# Hereditary species as monoidal decomposition spaces, comodule bialgebras,
and operadic categories

Louis Carlier

###### Abstract

We show that Schmitt’s hereditary species induce monoidal decomposition
spaces, and exhibit Schmitt’s bialgebra construction as an instance of the
general bialgebra construction on a monoidal decomposition space. We show
furthermore that this bialgebra structure coacts on the underlying
restriction-species bialgebra structure so as to form a comodule bialgebra.
Finally, we show that hereditary species induce a new family of examples of
operadic categories in the sense of Batanin and Markl [4].

## Introduction

The notion of decomposition space was introduced in combinatorics by Gálvez-
Carrillo, Kock, and Tonks [14, 15] as a generalisation of Möbius categories
[23] in turn generalising locally finite posets [25] and Cartier–Foata finite-
decomposition monoids [9], for the purpose of incidence (co)algebras and
Möbius inversion. The same notion was introduced by Dyckerhoff and Kapranov
[10] under the name unital 2-Segal space, for use in homological algebra and
representation theory. Decomposition spaces are simplicial groupoids that
satisfy an exactness condition weaker than the Segal condition: while the
Segal condition essentially characterises categories (the ability to compose),
the decomposition space axiom expresses the ability to decompose.

Not all coalgebras in combinatorics arise from posets, monoids or categories.
In a broader perspective important examples come from the Waldhausen
$S$-construction in topology and Hall algebras of various flavours. In
combinatorics, an important class of coalgebras are the coalgebras of
restriction species in the sense of Schmitt [26] (see also [2, §8.7]), and the
more general notion of directed restriction species of [16], shown to be
examples of decomposition spaces. Examples of these notions are given by the
chromatic Hopf algebra of graphs and the Butcher–Connes–Kreimer Hopf algebra
of rooted trees, as well as many related structures such as matroids, posets
(talking here about a Hopf algebra of all posets, not just a coalgebra of an
individual poset). Most of these examples are decomposition spaces that do not
satisfy the Segal condition.

In these examples, the comultiplication is given by splitting some structure
into two parts of equal status, constituting then the left and right tensor
factors. However, many important Hopf algebras in combinatorics are more
asymmetric, having on one side of the comultiplication a monomial instead of a
linear tensor factor. In the Segal case, the comultiplications with both
tensor factors linear are typical for incidence coalgebras of categories,
whereas the comultiplications with a monomial in the left-hand tensor factor
are typical for operads. Indeed, incidence coalgebras and bialgebras of
operads are another important class covered by the decomposition space
framework, see [17] and [22].

Just as there are many linear-linear coalgebras that do not come from
categories, there are important examples of multilinear-linear coalgebras that
do not come from operads. An important class of such coalgebras is given by
Schmitt’s hereditary species [26]. These are structures that admit restriction
(like restriction species) but also admit induction along quotient maps.
Formally these are functors $H\colon\mathbb{S}_{p}\to\text{{{Set}}}$, where
$\mathbb{S}_{p}$ denotes the category of finite sets and partially defined
surjections. The induced comultiplication works by summing over all partitions
of the underlying set, and then putting the monomial of all the blocks (with
restricted structure) on the left and putting the quotient structure on the
right. Schmitt [26] identified the properties needed for this to define a
coassociative coalgebra (in fact a bialgebra, and most often a Hopf algebra)
and exhibited important examples, such as in particular the hereditary species
of simple graphs.

The present paper shows that every hereditary species constitutes an example
of a monoidal decomposition space, and that Schmitt’s bialgebra construction
is a special case of the general incidence bialgebra construction for monoidal
decomposition spaces. These decomposition spaces are generally _not_ Segal
spaces (see Remark 3.7), and can therefore be seen as the first class of
examples of decomposition spaces filling the missing entry in the following
table.

| _linear-linear_ | _multilinear-linear_
---|---|---
_Segal-type_ | posets, monoids, categories | operads
_non-Segal type_ | restriction species, $S$-construction, |
Hall algebras |

The construction is similar to the two-sided bar construction for operads, see
[22].

After a brief review in Section 1 of needed notions about groupoids and
decomposition spaces, Section 2 summarises Schmitt’s hereditary species. In
Section 3, we realise the hereditary species of finite sets as a Segal space
$\mathbf{S}$ (hence a decomposition space) and establish some finiteness
conditions.

Proposition 3.5 and 3.6. The pseudosimplicial groupoid $\mathbf{S}$ is Segal,
and is complete, locally finite, locally discrete, and of locally finite
length.

In Section 4, exploiting the hereditary species of finite sets, we show that
every hereditary species $H$ defines a monoidal decomposition space
$\mathbf{H}$, and hence a bialgebra at the groupoid-slice level, and we
recover Schmitt’s construction by taking homotopy cardinality.

Proposition 4.2 and 4.5. For every hereditary species $H$, the simplicial
groupoid $\mathbf{H}$ is a monoidal decomposition space. The incidence
bialgebra obtained by taking homotopy cardinality coincides with the Schmitt
bialgebra associated to $H$.

Every hereditary species is also a restriction species, and the free algebra
on its incidence coalgebra is therefore a bialgebra. In Section 5, we show
that the incidence bialgebra of a hereditary species coacts on this bialgebra,
so as constitute a _comodule bialgebra_.

Proposition 5.1. The hereditary-species bialgebra $B$ coacts on the
restriction-species bialgebra $A$, so as to make $A$ a left comodule bialgebra
over $B$.

Comodule bialgebras have been found important recently in numerical analysis
[6] and in stochastic analysis [5], and there are general constructions based
on operads and trees [12], [21]. The incidence comodule bialgebras of
hereditary species introduced here constitute a new general class of comodule
bialgebras, not related to trees.

In Section 6, we describe a different construction on hereditary species,
showing that simple hereditary species induce operadic categories in the sense
of Batanin and Markl [4]. Precisely we define a functor from simple hereditary
species to operadic categories. This is interesting in its own right, as it
constitutes a new family of examples of operadic categories which had not been
observed before. The construction is not directly related to decomposition
spaces, but suggests that further connections are to be discovered.

### Acknowledgements

The author would like to thank Joachim Kock for his advice and support all
along the project. The author was supported by a PhD grant attached to
MTM2013-42293-P, and by the FEDER-MINECO grant MTM2016-80439-P from the
Spanish Ministry of Economy and Competitiveness.

##  1 Groupoids and decomposition spaces

We work in the $2$-category Grpd of groupoids, maps, and homotopies. All the
notions will be invariant under homotopy equivalences. We usually omit the
term homotopy and say ‘pullback’, ‘fibre’, etc. instead of ‘homotopy
pullback’, ‘homotopy fibre’, etc. The squares are commutative up-to-homotopy,
they come equipped with an (invertible) $2$-cell.

### Pullbacks and fibres

The main tool used throughout this paper are (homotopy) pullbacks. We use the
following lemmas many times.

######  Lemma 1.1 ([8],[24, Lemma 4.4.2.1]).

Given a prism diagram of 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 of the left-hand square is.

Given a map of groupoids $p\colon X\to S$ and 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}$

######  Lemma 1.2 ([8]).

A square of groupoids

${P}$${Y}$${X}$${S}$$\scriptstyle{u}$

is a pullback if and only if for each $x\in X$ the induced comparison map
$u_{x}\colon P_{x}\to Y_{fx}$ is an equivalence.

### Slices and linear functors

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

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

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

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

the functors
$p_{!}f^{*},g^{*}q_{!}\colon\text{{{Grpd}}}_{/I}\to\text{{{Grpd}}}_{/V}$ are
naturally homotopy equivalent (see [19] for the technical details regarding
coherence of these equivalences in the $\infty$-groupoids setting). By the
Beck-Chevalley rule, the composition of two linear functors is linear. The
slice groupoid $\text{{{Grpd}}}_{/I}$ plays the role of the vector space with
basis $I$.

We denote by LIN the monoidal $2$-category of slice categories and linear
functors, with the tensor product induced by the cartesian product:

$\text{{{Grpd}}}_{/I}\otimes\text{{{Grpd}}}_{/J}:=\text{{{Grpd}}}_{/I\times
J},$

with neutral object $\text{{{Grpd}}}\simeq\text{{{Grpd}}}_{/1}$. For an
extended treatment of linear functors and homotopy linear algebra, we refer to
[13].

### Decomposition spaces and incidence coalgebras

The notion of decomposition space was introduced by Gálvez-Carrillo, Kock, and
Tonks [14], in the general setting of simplicial $\infty$-groupoids, and
independently by Dyckerhoff and Kapranov [10] under the name unital $2$-Segal
space. The natural level of generality for decomposition spaces in
combinatorics is that of simplicial groupoids, because many combinatorial
objects have symmetries, which are taken care of by the groupoid formalism.
For a survey motivated by combinatorics, see [17].

The simplex category $\boldsymbol{\Delta}$ has an active-inert factorisation
system. An arrow is active when it preserves endpoints, and is inert if it is
distance preserving. In the present contribution, a _decomposition space_
$X\colon\boldsymbol{\Delta}^{\text{{\rm{op}}}}\to\text{{{Grpd}}}$ is a
simplicial groupoid such that the image of any active-inert pushout in
$\boldsymbol{\Delta}$ is a pullback of 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}}$

Decomposition spaces can be seen as an abstraction of posets. The pullback
condition is precisely the condition required to obtain a counital
coassociative comultiplication on $\text{{{Grpd}}}_{/X_{1}}$, called the
_incidence coalgebra_. Precisely, 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_ :

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

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

$\displaystyle\varepsilon\colon\text{{{Grpd}}}_{/X_{1}}$
$\displaystyle\to\text{{{Grpd}}}$ $\displaystyle(T\xrightarrow{t}X_{1})$
$\displaystyle\mapsto z_{!}\circ s_{0}^{*}(t).$

We obtain a coalgebra $(\text{{{Grpd}}}_{/X_{1}},\Delta,\varepsilon)$ called
the _incidence coalgebra_.

######  Proposition 1.3 ([10, Proposition 2.3.3], [14, Proposition 3.5]).

Every Segal space is a decomposition space. In particular, the nerve of a
poset is a decomposition space.

### Homotopy cardinality [17]

In order to be able to take homotopy cardinality to get a coalgebra at the
numerical level, we need to assume some finiteness conditions. A groupoid is
_locally finite_ if the automorphism groups are finite, and is _finite_ if
furthermore it has finitely many components. We denote by grpd the category of
finite groupoids. Note that every set is locally finite. A map of groupoids is
_finite_ if each fibre is finite.

The _homotopy cardinality_ of a finite groupoid $X$ is defined to be

$|X|=\sum_{x\in\pi_{0}X}\frac{1}{|\text{Aut}(x)|}\in\mathbb{Q},$

and the homotopy cardinality of a finite map of groupoid is

$|X\to
S|=\sum_{s\in\pi_{0}S}\frac{|X_{s}|}{|\text{Aut}(s)|}\delta_{s}\in\mathbb{Q}_{\pi_{0}S},$

where $X_{s}$ is the homotopy fibre, and $\mathbb{Q}_{\pi_{0}S}$ is the vector
space spanned by iso-classes, denoted by the formal symbol $\delta_{s}$, for
$s\in\pi_{0}S$. Homotopy equivalent groupoids have the same cardinality. For a
locally finite groupoid, there is a notion of cardinality
$|{\mathord{\text{--}}}|\colon\text{{{Grpd}}}_{/S}\to\mathbb{Q}_{\pi_{0}S}$
sending a basis element $\ulcorner s\urcorner$ to the basis element
$\delta_{s}=|\ulcorner s\urcorner|$.

A decomposition space $X$ is _locally finite_ if $X_{1}$ is a locally finite
groupoid, and the two maps $s_{0}$ and $d_{1}$ have finite fibres. If we
require furthermore the decomposition space to be _locally discrete_ , that is
the fibres of $s_{0}$ and $d_{1}$ are discrete groupoids, then the
comultiplication formula will be free of denominators, see [17].

For any locally finite decomposition space $X$, we can take the cardinality of
the linear functors $\Delta$ and $\varepsilon$ to obtain a coalgebra structure
$(\mathbb{Q}_{\pi_{0}{X_{1}}},|\Delta|,|\varepsilon|)$, called the _numerical
incidence coalgebra_ of $X$.

Under the completeness condition ($s_{0}$ is a monomorphism) and the locally
finite length condition (each edge has a finite length), there is a general
Möbius inversion principle, see [15].

### Culf maps

A map $f\colon X\to Y$ 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.

A simplicial map $f\colon X\to Y$ is _culf_ if it is cartesian on all active
maps. The culf functors induce coalgebra homomorphisms between the incidence
coalgebras [14, Lemma 8.2].

######  Proposition 1.4 ([14, Lemma 4.6]).

If $X$ is a decomposition space and $f\colon Y\to X$ is a culf map, then also
$Y$ is a decomposition space.

######  Proposition 1.5 ([16]).

If $X$ is a locally discrete (resp. locally finite length) decomposition space
and $f\colon Y\to X$ is a culf map, then also $Y$ is a locally discrete (resp.
locally finite length) decomposition space. This is also the case for locally
finite, but we must check moreover that $Y_{1}$ is locally finite.

### Monoidal decomposition spaces, bialgebras

A _monoidal structure_ [14, §9] on a decomposition space $X$ is given by the
data of simplicial maps $\eta\colon 1\to X$ and $\otimes\colon X\times X\to X$
required to be culf, and satisfying the standard associative and unital laws
of monoidal structures. This monoidal structure induces a monoid structure on
the incidence coalgebra $\text{{{Grpd}}}_{/X_{1}}$ and altogether a bialgebra
structure. In combinatorics, the monoidal structure is often given by disjoint
union.

##  2 Hereditary species

Let $\mathbb{B}$ denote the category of finite sets and bijections, let
$\mathbb{I}$ denote the category of finite sets and injective maps, and let
$\mathbb{S}$ denote the category of finite sets and surjective maps. We denote
by $\mathbb{S}_{p}$ the category of finite sets and partially defined
surjections. A partially defined surjection $V\to W$ consists of a subset
$U\subset V$ and a surjection $U\to W$. More formally, the arrows in
$\mathbb{S}_{p}$ are given by equivalence classes of spans

${U}$${V}$${W}$$\scriptstyle{i}$$\scriptstyle{p}$

where $i$ is injective and $p$ is surjective, and where two such spans are
equivalent if they are isomorphic as spans. Partially defined surjections are
composed by pullback composition of spans (in the category of sets). This is
meaningful since both injections and surjections are stable under pullbacks in
the category of sets. Note that the empty set is included here. Note also that
the category $\mathbb{S}_{p}$ contains the category $\mathbb{S}$ as a
subcategory (the spans in which the injection leg is an identity map) and also
contains the category $\mathbb{I}^{\text{{\rm{op}}}}$ as a subcategory (the
spans in which the surjection leg is an identity map).

### Species

Recall that a _species_ [20] is a functor $F\colon\mathbb{B}\to\text{{{Set}}}$
, $V\mapsto F[V]$. An element of $F[V]$ is called an $F$-structure on the
finite set $V$. A _restriction species_ [26] is a functor
$R\colon\mathbb{I}^{\text{{\rm{op}}}}\to\text{{{Set}}}$. An $R$-structure on a
set $V$ thus restricts to any subset $U\subset V$. Schmitt [26] further
defines a _hereditary species_ to be a functor
$H\colon\mathbb{S}_{p}\to\text{{{Set}}}$. An element $G\in H[V]$ is called a
_$H$ -structure_ on the set $V$. A hereditary species is thus covariantly
functorial (not only in bijections but also) in surjections, and also
contravariantly functorial in injections (that is, is a restriction species).
This means that a $H$-structure on a set $V$ induces also a $H$-structure on
any quotient set and on any subset. Furthermore, these functorialities are
compatible in the sense that for any pullback square

${U}$${U^{\prime}}$${V}$${V^{\prime}}$$\scriptstyle{p}$$\scriptstyle{i}$$\scriptstyle{p^{\prime}}$${\llcorner}$$\scriptstyle{j}$

we have

$H[p^{\prime}]\circ H[i]=H[j]\circ H[p].$

This ‘Beck-Chevalley’ law is a consequence of the fact that $H$ must respect
the composition of spans.

If $\pi$ is a partition of $V$, and $\rho_{V,\pi}\colon V\to\pi$ is the
canonical surjection, the _quotient_ $G/\pi$ is the $H$-structure on the set
$\pi$ defined by

$G/\pi=H[\rho_{V,\pi}](G).$

The _restriction_ $G|\pi$ is defined to be the family

$G|\pi=\\{G|B\\}_{B\in\pi}.$

A morphism of hereditary species is a natural transformation of functors. We
denote by HSp the category of hereditary species and natural transformations.

######  Example 2.1.

For a graph $G$ with vertex set $V$, and $\pi$ a partition of $V$, we define
$G|\pi$ to be the family of graphs whose vertex sets are blocks of $\pi$ and
with an edge between two elements of the same block if there is an edge in $G$
with both incident vertices in the block. We define $G/\pi$ to be the graph
with vertex set $\pi$ and with an edge between two vertices if there is a edge
in $G$ between the corresponding blocks.

Suppose that $\tau$ is a finer partition than $\sigma$ (denoted
$\tau\leq\sigma$), that is each block of $\sigma$ is a union of blocks of
$\tau$. We denote $\sigma/\tau$ the partition of the set $\tau$ induced by
$\sigma$.

The following proposition is a consequence of the functorialities in
surjections and injections, and the Beck-Chevalley law.

######  Proposition 2.2 ([26, Proposition 4.1]).

If $\tau$ and $\sigma$ are partitions of $V$ such that $\tau\leq\sigma$, then
the following identities hold:

$\displaystyle[(G|\sigma)|\tau]$ $\displaystyle=[G|\tau],$
$\displaystyle[(G/\tau)|(\sigma/\tau)]$ $\displaystyle=[(G|\sigma)/\tau],$
$\displaystyle[(G/\tau)/(\sigma/\tau)]$ $\displaystyle=[G/\sigma],$

where $[G]$ is the isomorphism class of $G$.

### Coalgebra

To any hereditary species, Schmitt associates a coalgebra $B$ on the vector
space spanned by all isomorphism classes of families of _non-empty_
$H$-structures. The comultiplication of a $H$-structure $G$ on the set $V$ is
defined by:

$\Delta(G)=\sum_{\sigma\in\Pi(V)}G|\sigma\otimes G/\sigma.$

The counit is defined by

$\varepsilon(G)=\begin{cases}1&\text{ if every member in the family is a
singleton,}\\\ 0&\text{ otherwise.}\end{cases}$

Note the importance of disallowing empty structures. With the empty
$H$-structure we would have $\Delta(\varnothing)=(\,)\otimes\varnothing$, and
the comultiplication would not be counital on the right. While not counital on
the right, we shall see later that it is still a left comodule over $B$.

### Hereditary species of non-empty sets

We consider the hereditary species of non-empty finite sets. The
comultiplication of a finite set $V$ is defined by summing over all partitions
of $V$ and putting on the left the family of blocks of the partition and on
the right the set whose elements are blocks of the partition:

$\Delta(V)=\sum_{\pi\in\Pi(V)}(V_{1},\dots,V_{k})\otimes\pi.$

##  3 A decomposition space of surjections

We work with groupoids instead of sets, to take into account symmetries (see
[3] and [17]) and define a _hereditary species_ to be a functor

$H\colon\mathbb{S}_{p}\to\text{{{Grpd}}}.$

In particular, a hereditary species is (covariantly) functorial in surjections
and contravariantly functorial in injections, and these two functorialities
interact via the Beck-Chevalley rule.

### Partitions and surjections

The groupoid of partitions (whose objects are sets with a partition into
blocks, and arrows are bijections between sets preserving blocks) is
equivalent to the groupoid of surjections (arrows are pairs of compatible
bijections). More precisely, a partition of a set $V$ can be given by a
surjection $V\twoheadrightarrow P$: a block of the partition of $V$ is the
preimage of an element of $P$. Refinement of partitions is rendered
conveniently in terms of composition of surjections. Precisely, the poset of
partitions of $V$ under refinement is equivalent to the coslice
$\mathbb{S}_{V/}$: to say that $\rho\leq\pi$ is precisely to say that there is
a commutative triangle of the corresponding surjections

${V}$${P}$${Q.}$

More generally, given $n$ composable surjections from $V$, we get $n$
partitions of $V$. A pair of composable surjections $V\stackrel{{\scriptstyle
f}}{{\twoheadrightarrow}}P\stackrel{{\scriptstyle g}}{{\twoheadrightarrow}}Q$
induces surjective maps between the fibres of $gf$ and $g$ over each element
of $Q$.

### Fat nerve of the category of finite sets and surjections

The _fat nerve_ [14] of $\mathbb{S}$ is the simplicial groupoid

$\displaystyle\mathbf{N}\mathbb{S}\colon\boldsymbol{\Delta}^{\text{{\rm{op}}}}$
$\displaystyle\to\text{{{Grpd}}}$ $\displaystyle[n]$
$\displaystyle\mapsto\operatorname{Map}([n],\mathbb{S}).$

Explicitly, $(\mathbf{N}\mathbb{S})_{0}$ is the groupoid of finite sets and
bijections, $(\mathbf{N}\mathbb{S})_{1}$ is the groupoid whose objects are
surjections and maps consist of a bijection between the sources, and a
compatible bijection between the targets. The objects of the groupoid
$(\mathbf{N}\mathbb{S})_{n}$ are $n$ composable surjections, and maps are
$(n{+}1)$-uplets of compatible bijections. The inner face maps are given by
composition, the outer face maps by forgetting the first, or the last set in
the chain. The degeneracy maps are given by inserting identity maps.

The skeleton $\mathbb{S}_{\text{ord}}$ of $\mathbb{S}$ consisting of ordinal
numbers and surjections is a full subcategory of $\mathbb{S}$, and
$\mathbf{N}\mathbb{S}_{\text{ord}}\simeq\mathbf{N}\mathbb{S}$.

### Symmetric monoidal category monad

The symmetric monoidal category monad
$\mathsf{S}\colon\text{{{Grpd}}}\to\text{{{Grpd}}}$ [22, §2.5] is the monad
represented by the polynomial

$1\leftarrow\mathbb{B}^{\prime}\to\mathbb{B}\to 1$

where $\mathbb{B}$ is the groupoid of finite ordinals and bijections (not
required to be monotone), and $\mathbb{B}^{\prime}$ is the groupoid of finite
pointed ordinals and basepoint-preserving bijections. It sends a groupoid $X$
to

$\int^{n\in\mathbb{B}}\operatorname{Map}(\mathbb{B}^{\prime}_{n},X)\simeq\int^{n\in\mathbb{B}}X^{\underline{n}},$

where $\underline{n}$ denotes the fibre over $n$, and the integral sign is a
homotopy sum [13]:

$\int^{k}X=\sum_{k}\frac{X}{\operatorname{Aut}k}.$

Given a groupoid $X$, on objects $\mathsf{S}X$ is the groupoid whose objects
are finite lists of objects of $X$, and a morphism
$(a_{1},\dots,a_{n})\to(b_{1},\dots,b_{m})$ consists of a bijection
$\underline{n}\to\underline{m}$ and morphisms $a_{i}\to b_{\sigma(i)}$ in $X$.
The algebras over $\mathsf{S}$ are symmetric monoidal categories. The unit
sends an element $l$ to the list with one element $(l)$, and the
multiplication concatenates the lists.

### Simplicial groupoid of surjections

Consider the hereditary species of non-empty sets. Associated to it we
construct a simplicial groupoid $\mathbf{S}$. Later this simplicial groupoid
$\mathbf{S}$ will be the base ingredient in the construction of a simplicial
groupoid $\mathbf{H}$ associated to each hereditary species $H$. The
simplicial groupoid $\mathbf{H}$ will be a monoidal decomposition space, and
therefore define a bialgebra, which will be shown to be the Schmitt bialgebra
construction. The basis elements of this bialgebra (i.e. the objects of the
groupoid $\mathbf{H}_{1}$), will be _families_ of non-empty $H$-structures,
not individual non-empty $H$-structures. Similarly for $\mathbf{S}$, the basis
elements, the objects of $\mathbf{S}_{1}$, will be _families_ of non-empty
finite sets, not just individual non-empty finite sets. Including families
rather than just individual structures is necessary in order to have a well-
defined comultiplication, since the left-hand tensor factor will be a monomial
rather than a linear factor, as explained in the introduction. At the same
time, working with families gives immediately the algebra structure (which is
free). Nevertheless, it will be technically important to consider also
individual structures, which we regard as _connected_ families.

Accordingly, we first describe a simplicial-groupoid-with-missing-top-face-
maps, which we call $\mathbf{C}$ for ‘connected’. We first consider the
groupoid $\mathbf{C}_{j}$ of $(j{-}1)$ composable surjections between non-
empty finite sets. The objects of $\mathbf{C}_{2}$ are surjections, the
objects of $\mathbf{C}_{1}$ are non-empty finite sets, and $\mathbf{C}_{0}$ is
the terminal groupoid, that is equivalent to a point. Face maps are given by:

  * •

$d_{0}$ forgets the first set in the chain of surjections;

  * •

$d_{i}\colon\mathbf{C}_{j}\to\mathbf{C}_{j-1}$ compose the $i$th and
$(i{+}1)$st surjection, for $0<i<j-1$;

  * •

$d_{j-1}$ forgets the last set in the chain of surjections.

The degeneracy maps $s_{i}\colon\mathbf{C}_{j}\to\mathbf{C}_{j+1}$, for $0\leq
i\leq j-1$ are given by inserting an identity arrow at object number $i$. The
degeneracy map $s_{\top}\colon\mathbf{C}_{j}\to\mathbf{C}_{j+1}$ is given by
appending with the map whose target is the terminal set $1$.

######  Remark 3.1.

The map to $1$ is a surjection since the sets were required non-empty. With
possibly empty sets, it would not be possible to define the top degeneracy
map.

To obtain top face maps, it is necessary to introduce families: the top face
map of a surjections chain must be the family of surjections chain shorter by
one, obtained as the fibre over each element in the last set. We define
$\mathbf{S}$ to be the symmetric monoidal category monad $\mathsf{S}$ applied
to $\mathbf{C}$. All the face maps (except the missing top ones) and all the
degeneracy maps are just $\mathsf{S}$ applied to the face and degeneracy maps
of $\mathbf{C}$. The missing top face map $d_{\top}$ is now given by fibres:
given $(j{-}1)$ composable surjections, for each element $k$ of the last
target set, we can form the fibres over $k$ of the different source sets. It
also induces surjective maps between these different fibres. We end up with a
family (indexed by elements of the last target set) of $(j{-}2)$ composable
surjections between the fibres. Note that the fibres are non-empty since we
only consider surjections, not arbitrary maps.

######  Remark 3.2.

Simplicially, the multi aspect is localised to the top face maps. This is
already a feature well known from operads [22]: the ‘domain’ (given by the
simplicial map $d_{1}$) of a single operation of an operad is not a single
object but a family of objects. In the present situation, beyond operads, a
new feature is that the top face maps do not satisfy the simplicial identities
on the nose, only up to coherent homotopy, making altogether the simplicial
groupoids pseudosimplicial (as already allowed for in the homotopy setting in
which the theory of decomposition spaces is staged). This is caused by
symmetries acting on blocks of partitions, and hence on factors in monomials,
and seems to be an unavoidable nuisance, except in fully rigid situations such
as $L$-species with monotone surjections.

######  Proposition 3.3.

The groupoids $\mathbf{S}_{j}$ and the degeneracy and face maps given above
form a pseudosimplicial groupoid $\mathbf{S}$.

###### Proof.

The only pseudosimplicial identity is $d_{\top}\circ d_{\top}\simeq
d_{\top}\circ d_{\top-1}$. The other simplicial identities are strict and
straightforward to check. The ones involving top face and top degeneracy maps
are:

$\displaystyle d_{i}\circ d_{\top}$ $\displaystyle=d_{\top}\circ d_{i}$
$\displaystyle s_{i}\circ d_{\top}$ $\displaystyle=d_{\top}\circ s_{i}$
$\displaystyle d_{\top}\circ s_{\top}$ $\displaystyle=\operatorname{id}$
$\displaystyle d_{\top}\circ d_{\bot}$ $\displaystyle=d_{\bot}\circ d_{\top}$
$\displaystyle d_{i}\circ s_{\top}$ $\displaystyle=s_{\top}\circ d_{i}$
$\displaystyle s_{\top}\circ s_{i}$ $\displaystyle=s_{i}\circ s_{\top}$

It remains to verify the functor is pseudosimplicial for the top face map
$d_{\top}$. Given two composable surjections $V\stackrel{{\scriptstyle
f}}{{\twoheadrightarrow}}P\stackrel{{\scriptstyle g}}{{\twoheadrightarrow}}Q$,
we can consider the family $\\{V_{p}\\}_{p\in P}$ of fibres of $f$ over
elements of $P$, and we can also consider the family of families
$\left\\{\\{V_{p}\\}_{p\in P_{q}}\right\\}_{q\in Q}$. There is a canonical
isomorphism $\left\\{\\{V_{p}\\}_{p\in P_{q}}\right\\}_{q\in
Q}\to\\{V_{p}\\}_{p\in P}$. We want to show the following square is
commutative:

${((V_{k})_{l})_{m}}$${(V_{k})_{l}}$${(V_{k})_{p}}$${V_{p}.}$

The isomorphisms are compatible with the injections of fibres into $V$, by the
universal property of pullback: given $p\in P$ such that $g(p)=q$, the
following square is a pullback by definition:

${V_{p}}$${V}$${1}$${P.}$$\scriptstyle{\ulcorner p\urcorner}$

We also have the following commutative diagram, given by pullbacks

${(V_{q})_{p}\ }$${V_{q}}$${V}$${1}$${P_{q}\
}$${P}$${1}$${Q.}$$\scriptstyle{\ulcorner p\urcorner}$$\scriptstyle{\ulcorner
q\urcorner}$

Hence by the universal property of pullbacks, there exists a unique
isomorphism $(V_{q})_{p}\to V_{p}$ such that the following triangle commutes:

${(V_{q})_{p}}$${V_{p}}$${V.}$$\scriptstyle{\simeq}$

In a similar way, we obtain the three other isomorphisms between the fibres
compatible with the injections into $V$. Since all four isomorphisms in the
square are compatible with the injections into $V$, the commutativity of the
square is ensured by the fact that each $V_{i}\to V$ is a monomorphism. ∎

Considering only non-empty ordinals $\underline{n}$ instead of all non-empty
finite sets, we get an equivalent pseudosimplicial groupoid, which we denote
$\mathbf{S}_{\text{ord}}$. The objects of $(\mathbf{S}_{\text{ord}})_{1}$ are
lists of non-empty ordinals, and a map between two lists
$(\underline{n}_{1},\dots,\underline{n}_{p})$ and
$(\underline{m}_{1},\dots,\underline{m}_{p^{\prime}})$ consists of a bijection
$\sigma\colon p\to p^{\prime}$ and, for all $i\in p$, a map
$\rho_{i}\colon\underline{n}_{i}\to\underline{m}_{\sigma(i)}$.

######  Proposition 3.4.

We have a natural equivalence
$\mathbf{N}\mathbb{S}_{\text{ord}}\simeq\mathbf{S}_{\text{ord}}$.

###### Proof.

The functor $\mathbf{N}\mathbb{S}_{\text{ord}}\to\mathbf{S}_{\text{ord}}$
sends a surjection $\underline{n}\twoheadrightarrow\underline{p}$ to the list
of fibres $(\underline{n}_{1},\dots,\underline{n}_{p})$, and a map
$(\underline{n}\xrightarrow{\rho}\underline{n}^{\prime},\underline{p}\xrightarrow{\sigma}\underline{p}^{\prime})$
between two surjections $\underline{n}\twoheadrightarrow\underline{p}$ and
$\underline{n}^{\prime}\twoheadrightarrow\underline{p}^{\prime}$ to the map
$(\underline{n}_{1},\dots,\underline{n}_{p})\xrightarrow{(\sigma,\rho_{1},\dots,\rho_{p})}(\underline{n}^{\prime}_{1},\dots,\underline{n}^{\prime}_{p^{\prime}})$.
Note that the surjection $\varnothing\twoheadrightarrow\varnothing$ gives the
empty list, which is allowed. Similarly, it sends a family of $n$ composable
surjections to the list of $(n{-}1)$ composable surjections given by the
fibres. The maps are given in a similar way.

There is also a functor
$\mathbf{S}_{\text{ord}}\to\mathbf{N}\mathbb{S}_{\text{ord}}$. Given a list of
non-empty ordinals $(\underline{n}_{1},\dots,\underline{n}_{p})$, we obtain a
surjection
$\sum_{i\in\underline{p}}\underline{n}_{i}\twoheadrightarrow\underline{p}$
from the disjoint union of the elements of the list to the indexing set. (This
map is a surjection because all the $n_{i}$ are non-empty.) Given a map
between two lists $(\underline{n}_{1},\dots,\underline{n}_{p})$ and
$(\underline{m}_{1},\dots,\underline{m}_{p^{\prime}})$, we obtain a map
$\sum_{i\in\underline{p}}\underline{n}_{i}\to\sum_{i\in\underline{p}^{\prime}}\underline{m}_{i}$
by sending $\underline{n}_{i}$ to $\underline{m}_{\sigma(i)}$. Similarly,
given a list of $(j{-}1)$ composable surjections, we obtain $j$ composable
surjections by disjoint union, and the last one is given as before, using the
target sets of the surjections.

It is easy to check they form an equivalence since the disjoint union of
fibres of a surjection is isomorphic to the source set of this surjection. ∎

######  Proposition 3.5.

The pseudosimplicial groupoid $\mathbf{S}$ is Segal, and hence a decomposition
space.

###### Proof.

It follows from the equivalence
$\mathbf{N}\mathbb{S}_{\text{ord}}\simeq\mathbf{S}_{\text{ord}}\simeq\mathbf{S}$
since the fat nerve of a small category is always Segal [14]. ∎

######  Proposition 3.6.

The Segal space $\mathbf{S}$ is complete, locally finite, locally discrete,
and of locally finite length.

###### Proof.

The map $s_{0}$ is a monomorphism because the fibre is empty if the first
surjection is not the identity, and is contractible else. The groupoid
$\mathbf{S}_{1}$ is locally finite, because elements of the families are non-
empty finite sets, and each finite set has only a finite number of
automorphisms. We have seen $s_{0}$ is finite and discrete, the map $d_{1}$ is
also finite and discrete: the fibre of $d_{1}$ over $(n{-}1)$ composable
surjections $f\in\mathbf{S}_{n}$ is either empty or the finite discrete
groupoid of $n$ composable surjections where the first one is the identity and
the other surjections are given by $f$. Finally, $\mathbf{S}$ is of locally
finite length (every edge $f$ has finite length): the degenerate simplices are
families where one of the surjections is an identity, or the last set is a
singleton. The fibre of $f$ has no nondegenerate simplices for $n$ greater
than the total number of elements of the source sets of the family. ∎

######  Remark 3.7.

The decomposition space $\mathbf{H}$ is not usually a Segal space. The base
case of the Segal condition stipulates that the square

${\mathbf{H}_{2}}$${\mathbf{H}_{1}}$${\mathbf{H}_{1}}$${\mathbf{H}_{0}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$

is a pullback. This would mean that one should be able to reconstruct a
$H$-structure on a surjection $V\twoheadrightarrow P$ by knowing it on $P$ and
on the fibres $V_{1},\dots,V_{p}$. In other words, one could substitute the
$H$-structures on $V_{1},\ldots,V_{p}$ into the elements of $P$ of another
$H$-structure, as if those elements were ‘input slots’ of the operation of an
operad.

Consider for example the case of simple graphs (Example 2.1). Given $p$ simple
graphs with vertex sets $V_{1},\dots,V_{p}$ and another graph with $p$
vertices, there is no canonical prescription for substituting the $p$ graphs
into those vertices.

Since in every degree, the groupoid is given by applying $\mathsf{S}$, the
decomposition space $\mathbf{S}$ is automatically a monoidal decomposition
space. The associated incidence bialgebra has the property that the
comultiplication applied to a connected element gives a monomial in the left-
hand tensor factor and a connected element (linear factor) in the right-hand
tensor factor. That’s the immediate conclusion of the fact that $d_{2}$
requires $\mathsf{S}$ whereas $d_{0}$ does not. This observation can be
formalised at the simplicial level by the following result.

Recall from [7, Proposition 2.1.1] (see also [27] and [28]) that the
decomposition space analog of (left) comodule is given by a simplicial map
$f\colon C\to X$ between two simplicial groupoids such that $C$ is Segal, $X$
is a decomposition space and the map $f\colon C\to X$ 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 category $\text{{{Grpd}}}_{/C_{0}}$ the structure of a
left $\text{{{Grpd}}}_{/X_{1}}$-comodule.

######  Lemma 3.8.

The slice category $\text{{{Grpd}}}_{/\mathbf{C}_{1}}$ is a left comodule over
$\text{{{Grpd}}}_{/\mathbf{S}_{1}}$.

###### Proof.

Note that $\mathbf{C}$ is lacking top face maps, but
$\operatorname{Dec}_{\top}\mathbf{C}$ is a genuine simplicial groupoid as
required by the notion of comodule. Since
$\operatorname{Dec}_{\top}(\mathbf{C})$ is a Segal space (the décalage of a
decomposition space is always a Segal space [14, Proposition 4.9]), and
$\mathbf{S}$ is a decomposition space, we just need to exhibit a culf map
$\operatorname{Dec}_{\top}(\mathbf{C})\to\mathbf{S}$, which is given by
$d_{\top}$. Note that it is essentially the unit for the monad $\mathsf{S}$,
therefore it is cartesian, and in particular culf. ∎

In fact, we can consider all (possibly empty) finite sets, and still obtain a
comodule structure. (The surjection $\varnothing\to\varnothing$ will be sent
to the empty family.)

######  Lemma 3.9.

The slice category $\text{{{Grpd}}}_{/\mathbf{N}{\mathbb{S}}_{0}}$ is a left
comodule over $\text{{{Grpd}}}_{/\mathbf{S}_{1}}$.

###### Proof.

The fat nerve of a category is always a Segal space [14, §2.14]. We need to
exhibit a culf map $\mathbf{N}{\mathbb{S}}\to\mathbf{S}$, which is given by
sending a surjection to the family of fibres (which are non-empty), as the map
$d_{\top}$ of the decomposition space $\mathbf{S}$. ∎

##  4 Hereditary species and decomposition spaces

We can now add a hereditary structure on the source of each member of the
family. Given a hereditary species $H$, define $\mathbf{H}_{1}$ to be the
groupoid of families of non-empty $H$-structures. More formally,
$\mathbf{H}_{1}$ is defined as families of the Grothendieck construction of
the underlying ordinary species $H\colon\mathbb{B}_{+}\to\text{{{Grpd}}}$,
where $\mathbb{B}_{+}$ denote the category of non-empty finite sets and
bijections. The groupoid $\mathbf{H}_{n}$ is given by the pullback

${\mathbf{H}_{n}}$${\mathbf{H}_{1}}$${\mathbf{S}_{n}}$${\mathbf{S}_{1}.}$${\lrcorner}$source

Inner face maps and degeneracy maps are defined by pullback. For example, in
the following diagram, the right-hand square and the rectangle are pullbacks
by definition

${\mathbf{H}_{3}}$${\mathbf{H}_{2}}$${\mathbf{H}_{1}}$${\mathbf{S}_{3}}$${\mathbf{S}_{2}}$${\mathbf{S}_{1}.}$${\lrcorner}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{1}}$

Thus the left-hand square is a pullback and it induces a map
$d_{2}\colon\mathbf{H}_{3}\to\mathbf{H}_{2}$. For the bottom face maps we use
that $H$ is functorial in surjections: given an object in $\mathbf{H}_{n}$,
that is a chain of surjections $V_{1}\twoheadrightarrow
V_{2}\twoheadrightarrow\dots\twoheadrightarrow V_{n}$ with a $H$-structure on
$V_{1}$, we get a $H$-structure on $V_{2}$ by functoriality, and thus an
object $V_{2}\twoheadrightarrow\dots\twoheadrightarrow V_{n}$ of
$\mathbf{H}_{n-1}$. For the top face maps, we use (contravariant)
functoriality in injections, this is restriction to each of the fibres, and
produces thus a family, even if the input is a single chain of surjections.

######  Proposition 4.1.

The groupoids $\mathbf{H}_{j}$ form a simplicial groupoid $\mathbf{H}$.

###### Proof.

Checking the simplicial identities requires precisely the three
functorialities of the notion of hereditary species (covariant in surjections,
contravariant in injections, and Beck–Chevalley condition). For example, the
maps $d_{\top}$ and $d_{\bot}$ are given by the following pullbacks

${\mathbf{H}_{n+1}}$${\mathbf{H}_{n}}$${\mathbf{H}_{n-1}}$${\mathbf{S}_{n+1}}$${\mathbf{S}_{n}}$${\mathbf{S}_{n-1}}$$\scriptstyle{d_{\top}}$${\lrcorner}$${\lrcorner}$$\scriptstyle{f}$$\scriptstyle{d_{\bot}}$$\scriptstyle{g}$$\scriptstyle{d_{\top}}$$\scriptstyle{d_{\bot}}$

and by Beck–Chevalley:

$\displaystyle(d_{\bot})_{!}f^{*}d_{\top}$ $\displaystyle\simeq
g^{*}((d_{\bot})_{!}d_{\top})$ $\displaystyle(d_{\bot})!d_{\top}$
$\displaystyle\simeq g^{*}(d_{\bot}d_{\top})$ $\displaystyle d_{\bot}d_{\top}$
$\displaystyle\simeq g^{*}(d_{\top}d_{\bot})$ $\displaystyle d_{\bot}d_{\top}$
$\displaystyle\simeq d_{\top}d_{\bot}.$

∎

######  Proposition 4.2.

The simplicial groupoid $\mathbf{H}$ is a monoidal decomposition space.

###### Proof.

This follows from Proposition 1.4 because there is a culf map to the
decomposition space $\mathbf{S}$ (in fact even a Segal space) by construction.
The monoidal structure is obtained by concatenation of families. ∎

######  Lemma 4.3.

The groupoid $\mathbf{H}_{1}$ is locally finite.

###### Proof.

The objects of the groupoid $\mathbf{H}_{1}$ are families of non-empty
$H$-structures, and elements of the families are non-empty finite sets. Since
a finite set has only a finite number of automorphism, the automorphisms
groups are finite. ∎

######  Proposition 4.4.

The decomposition space $\mathbf{H}$ is complete, locally finite, locally
discrete, and of locally finite length.

###### Proof.

It follows from Lemmas 1.5 and 4.3, and Proposition 3.6 since there is a
functor $\mathbf{H}\to\mathbf{S}$ which is culf by construction. ∎

######  Proposition 4.5.

The incidence bialgebra $B$ obtained by taking homotopy cardinality of
$\text{{{Grpd}}}_{/\mathbf{H}_{1}}$ coincides with the Schmitt bialgebra
associated to a hereditary species $H$.

###### Proof.

The comultiplication is given by pullback along $d_{1}$ followed by
composition with $(d_{2},d_{0})$. Given a $H$-structure $G$ as a morphism
$1\xrightarrow{\ulcorner G\urcorner}\mathbf{H}_{1}$, the pullback along
$d_{1}$ is the groupoid $(\mathbf{H}_{2})_{G}$ of families of surjections with
the source sets given by $G$ (thus having $H$-structure). Composing with
$(d_{2},d_{0})$ amounts to returning for each surjection $V\twoheadrightarrow
P$ the family of fibres over the element of $P$ (having $H$-structure given by
restriction), and the target set $P$ (having $H$-structure given by quotient).
We can then take cardinality since $\mathbf{H}$ is locally finite. Since
$\mathbf{H}$ is locally discrete, $(\mathbf{H}_{2})_{G}$ is discrete and the
homotopy cardinality counts isomorphisms classes, giving Schmitt’s
comultiplication. ∎

### Comodule structure

Given a hereditary species $H$, define $\mathbf{M}_{0}$ to be the groupoid of
(possibly empty) $H$-structures. More formally, $\mathbf{M}_{0}$ is defined as
the Grothendieck construction of the underlying ordinary species
$H\colon\mathbb{B}\to\text{{{Grpd}}}$. The groupoid $\mathbf{M}_{n}$ is given
by the pullback

${\mathbf{M}_{n}}$${\mathbf{M}_{0}}$${\mathbf{N}\mathbb{S}_{n}}$${\mathbf{N}\mathbb{S}_{0}.}$${\lrcorner}$source

The objects of the groupoid $\mathbf{M}_{1}$ are surjections with a
$H$-structure on the source; the objects of $\mathbf{M}_{n}$ are composable
surjections with a $H$-structure on the source. In the same way as for
$\mathbf{H}$, the face and degeneracy maps are defined by pullback.

######  Lemma 4.6.

The groupoids $\mathbf{M}_{n}$ form a Segal space $\mathbf{M}$.

###### Proof.

It is easy to check that they assemble into a simplicial groupoid, using the
three functorialities of the notion of hereditary species. The simplicial
groupoid is equivalent to the fat nerve of the Grothendieck construction of
the underlying species $H\colon\mathbb{S}\to\text{{{Grpd}}}$. Whereas
$\mathbf{H}$ is defined as a chain of surjections with a $H$-structure on the
source, in the Grothendieck construction, an $n$-simplex is a chain with an
$H$-structure on _each_ set, and with specified isomorphisms with the
$H$-structures pushed forward along the surjections. The presence of these
specified isomorphisms readily shows that the two simplicial groupoids are
level-wise equivalent. Thus $\mathbf{M}$ is Segal since the fat nerve of a
small category is always Segal [14]. ∎

######  Lemma 4.7.

The slice category $\text{{{Grpd}}}_{/\mathbf{M}_{0}}$ is a left comodule over
$\text{{{Grpd}}}_{/\mathbf{H}_{1}}$.

###### Proof.

The culf map $f\colon\mathbf{M}\to\mathbf{H}$ is given by taking fibres, as
the map $d_{\top}$ of $\mathbf{H}$. Thus the span

$\mathbf{M}_{0}\xleftarrow{d_{1}}\mathbf{M}_{1}\xrightarrow{(f_{1},d_{0})}\mathbf{H}_{1}\times\mathbf{M}_{0}$

induces on $\text{{{Grpd}}}_{/\mathbf{M}_{0}}$ the structure of a comodule
over $\text{{{Grpd}}}_{/\mathbf{H}_{1}}$. ∎

##  5 The incidence comodule bialgebra of a hereditary species

We have constructed, via a monoidal decomposition space, the incidence
bialgebra $B$ of a hereditary species $H$. It is the vector space spanned by
all families of non-empty $H$-structures. But every hereditary species is in
particular a restriction species, by precomposition with the inclusion
$\mathbb{I}^{\text{{\rm{op}}}}\to\mathbb{S}_{p}$. Therefore there is another
coalgebra, linearly spanned by the (possibly empty) $H$-structures themselves.
The free algebra on this coalgebra is therefore the bialgebra $A$ linearly
spanned by the families of $H$-structures. So now we have two different
bialgebra structures on closely related vector spaces, and the two share the
same multiplication. The main result of this section relates these two
structures.

######  Proposition 5.1.

The hereditary-species bialgebra $B$ coacts on the restriction-species
bialgebra $A$, so as to make $A$ a left comodule bialgebra over $B$.

The proof is a nice illustration of the objective method: after unpacking the
definitions, the proof consists in computing a few pullbacks. Let us first
recall some definitions and set notation.

### Hereditary species and decomposition spaces

Given a hereditary species $H\colon\mathbb{S}_{p}\to\text{{{Grpd}}}$, we get a
decomposition space $\mathbf{H}$ where an $n$-simplex is a family of $(n{-}1)$
composable surjections between non-empty finite sets, with a $H$-structure on
each source set. The comultiplication
$\text{{{Grpd}}}_{/\mathbf{H}_{1}}\to\text{{{Grpd}}}_{/\mathbf{H}_{1}\times\mathbf{H}_{1}}$
is given by the span

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

where $d_{1}$ returns the family of source sets, $d_{0}$ returns the family of
target sets, and $d_{2}$ returns the family of families of fibres over each
element of the target sets. Let $B$ denote the homotopy cardinality, i.e. the
numerical incidence bialgebra of $\mathbf{H}$.

### Restriction species and decomposition spaces

Since $\mathbb{I}^{\text{{\rm{op}}}}$ is a subcategory of $\mathbb{S}_{p}$,
every hereditary species $H$ induces a restriction species
$\colon\mathbb{I}^{\text{{\rm{op}}}}\to\text{{{Grpd}}}$. Since the hereditary
species $H$ is fixed throughout this section, we denote the underlying
restriction species simply by $R$. Recall from [16] that every restriction
species $R$ induces a decomposition space $\mathbf{R}$ where an $n$-simplex is
an $n$-layered set with an $R$-structure on the underlying set. The
comultiplication
$\Delta\colon\text{{{Grpd}}}_{/\mathbf{R}_{1}}\to\text{{{Grpd}}}_{/\mathbf{R}_{1}\times\mathbf{R}_{1}}$
is given by the span

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

where $d_{1}$ joins the two layers of the $2$-simplex, and $d_{2}$ and $d_{0}$
return the first and second layers respectively. Note that
$\mathbf{R}_{1}=\mathbf{M}_{0}$ and by Lemma 4.7 the slice category
$\text{{{Grpd}}}_{/\mathbf{R}_{1}}$ is a left
$\text{{{Grpd}}}_{/\mathbf{H}_{1}}$-comodule.

### Comodule bialgebra

For background on comodule bialgebras, see [1, §3.2]. Let $B$ be a bialgebra.
Recall that a _(left) $B$-comodule bialgebra_ is a bialgebra object in the
braided monoidal category of left $B$-comodules. For any coalgebra $B$ we have
the category of left $B$-comodules. A left $B$-comodule is a vector space $M$
equipped with a coaction $\gamma\colon M\to B\otimes M$ satisfying the usual
axioms. So far only the coalgebra structure of $B$ is needed. The algebra
structure of $B$ comes in to provide a (braided) monoidal structure on the
category of left $B$-comodules. It is given as follows. If $M$ and $N$ are
left $B$-comodules, then $M\otimes N$ is given a left $B$-comodule structure
by the composite map

$M\otimes N\to B\otimes M\otimes B\otimes
N\stackrel{{\scriptstyle\omega}}{{\to}}B\otimes M\otimes N,$

where the map $\omega$ is given by first swapping the two middle tensor
factors, and then using the multiplication of $B$ in the two now adjacent
$B$-factors. It follows from the bialgebra axioms that this is a valid left
$B$-coalgebra structure. This defines the monoidal structure on the category
of left $B$-comodules. The unit object for this monoidal structure is the
$B$-comodule $\mathbb{Q}$ (with structure map the unit of $B$). It is easy to
check that the underlying braiding of the category of vector spaces lifts to a
braiding on this monoidal structure.

We now have a braided monoidal structure on the category of left
$B$-comodules, and it makes sense to consider bialgebras in here. For
reference, let us recall that a bialgebra in the braided monoidal category of
left $B$-comodules is a $B$-comodule $M$ together with structure maps

$\displaystyle\Delta_{M}\colon M$ $\displaystyle\to M\otimes M$
$\displaystyle\varepsilon_{M}\colon M$ $\displaystyle\to\mathbb{Q}$
$\displaystyle\mu_{M}\colon M\otimes M$ $\displaystyle\to M$
$\displaystyle\eta_{M}\colon\mathbb{Q}$ $\displaystyle\to M$

which are all required to be $B$-comodule maps and to satisfy the usual
bialgebra axioms. We shall be concerned in particular with the requirement
that $\Delta$ and $\varepsilon$ be $B$-comodule maps:

${M}$${M\otimes M}$${B\otimes M\otimes B\otimes M}$${B\otimes M}$${B\otimes
M\otimes
M}$$\scriptstyle{\Delta_{M}}$$\scriptstyle{\gamma}$$\scriptstyle{\gamma\otimes\gamma}$$\scriptstyle{\omega}$$\scriptstyle{B\otimes\Delta_{M}}$${M}$${\mathbb{Q}}$${B\otimes
M}$${B}$$\scriptstyle{\gamma}$$\scriptstyle{\varepsilon_{M}}$$\scriptstyle{\eta_{B}}$$\scriptstyle{B\otimes\varepsilon}$

To simplify the proof of Proposition 5.1, we shall invoke the following
general result.

######  Lemma 5.2.

If $M$ is a comodule coalgebra over $B$, then the free algebra $\mathsf{S}M$
is naturally a comodule bialgebra over $B$.

###### Proof.

If $\gamma\colon M\to B\otimes M$ is the coaction for $M$, then the new
coaction $\overline{\gamma}\colon\mathsf{S}M\to B\otimes\mathsf{S}M$ is given
by extending multiplicatively, and using the algebra structure of $B$:

$\mathsf{S}M\xrightarrow{\mathsf{S}(\gamma)}\mathsf{S}(B\otimes
M)\longrightarrow\mathsf{S}B\otimes\mathsf{S}M\xrightarrow{\mu_{B}\otimes\operatorname{Id}_{\mathsf{S}M}}B\otimes\mathsf{S}M.$

Here the middle map is the oplax-monoidal structure of $\mathsf{S}$. If
$\Delta_{M}\colon M\to M\otimes M$ is the comultiplication of $M$, then the
new comultiplication
$\overline{\Delta}_{M}\colon\mathsf{S}M\to\mathsf{S}M\otimes\mathsf{S}M$ is
given by extending multiplicatively in the usual way:

$\mathsf{S}M\xrightarrow{\mathsf{S}(\Delta_{M})}\mathsf{S}(M\otimes
M)\longrightarrow\mathsf{S}M\otimes\mathsf{S}M.$

It is now straightforward to check that $\overline{\Delta}_{M}$ and the new
free multiplication are $B$-comodule maps for $\overline{\gamma}$. ∎

Proposition 5.1 now follows from the following result, together with Lemma
5.2.

######  Proposition 5.3.

Let $A$ be the incidence coalgebra of the ordinary restriction species
underlying $H$. Then $A$ is naturally a left $B$-comodule coalgebra (where $B$
is the incidence bialgebra of the hereditary species as in Section 4).

###### Proof.

The underlying vector space of $A$ is the homotopy cardinality of the comodule
of Lemma 4.7. It remains to check that the structure maps $\Delta$ and
$\varepsilon$ of the incidence coalgebra of the ordinary restriction species
are $B$-comodule maps. We need to check that the two above squares are
commutative.

The composition of comultiplications is given by composition of spans. We need
to exhibit a commutative diagram as follows, such that the bottom left-hand
square, and the top right-hand squares are pullbacks:

${\mathbf{R}_{1}}$${\mathbf{R}_{2}}$${\mathbf{R}_{1}\times\mathbf{R}_{1}}$${\mathbf{M}_{1}}$${X_{3}}$${\mathbf{M}_{1}\times\mathbf{M}_{1}}$${\mathbf{H}_{1}\times\mathbf{R}_{1}\times\mathbf{H}_{1}\times\mathbf{R}_{1}}$${\mathbf{H}_{1}\times\mathbf{R}_{1}}$${\mathbf{H}_{1}\times\mathbf{R}_{2}}$${\mathbf{H}_{1}\times\mathbf{R}_{1}\times\mathbf{R}_{1}.}$$\scriptstyle{d_{1}}$$\scriptstyle{(d_{2},d_{0})}$$\scriptstyle{d_{1}}$$\scriptstyle{(f,d_{0})}$$\scriptstyle{\bar{d}_{2}}$$\scriptstyle{(g,\bar{d}_{0})}$$\scriptstyle{\bar{d}_{3}}$$\scriptstyle{\bar{d}_{1}}$${\urcorner}$${\llcorner}$$\scriptstyle{d_{1}\otimes
d_{1}}$$\scriptstyle{\operatorname{id}\otimes
d_{1}}$$\scriptstyle{\operatorname{id}\otimes{(d_{2},d_{0})}}$

The objects of $\mathbf{R}_{1}$ are $H$-structures. The groupoid $X_{3}$
consists of pairs of composable maps $V\twoheadrightarrow P\to 2$, such that
the first one is a surjection, and with a $H$-structure on $V$. The map
$\overline{d}_{0}$ sends $(V\twoheadrightarrow P\to\underline{2})$ to
$(P\to\underline{2})$, the map $\overline{d}_{1}$ sends it to
$(V\to\underline{2})$, the map $\overline{d}_{2}$ to $(V\twoheadrightarrow
P)$, the map $\overline{d}_{3}$ to the pair of surjections between the fibres
$(V_{1}\twoheadrightarrow P_{1},V_{2}\twoheadrightarrow P_{2})$, and the map
$g$ to the family $\\{V_{i}\\}_{i\in P}$ of fibres of the surjection
$V\twoheadrightarrow P$ over all the elements of $P$.

Recall that objects of $\mathbf{R}_{2}$ are maps of sets $V\to\underline{2}$,
with a $H$-structure on the source. Objects of $\mathbf{H}_{1}$ are
surjections with a $H$-structure on the source.

It is straightforward to verify the four squares are commutative, using the
functoriality of $H$ and the Beck-Chevalley rule. The structure on
$\mathbf{H}_{1}\times\mathbf{R}_{1}\times\mathbf{R}_{1}$ is obtained as
follow. On $\mathbf{H}_{1}$, the structure is given by restriction on the
fibres, and the different paths give equivalent output since $H$ is
contravariantly functorial in injections. For the two $\mathbf{R}_{1}$, the
structure is given by quotient (functoriality in surjections) then restriction
taking the left then down path, or by restriction then quotient taking the top
then right path; this gives equivalent output by the Beck-Chevalley rule.

The lower left-hand square is a pullback. Indeed after projecting away
$\mathbf{H}_{1}$, it is enough to verify, by Lemma 1.1, that the bottom square
of the following diagram is a pullback:

${\mathbf{M}_{1}}$${X_{3}}$${\mathbf{H}_{1}\times\mathbf{R}_{1}}$${\mathbf{H}_{1}\times\mathbf{R}_{2}}$${\mathbf{R}_{1}}$${\mathbf{R}_{2}.}$$\scriptstyle{(f,d_{0})}$$\scriptstyle{d_{0}}$$\scriptstyle{\overline{d}_{2}}$$\scriptstyle{(g,\overline{d}_{0})}$$\scriptstyle{\overline{d}_{0}}$$\scriptstyle{\operatorname{id}\times
d_{1}}$$\scriptstyle{d_{1}}$

The fibre of $\overline{d}_{0}$ over a element $P\to\underline{2}$ of
$\mathbf{R}_{2}$, or the fibre of $\overline{d}_{0}$ over the element
$d_{1}(P\to\underline{2})=P$ consist both of pairs with a surjection onto $P$:
$V\twoheadrightarrow P$ and the map $P\to\underline{2}$. Thus by Lemma 1.2,
the bottom square is a pullback

The top right-hand square is also a pullback, using Lemma 1.2 one more time:
the fibre of $\overline{d}_{1}$ over a object $V\to\underline{2}$ of
$\mathbf{R}_{2}$ consists of pairs of composable maps, where the first one is
a surjection from $V$, the second one is a map to $\underline{2}$, such that
the composition is $V\to\underline{2}$. The fibre of $\overline{d}_{1}$ over
the object $(d_{2},d_{0})(V\to\underline{2})$ is a pair of surjections with
sources $V_{1}$ and $V_{2}$. This is equivalent to the fibre of
$\overline{d}_{0}$ over $P\to\underline{2}$ since we can take the disjoint
union of $V_{1}\twoheadrightarrow P_{1}$ and $V_{2}\twoheadrightarrow P_{2}$
to get a surjection, and we obtain a map to $\underline{2}$ by sending
elements of $P_{1}$ to $1$, and elements of $P_{2}$ to $2$.

For the counit condition, it is easy to verify that the following diagram is
commutative and the two marked squares are pullbacks

${\mathbf{R}_{1}}$${\mathbf{R}_{0}}$${1}$${\mathbf{M}_{1}}$${\mathbf{R}_{0}}$${1}$${\mathbf{H}_{1}\times\mathbf{R}_{1}}$${\mathbf{H}_{1}\times\mathbf{R}_{0}}$${\mathbf{H}_{1}.}$$\scriptstyle{s_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{(f,d_{0})}$${\urcorner}$${\llcorner}$$\scriptstyle{\operatorname{id}\otimes
s_{0}}$

Indeed, the bottom left pullback is the groupoid of surjections
$V\twoheadrightarrow P$ with a $H$-structure on $V$, such that the induced
$H$-structure on $P$ is an empty $H$-structure. This implies that both $P$ and
$V$ are empty, so it is just any $H$-structure on the empty set, which is
$\mathbf{R}_{0}$.

∎

######  Example 5.4.

The hereditary species of simple graphs, described in Example 2.1, induces a
comodule bialgebra. The first comultiplication is given by the hereditary
structure as in Section 2. The secondary comultiplication is given by plain
restriction species structure. By Proposition 5.1, this is a comodule
bialgebra. It has been studied deeply by Foissy [11]. It is interesting here
to see this example as an instance of a general construction.

##  6 Hereditary species and operadic categories

As we have seen, hereditary species are operad-like without being operads, in
the sense that they admit a kind of two-sided bar construction, which is not
in general a Segal space. A relationship between operadic categories and
decomposition spaces was established recently be Garner, Kock, and Weber [18].
They show that certain unary operadic categories are decomposition spaces. The
following construction shows that certain _non-unary_ operadic categories are
decomposition spaces, namely those that come from hereditary species.

Since we are going to verify the axioms of operadic category in detail, we
list them here, following the formulation of [18].

Let $\mathbb{F}$ denote the category whose objects are the sets
$n=\\{1,\dots,n\\}$ for $n\in\mathbb{N}$ and whose maps are arbitrary
functions. We denote by $1\in 1$ the unique element in the terminal object. In
any category with terminal object $1$, we write $\tau_{X}\colon X\to 1$ for
the unique map from an object to the terminal.

Given a function $\varphi\colon m\to n$ in $\mathbb{F}$ and $i\in n$, there is
a unique monotone injection

$\varepsilon_{\varphi,i}\colon\varphi^{-1}(i)\rightarrowtail m$ (1)

in $\mathbb{F}$ whose image is $\\{\,j\in m\colon\varphi(j)=i\,\\}$; the
object $\varphi^{-1}(i)$ is called the _fibre of $\varphi$ at $i$_. Often the
map $\varphi$ is clear from the context, and we write simply

$\varepsilon_{i}\colon m_{i}\rightarrowtail m.$

If we are given two maps in $\mathbb{F}$,
$\ell\stackrel{{\scriptstyle\psi}}{{\to}}m\stackrel{{\scriptstyle\varphi}}{{\to}}n$,
then we denote by $\psi^{\varphi}_{i}$ the unique map comparing the fibres,
given by the universal property of pullback:

${\ell_{i}}$${m_{i}}$${\\{i\\}}$${\ell}$${m}$${n}$${\lrcorner}$$\scriptstyle{\varepsilon_{i}}$$\scriptstyle{\psi^{\varphi}_{i}}$${\lrcorner}$$\scriptstyle{\varepsilon_{i}}$$\scriptstyle{\psi}$$\scriptstyle{\varphi}$
(2)

and call it the _fibre map of $\psi$ with respect to $\varphi$ at $i$_. We
usually omit the $\psi$-decoration.

### Operadic categories

An _operadic category_ [4] is given by the following data:

  1. (D1)

a category $\mathcal{C}$ endowed with chosen local terminal objects (i.e. a
chosen terminal object in each connected component);

  2. (D2)

a _cardinality functor_
$\lvert{\mathord{\text{--}}}\rvert\colon\mathcal{C}\to\mathbb{F}$;

  3. (D3)

for each object $X\in\mathcal{C}$ and each $i\in\lvert X\rvert$ a _fibre
functor_

$\phi_{X,i}:\mathcal{C}/X\to\mathcal{C}$

whose action on objects and morphisms we denote as follows:

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X}$
$\displaystyle\mapsto\qquad f^{-1}(i)$
---
$\textstyle{{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{fg}$$\textstyle{{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X}$
$\displaystyle\mapsto\qquad g^{f}_{i}\colon(fg)^{-1}(i)\to f^{-1}(i)\hbox
to0.0pt{ ,\hss}$

referring to the object $f^{-1}(i)$ as the _fibre of $f$ at $i$_, and the
morphism
$g^{f}_{i}\colon(fg)^{-1}(i)\to f^{-1}(i)$ as the _fibre map of $g$ with
respect to $f$ at $i$_;

all subject to the following axioms, where in (A5), we write $\varepsilon j$
for the image of $j\in{\lvert f\rvert}^{-1}(i)$ under the map
$\varepsilon_{\lvert f\rvert,i}\colon{\lvert f\rvert}^{-1}(i)\to\lvert
Y\rvert$ of Equation (1):

  1. (A1)

if $X$ is a local terminal then $\lvert X\rvert=1$;

  2. (A2)

for all $X\in\mathcal{C}$ and $i\in\lvert X\rvert$, the object
$(\operatorname{id}_{X})^{-1}(i)$ is chosen terminal;

  3. (A3)

for all $f\in\mathcal{C}/X$ and $i\in\lvert X\rvert$, we have
$\lvert\smash{f^{-1}(i)}\rvert={\lvert f\rvert}^{-1}(i)$, while for all
$g\colon fg\to f$ in $\mathcal{C}/X$ and $i\in\lvert X\rvert$, we have
$\lvert\smash{g^{f}_{i}}\rvert=\smash{\lvert g\rvert^{\lvert f\rvert}_{i}}$;

  4. (A4)

for $Y\in\mathcal{C}$, we have $\tau_{Y}^{-1}(1)=Y$, and for $g\colon Z\to Y$,
we have $g^{\tau_{Y}}_{1}=g$;

  5. (A5)

for $g\colon fg\to f$ in $\mathcal{C}/X$, $i\in\lvert X\rvert$ and $j\in\lvert
f\rvert^{-1}(i)$, we have that $(g^{f}_{i})^{-1}(j)=g^{-1}(\varepsilon j)$,
and given also $h\colon fgh\to fg$ in $\mathcal{C}/X$, we have
$(h^{fg}_{i})^{g^{f}_{i}}_{j}=h^{g}_{\varepsilon j}$.

######  Example 6.1.

The terminal operadic category is the category $\mathbb{F}$ of finite ordered
sets and arbitrary maps. The cardinality functor is the identity, the fibres
are the ‘true’ fibres (as in Equation (1)).

######  Example 6.2.

For the present purposes the key example is the category
$\mathbb{S}_{\text{ord}}$ of finite ordinals and arbitrary surjections. The
cardinality functor is the inclusion functor
$\mathbb{S}_{\text{ord}}\to\mathbb{F}$. The fibre functor is the same as that
from $\mathbb{F}$, but note that these fibres are not true fibres in the
strict sense of the word, because they are not given by pullback. Indeed, the
category of surjections does not have pullbacks. And the ‘inclusion of a
fibre’ is not a morphism in the category. It is important nevertheless that
many constructions with surjections can be interpreted as taking place in
$\mathbb{F}$. The axioms are easily verified.

### The construction

We now work with Set-valued species as in the classical theory. This is needed
to achieve the strictness characteristic for operadic categories. We also need
to assume that the hereditary species have the property that $H[1]=1$. Schmitt
[26] calls such hereditary species _simple_. This is true for example for
simple graphs.

Given a simple hereditary species $H\colon\mathbb{S}_{p}\to\text{{{Set}}}$, we
consider first its Grothendieck construction. It is a left fibration (discrete
opfibration) $\int H\to\mathbb{S}_{p}$. The objects of $\int H$ are pairs
$(n,x)$ where $n\in\mathbb{S}_{p}$ and $x\in H[n]$. We will denote such an
object $X$. The morphisms in $\int H$ are described in the usual way. They
have an underlying span as in $\mathbb{S}_{p}$. We are interested in a
subcategory, namely the subcategory obtained by pullback along the inclusion
$\mathbb{S}_{\text{ord}}\to\mathbb{S}_{p}$ (from the category of finite
ordinals and genuine surjections, not all partial surjections). We denote this
category by $\mathcal{H}$. Its objects are $X=(n,x)$ as before, and an arrow
from $Y=(m,y)$ to $X=(n,x)$ is given by a genuine surjection $s\colon
m\twoheadrightarrow n$ such that $H[s](y)=x$.

We now work towards equipping $\mathcal{H}$ with the structure of operadic
category. The category $\mathcal{H}$ is clearly connected. So to choose local
terminal objects is to choose a global terminal object. By our assumption
$H[1]=1$, there is a unique such, namely $(1,1)$, easily seen to be terminal
in $\mathcal{H}$.

We define the cardinality functor to be the composite functor
$\mathcal{H}\to\mathbb{S}_{\text{ord}}\to\mathbb{F}$.

We define the fibre functor, for each $X=(n,x)\in\mathcal{H}$ and each $i\in
n$, to be the assignment

$\displaystyle\mathcal{H}_{/X}$ $\displaystyle\longrightarrow$
$\displaystyle\mathcal{H}$ $\displaystyle f\colon Y\to X$
$\displaystyle\longmapsto$ $\displaystyle Y_{i}:=(\lvert
f\rvert^{-1}(i),H[\varepsilon_{i}](y)).$

Since this will be needed in all the checks, let us spell this out in more
detail. We assume $X=(n,x)$ and $Y=(m,y)$, and the morphism $f\colon Y\to X$
is given by an underlying surjection $\lvert f\rvert\colon m\twoheadrightarrow
n$ such that $H[\lvert f\rvert](y)=x$. The fibre $Y_{i}$ is defined to be the
pair $(m_{i},y_{i})$, where $m_{i}=\lvert f\rvert^{-1}(i)$ is the fibre of the
map in $\mathbb{F}$:

${m_{i}}$${\\{i\\}}$${m}$${n}$${\lrcorner}$$\scriptstyle{\varepsilon_{i}}$$\scriptstyle{\lvert
f\rvert}$

and $y_{i}$ is defined as $y_{i}=H[\varepsilon_{i}](y)$, the restriction of
the $H$-structure $y$ along the injection $\varepsilon_{i}\colon
m_{i}\rightarrowtail m$.

We must also provide the assignment on arrows. So given

$Z\stackrel{{\scriptstyle g}}{{\to}}Y\stackrel{{\scriptstyle f}}{{\to}}X$

considered as a morphism in $\mathcal{H}_{/X}$ from $fg$ to $f$, we need to
provide a morphism

$Z_{i}\to Y_{i}.$

If we let $Z=(\ell,z)$ then we have $H[\lvert g\rvert](z)=y$. The morphism
must be constituted by a surjection $g_{i}\colon\ell_{i}\twoheadrightarrow
m_{i}$, such that $H[g_{i}](z_{i})=y_{i}$, where $z_{i}=H[\varepsilon_{i}](z)$
is the point in $H[\ell_{i}]$ representing $Z_{i}$ (that is, restriction of
the $H$-structure $z$ along the injection
$\varepsilon_{i}\colon\ell_{i}\rightarrowtail\ell$), and
$y_{i}=H[\varepsilon_{i}](y)$ is the point in $H[m_{i}]$ representing $Y_{i}$.
For the surjection $g_{i}\colon\ell_{i}\twoheadrightarrow m_{i}$ to be valid,
we need to check that $H[g_{i}](z_{i})=y_{i}$. But this is precisely the pull-
push formula for hereditary species on the pullback square from (2):

${\ell_{i}}$${m_{i}}$${\ell}$${m.}$${\lrcorner}$$\scriptstyle{g_{i}}$$\scriptstyle{\varepsilon_{i}}$$\scriptstyle{g}$

(This shows that the assignment extends to arrows. The check that this
assignment on arrows respects composition and identity arrows is routine, and
depends on transitivity of pullbacks in the skeletal category $\mathbb{F}$.)

We have now exhibited all the data required for an operadic category.

######  Proposition 6.3.

The structures on $\mathcal{H}$ given above satisfy the axioms for an operadic
category.

Since this is a new class of operadic categories, not considered previously,
and since the operadic category axioms can be a bit subtle, we include the
details of the checks.

###### Proof.

  1. (A1)

The chosen terminal $(1,1)$ clearly has cardinality $1$.

  2. (A2)

We must check that all fibres of an identity map
$\operatorname{id}_{X}\colon(n,x)\to(n,x)$ are the chosen terminal. By
definition, for $i\in n$, the fibre is $(1,?)$ where $1$ is the
$\mathbb{F}$-fibre of the identity $n\to n$, and $?$ can be no other than
$1\in H[1]$.

  3. (A3)

We need to compute the cardinality of a fibre $Y_{i}=(m_{i},y_{i})$ of a
morphism $Y\to X$ and $i\in\lvert X\rvert=n$. But by construction this is
$m_{i}$, the fibre of the underlying surjection $m\twoheadrightarrow n$. We
must also verify that for a triangle $Z\to Y\to X$, the cardinality of the
fibre map (over $i\in\lvert X\rvert=n$) is the fibre map in $\mathbb{F}$. But
this is clear from the definition of fibre map: it was defined to have as
underlying surjection $\ell_{i}\twoheadrightarrow m_{i}$, the fibre map in
$\mathbb{F}$.

  4. (A4)

We must check that for any object $Y=(m,y)$, the fibre of the unique map
$(m,y)\to(1,1)$ has unique fibre $Y$. For the underlying map in $\mathbb{F}$
this is clear: the unique fibre of $m\twoheadrightarrow 1$ is $m$. And the new
point must be $H[\operatorname{id}](y)=y$, so altogether we find $Y$ again as
required. We must also check that given $g\colon Z\to Y$ (given by
$(\ell,z)\to(m,y)$), then the fibre map $g_{1}^{\tau}\colon Z_{1}\to Y_{1}$
over the unique point in $(1,1)$ coincides with $g$ itself. On the
$\mathbb{F}$-level, this is clear, as we get $\ell\twoheadrightarrow m$ again.
The points $z_{i}\in H[\ell]$ and $y_{i}\in H[m]$ are given, by construction
of the fibre functor, by contravariant functoriality in the injections (fibre
inclusions in $\mathbb{F}$) $\ell_{i}\rightarrowtail\ell$ and
$m_{i}\rightarrowtail m$. But these are the identity maps, so $z_{i}=z$ and
$y_{i}=y$ as required. Note that axiom (A4) just says that the fibre functor
$\phi_{1,1}\colon\mathcal{C}_{/1}\to\mathcal{C}$ must coincide with the
canonical projection functor.

  5. (A5)

Given morphisms $Z\stackrel{{\scriptstyle g}}{{\to}}Y\stackrel{{\scriptstyle
f}}{{\to}}X$ and elements $i\in\lvert X\rvert$ and $j\in\lvert
f\rvert^{-1}(i)=m_{i}$, we need to establish that
$(g^{f}_{i})^{-1}(j)=g^{-1}(\varepsilon j)$. In detail, if the objects and
maps are given by

$(\ell,z)\stackrel{{\scriptstyle
g}}{{\longrightarrow}}(m,y)\stackrel{{\scriptstyle
f}}{{\longrightarrow}}(n,x)$

and we have $i\in\lvert X\rvert=n$ and $j\in m_{i}$, then we first form the
diagram of pullbacks in $\mathbb{F}$:

${(\ell_{i})_{j}}$${\\{j\\}}$${\ell_{i}}$${m_{i}}$${\\{i\\}}$${\ell}$${m}$${n.}$${\lrcorner}$$\scriptstyle{\varepsilon_{j}}$$\scriptstyle{\varepsilon_{ij}}$${\lrcorner}$$\scriptstyle{\varepsilon_{i}}$$\scriptstyle{\lvert
g\rvert_{i}}$${\lrcorner}$$\scriptstyle{\lvert g\rvert}$$\scriptstyle{\lvert
f\rvert}$

Note that the set $(\ell_{i})_{j}$ has two interpretations: it is at the same
time the fibre of $\lvert g\rvert$ over $\varepsilon j$, and the fibre of
$\lvert g\rvert_{i}$ over $j$. This shows that the two objects
$(g^{f}_{i})^{-1}(j)$ and $g^{-1}(\varepsilon j)$ have the same underlying
set. We just need to check their $H$-structures are the same. According to the
definitions, the point in $(g_{i}^{f})^{-1}(j)$ is given by
$H[\varepsilon_{j}](z_{i})$, where $z_{i}=H[\varepsilon_{i}](z)$. On the other
hand, the point in $g^{-1}(\varepsilon j)$ is given by
$H[\varepsilon_{ij}](z)$. But these two are the same, by contravariant
functoriality of $H$ in injections:

$H[\varepsilon_{j}](z_{i})=H[\varepsilon_{j}]\big{(}H[\varepsilon_{i}](z)\big{)}=H[\varepsilon_{i}\circ\varepsilon_{j}](z)=H[\varepsilon_{ij}](z).$

For the second part of (A5), given morphisms $W\stackrel{{\scriptstyle
h}}{{\to}}Z\stackrel{{\scriptstyle g}}{{\to}}Y\stackrel{{\scriptstyle
f}}{{\to}}X$ and elements $i\in\lvert X\rvert$ and $j\in\lvert
f\rvert^{-1}(i)=m_{i}$, we need to establish that
$(h^{fg}_{i})^{g^{f}_{i}}_{j}=h^{g}_{\varepsilon j}$. These morphisms have the
same source and target thanks to the first item in (A5). More precisely the
second part of the axiom can be formulated as saying that this square
commutes:

${((gh)_{i}^{f})^{-1}(j)}$${(g_{i}^{f})^{-1}(j)}$${(gh)^{-1}(\varepsilon
j)}$${g^{-1}(\varepsilon
j).}$$\scriptstyle{(h_{i}^{fg})^{g_{i}^{f}}}$$\scriptstyle{h_{\varepsilon
j}^{g}}$

Checking this is only a question of unpacking. At the level of underlying
sets, we have the pullback diagram

${(k_{i})_{j}}$${(\ell_{i})_{j}}$${\\{j\\}}$${k_{i}}$${\ell_{i}}$${m_{i}}$${\\{i\\}}$${k}$${\ell}$${m}$${n.}$${\lrcorner}$$\scriptstyle{\varepsilon_{j}}$${\lrcorner}$$\scriptstyle{\varepsilon_{j}}$${\lrcorner}$$\scriptstyle{\varepsilon_{i}}$$\scriptstyle{\lvert
h\rvert_{i}}$${\lrcorner}$$\scriptstyle{\varepsilon_{i}}$$\scriptstyle{\lvert
g\rvert_{i}}$${\lrcorner}$$\scriptstyle{\lvert h\rvert}$$\scriptstyle{\lvert
g\rvert}$$\scriptstyle{\lvert f\rvert}$

The point is that the surjection $(k_{i})_{j}\twoheadrightarrow(\ell_{i})_{j}$
has two interpretations, namely as the $j$-fibre map of $\lvert h\rvert_{i}$
or as the $\varepsilon j$-fibre map of $\lvert h\rvert$. But this is precisely
to say that the two morphisms $(h^{fg}_{i})^{g^{f}_{i}}_{j}$ and
$h^{g}_{\varepsilon j}$ have the same underlying surjection. But they also
have the same source (and the same target), by the first part of A5. It
follows that they agree, because the underlying map is a surjection and the
projection $\mathcal{H}\to\mathbb{S}_{\text{ord}}$ is a discrete opfibration
by construction.

∎

### Operadic functors

A functor $F\colon\mathcal{C}\to\mathcal{D}$ between operadic categories is
called an _operadic functor_ if it strictly preserves local terminal objects,
strictly commutes with the cardinality functors to $\mathbb{F}$, and preserves
fibres and fibre maps in the sense that

$F(f^{-1}(i))=(Ff)^{-1}(i)\qquad\text{and}\qquad F(g^{f}_{i})=(Fg)^{Ff}_{i}$

for all $g\colon fg\to f$ in $\mathcal{C}/X$ and $i\in\lvert X\rvert$. We
denote by OpCat the category of operadic categories and operadic functors.

######  Proposition 6.4.

The construction given above is the object part of a functor
$\text{{{HSp}}}_{\operatorname{simple}}\to\text{{{OpCat}}}$.

###### Proof.

A morphism of hereditary species is by definition a natural transformation
$F\colon H^{\prime}\Rightarrow H$, or equivalently a morphism of discrete
opfibrations over $\mathbb{S}_{p}$. Clearly this defines also a morphism
$F\colon\mathcal{H}^{\prime}\to\mathcal{H}$ of discrete opfibrations over
$\mathbb{S}_{\text{ord}}$, and in particular a functor. We just need to check
that this functor is operadic. It is clear that it preserves the chosen
terminal objects. It is also clear that it is compatible with cardinality,
since a morphism of discrete opfibrations over $\mathbb{S}_{\text{ord}}$
obviously induces a functor over $\mathbb{F}$. To check compatibility with
fibres, consider a morphism $f\colon(m,y)\to(n,x)$ in $\mathcal{H}^{\prime}$
and an element $i\in n$. The fibre over $i$ is by definition
$(m_{i},H^{\prime}[\varepsilon_{i}](y))$, and applying $F$ to that gives

$F(f^{-1}(i))=F\big{(}m_{i},H^{\prime}[\varepsilon_{i}](y)\big{)}=\big{(}m_{i},F(H^{\prime}[\varepsilon_{i}](y)\big{)}=\big{(}m_{i},H[\varepsilon_{i}](F(y))\big{)},$

the last equality by naturality of $F$ with respect to the arrow
$\varepsilon_{i}\colon m_{i}\rightarrowtail m$. But the last object is
precisely the fibre of $F(f)$ over $i$, as required.

We also have to show that given $(\ell,z)\stackrel{{\scriptstyle
g}}{{\to}}(m,y)\stackrel{{\scriptstyle f}}{{\to}}(n,x)$ and an element $i\in
n$, we have

$F(g_{i}^{f})=(F(g))_{i}^{Ff}.$

This is well typed in view of the first part of the proof. More precisely the
assertion is that this diagram commutes:

${F(fg)^{-1}(i)}$${F(f^{-1}(i))}$${F(fg)^{-1}(i)}$${F(f)^{-1}(i).}$$\scriptstyle{F(g_{i}^{f})}$$\scriptstyle{F(g)_{i}^{Ff}}$

Checking this is only a question of unpacking. Both morphisms have the same
underlying surjection, namely $\lvert
F(g)\rvert_{i}\colon\ell_{i}\twoheadrightarrow m_{i}$. Since they also have
the same source (and target), the fact that
$\mathcal{H}\to\mathbb{S}_{\text{ord}}$ is a discrete opfibration ensures that
they are also equal as morphisms in $\mathcal{H}$, as required. ∎

######  Example 6.5.

Let $H\colon\mathbb{S}_{p}\to\text{{{Set}}}$ be the hereditary species of
simple graphs (see Example 2.1). The associated operadic category is the
category $\mathcal{H}$ whose objects are simple graphs (with vertex set some
ordinal $n$), and whose morphisms are graph contractions $G\twoheadrightarrow
Q$. This means it is a surjective map on vertices, and also on edges, and
edges are allowed to map to a vertex. The chosen terminal graph is the one-
vertex graph. The cardinality of a graph is the set of vertices, and the fibre
of a contraction $G\twoheadrightarrow Q$ over some vertex in $Q$ is the
preimage of that vertex (the graph contracted onto the vertex).

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Departament de Matemàtiques
Universitat Autònoma de Barcelona

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