higher algebra – Understanding the disintegration of unital $infty$-operads

In section 2.3.4 of Higher
Lurie shows that any unital $infty$-operad (whose underlying
$infty$-category is an $infty$-groupoid) can be obtained by gluing
together a family of reduced $infty$-operads. This question is about
understanding this construction in more intuitive operadic language.

Context on generalised $infty$-operads

A convenient algebraic formulation of the notion of family of
$infty$-operads is given by generalised $infty$-operads, which
are defined by relaxing the conditions for a functor
${cal O}^{otimes}tomathbb{F}_{ast}$ (where $mathbb{F}_{ast}$ is
the category of pointed finite sets, opposite to Segal’s category
$Gamma$) to describe the category of operators of an $infty$-operad,
allowing the fibre ${cal O}^{otimes}_{langle0rangle}$ over
$langle0rangle$ to be a non-trivial $infty$-category (this
definition can be systematised using
algebraic patterns: just as $infty$-operads are weak Segal fibrations
over the pattern $mathbb{F}_{ast}^{flat}$, generalised
$infty$-operads are weak Segal fibrations over
$mathbb{F}_{ast}^{natural}$). As noticed by Gepner–Haugseng,
generalised $infty$-operads are a symmetric version of virtual double
$infty$-categories, and as such I will use an analogous diagrammatic
language to discuss them more intuitively, and which I start by
presenting (as I understand it).

If ${cal O}^{otimes}tomathbb{F}_{ast}$ is a generalised
$infty$-operad, the fibre ${cal O}^{otimes}_{langle0rangle}$ can
be seen as consisting of objects and vertical arrows, while the
operadic structure of the other fibres is stored horizontally: the
objects of ${cal O}^{otimes}_{langle1rangle}$ (the “colours”) are
horizontal arrows; more generally the objects of
${cal O}^{otimes}_{langle nrangle}$ are suitably composable (see
after) sequences of $n$ horizontal arrows, and the morphisms are
cells. There is one major difference in comparison to virtual double
categories, owing to the symmetry (equivalently, to the unicity of the
morphism $langle1rangletolangle0rangle$ in $mathbb{F}_{ast}$):
a horizontal arrow does not have distinct source and target (so I
shall denote such arrows as $leftrightarrow$), and so must the
(external) source and target vertical arrows of a cell coincide as
well. Hence a generic cell $m$ can be visualised as
& & O_{1} & & O_{n} & & \
& C & leftrightarrow & cdots & leftrightarrow & C & \
f & downarrow & m & Downarrow & & downarrow & f \
& D & & leftrightarrow & & D &\
& & & P & & &

(or see a more legible version on

There is a clear inclusion $infty$-functor from $infty$-operads into
generalised $infty$-operads, and by formal nonsense on localisations
and categorical patterns (Remark it admits a left-adjoint
called the functor of assembly. This means that for any
generalised $infty$-operad ${cal O}^{otimes}$ there is a unit map
${cal O}^{otimes}tooperatorname{Assemb}({cal O}^{otimes})$ inducing (by
composition) an equivalence
O}^{otimes},{cal P}^{otimes})$
for any $infty$-operad
${cal P}^{otimes}$.


There is an $infty$-functor
$mathfrak{GenOpd}_{infty}tomathfrak{Cat}_{infty}$ (measuring how
far a generalised $infty$-operad is from being an $infty$-operad),
which sends ${cal O}^{otimes}tomathbb{F}_{ast}$ to
${cal O}^{otimes}_{langle0rangle}$. It admits (Proposition a right-adjoint given by
$mathfrak{C}mapstomathfrak{C}timesmathbb{F}_{ast}$. In the
language laid out above, the vertical $infty$-category is
$mathfrak{C}$, for any object $Cinmathfrak{C}$ there is a unique
horizontal arrow $O_{C}$, and for any arrow $fcolon Cto D$ in
$mathfrak{C}$ and any arity $n$ there is a unique $n$-ary cell
& & O_{C} & & O_{C} & & \
& C & leftrightarrow & cdots & leftrightarrow & C & \
f & downarrow & ast_{f}^{n} & Downarrow & & downarrow & f \
& D & & leftrightarrow & & D &\
& & & O_{D} & & &

This construction can also be used to relate the notion of generalised
$infty$-operad to that of family of $infty$-operads, in the form
of a criterion (Proposition for an $infty$-functor
${cal P}^{otimes}tomathfrak{C}timesmathbb{F}_{ast}$ to define a
fibration of generalised $infty$-operads with an equivalence
$mathfrak{C}simeq{cal P}^{otimes}_{langle1rangle}$. This allows
one to define easily a reduced generalised $infty$-operad as one
corresponding to a family whose fibres (at each object
$Cinmathfrak{C}$) are reduced $infty$-operads.

The main result I am interested in (Theorem is that the
functor $operatorname{Assemb}$, restricted to families of reduced
$infty$-operads and astricted to unital $infty$-operads with
groupoidal underlying $infty$-category, is essentially
surjective. For any such $infty$-operad ${cal O}^{otimes}$, we must
find a (reduced) generalised $infty$-operad
$operatorname{Disint}({cal O}^{otimes})$, its disintegration,
equipped with a morphism
$operatorname{Disint}({cal O}^{otimes})to{cal O}^{otimes}$
$operatorname{Assemb}(operatorname{Disint}({cal O}^{otimes}))
simeq{cal O}^{otimes}$

To construct it, we first consider the functor
$mathfrak{Opd}_{infty}tomathfrak{Cat}_{infty}$ sending an
$infty$-operad to its underlying $infty$-category; its restriction
to unital $infty$-operads admits a right-adjoint, forming the
cocartesian $infty$-operad $mathfrak{C}^{amalg}$ associated with an
$infty$-category $mathfrak{C}$. It can be described explicitly as
follows: its colours are the objects of $mathfrak{C}$, and the space
of multimorphisms $(C_{1},cdots,C_{n})to D$ is
$prod_{i=1}^{n}hom_{mathfrak{C}}(C_{i},D)$. The adjunction means
that it is characterised by the universal property (Proposition that, for any unital $infty$-operad ${cal O}^{otimes}$,
morphisms ${cal O}^{otimes}tomathfrak{C}^{amalg}$ are the same as
${cal O}^{otimes}_{langle1rangle}tomathfrak{C}$. Given an
$Fcolon{cal O}^{otimes}_{langle1rangle}tomathfrak{C}$, its
extension to a morphism
$widetilde{F}colon{cal O}^{otimes}tomathfrak{C}^{amalg}$ of
$infty$-operads sends each colour $O$ to $F(O)$; a multimorphism
$mcolon(O_{1},cdots,O_{n})to P$ is mapped to the family of images
of (unary) arrows
$F(m(emptyset,cdots,emptyset,-,emptyset,cdots,emptyset))$ ($m$
with all inputs but the $i$th composed with the nullary operation of
the relevant colour, which makes sense because ${cal O}^{otimes}$ is

For any $infty$-category $mathfrak{C}$, there is (Example a
morphism of generalised $infty$-operads
$mathfrak{C}timesmathbb{F}_{ast}tomathfrak{C}^{amalg}$. Since
$mathfrak{C}^{amalg}$ is a non-generalised $infty$-operad, this
morphism must send the vertical structure of
$mathfrak{C}timesmathbb{F}_{ast}$ to the trivial one. Then, a
horizontal arrow $O_{C}$ corresponding to the colour $C$ is mapped to
$C$, while the cell $ast^{n}_{f}$ is mapped the $n$-uple

Now if ${cal O}tomathbb{F}$ is a unital $infty$-operad, we can
consider the identity functor
${cal O}_{langle1rangle}to{cal O}_{langle1rangle}$, inducing
${cal O}^{otimes}to({cal O}^{otimes}_{langle1rangle})^{amalg}$, and the
disintegration of ${cal O}^{otimes}$ may be defined as the fibre
product of generalised $infty$-operads
operatorname{Disint}({cal O}^{otimes})={cal

We can describe explicitly its components using the remarks above: a
cell in $operatorname{Disint}({cal O}^{otimes})$ is of the form
& & O_{C} & & O_{C} & & \
& C & leftrightarrow & cdots & leftrightarrow & C & \
f & downarrow & m & Downarrow & & downarrow & f \
& D & & leftrightarrow & & D &\
& & & O_{D} & & &

where $m$ is an iterated extension of $f$, i.e. plugging the unique
nullary operation with target $C$ in all its inputs bar one must
recover $f$.

The question

The thing that bothers me in this description is that the
multimorphisms of ${cal O}^{otimes}$ with non-homogeneous input
colours do not seem to be anywhere to be found in
$operatorname{Disint}({cal O}^{otimes})$. In particular, if
$Fcolonoperatorname{Disint}({cal O}^{otimes})to{cal
is a morphism to an $infty$-operad, the theorem says
that there should be a unique
$widetilde{F}colon{cal O}^{otimes}to{cal P}^{otimes}$ factoring
$F$ through
$operatorname{Disint}({cal O}^{otimes})to{cal O}^{otimes}$, but
I cannot see any information on what this $widetilde{F}$ should send
the aforementioned non-monochromatic morphisms of ${cal O}^{otimes}$
to. So I am wondering how the construction works, that is how
$widetilde{F}$ is actually determined by $F$, in this description, or
if somewhere the interpretation I gave is just plain wrong and
misrepresents what Lurie is actually doing.

N.B.: The actual technical proof of the result rests on the fact
that the morphism
$mathfrak{C}timesmathbb{F}_{ast}tomathfrak{C}^{amalg}$ is a
weak approximation, that weak approximations are stable by pullback
along fibrations, and another result (Proposition that weak
approximations give assemblies. However, this does not help me get the
operadic picture of how the disintegration works, so what I am really
asking for is an explanation which, in the spirit of my descriptions
above, may not be strictly correct, but emphasises the operadic picture
rather than the homotopy-theoretic one.