}$ is a
monomorphism, we can identify $C_{-1}$ with a $\infty$-subgroupoid of $C_{0}$.
We denote by $C_{b}$ its complement: $C_{0}=C_{-1}+C_{b}$. Denote by $C_{vw}$
the $\infty$-groupoid of simplices whose principal edges have the type
indicated in the word $vw$, where $v\in\\{-1,0,b\\}$ and $w$ is a word in the
alphabet $\\{0,1,a\\}$, that is the full $\infty$-subgroupoid of $C_{n}$ given
by the pullback

${C_{vw}}$${C_{n}}$${C_{v}\times
Y^{w}}$${C_{0}\times(Y_{1})^{n},}$${\lrcorner}$

where $n=|w|\geq 0$. The principal edges of the $\infty$-groupoid $C_{n}$
consist of an element in $C_{0}$ given by $(d_{\top})^{n}$, and $n$ edges in
$Y_{1}$, the principal edges of the image of $C_{n}$ by $f$. In this
situation, we define $\overrightarrow{C}_{n}=C_{ba\dots a}\subset C_{n}$ to be
the full subgroupoid of simplices with all principal edges nondegenerate. It
is given by the pullback diagram

${C_{ba\dots a}}$${C_{n}}$${C_{b}\times Y^{a\dots a}}$${C_{0}\times
Y_{1}^{n}.}$${\lrcorner}$

Define

$\delta^{R}:{\mathcal{S}}_{/C_{0}}\to{\mathcal{S}}$

to be the linear functor given by the span

$C_{0}\xleftarrow{s_{-1}}C_{-1}\xrightarrow{}1$

and define the _right Phi functors_

$\Phi_{n}^{R}:{\mathcal{S}}_{/C_{0}}\to{\mathcal{S}}$

to be the linear functors given by the spans

$C_{0}\xleftarrow{}\overrightarrow{C}_{n}\xrightarrow{}1.$

If $n=-1$, $\overrightarrow{C}_{-1}=C_{-1}$ (by convention) and
$\Phi_{-1}^{R}$ is the linear functor $\delta^{R}$.

######  Lemma 4.4.2.

For every word $w$ in the alphabet $\\{0,1,a\\}$, the following square is a
pullback:

${C_{0w}}$${C_{1}}$${C_{0}\times Y_{w}}$${C_{0}\times
Y_{1}.}$${\lrcorner}$$\scriptstyle{(d_{\top}{,}f)}$

###### Proof.

Let $n=|w|$. The square is the top rectangle of the diagram

${C_{0w}}$${C_{n}}$${C_{1}}$${C_{0}\times Y_{w}}$${C_{0}\times
Y_{n}}$${C_{0}\times Y_{1}}$${C_{0}\times Y^{w}}$${C_{0}\times
Y_{1}^{n}.}$${\lrcorner}$$\scriptstyle{(d_{\top}{,}f)}$${\lrcorner}$

The bottom square and left-hand rectangle are pullbacks by definition of
$Y_{w}$ and $C_{0w}$, hence the top left-hand square is a pullback. The right-
hand square is a pullback because the augmentation map $C\to Y$ is culf. Hence
the top rectangle, which is the desired square, is a pullback. ∎

Given a complete decomposition space $Y$, we denote
$\Phi_{n}^{Y}:{\mathcal{S}}_{/Y_{1}}\to{\mathcal{S}}$ the usual Phi functors,
see 4.1 above.

######  Proposition 4.4.3.

The right Phi functors satisfy

$\zeta^{C}\star_{r}\Phi_{n}^{Y}\simeq\Phi_{n-1}^{R}+\Phi_{n}^{R}.$

###### Proof.

Compute the convolution action $\zeta^{C}\star_{r}\Phi_{n}^{Y}$ by Lemma 4.4.2
as:

${C_{0}}$${C_{1}}$${C_{0a\dots a}}$${C_{0}\times
Y_{1}}$${C_{0}\times\overrightarrow{Y_{n}}}$${1.}$${\llcorner}$

But $C_{0a\dots a}\simeq C_{-1a\dots a}+C_{ba\dots
a}\simeq\overrightarrow{C}_{n-1}+\overrightarrow{C}_{n}$. This is an
equivalence of $\infty$-groupoids over $C_{0}$ and the resulting span is
$\Phi_{n-1}^{R}+\Phi_{n}^{R}$. ∎

Denote

$\Phi_{\text{even}}^{Y}:=\sum_{n\text{
even}}\Phi_{n}^{Y},\quad\Phi_{\text{odd}}^{Y}:=\sum_{n\text{
odd}}\Phi_{n}^{Y}.$

The previous proposition implies the following Möbius inversion formula.

######  Theorem 4.4.4.

Given $C\to Y$ a complete right pointed comodule configuration,

$\zeta^{C}\star_{r}\Phi_{\text{even}}^{Y}\simeq\delta^{R}+\zeta^{C}\star_{r}\Phi_{\text{odd}}^{Y}.$

###### Proof.

The two linear functors are equivalent to the sum of the right Phi functors:

$\zeta^{C}\star_{r}\Phi_{\text{even}}^{Y}\simeq\Phi_{-1}^{R}+\Phi_{0}^{R}+\Phi_{1}^{R}+\cdots\simeq\delta^{R}+\zeta^{C}\star_{r}\Phi_{\text{odd}}^{Y}.$

∎

We can also define a _left pointed_ comodule configuration $D\to X$, with new
top sections instead of bottom: we consider instead the mapping cylinder of
$\boldsymbol{\Delta}\to\boldsymbol{\Delta}_{\text{top}}$, where
$\boldsymbol{\Delta}_{\text{top}}$ is the simplex category of finite linear
orders with a specified top element, and with monotone maps preserving the top
element. A left pointed comodule configuration is _complete_ if the new
degeneracies $t_{\top+1}:D_{n-1}\to D_{n}$ are monomorphisms. Similarly, we
define the _left Phi functors_ and $\delta^{L}$ using $t_{\top+1}$ and
$e_{\top}$ and we obtain the following formula.

######  Theorem 4.4.5.

Given $D\to X$ a complete left pointed comodule configuration,

$\Phi_{\text{even}}^{X}\star_{l}\zeta^{D}\simeq\delta^{L}+\Phi_{\text{odd}}^{X}\star_{l}\zeta^{D}.$

###  4.5 Möbius bicomodule configurations and the Rota formula

In order to take homotopy cardinality to recover the usual Möbius inversions,
we need to impose some finiteness conditions. We adapt the approach of [14]
summarised in 4.1 and 4.2 above.

A _right Möbius comodule configuration_ is a complete right pointed comodule
configuration $C\to Y$ such that the decomposition space $Y$ is Möbius and the
augmented comodule is Möbius, that is

  * •

$C$ is locally finite: the $\infty$-groupoid $C_{0}$ is locally finite and
both $s_{-1}$ and $d_{0}$ are finite maps;

  * •

$C$ is of locally finite length: every edge has a finite length, that is for
all $a\in C_{0}$, the fibres of $d_{0}^{(n)}:\overrightarrow{C}_{n}\to C_{0}$
over $a$ are empty for $n$ sufficiently large.

Under these conditions, the Phi functors descend to

$\Phi_{n}^{R}:{\mathcal{F}}_{/C_{0}}\to{\mathcal{F}}$

and we can now take the cardinality of the “Möbius formulas” (Theorems 4.4.4
and 4.4.5).

Similarly we define a _left Möbius comodule configuration_ to be a complete
left pointed comodule configuration $D\to X$ such that the decomposition space
$X$ is Möbius and the augmented comodule is Möbius, using $t_{\top+1}$ and
$d_{\top}$.

######  Theorem 4.5.1.

Given $C\to Y$ a right Möbius comodule configuration and $D\to X$ a left
Möbius comodule configuration,

$|\zeta^{C}|\star_{r}|\mu^{Y}|=|\delta^{R}|,\qquad|\mu^{X}|\star_{l}|\zeta^{D}|=|\delta^{L}|,$

where $|\mu^{Y}|:=|\Phi^{Y}_{\text{even}}|-|\Phi^{Y}_{\text{odd}}|$ and
$|\mu^{X}|:=|\Phi^{X}_{\text{even}}|-|\Phi^{X}_{\text{odd}}|$.

A _Möbius bicomodule configuration_ is a bicomodule configuration with two
pointings such that both left and right comodule configurations are Möbius. It
hence has extra degeneracy maps in both directions, extra bottom degeneracy
maps in horizontal direction and extra top degeneracy maps in vertical
direction.

Note that given a Möbius bicomodule configuration $B$, the zeta functors are
defined only on the $\infty$-groupoid $B_{0,0}$ and then are the same for the
two comodules. In both cases it is given by the span

$B_{0,0}\xleftarrow{=}B_{0,0}\xrightarrow{}1.$

######  Theorem 4.5.2.

Given a Möbius bicomodule configuration $B$ with $X:=B_{\bullet,-1}$ and
$Y:=B_{-1,\bullet}$, we have

$|\mu^{X}|\star_{l}|\delta^{R}|=|\delta^{L}|\star_{r}|\mu^{Y}|,$

where $\delta^{R}$ is the linear functor given by the span

$B_{0,0}\xleftarrow{}X_{0}\xrightarrow{}1,$

and $\delta^{L}$ is the linear functor given by the span

$B_{0,0}\xleftarrow{}Y_{0}\xrightarrow{}1.$

###### Proof.

Using the Möbius formulas at the algebraic level from Theorem 4.5.1, and the
associativity of the convolution actions from Proposition 4.3.1, we compute

$\displaystyle|\mu^{X}|\star_{l}|\delta^{R}|$
$\displaystyle=|\mu^{X}|\star_{l}(|\zeta|\star_{r}|\mu^{Y}|)$
$\displaystyle=(|\mu^{X}|\star_{l}|\zeta|)\star_{r}|\mu^{Y}|$
$\displaystyle=|\delta^{L}|\star_{r}|\mu^{Y}|.$

∎

###  4.6 Möbius bicomodule configurations from adjunctions of Möbius
decomposition spaces

We saw in Section 3.2 that given a cocartesian fibration
$p:\mathcal{M}\to\Delta^{1}$ between decomposition spaces, we obtain a right
comodule configuration $B$, with diagonal maps $B_{i,j-1}\to B_{i-1,j}$ and
new sections $s_{-1}:B_{i,j-1}\to B_{i,j}$, for $i\geq 0$ given by the
composition with a diagonal map.

######  Lemma 4.6.1.

Given a cocartesian fibration $p:\mathcal{M}\to\Delta^{1}$ between
decomposition spaces, suppose moreover that $\mathcal{M}$ is complete. Then
the associated right pointed comodule configuration is complete.

###### Proof.

The new sections will be monomorphisms if the following square is a pullback:

${B_{i,j-1}}$${B_{i,j-1}}$${B_{i,j-1}}$${B_{i,j}.}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{s_{-1}}$$\scriptstyle{s_{-1}}$

By assumption, $\mathcal{M}$ is a complete decomposition space, hence all
degeneracy maps are monomorphisms, and we can apply Lemma 3.1.3, to obtain the
desired pullbacks. ∎

Instantiating the general definitions from Section 4.4, the zeta functor

$\zeta:{\mathcal{S}}_{/B_{0,0}}\to{\mathcal{S}}$

is given by the span

$B_{0,0}\xleftarrow{=}B_{0,0}\xrightarrow{}1,$

and the functor

$\delta^{R}:{\mathcal{S}}_{/B_{0,0}}\to{\mathcal{S}}$

is defined by the span

$B_{0,0}\xleftarrow{s_{-1}}B_{0,-1}\xrightarrow{}1.$

The right comodule configuration being complete, we get a Möbius inversion
formula (Theorem 4.4.4):

$\zeta\star_{r}\Phi_{\text{even}}^{Y}\simeq\delta^{R}+\zeta\star_{r}\Phi_{\text{odd}}^{Y},$

where $Y:=B_{-1,\bullet}$.

Similarly, given a cartesian fibration $p:\mathcal{M}\to\Delta^{1}$ between
decomposition spaces, we obtain a left pointed comodule configuration.

######  Lemma 4.6.2.

Given a cartesian fibration $p:\mathcal{M}\to\Delta^{1}$ between decomposition
spaces, suppose moreover that $\mathcal{M}$ is complete. Then the left pointed
comodule configuration is complete.

The functor

$\delta^{L}:{\mathcal{S}}_{/B_{0,0}}\to{\mathcal{S}}$

is given by the span

$B_{0,0}\xleftarrow{}B_{-1,0}\xrightarrow{}1.$

This leads to the Möbius inversion formula

$\Phi_{\text{even}}^{X}\star_{l}\zeta\simeq\delta^{L}+\Phi_{\text{odd}}^{X}\star_{l}\zeta.$

Given an adjunction between decomposition spaces, that is a simplicial map
$\mathcal{M}\to\Delta^{1}$ which is both cartesian and cocartesian, and
suppose that $\mathcal{M}$ is complete, then we just obtained two Möbius
inversion formulas.

######  Theorem 4.6.3.

Given an adjunction of decomposition spaces in the form of a bicartesian
fibration $p:\mathcal{M}\to\Delta^{1}$, suppose moreover that $\mathcal{M}$ is
a Möbius decomposition space. Then the bicomodule configuration extracted from
this data is Möbius. In particular, we have the Rota formula for the
adjunction $p$:

$|\mu^{X}|\star_{l}|\delta^{R}|=|\delta^{L}|\star_{r}|\mu^{Y}|.$

###### Proof.

First observe that $B_{0,0}$, and in fact all $B_{i,j}$, are locally finite
since they are given by pullback (see page 3.1) of locally finite spaces.
Second, note that $e_{\top}\colon B_{i+1,j}\to B_{i,j}$ is induced in the same
way from the face map $d_{i}\colon\mathcal{M}_{i+2+j}\to\mathcal{M}_{i+1+j}$,
which is an inner face map, and is therefore finite since $\mathcal{M}$ is
Möbius. Similarly $d_{0}\colon B_{i,j+1}\to B_{i,j}$ is obtained from
$d_{i+1}\colon\mathcal{M}_{i+2+j}\to\mathcal{M}_{i+1+j}$ which is also an
inner face map. Finally the fibres of $e_{\top}^{(n)}$ are empty for $n$
sufficiently large because the fibres of $d_{i-n+1}\circ\dots\circ
d_{i}\colon\mathcal{M}_{i+2+j}\to\mathcal{M}_{i+2-n+j}$ are empty for $n$
sufficiently large since $\mathcal{M}$ is Möbius. Similarly, the fibres of
$d_{0}^{(n)}$ are empty for $n$ sufficiently large. ∎

###  4.7 The Möbius function of the incidence algebra of the decomposition
space of finite posets

We come back to the bicomodule configuration $\mathbf{B}$ of layered sets and
posets given in Section 2.5.

######  Proposition 4.7.1 ([7, Proposition 3.3]).

The bicomodule configuration $\mathbf{B}$ is Möbius.

######  Remark 4.7.2.

This Möbius bicomodule configuration does not come from an adjunction of
decomposition spaces.

We can now apply the generalised Rota formula of Theorem 4.5.2 to compute the
Möbius function of the bialgebra of finite posets. The result is known [1],
but its derivation from the generalised Rota formula is new and interesting.

######  Theorem 4.7.3 ([7, Theorem 3.4]).

The Möbius function of the incidence algebra of the decomposition space
$\mathbf{C}$ of finite posets is

$\mu(P)=\begin{cases}(-1)^{n}&\text{ if $P\in\mathbf{C}_{1}$ is a discrete
poset with $n$ elements,}\\\ 0&\text{ else.}\end{cases}$

This result can be extended to the incidence algebra of any directed
restriction species, including rooted forests, and free operads [7].

## References

  * [1] M. Aguiar, N. Bergeron, and F. Sottile.  Combinatorial Hopf algebras and generalized Dehn–Sommerville relations.  Compositio Mathematica, 142(1):1–30, 2006.
  * [2] M. Aguiar and W. Ferrer Santos.  Galois connections for incidence Hopf algebras of partially ordered sets.  Advances in Mathematics, 151(1):71–100, 2000.
  * [3] M. Aigner.  Combinatorial theory.  Classics in Mathematics. Springer-Verlag Berlin Heidelberg, 1997.
  * [4] D. Ayala and J. Francis.  Fibrations of $\infty$-categories.  Preprint, arxiv:1702.02681.
  * [5] J. C. Baez and J. Dolan.  From finite sets to Feynman diagrams.  In Mathematics unlimited—2001 and beyond, pages 29–50. Springer, Berlin, 2001.
  * [6] J. E. Bergner, A. M. Osorno, V. Ozornova, M. Rovelli, and C. I. Scheimbauer.  2-Segal sets and the Waldhausen construction.  Topology and its Applications, 235:445–484, 2018.
  * [7] L. Carlier.  Möbius functions of directed restriction species and free operads, via the generalised Rota formula.  Preprint, arxiv:1812.09915.
  * [8] P. Cartier and D. Foata.  Problèmes combinatoires de commutation et réarrangements.  Lecture Notes in Mathematics, No. 85. Springer-Verlag, Berlin-New York, 1969.
  * [9] M. Content, F. Lemay, and P. Leroux.  Catégories de Möbius et fonctorialités : un cadre général pour l’inversion de Möbius.  Journal of Combinatorial Theory, Series A, 28(2):169–190, 1980\.
  * [10] T. Dyckerhoff and M. Kapranov.  Higher Segal spaces I.  Preprint, arxiv:1212.3563. To appear in _Springer Lecture Notes_.
  * [11] D. Gepner, R. Haugseng, and J. Kock.  $\infty$-operads as analytic monads.  Preprint, arxiv:1712.06469.
  * [12] I. Gálvez-Carrillo, J. Kock, and A. Tonks.  Homotopy linear algebra.  Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 148(2):293–325, 2018.
  * [13] I. Gálvez-Carrillo, J. Kock, and A. Tonks.  Decomposition spaces, incidence algebras and Möbius inversion I: basic theory.  Advances in Mathematics, 331:952–1015, 2018.
  * [14] I. Gálvez-Carrillo, J. Kock, and A. Tonks.  Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration, and finiteness.  Advances in Mathematics, 333:1242–1292, 2018.
  * [15] I. Gálvez-Carrillo, J. Kock, and A. Tonks.  Decomposition spaces and restriction species.  International Mathematics Research Notices, 2018.  doi:10.1093/imrn/rny089.
  * [16] I. Gálvez-Carrillo, J. Kock, and A. Tonks.  Decomposition spaces in combinatorics.  Preprint, arxiv:1612.09225.
  * [17] L. Illusie.  Complexe cotangent et déformations II, volume 283 of Lecture Notes in Mathematics, No. 85.  Springer-Verlag, Berlin-New York, 1972.
  * [18] A. Joyal.  Quasi-categories and Kan complexes.  Journal of Pure and Applied Algebra, 175(1):207–222, 2002.  Special Volume celebrating the 70th birthday of Professor Max Kelly.
  * [19] A. Joyal.  The theory of quasi-categories and its applications, 2008.  Available at http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf.
  * [20] A. Joyal.  Distributors and barrels, 2012.  Available at Joyal’s CatLab, or http://mat.uab.cat/~kock/crm/hocat/Joyal-Distributors-and-barrels.pdf.
  * [21] P. Leroux.  Les catégories de Möbius.  Cahiers de topologie et géométrie différentielle, 16:280–282, 1975.
  * [22] J. Lurie.  Higher Topos Theory.  Annals of Mathematics Studies. Princeton University Press, 2009.  Available at http://www.math.harvard.edu/~lurie/papers/HTT.pdf.
  * [23] J. Lurie.  Higher Algebra, 2017.  Available at http://www.math.harvard.edu/~lurie/papers/HA.pdf.
  * [24] M. D. Penney.  Simplicial spaces, lax algebras and the 2-Segal condition.  Preprint, arxiv:1710.02742.
  * [25] G.-C. Rota.  On the foundations of combinatorial theory I. Theory of Möbius functions.  Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 2:340–368, 1964.
  * [26] R. P. Stanley.  Enumerative combinatorics. Vol. I.  The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1986.
  * [27] T. Walde.  Hall monoidal categories and categorical modules.  Preprint, arxiv:1611.08241.
  * [28] M. B. Young.  Relative $2$-Segal spaces.  Algebraic and Geometric Topology, 18:975–1039, 2018.

Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra (Barcelona), Spain

E-mail address<EMAIL_ADDRESS>