In section 2.3.4 of *Higher
Algebra*,

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

Chu–Haugseng’s

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

$$

begin{array}{ccccc}

& & O_{1} & & O_{n} & & \

& C & leftrightarrow & cdots & leftrightarrow & C & \

f & downarrow & m & Downarrow & & downarrow & f \

& D & & leftrightarrow & & D &\

& & & P & & &

end{array}

$$

(or see a more legible version on

quiver).

There is a clear inclusion $infty$-functor from $infty$-operads into

generalised $infty$-operads, and by formal nonsense on localisations

and categorical patterns (Remark 2.3.3.2) it admits a left-adjoint

$operatorname{Assemb}colonmathfrak{GenOpd}_{infty}tomathfrak{Opd}_{infty}$,

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

$hom_{mathfrak{GenOpd}_{infty}}(operatorname{Assemb}({cal

O}^{otimes}),{cal

P}^{otimes})xrightarrow{simeq}hom_{mathfrak{Opd}_{infty}}({cal

O}^{otimes},{cal P}^{otimes})$ for any $infty$-operad

${cal P}^{otimes}$.

### Construction(s)

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

2.3.2.9) 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

$$

begin{array}{ccccc}

& & O_{C} & & O_{C} & & \

& C & leftrightarrow & cdots & leftrightarrow & C & \

f & downarrow & ast_{f}^{n} & Downarrow & & downarrow & f \

& D & & leftrightarrow & & D &\

& & & O_{D} & & &

end{array}

$$

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 2.3.2.11) 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 2.3.4.4) 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

$operatorname{Disint}({cal O}^{otimes})to{cal O}^{otimes}$

exhibiting

$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

2.4.3.16) that, for any unital $infty$-operad ${cal O}^{otimes}$,

morphisms ${cal O}^{otimes}tomathfrak{C}^{amalg}$ are the same as

$infty$-functors

${cal O}^{otimes}_{langle1rangle}tomathfrak{C}$. Given an

$infty$-functor

$Fcolon{cal O}^{otimes}_{langle1rangle}tomathfrak{C}$, 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

unital).

For any $infty$-category $mathfrak{C}$, there is (Example 2.4.3.5) 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

$(f,dots,f)$.

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

O}^{otimes}mathbin{mathop{times}limits_{({cal

O}^{otimes}_{langle1rangle})^{amalg}}}{cal

O}^{otimes}_{langle1rangle}timesmathbb{F}_{ast}text{.}

$$

We can describe explicitly its components using the remarks above: a

cell in $operatorname{Disint}({cal O}^{otimes})$ is of the form

$$

begin{array}{ccccc}

& & O_{C} & & O_{C} & & \

& C & leftrightarrow & cdots & leftrightarrow & C & \

f & downarrow & m & Downarrow & & downarrow & f \

& D & & leftrightarrow & & D &\

& & & O_{D} & & &

end{array}

$$

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

P}^{otimes}$ 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 2.3.4.5) 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.