# Monoidal envelopes and Grothendieck construction for dendroidal Segal
objects

David Kern

###### Abstract

We propose a construction of the monoidal envelope of $\infty$-operads in the
model of Segal dendroidal spaces, and use it to define cocartesian fibrations
of such. We achieve this by viewing the dendroidal category as a “plus
construction” of the category of pointed finite sets, and work in the more
general language of algebraic patterns for Segal conditions. Finally, we
rephrase Lurie’s definition of cartesian structures as exhibiting the
categorical fibrations coming from envelopes, and deduce a
straightening/unstraightening equivalence for dendroidal spaces.

###### Contents

  1. 1 Introduction
  2. 2 Review of the language of algebraic patterns
    1. 2.1 Algebraic patterns and Segal objects
    2. 2.2 Morphisms of algebraic patterns I: (weak) Segal fibrations
    3. 2.3 Morphisms of algebraic patterns II: Combinatorics of enrichable structures
  3. 3 The plus construction and its Segal objects
    1. 3.1 Categories of patterned trees
    2. 3.2 The pattern structure on the plus construction
  4. 4 Monoidal envelopes and Grothendieck opfibrations
    1. 4.1 Construction of the monoidal envelope functor
    2. 4.2 Opfibrations and representable monoidality
  5. 5 Cartesian monoidal structures and application to the Grothendieck construction
    1. 5.1 Cartesian monoidal structures
    2. 5.2 Straightening cocartesian fibrations

##  1 Introduction

Any monoidal category $\mathfrak{V}^{\otimes}$ defines an operad whose colours
are the objects of $\mathfrak{V}$ and whose multimorphisms
$C_{1},\cdots,C_{n}\to D$ are given by the morphisms
$C_{1}\otimes\cdots\otimes C_{n}\to D$. The operads which arise in this way
are said to be representable; to avoid confusion — since we shall model
operads as certain presheaves — we will call them _representably monoidal_.
Indeed, the tautological multimorphism $C_{1},\cdots,C_{n}\to
C_{1}\otimes\cdots\otimes C_{n}$ (corresponding to
$\operatorname{id}_{C_{1}\otimes\cdots\otimes C_{n}}$) carries the universal
property that every multimorphism with source $(C_{1},\cdots,C_{n})$ must
factor through it, by a unique unary morphism of source
$C_{1}\otimes\cdots\otimes C_{n}$. This universal property can be seen as a
cocartesianity condition.

Recall indeed that, if $\mathcal{p\\!}\,\colon\mathfrak{E}\to\mathfrak{B}$ is
an opfibration (or cocartesian fibration), there is a factorisation system on
$\mathfrak{E}$ whose left class consists of $\mathcal{p\\!}\,$-cocartesian
morphisms and whose right class consists of purely $\mathcal{p\\!}\,$-vertical
morphisms. One can also define a notion of opfibration of multicategories, as
done for example in [Her04] for generalised multicategories, and more
generally of cocartesian (multi)morphisms therein, so that a multicategory is
representably monoidal if and only if its morphism to the terminal operad is
an opfibration. In general, an opfibration of multicagories can be thought of
as a morphism whose fibres are monoidal categories: since the selection of a
colour comes from the operad generated by the nodeless edge $\eta$, which only
has unary morphisms, the vertical arrows are always unary ones, while the
cocartesian arrows are those exhibiting tensor products.

In the homotopical setting, there are various ways of modelling
$\infty$-operads, each coming with its advantages and drawbacks. The model
preferred by [Lur17], which represents an $\infty$-operad by its
$\infty$-category of operators, is biased to make cocartesian fibrations and
monoidal envelopes immediately accessible, but requires entangling
combinatorics to recover an operadic intuition on any construction. On the
other hand, the dendroidal models, based on categories of trees as the shapes
for $\infty$-operads, is closer to the diagramatic operadic intuition, but
generally requires more work to make any construction. For exemple,
cocartesian fibrations and their straightening were studied in [Heu11] in the
model of dendroidal sets, a point-set model which is not manifestly model-
invariant.

A more homotopically robust model is that of Segal dendroidal spaces, or more
generally Segal dendroidal objects in any complete $(\infty,1)$-category. This
model makes it clear that all that is needed to model algebraic objects such
as operads is a general category of “shapes” and their Segal decompositions.
This philosophy was realised in [CH21] which defines a notion of algebraic
pattern, a base shape category over which can be defined Segal presheaves.

Thus one would like to define a notion of cocartesian operations and
fibrations for Segal objects over algebraic patterns, and in addition
construct the free fibration generated by an arbitrary morphism. Here, we will
not achieve this generality, but take a simpler route to defining
opfibrations. In [Her04], opfibrations of multicategories were characterised
as the morphisms whose image under the functor taking monoidal envelopes is an
opfibration (or cocartesian fibration) of underlying categories. We will
follow this idea, and focus on constructing an envelope $\infty$-functor and
exploiting it to define cocartesian fibrations.

### The direction problem and the plus construction

To perform the above requires a notion of direction for the operations of
Segal objects: in categories, there are two directions for cartesianity, from
the source or from the target (giving rise to cartesian and cocartesian
morphisms), while for operads or their “many-objects” generalisation, virtual
double categories, the notion of (co)cartesian morphisms which has so far been
explored looks from the target (as in [CS10, Definition 5.1]), but we expect
that any choice of input to separate may give a notion of cartesianity. Not
all choices of direction, however, are as good as others: for operads, the
choice of the output to keep apart leads to functoriality for composition,
while the other choices do not (due to the absence of a duality operation as
for categories). As a consequence, one may need to restrict the study to
“good” orientations; this seems likely to be impossible to find for modular
operads.

In this work, we have elected to eschew the problem by replacing a perhaps
more canonical definition of directibility by a more practical one. The
presence of a notion of direction for operations means that we can think of
any composite of them as having an order of progression. This is what is
captured by the objects of the simplex category $\textstyle\Delta$, chains of
morphisms going from a beginning to an end. Hence we will base our notion of
direction on this category, declaring an algebraic pattern to be _well-
directed_ if it can be written as the output of a certain construction
involving $\textstyle\Delta$. The appropriate construction to consider turns
out to be a variant of the plus construction suggested by Baez–Dolan and
studied by, among others, [Bar18] and [Ber22].

The plus construction of a pattern $\mathfrak{O}$ is characterised by the fact
that its Segal objects are the weak Segal $\mathfrak{O}$-fibrations, or
“$\mathfrak{O}$-operads”. In this language, envelopes for weak Segal
fibrations of patterns have been recently constructed in [BHS22], building on
the construction of [Lur17] and the properties exhibited in [HK21]. While in a
context equivalent to ours, this construction has orthogonal goals to the one
performed here, as we wish to always remain in the explicit setting of Segal
objects.

### Idea of the construction

In order to understand the idea of our construction of the monoidal envelope
functor, we describe it here in the case of the pattern
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$, whose
Segal objects are the commutative monoids and whose plus construction
parameterises operads.

Let $\mathcal{O}$ be an operad. Its monoidal envelope is constructed as a
monoidal category $\mathcal{E\\!n\\!v\\!}\,(\mathcal{O})$ which has as set of
objects the free monoid generated by the colours of $\mathcal{O}$, whose
elements are denoted as $C_{1}\otimes\cdots\otimes C_{n}$ or simply
$C_{1}\cdots C_{n}$. If $C_{1}\otimes\cdots\otimes C_{n}$ is such a string of
colours of $\mathcal{O}$ and $D$ is one colour, a morphism
$C_{1}\otimes\cdots\otimes C_{n}\to D$ is given by a multimorphism
$C_{1},\dots,C_{n}\to D$ in $\mathcal{O}$. If $C_{1}\otimes\cdots\otimes
C_{n}$ and $D_{1}\otimes\cdots\otimes D_{m}$ are two such strings of colours
of $\mathcal{O}$, to defines a morphism $C_{1}\otimes\cdots\otimes C_{n}\to
D_{1}\otimes\cdots\otimes D_{m}$ one needs to further select of partition of
the inputs $(C_{1},\dots,C_{n})$ into $m$ (possibly empty) parts.

The above is so far just a description of the underlying category
$\mathfrak{Env}(\mathcal{O})$ of the envelope of $\mathcal{O}$; to define it
as a monoidal category, or representably monoidal operad, one must also define
multimorphisms of higher arity in the operadic structure. Let
$(C^{i}_{1}\otimes\cdots\otimes C^{i}_{p_{i}})_{i\in[\\![1,n]\\!]}$ be $n$
colours of $\mathcal{E\\!n\\!v\\!}\,(\mathcal{O})$ and let
$D_{1}\otimes\cdots\otimes D_{m}$ be a further colour. By the representability
condition, a multimorphism $(C^{1}_{1}\cdots
C^{1}_{p_{1}},\dots,C^{n}_{1}\cdots C^{n}_{p_{n}})\to
D_{1}\otimes\cdots\otimes D_{m}$ is given by a morphism
$C^{1}_{1}\otimes\cdots\otimes C^{n}_{p_{n}}\to D_{1}\otimes\cdots\otimes
D_{m}$, that is a partition of the entries and a collection of multimorphisms
to each $D_{i}$.

In the dendroidal model, one simply defines the object
$\mathcal{E\\!n\\!v\\!}\,(\mathcal{O})(\star_{n})$ of all $n$-ary morphisms
without specifying their sources and targets. To describe this, it becomes
useful to reverse the thinking: taking a family of multimorphisms of
$\mathcal{O}$, we can ask how to interpret it as a multimorphism in
$\mathcal{E\\!n\\!v\\!}\,(\mathcal{O})$. If $C_{1},\cdots,C_{r}$ is the union
of the domains of the multimorphisms in the family considered, the
decomposition in family provides a partition of $p$ indexed by the targets;
however, from the point of view of $\mathcal{E\\!n\\!v\\!}\,(\mathcal{O})$,
this partition is completely artificial as it is only used to construct a
morphism whose target may consist of several colours. Thus it must be
forgotten. Meanwhile, if the family is to be interpreted as a multimorphism of
specified arity $n$ in $\mathcal{E\\!n\\!v\\!}\,(\mathcal{O})$, the set of
colours $(C_{i})_{i\in[\\![1,r]\\!]}$ must be endowed with a partition into
$n$ parts.

In a formula, we have that

$\mathcal{E\\!n\\!v\\!}\,(\mathcal{O})(\star_{n})=\coprod_{\begin{subarray}{c}r\in\mathbb{N}\\\
\lambda\text{ partition of $r$ in
$n$}\end{subarray}}\prod_{i=1}^{m}\mathcal{O}(\star_{\lambda(i)})\text{.}$ (1)

Recall that partitions are the same thing as active morphisms in
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$ (which is
where the power of the plus construction, relating ${\Gamma}^{\mathrm{op}}$ to
the pattern for operads, comes into play). We can then interpret the coproduct
on partitions as a colimit over morphisms in
$({{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}})^{\mathrm{act}}$.

Viewed in this way, this formula is very reminiscent to the one computing
(pointwise) oplax extensions111Oft known as left Kan extensions, with one
difference: the colimit is taken only of the trees whose height is that of a
corolla, _i.e._ compatibly with the projection to $\textstyle\Delta$. To that
end, the notion of extension will need to be refined to a “fibrewise” one.

### Outline of the paper

In section 3, we define the plus construction for appropriate algebraic
patterns, and establish its main properties, in a manner very similar to the
study of the plus construction of
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}}^{\natural}$
carried out in [CHH18] or [CH20]. Then, in section 4, we will use it to
construct the “representably monoidal” envelope of a Segal object. While the
construction makes sense for general well-directed patterns, we are only able
to exhibit its good monoidal properties by restricting patterns such as the
one
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\flat}$ for operads. Finally, in section 5 we
further use this envelope functor to define a straightening/unstraightening
Grothendieck construction for $\infty$-operads; the results of this section
are not new and are simply variants of results appearing in [Lur17], the
combinatorics of whose proofs are in fact directly reused, but this language
provides a new, more operadic, point of view on them.

### Acknowledgements

The core results of this note were worked out as part of my PhD with Étienne
Mann, and I thank him for many useful discussions.

I acknowledge funding from the grant of the Agence Nationale de la Recherche
“Categorification in Algebraic Geometry” ANR-17-CE40-0014 and the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (Grant “Derived Symplectic Geometry and Applications”
Agreement No. 768679)

##  2 Review of the language of algebraic patterns

###  2.1 Algebraic patterns and Segal objects

######  Definition 2.1.1 (Algebraic pattern).

An algebraic pattern is an $(\infty,1)$-category endowed with a unique
factorisation system and a selected class of objects called elementary. The
morphisms in the left class of the factorisation system are called inert, and
those in the right class active.

######  Notation 2.1.2.

Inert maps in an algebraic pattern are usually denoted as $\rightarrowtail$,
while active maps are denoted as $\rightsquigarrow$.

If $\mathfrak{O}$ is the underlying $(\infty,1)$-category of the algebraic
pattern considered, one writes $\mathfrak{O}^{\mathrm{inrt}}$ and
$\mathfrak{O}^{\mathrm{act}}$ for the wide and locally full
sub-$(\infty,1)$-categories whose morphisms are respectively the inert and the
active morphisms.

We also denote $\mathfrak{O}^{\mathrm{el}}$ the full sub-$(\infty,1)$-category
of $\mathfrak{O}^{\mathrm{inrt}}$ on the selected elementary objects. For any
object $O\in\mathfrak{O}$, we further write
$\mathfrak{O}^{\mathrm{el}}_{O/}\coloneqq\mathfrak{O}^{\mathrm{el}}\times_{\mathfrak{O}^{\mathrm{inrt}}}\mathfrak{O}^{\mathrm{inrt}}_{O/}$,
the $(\infty,1)$-category of inert morphisms from $O$ to an elementary object.

######  Example 2.1.3.

Segal’s category $\textstyle\Gamma$, (a skeleton of) the opposite of the
category $\mathfrak{FinSet}_{\ast}$ of pointed finite sets, admits an active-
inert factorisation system where a morphism of pointed finite sets
$f\colon(S,s_{0})\to(T,t_{0})$ is inert if for every $t\in
T\setminus\\{t_{0}\\}$, the preimage $f^{-1}(t)$ consists of exactly one
element, and active if the only element of $S$ mapped to $t_{0}$ is $s_{0}$ —
so that, in particular, the subcategory of active morphisms
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\mathrm{act}}$ identifies with the category
of finite sets (by mapping $(S,s_{0})$ to $S\setminus\\{s_{0}\\}$).

The induced inert-active factorisation system on
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$ gives rise
to two algebraic pattern structures, from two choices of elementary objects.
The algebraic pattern
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}}^{\flat}$
has as elementaries the two-element sets (isomorphic to $\langle 1\rangle$),
while the pattern
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}}^{\natural}$
has as elementaries the singletons (isomorphic to $\langle 0\rangle$) and the
two-element sets.

######  Example 2.1.4.

The (non-augmented) simplicial indexing category $\textstyle\Delta$,
identified with the category of ordered non-empty finite sets and order-
preserving maps between them, admits a factorisation system in which a map is
active if it preserves the top and bottom elements, and inert if it
corresponds to the inclusion of a linear subset.

The induced inert-active factorisation system on
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}}^{\mathrm{op}}$ gives rise
to two algebraic pattern structures: the algebraic pattern
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}}^{\mathrm{op}}}^{\flat}$
has as elementaries the two-element sets (isomorphic to $[1]$), while the
pattern
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}}^{\natural}$
has as elementaries the singletons (isomorphic to $[0]$) and the two-element
sets.

######  Example 2.1.5.

Of particular interest to us, the dendroidal category $\textstyle\Omega$ of
Moerdijk–Weiss can be described as a category of (non-planar) rooted trees,
with morphisms the morphisms of free (coloured, symmetric) operads generated
by these trees. We shall give an alternate construction of (a sufficient
subcategory of) it in section 3.

Certain particularly interesting trees can be distinguished:

the free-living edge

is the tree, denoted $\eta$, consisting of one edge but no vertex (so the
operad it freely generates has one colour, and only its identity unary
morphism);

the corollas

are the trees, denoted $\star_{n}$ with $n\in\mathbb{N}$, with a single vertex
and $n+1$ edges (one of which is the root) attached to it. Note that the
corolla $\star_{n}$ with $n$ leaves has as automorphism group the symmetric
group $\mathfrak{S}_{n}$.

The category $\textstyle\Omega$ admits a factorisation system in which a
morphism is active if it is boundary-preserving, and inert if it corresponds
to a subtree inclusion (which, in particular, is valence-preserving on the
vertices).

The induced inert-active factorisation system on
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}$ gives again rise to two algebraic pattern
structures: the algebraic pattern ${{\Omega}^{\mathrm{op}}}^{\flat}$ has as
elementaries the corollas $\star_{n}$, while the pattern
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$ has as elementaries the corollas
and the free-living edge $\eta$.

######  Definition 2.1.6 (Morphisms of algebraic patterns).

Let $\mathfrak{O}$ and $\mathfrak{P}$ be algebraic patterns. A morphism of
algebraic patterns from $\mathfrak{O}$ to $\mathfrak{P}$ is an
$(\infty,1)$-functor $\mathfrak{O}\to\mathfrak{P}$ which preserves active and
inert morphisms and elementary objects.

######  Example 2.1.7.

There is a functor
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}\to{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}$ which, taking the standard skeleton of $\textstyle\Delta$,
sends $[n]$ to the linear tree with $n$ nodes and $n+1$ edges. One may remark
that it is fully faitful, and can also be identified with the canonical
projection functor
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}_{/\eta}\to{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}$. By translating the definition of inert and active morphisms
for the given factorisation system on
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}}^{\mathrm{op}}$, one
immediately sees that this functor induces morphisms of algebraic patterns
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}\to{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$ and
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\flat}\to{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\flat}$.

######  Definition 2.1.8 (Segal objects).

Let $\mathfrak{O}$ be an algebraic pattern. An $(\infty,1)$-category
$\mathfrak{C}$ is said to be $\mathfrak{O}$-complete if it admits limits of
diagrams with shape $\mathfrak{O}^{\mathrm{el}}_{O/}$ for any
$O\in\mathfrak{O}$.

Let $\mathfrak{C}$ be an $\mathfrak{O}$-complete $(\infty,1)$-category. A
Segal $\mathfrak{O}$-object in $\mathfrak{C}$ is an $(\infty,1)$-functor
$\mathcal{X\\!}\,\colon\mathfrak{O}\to\mathfrak{C}$ such that
${}\mathcal{X\\!}\,\rvert_{\mathfrak{O}^{\mathrm{inrt}}}$ is a lax extension
of ${}\mathcal{X\\!}\,\rvert_{\mathfrak{O}^{\mathrm{el}}}$ (along the
inclusion). Explicitly, this means that for any $O\in\mathfrak{O}$ the
canonical map

$\mathcal{X\\!}\,(O)\to\lim_{E\in\mathfrak{O}^{\mathrm{el}}_{O/}}\mathcal{X\\!}\,(E)$
(2)

is invertible.

The $(\infty,1)$-category $\mathfrak{Seg}_{\mathfrak{O}}(\mathfrak{C})$ of
Segal $\mathfrak{O}$-objects in $\mathfrak{C}$ is the full
sub-$(\infty,1)$-category of $\mathfrak{C}^{\mathfrak{O}}$ spanned by the
Segal objects.

######  Example 2.1.9.

  * •

A Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\flat}$-object is a commutative algebra
object (or $\mathcal{E}_{\infty}$-algebra object).

  * •

A Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\flat}$-object is an associative algebra
object (or $\mathcal{A}_{\infty}$-algebra object, or
$\mathcal{E}_{1}$-algebra), while a Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$-object is an internal category.

  * •

A Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\flat}$-object is an internal monochromatic
operad, while a Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$-object is an internal (coloured)
operad.

######  Remark 2.1.10.

Let $\mathfrak{O}$ be an algebraic pattern such that the inclusion
$\mathfrak{O}^{\mathrm{el}}\hookrightarrow\mathfrak{O}^{\mathrm{inrt}}$ is
codense. Then for any $O\in\mathfrak{O}$, the corepresentable
$\mathcal{y\\!}\,_{{\mathfrak{O}}^{\mathrm{op}}}(O)\colon\mathfrak{O}\to\mathfrak{\infty\textnormal{-}Grpd}$
is a Segal $\mathfrak{O}$-$\infty$-groupoid — this is an immediate consequence
of the limit-preservation property of hom $\infty$-functors. It should be
viewed as the Segal object generated by $O$.

######  Example 2.1.11.

Over
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}}^{\mathrm{op}}$, the Segal
object $\mathcal{y\\!}\,{[n]}$ corresponds to the linear category
$\mathbb{n+1}$ with $n$ successive arrows.

######  Example 2.1.12.

Over
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}$, the Segal object
$\mathcal{y\\!}\,{\star_{n}}$ generated by the corolla with $n+1$ flags is
also denoted $\star_{n}$ and called the corolla. It corresponds to the operad
with $n+1$ colours $O_{1},\dots,O_{n+1}$ and, for each permutation
$\sigma\in\mathfrak{S}_{n}$, a single operation of signature
$(O_{\sigma(1)},\dots,O_{\sigma(n)};O_{n+1})$.

###  2.2 Morphisms of algebraic patterns I: (weak) Segal fibrations

######  Definition 2.2.1 (Segal morphism of algebraic patterns).

A morphism of algebraic patterns
$\mathcal{F\\!}\,\colon\mathfrak{O}\to\mathfrak{P}$ is said to be a Segal
morphism if “it preserves Segal conditions”, that is if for any
$\mathfrak{P}$-complete $(\infty,1)$-category $\mathfrak{C}$, the induced
$(\infty,1)$-functor
${}\mathcal{F\\!}\,^{\ast}\rvert_{\mathfrak{Seg}_{\mathfrak{P}}}\colon\mathfrak{Seg}_{\mathfrak{P}}(\mathfrak{C})\subset\mathfrak{C}^{\mathfrak{P}}\to\mathfrak{C}^{\mathfrak{O}}$
factors through
$\mathfrak{Seg}_{\mathfrak{O}}(\mathfrak{C})\hookrightarrow\mathfrak{C}^{\mathfrak{O}}$.

######  Remark 2.2.2.

By [CH21, Lemma 4.5], it is enough to check Segality of a morphism with
$\mathfrak{C}=\mathfrak{\infty\textnormal{-}Grpd}$, that is to check
preservation of Segal $\infty$-groupoids. Then the condition for
$\mathcal{F\\!}\,$ to be a Segal morphism can be written in a formula as: for
every $O\in\mathfrak{O}$, for every Segal $\mathfrak{P}$-$\infty$-groupoid
$\mathcal{X\\!}\,$, the morphism of $\infty$-groupoids

$\lim_{\mathfrak{P}^{\mathrm{el}}_{\mathcal{F\\!}\,(O)/}}\mathcal{X\\!}\,\to\lim_{\mathfrak{O}^{\mathrm{el}}_{O/}}\mathcal{X\\!}\,\circ\mathcal{F\\!}\,^{\mathrm{el}}$
(3)

induced by the $(\infty,1)$-functor
$\mathfrak{O}^{\mathrm{el}}_{O/}\to\mathfrak{P}^{\mathrm{el}}_{\mathcal{F\\!}\,(O)/}$
is an equivalence.

######  Construction 2.2.3.

Suppose
$\mathcal{V\\!}\,\colon\mathfrak{O}\to\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}$
is an $\mathfrak{O}$-monoidal $(\infty,1)$-category. Passing to the
Grothendieck construction of the functor produces a cocartesian fibration
$\int^{\mathrm{co}}\mathcal{V\\!}\,\to\mathfrak{O}$. We shall refer to a
cocartesian fibration over $\mathfrak{O}$ whose associated
$(\infty,1)$-functor
$\mathfrak{O}\to\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}$ satisfies
the Segal conditions as a Segal fibration over $\mathfrak{O}$.

By [Lur17, Proposition 2.1.2.5], if
$\int^{\mathrm{co}}\mathcal{V\\!}\,\to\mathfrak{O}$ is a Segal
$\mathfrak{O}$-fibration, the inert-active factorisation system on
$\mathfrak{O}$ lifts to one on $\int^{\mathrm{co}}\mathcal{V\\!}\,$, endowing
it with a structure of algebraic pattern. If $\mathfrak{P}\to\mathfrak{O}$ is
another algebraic pattern over $\mathfrak{O}$, we shall define a
$\mathfrak{P}$-algebra in $\int^{\mathrm{co}}\mathcal{V\\!}\,$ to be a
morphism of algbraic pattern over $\mathfrak{O}$ from $\mathfrak{P}$ to
$\int^{\mathrm{co}}\mathcal{V\\!}\,$.

######  Definition 2.2.4 (Weak Segal fibration).

Let $\mathfrak{O}$ be an algebraic pattern. A weak Segal
$\mathfrak{O}$-fibration (also called $\mathfrak{O}$-operad) is an
$(\infty,1)$-functor $\mathcal{p\\!}\,\colon\mathfrak{X}\to\mathfrak{O}$ such
that:

  1. 1.

for every object $X\in\mathfrak{X}$, every inert arrow
$i\colon\mathcal{p\\!}\,X\to O$ in $\mathfrak{O}$ admits a
$\mathcal{p\\!}\,$-cocartesian lift $i_{!}\colon X\to i_{!}X$;

  2. 2.

for every object $O\in\mathfrak{O}$, the $(\infty,1)$-functor
$\mathfrak{X}_{O}\to\lim_{E\in\mathfrak{O}^{\mathrm{el}}_{O/}}\mathfrak{X}_{E}$
induced by the cocartesian morphisms over inert arrows is invertible;

  3. 3.

for every $X\in\mathfrak{X}$ and every choice of
$\mathcal{p\\!}\,$-cocartesian lift of the tautological diagram
$\mathcal{i\\!}\,\colon\mathfrak{O}^{\mathrm{el}}_{\mathcal{p\\!}\,X/}\to\mathfrak{O}$
(of inert morphisms from $\mathcal{p\\!}\,X$) to an
$\mathcal{i\\!}\,_{!}\colon(\mathfrak{O}^{\mathrm{el}}_{\mathcal{p\\!}\,X/})^{\lhd}\to\mathfrak{X}$
taking the cone point to $X$, for every $Y\in\mathfrak{X}$, the commutative
square

${\mathfrak{X}(Y,X)}$${\lim_{E\in\mathfrak{O}^{\mathrm{el}}_{\mathcal{p\\!}\,X/}}\mathfrak{X}(Y,\mathcal{i\\!}\,_{!}(E))}$${\mathfrak{O}(\mathcal{p\\!}\,Y,\mathcal{p\\!}\,X)}$${\lim_{E\in\mathfrak{O}^{\mathrm{el}}_{\mathcal{p\\!}\,X/}}\mathfrak{O}(\mathcal{p\\!}\,Y,\mathcal{i\\!}\,(E)=E)}$
(4)

is cartesian.

######  Example 2.2.5.

  * •

A weak Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\flat}$-fibration is the
$(\infty,1)$-category of operators of an $\infty$-operad in the sense of
[Lur17, Definition 2.1.1.10], while a weak Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$-fibration is a generalised
$\infty$-operad in the sense of [Lur17, Definition 2.3.2.1].

  * •

A weak Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\flat}$-fibration is the
$(\infty,1)$-category of operators of a non-symmetric $\infty$-operad in the
sense of [GH15, Definition 2.2.6, Definition 3.1.3] while a weak Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$-fibration is virtual double
$\infty$-category, or generalised $\infty$-operad in [GH15, Definition 2.4.1,
Definition 3.1.13].

######  Definition 2.2.6 (Morphisms of weak Segal fibrations).

By [CH21], the source $\mathfrak{X}$ of a weak Segal $\mathfrak{O}$-fibration
$\mathcal{p\\!}\,\colon\mathfrak{X}\to\mathfrak{O}$ inherits an algebraic
pattern structure in which active morphisms are those lying over an active
morphism in $\mathfrak{O}$, inert morphisms are the
$\mathcal{p\\!}\,$-cocartesian morphisms lying over inert arrows of
$\mathcal{O\\!}\,$, and elementaries are the objects lying over elementary
objects.

The $(\infty,2)$-category of weak Segal $\mathfrak{O}$-fibrations
$\mathfrak{WkSegFib}_{/\mathfrak{O}}$ is the locally full
sub-$(\infty,2)$-category of
$\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}_{/\mathfrak{O}}$ spanned
by the weak Segal $\mathfrak{O}$-fibrations and Segal morphisms thereof.

It can be checked directly that any Segal fibration as in 2.2.3 is in
particular a weak Segal fibration. In fact Segal $\mathfrak{O}$-fibrations are
exactly those $(\infty,1)$-functors to $\mathfrak{O}$ which are both weak
Segal fibrations and cocartesian fibrations. This provides a (non-full)
inclusion $(\infty,2)$-functor from the $(\infty,2)$-category of Segal
fibrations into that of weak Segal fibrations.

###  2.3 Morphisms of algebraic patterns II: Combinatorics of enrichable
structures

To finish this preliminary section, we state (a variant of) some
definitions222which were presented in the seminar talk available at
https://www.msri.org/seminars/25057 of the forthcoming paper [CH22].

######  Definition 2.3.1 (Combinatorial structure).

A combinatorial pattern is an algebraic pattern $\mathfrak{O}$ equipped with a
morphism
$\lvert-\rvert\colon\mathfrak{O}\to{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$.

When $(\mathfrak{O},\lvert-\rvert)$ is a combinatorial pattern, we let
$\mathfrak{O}^{\flat}\to{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}}^{\flat}$
denote the base-change

$\mathfrak{O}^{\flat}\coloneqq\mathfrak{O}\times_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}}{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\flat}\text{.}$ (5)

Thanks to [CH21, Corollary 5.5], we see that the fibre product appearing in
eq. 5 has a very simple description: it consists of the category
$\mathfrak{O}$ equipped with its same factorisation system, and the choice of
only those elementary objects living over $\langle
1\rangle\in{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$ (_i.e._
excluding those over $\langle 0\rangle$).

######  Definition 2.3.2 (Enrichable pattern).

A combinatorial structure $\lvert-\rvert$ on an algebraic pattern
$\mathfrak{O}$ is enrichable if, for any $O\in\mathfrak{O}$, the morphism
$\mathfrak{O}^{\flat,\mathrm{el}}_{O/}\to{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\flat,\mathrm{el}}_{\lvert O\rvert/}$ is an
equivalence.

######  Remark 2.3.3.

Viewing
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$ as (the
standard skeleton of) the category of pointed finite sets, one verifies that
any $\langle
n\rangle\in{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$ admits
exactly $n$ inert morphisms to the unique elementary $\langle 1\rangle$, the
pointed morphisms $\rho_{i}\colon\langle n\rangle\to\langle 1\rangle$ mapping
$i$ to $1$ and every other element of $\langle n\rangle$ to the base-point.
The condition of being an enrichable pattern then means that, for any
$O\in\mathfrak{O}$, the $(\infty,1)$-category
$\mathfrak{O}^{\flat,\mathrm{el}}_{O/}$ must be equivalent to the discrete set
of the (essentially unique) lifts $\rho_{i,!}$ of the $\rho_{i}$. In
particular, the $\mathfrak{O}^{\flat}$-Segal condition for a precosheaf
$\mathcal{X\\!}\,$ on $\mathfrak{O}$ is constrained to being the finite
product condition

$\mathcal{X\\!}\,(O)\simeq\prod_{i=1}^{\lvert
O\rvert}\mathcal{X\\!}\,(\rho_{i,!}O)\text{.}$ (6)

######  Example 2.3.4.

The functor
$\ast\to{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$ picking the
object $\langle 1\rangle$ defines a structure of enrichable pattern on the
terminal algebraic pattern.

######  Example 2.3.5.

There is a functor
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}\to{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$ mapping
$[n]$ to $\langle n\rangle$ and sending an arrow of
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}}^{\mathrm{op}}$
corresponding to $\phi\colon[n]\to[m]$ in $\textstyle\Delta$ to
$\lvert\phi\rvert\colon\langle m\rangle\to\langle n\rangle$ given by

$\lvert\phi\rvert(i)=\begin{cases}j&\text{if $\phi(j-1)<i\leq\phi(j)$}\\\
\ast&\text{otherwise.}\end{cases}$ (7)

It can be checked directly that this is a structure of enrichable pattern on
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$.

######  Example 2.3.6.

A functor
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}\to{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}\simeq\mathfrak{FinSet}_{\ast}$ is defined in
[CH20, Definition 4.1.16] in the following way. A tree $T$ with set of
vertices $V(T)$ is mapped to the freely pointed set $V(T)_{+}$. A morphism
$T^{\prime}\leftarrow T$ in
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}$ is mapped to the pointed morphism
$V(T^{\prime})_{+}\to V(T)_{+}$ which sends a vertex $v\in V(T^{\prime})$ to
the unique vertex of $T$ whose image subtree contains $v$, or to the basepoint
if there is no such vertex. By [loc. cit., Lemma 4.1.18], this functor
preserves the inert-active factorisation system, and thus defines a morphism
of algebraic patterns. One checks by inspection that it is an enrichable
structure.

In 3.2.9 we will provide another construction of this enrichable structure
from the point of view of $\textstyle\Omega$ as obtained from the plus
construction.

We recall from [Lur17] the definition of semi-inert arrows in
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$. A map of
pointed finite sets $f\colon(S,s_{0})\to(T,t_{0})$ is semi-inert if for any
$t\in T\setminus\\{t_{0}\\}$, there is at most one element in $f^{-1}(t)$.

######  Definition 2.3.7.

Let $(\mathfrak{O},\lvert\cdot\rvert)$ be a combinatorial algebraic pattern.
An arrow $f$ of $\mathfrak{O}$ is semi-inert if $\lvert f\rvert$ is a semi-
inert arrow of
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$.

######  Example 2.3.8.

In
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$, the semi-inert morphisms are
those corresponding to the cellular morphisms of $\textstyle\Delta$ as defined
in [Hau17] and [Hau16], that is maps of totally ordered sets $f\colon S\to
S^{\prime}$ such that for all $s\in S$,
$f(\operatorname{succ}s)\leq\operatorname{succ}(f(s))$.

For the product pattern
$({{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\natural})^{n}$ equipped with
$({{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\natural})^{n}\xrightarrow{\lvert-\rvert}({{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\natural})^{n}\xrightarrow{\wedge}{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$, we also recover the cellular
morphisms of [Hau17], those maps all of whose components are cellular in
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}}^{\mathrm{op}}$.

######  Remark 2.3.9.

As observed in [Hau17], an arrow $f\colon O\to O^{\prime}$ of $\mathfrak{O}$
is semi-inert if and only if, for any elementary $E$ and any inert morphism
$k\colon O^{\prime}\rightarrowtail E$, the composite $O\xrightarrow{k\circ
f}E$ is semi-inert.

######  Example 2.3.10.

In ${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$, any object
$\langle n\rangle$ admits precisely $n$ active morphisms from the unique
elementary object $\langle 1\rangle$, the functions $\lambda_{i}\colon\langle
1\rangle\rightsquigarrow\langle n\rangle$ sending $1\in\langle 1\rangle$ to
$i\in\langle n\rangle$ for any $1\leq i\leq n$. They are semi-inert. Likewise,
the unique map $\langle 0\rangle\to\langle n\rangle$ is also semi-inert.

It follows that, in any combinatorial pattern, any active arrow from an
elementary object is semi-inert.

######  Definition 2.3.11 (Momentous pattern).

A combinatorial algebraic pattern $(\mathfrak{O},\lvert-\rvert)$ is momentous
if any active morphism from a $\flat$-elementary admits an essentially unique
inert retraction.

Momentous patterns which are enrichable are closely related to (an
$\infty$-categorical version of) the hypermoment categories of [Ber22].

######  Example 2.3.12.

The pattern
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}}^{\natural}$
itself is momentous: for any object $\langle n\rangle$, the function
$\lambda_{i}\colon\langle 1\rangle\to\langle n\rangle$ of 2.3.10 admits as
unique inert retraction $\rho_{i}$. In fact $\lambda_{i}$ is also the unique
active section of $\rho_{i}$, so there is a one-to-one correspondence between
inert morphisms to $\flat$-elementaries and (automaticaly semi-inert) active
morphisms from $\flat$-elementaries.

As a consequence, for any momentous enrichable pattern $\mathfrak{O}$ and any
object $O\in\mathfrak{O}$, the (semi-inert) active morphisms from
$\flat$-elementary objects to $O$ can be identified with a subset of
$\mathfrak{O}^{\mathrm{el}}_{O/}\simeq{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\natural,\mathrm{el}}_{\lvert O\rvert/}$.

##  3 The plus construction and its Segal objects

###  3.1 Categories of patterned trees

The following construction is a variant of one due to [Bar18] in the setting
of operator categories, inspired by the plus construction or “slice operads”
of [BD98], and which was also used in [CHH18] and studied in [Ber22] in the
setting of hypermoment categories — as well as [BM21], in a different form,
for operadic categories.

######  Construction 3.1.1.

Let $\mathfrak{O}$ be a combinatorial algebraic pattern. Consider the
restriction
${}\underline{\mathcal{y\\!}\,}_{\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}}(\mathfrak{O}^{\mathrm{act}})\rvert_{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}\colon[n]\mapsto\mathfrak{\mathfrak{O}^{\mathrm{act}}}^{\mathfrak{[\mathnormal{n}]}}$
to ${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}\subset\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}$ of
the $(\infty,2)$-functor represented by $\mathfrak{O}^{\mathrm{act}}$ and let
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}^{\mathrm{pre}}\to{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}$ be its Grothendieck
construction. Note that
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}^{\mathrm{pre}}$ is equivalent to
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{/\mathfrak{O}^{\mathrm{act}}}$ — where $\textstyle\Delta$ is
seen as a subcategory of $\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}$
— or also to
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{/\operatorname{\mathcal{N\\!}\,\\!\\!_{\bullet}}\mathfrak{O}^{\mathrm{act}}}$,
where
$\operatorname{\mathcal{N\\!}\,\\!\\!_{\bullet}}\colon\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}\hookrightarrow\mathfrak{\mathfrak{\infty\textnormal{-}Grpd}}^{\mathfrak{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}}^{\mathrm{op}}}}$ is a
nerve functor (for example incarnating $(\infty,1)$-categories as complete
Segal $\infty$-groupoids, or as quasicategories, to the reader’s preference).
Thus an

object

of ${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}^{\mathrm{pre}}$ consists of a pair
$([n],O_{\bullet})$ where
$[n]\in{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}$ and
$O_{\bullet}\colon[n]\to\mathfrak{O}^{\mathrm{act}}$ is a linear diagram in
$\mathfrak{O}^{\mathrm{act}}$, that is a sequence $O_{0}\rightsquigarrow
O_{1}\rightsquigarrow\dots\rightsquigarrow O_{n}$ of active morphisms in
$\mathfrak{O}$, while a

morphism

$([n],O_{\bullet})\to([n^{\prime}],O^{\prime}_{\bullet})$ consists of a pair
$(\phi,f_{\bullet})$ where $\phi\colon[n]\to[n^{\prime}]$ is a map in
$\textstyle\Delta$ and $f_{\bullet}\colon O_{\bullet}\Rightarrow
O^{\prime}_{\phi(\bullet)}=O^{\prime}_{\bullet}\circ\phi$ is a natural
transformation of $[n]$-shaped diagrams in $\mathfrak{O}^{\mathrm{act}}$.

We define
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ to be the
wide and locally full (_i.e._ containing all objects and all higher morphisms
between a selection of $1$-morphisms) sub-$(\infty,1)$-category of
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}^{\mathrm{pre}}$ on those morphisms
$(\phi,f_{\bullet})$ such that

  * •

$f_{\bullet}$ is component-wise semi-inert (in addition to active), that is
each $f_{i}\colon O_{i}\rightsquigarrow O^{\prime}_{\phi(i)}$, for $i\in[n]$,
is semi-inert, and

  * •

$f_{\bullet}$ is a cartesian (or equifibred) natural transformation, that is
for any morphism $i<j$ in $[n]$ the naturality square

${O_{i}}$${O^{\prime}_{\phi(i)}}$${O_{j}}$${O^{\prime}_{\phi(j)}}$$\scriptstyle{f_{i}}$$\scriptstyle{O_{i<j}}$$\scriptstyle{O^{\prime}_{\phi(i<j)}}$$\scriptstyle{f_{j}}$
(8)

is cartesian in $\mathfrak{O}^{\mathrm{act}}$.

######  Remark 3.1.2.

Given fixed $\phi\colon[m]\to[n]$ in $\textstyle\Delta$ and
$([n],P_{\bullet})$ in
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ over
$[n]$, morphisms
$(\phi,f_{\bullet})\colon([m],O_{\bullet})\to([n],P_{\bullet})$ in
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ lifting
$\phi$ are essentially determined, if they exist, by their underlying arrow
$f_{m}\colon O_{m}\to P_{\phi(m)}$ at the terminal object $m\in[m]$. Indeed,
for each $i\in[m]$ the object $O_{i}$ and arrow $f_{i}$ are required by the
pullback condition in eq. 8 to be the base-change of $O_{m}$ and $f_{m}$ along
$i\leq m$.

######  Definition 3.1.3.

Let $\mathfrak{O}$ be a combinatorial algebraic pattern. The
$(\infty,1)$-category
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ is called
the $(\infty,1)$-category of $\mathfrak{O}$-forests.

The full sub-$(\infty,1)$-category
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}^{(1)}_{\mathfrak{O}}$ of
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ on the
$([n],O_{\bullet})$ such that $O_{n}\in\mathfrak{O}^{\flat,\mathrm{el}}$ is
called the $(\infty,1)$-category of misodendric $\mathfrak{O}$-forests, or
$\mathfrak{O}$-trees.

######  Example 3.1.4.

For the terminal algebraic pattern $\ast$, the $(\infty,1)$-categories of
$\ast$-forests and of $\ast$-trees both recover the simplex category
$\textstyle\Delta$.

######  Example 3.1.5.

For
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}}^{\flat}$,
we obtain a category of forests, _i.e._ disjoint unions of trees, with level
structures, whose misodendric subcategory is identified in [Ber22] with a full
subcategory of $\textstyle\Omega$. Indeed, recalling that the active
subcategory of
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$ is
equivalent to the category of finite sets, the object $O_{n}$ is to be thought
of as the set of roots of the forest, and each $O_{i}$ is the set of leaves at
level $n-i$. The morphisms in
$({{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}})^{\mathrm{act}}$ give partitions of the leaves
at levels $\ell$ corresponding to the node (recognised by its unique output
leaf at level $\ell+1$) to which they lead.

######  Example 3.1.6.

For
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}}^{\mathrm{op}}}^{\flat}$,
we similarly obtain a category of planar trees (or rather, forests thereof)
with level structures.

###  3.2 The pattern structure on the plus construction

######  Lemma 3.2.1.

The $\infty$-functor
$\mathcal{d\\!}\,_{\mathfrak{O}}\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}\subset{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}^{\mathrm{pre}}_{\mathfrak{O}}\to{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}$ is a cartesian fibration.

###### Proof.

We first observe as in [CHH18] that, due to the pullback condition in eq. 8, a
morphism $(\phi,f_{\bullet})\colon([m],O_{\bullet})\to([n],P_{\bullet})$ in
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ is
$\mathcal{d\\!}\,_{\mathfrak{O}}$-cartesian if and only if $f_{\bullet}$ is a
natural equivalence. Indeed, to say that
$(\phi,f_{\bullet})\colon([m],O_{\bullet})\to([n],P_{\bullet})$ is
$\mathcal{d\\!}\,_{\mathfrak{O}}$-cartesian is to say that

${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O},/([m],O_{\bullet})}}$${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O},/([n],P_{\bullet})}}$${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{/[m]}}$${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{/[n]}}$$\scriptstyle{(\phi,f_{\bullet})\circ-}$$\scriptstyle{\phi\circ-}$
(9)

is a cartesian square.

Let then $([k],L_{\bullet})$ be any object of
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$, and take
the fibre of eq. 9 at $([k],L_{\bullet})$, which we decompose as follows:

${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}\bigl{(}([k],L_{\bullet}),([m],O_{\bullet})\bigr{)}}$${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}\bigl{(}([k],L_{\bullet}),([n],P_{\bullet})\bigr{)}}$${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}^{\mathrm{pre}}\bigl{(}([k],L_{\bullet}),([m],O_{\bullet})\bigr{)}}$${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}^{\mathrm{pre}}\bigl{(}([k],L_{\bullet}),([n],P_{\bullet})\bigr{)}}$${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}([k],[m])}$${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}([k],[n])\text{.}}$${\lrcorner_{\text{ iff $f_{\bullet}$
invertible}}}$ (10)

Since
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}^{\mathrm{pre}}\to{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}$ is constructed — and
described — as a Grothendieck construction, the lower square is cartesian if
and only if $f_{\bullet}$ invertible. It is then enough to observe that the
condition for a morphism in
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}^{\mathrm{pre}}_{\mathfrak{O}}$ to be in
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ is
detected by postcomposition with a componentwise invertible morphism of
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$, which is
clear.

This characterisation now makes it easy to determine cartesian lifts. Let
$\phi\colon[m]\to[n]$ be a morphism in $\textstyle\Delta$, and let
$([n],O_{\bullet})$ be a lift of $[n]$ in
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$. We define
a cartesian lift of $([n],O_{\bullet})$ along $\phi$ to be
$([m],O_{\phi(\bullet)})$ with
$(\phi,\operatorname{id}_{O_{\bullet}\circ\phi})$. ∎

######  Corollary 3.2.2.

Say that an arrow $(\phi,f_{\bullet})$ in
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ is inert
if $\phi$ is inert in $\textstyle\Delta$ and active if $\phi$ is active in
$\textstyle\Delta$ and $f_{\bullet}$ is an equivalence. Then
$(({{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}})^{\mathrm{inrt}},({{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}})^{\mathrm{act}})$ defines a
factorisation system on
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}$.

###### Proof.

By [Lur17, Proposition 2.1.2.5], since
$\mathcal{d\\!}\,_{\mathfrak{O}}\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}\subset{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}^{\mathrm{pre}}_{\mathfrak{O}}\to{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}$ is a cartesian fibration,
the factorisation system on its base can be lifted as required. ∎

######  Definition 3.2.3.

Let $\mathfrak{O}$ be a combinatorial algebraic pattern. Its plus construction
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}}^{\natural}$ is the
$(\infty,1)$-category
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}$ equipped with the inert-active
factorisation of 3.2.2 and as elementary objects those $([n],O_{\bullet})$
with
$[n]\in{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\natural,\mathrm{el}}$ (_i.e._ either $[0]$
or $[1]$) and $O_{n}\in\mathfrak{O}^{\flat,\mathrm{el}}$.

Its misodendric plus construction is the algebraic pattern induced on
${{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}^{(1)}_{\mathfrak{O}}}^{\mathrm{op}}$.

We shall say that an algebraic pattern is well-directed if it can be written
as the plus construction of some combinatorial algebraic pattern.

######  Example 3.2.4.

The main result of [CHH18] shows that
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}}}^{\mathrm{op}}}^{\natural}$ is
Morita-equivalent to
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Omega$}
\put(0.2925,0.167){\line(0,1){0.72}} \put(0.6825,0.17){\line(0,1){0.72}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$ in the sense that their
$(\infty,1)$-categories of Segal $\infty$-groupoids are equivalent. We will
henceforth conflate operad objects with Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}}}^{\mathrm{op}}}^{\natural}$-objects.

######  Scholium 3.2.5.

The Segal condition for a Segal
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}}^{\natural}$-object
$\mathcal{X\\!}\,$ can be explicitly written as the following set of
conditions:

level decomposition

For any $([n],O_{\bullet})$, the canonical arrow

$\mathcal{X\\!}\,\bigl{(}[n],O_{\bullet}\bigr{)}\to\mathcal{X\\!}\,\bigl{(}[1],(O_{0}\rightsquigarrow
O_{1})\bigr{)}\mathop{\times}\limits_{\mathcal{X\\!}\,([0],(O_{1}))}\cdots\mathop{\times}\limits_{\mathcal{X\\!}\,([0],(O_{n-1}))}\mathcal{X\\!}\,\bigl{(}[1],(O_{n-1}\rightsquigarrow
O_{n})\bigr{)}$ (11)

is an equivalence.

root decomposition

For any height-$0$ forest of the form $([0],O_{0})$, the canonical map

$\mathcal{X\\!}\,\bigl{(}[0],(O_{0})\bigr{)}\to\lim_{E\rightsquigarrow
O_{0}}\mathcal{X\\!}\,\bigl{(}[0],(E)\bigr{)}$ (12)

is an equivalence (where the limit is taken over
$\mathfrak{O}^{\mathrm{act}}_{\mathrm{el/O}}$)

shrub decomposition

For any $\bigl{(}[1],(O_{0}\rightsquigarrow O_{1})\bigr{)}$, the canonical
arrow

$\mathcal{X\\!}\,\bigl{(}[1],(O_{0}\rightsquigarrow
O_{1})\bigr{)}\to\lim_{E\rightsquigarrow
O_{1}}\mathcal{X\\!}\,\bigl{(}[1],(O_{0,E}\rightsquigarrow E)\bigr{)}$ (13)

is an equivalence, where $O_{0,E}$ is the fibre product
$O_{0}\times^{\mathrm{act}}_{O_{1}}E$ — with $\times^{\mathrm{act}}$
indicating that it is a fibre product in $\mathfrak{O}^{\mathrm{act}}$.

###### Proof.

The only part that bears commenting upon is the shrub decomposition. Recall
that the inert maps in
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ are simply
those whose projection to $\textstyle\Delta$ is inert, so the relevant inert
morphisms in
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ are those
from the elementaries over
$[1]\in{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}$. If we consider such a
morphism, represented by

${P}$${O_{0}}$${E}$${O_{1}\text{,}}$ (14)

the equifibred condition implies that $P$ is the active pullback
$O_{0}\times^{\mathrm{act}}_{O_{1}}E$. ∎

######  Example 3.2.6.

When $\mathfrak{O}$ is
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$, or more generally a momentous
enrichable patter, the limits appearing in the root and shrub decompositions
take on a simpler form: the root decomposition now becomes the condition that

$\mathcal{X\\!}\,\bigl{(}[0],(O_{0})\bigr{)}\to\lim_{E\rightsquigarrow
O_{0}}\mathcal{X\\!}\,\bigl{(}[0],(E^{O_{0}}_{i})\bigr{)}$ (15)

be an equivalence (where $E^{O}_{i}\rightsquigarrow O$ is the unique lift of
$O$ along the map $\lambda_{i}$ of 2.3.12, and we put a trivial factor if this
lift is inexistent), and the shrub decomposition is the condition that

$\mathcal{X\\!}\,\bigl{(}[1],(O_{0}\rightsquigarrow
O_{1})\bigr{)}\to\prod_{i=1}^{\lvert
O_{1}\rvert}\mathcal{X\\!}\,\bigl{(}[1],(O_{0,i}\rightsquigarrow
E^{O_{1}}_{i})\bigr{)}$ (16)

be an equivalence, where $O_{0,i}$ is the fibre product
$O_{0}\times^{\mathrm{act}}_{O_{1}}E^{O_{1}}_{i}$, as since $\mathfrak{O}$ is
momentous and enrichable, $E\rightsquigarrow O_{1}$ must be one of the lifts
$E^{O_{1}}_{i}$.

######  Proposition 3.2.7 ([CHH18, Lemma 2.11]).

Forests and trees have the same Segal objects: the $(\infty,1)$-functor
$\mathfrak{Seg}_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}}^{\natural}}(\mathfrak{\infty\textnormal{-}Grpd})\to\mathfrak{Seg}_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}^{(1)}_{\mathfrak{O}}}^{\mathrm{op}}}^{\natural}}(\mathfrak{\infty\textnormal{-}Grpd})$
of restriction along the inclusion
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}^{(1)}_{\mathfrak{O}}\hookrightarrow{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ is an
equivalence of $(\infty,1)$-categories.

###### Proof.

Let $([n],O_{\bullet})$ be an $\mathfrak{O}$-forest, and denote the
decomposition of $\lvert O_{n}\rvert$ into its fibres. We then write any
forest as a union of trees, and use the forest decomposition condition of
3.2.5. ∎

######  Construction 3.2.8.

We define a corolla of an $\mathfrak{O}$-forest
$F\in{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ to be an
equivalence class of inert morphisms $F\to E$ where $E$ is elementary lying
over
$[1]\in{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}$.

We can now define a functor
${\lvert-\rvert}^{\mathrm{op}}\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}\to{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}$ by counting the numbers of
corollas in an $\mathfrak{O}$-forest, as in [CH20, Definition 2.2.10].

######  Proposition 3.2.9.

The functor
$\lvert-\rvert\colon{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}\to{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}$ gives a
structure of enrichable pattern on
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}}^{\natural}$.

###### Proof.

By direct verification; this follows essentially by definition of corollas. ∎

######  Proposition 3.2.10 ([Bar18, Theorem 10.16]).

There is an equivalence of $(\infty,1)$-categories between
${{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}}^{\flat}$-Segal objects and weak
Segal $\mathfrak{O}$-fibrations.

###### Proof (Sketch of construction).

The proof is a direct generalisation of that of [HM22, Theorem 3.1.1], which
is quite technically involved, so we do not reproduce it here. However, since
it will be useful later on, we recall the construction of the adjunction.

We first explain what is, for our purposes, the most important part: how to
get from Segal objects to weak fibrations.

We start by constructing an $\infty$-functor
$\omega^{\ast}\colon\mathfrak{\mathfrak{\infty\textnormal{-}Grpd}}^{\mathfrak{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}}}\to\mathfrak{\mathfrak{\infty\textnormal{-}Grpd}}^{\mathfrak{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{/\mathfrak{O}}}^{\mathrm{op}}}}$. Recall that presheaves on
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{\mathfrak{O}}$ can be
seen equivalently as presheaves on
$\mathfrak{\mathfrak{\infty\textnormal{-}Grpd}}^{\mathfrak{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}}}$ satisfying the
representability condition of being colimit-preserving. We now define an
$\infty$-functor
$\omega\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{/\mathfrak{O}}\to\mathfrak{\mathfrak{\infty\textnormal{-}Grpd}}^{\mathfrak{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}}}$ as the composition
$\mathcal{i\\!}\,^{\ast}\circ\mathcal{y\\!}\,_{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{/\mathfrak{O}}}$ where
$\mathcal{i\\!}\,^{\ast}$ is restriction (or inverse image) of presheaves
along the canonical $\infty$-functor
$\mathcal{i\\!}\,\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}\hookrightarrow{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{/\mathfrak{O}^{\mathrm{act}}}\xrightarrow{(\mathfrak{O}^{\mathrm{act}}\to\mathfrak{O})_{!}}{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}} \end{picture}}}_{/\mathfrak{O}}$.

We thus get an inverse image $\infty$-functor
$\omega^{\ast}\colon\mathfrak{\mathfrak{\infty\textnormal{-}Grpd}}^{\mathfrak{{\mathfrak{\mathfrak{\infty\textnormal{-}Grpd}}^{\mathfrak{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}}}}^{\mathrm{op}}}}\to\mathfrak{\mathfrak{\infty\textnormal{-}Grpd}}^{\mathfrak{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{/\mathfrak{O}}}^{\mathrm{op}}}}$, which we can restrict to
the (full) sub-$(\infty,1)$-category of those presheaves which are
representable and satisfy the Segal condition. To finish, recall also (from
[Lur09, Corollary 5.1.6.12]) that
$\mathfrak{\mathfrak{\infty\textnormal{-}Grpd}}^{\mathfrak{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{/\mathfrak{O}}}^{\mathrm{op}}}}\simeq{\mathfrak{\mathfrak{\infty\textnormal{-}Grpd}}^{\mathfrak{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}}}_{/\operatorname{\mathcal{N\\!}\,\\!\\!_{\bullet}}\mathfrak{O}}$,
so we eventually have

$\omega^{\ast}\colon\mathfrak{Seg}_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}}^{\natural}}(\mathfrak{\infty\textnormal{-}Grpd})\to{\mathfrak{\mathfrak{\infty\textnormal{-}Grpd}}^{\mathfrak{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}}}_{/\operatorname{\mathcal{N\\!}\,\\!\\!_{\bullet}}\mathfrak{O}}\text{.}$
(17)

In the case where
$\mathfrak{O}={{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}$, our functor $\omega$ corresponds
to the one described in [HHM16, §5.1]. By [HM22, §3.3], this functor does land
first in the subcategory
$\mathfrak{Seg}_{{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}}^{\mathrm{op}}}^{\natural}}(\mathfrak{\infty\textnormal{-}Grpd})_{/\operatorname{\mathcal{N\\!}\,\\!\\!_{\bullet}}\mathfrak{O}}$
which we interpret as
$\mathfrak{\mathnormal{(\infty,1)}\textnormal{-}Cat}_{/\mathfrak{O}}$, and in
fact in the further subcategory of those $\infty$-functors to $\mathfrak{O}$
which are weak Segal fibrations.

To go the other way, we consider
$\varpi\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}\xrightarrow{\mathcal{y\\!}\,}\mathfrak{\mathfrak{\infty\textnormal{-}Grpd}}^{\mathfrak{{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}}^{\mathrm{op}}}}\to\mathfrak{WkSegFib}_{/\mathfrak{O}}$
and the “conjoint” profunctor it corepresents: then $(\omega^{\ast})^{-1}$ is
the $\infty$-functor mapping $(\mathfrak{E}\to\mathfrak{O})$ to the presheaf
$\mathfrak{WkSegFib}_{/\mathfrak{O}}(\varpi(-),\mathfrak{E})\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.42,0.7){\line(10,-26){0.25}}
\end{picture}}}_{\mathfrak{O}}\to\mathfrak{\infty\textnormal{-}Grpd}$. ∎

Note that — again for
$\mathfrak{O}={{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Gamma$}
\put(0.47,0.025){\line(0,1){0.95}} \end{picture}}}}^{\mathrm{op}}}^{\natural}$
— an alternate proof, using the monoidal envelope $\infty$-functor, appears
in[HK21, Corollary 4.2.8]. However, this proof uses crucially an explicit
description of the Grothendieck construction of
$\mathcal{E\\!n\\!v\\!}\,(\mathbb{1})$ seen as a Segal
$\mathfrak{O}$-$\infty$-category, which do not have in general.

##  4 Monoidal envelopes and Grothendieck opfibrations

###  4.1 Construction of the monoidal envelope functor

As explained in the introduction, we wish to interpret the usual formula
computing monoidal envelopes of operads as a fibrewise oplax extension along a
certain functor appropriately mixing in partitions. To define the necessary
fibrewise extensions, we need to briefly work in the generality of formal
$\infty$-category theory introduced in [RV22], that is in the framework of
$\infty$-cosmoi, modelling the $(\infty,2)$-category of $\infty$-categories.
Recall from [RV22, Proposition 9.1.8, Theorem 9.3.3] that (pointwise) oplax
extensions along an $\infty$-functor can be expressed as colimits weighted by
the conjoint of this $\infty$-functor. We will define fibrewise extensions
similarly, replacing this conjoint by a relative version.

The reader who wishes to limit their intake of abstract nonsense may use the
formula of eq. 23 as a definition, and take its functoriality properties on
faith.

######  Construction 4.1.1 (Relative comma $\infty$-category).

In an $\infty$-cosmos $\mathfrak{K}$, consider a cocorrespondence in the
sliced $\infty$-cosmos $\mathfrak{K}_{/B}$: