I'm just trying to understand the following definition:


Definition 3.3.3.8 in Higher Algebra by J. Lurie defines the $\infty$-operad of $O$-module objects, and says the following:


Let $O^\otimes$ be a unital $\infty$-operad and $C^\otimes \to O^\otimes$ a fibration of generalized $\infty$-operads. We let $Mod^O(C)^\otimes$ denote the fiber product $$\overline{Mod}^O(C)^\otimes \times\_{^p Alg\_{/O}(C)} (O^\otimes \times Alg\_{/O}(C))$$


My question is what does an object in here look like?




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Here is my rough attempt at understanding what is going on:
First an object of $\overline{Mod}^{O}(C)^\otimes$ consists of the data $(v, F)$
of an object $v \in O^\otimes$ and a functor $F:$ $ \_{v}K\_O \to C^\otimes$ (where $\_{v}K\_O$ denote those semi-inert maps in $O^\otimes$ that start at $v$) sending a semi-inert map $f:v \to y$ in $O^\otimes$ to an object $F(f$) of $C^\otimes\_y$, such that if $y \to z$ is inert, then $F(f) \to F(v \to z)$ is inert in $C^\otimes$. 


So my guess is that an object of $Mod^O(C)^\otimes$ is a pair $(v, A)$ where $v \in O^\otimes$ and $A \in Alg\_{/O}(C)$. Or perhaps it is something like $(v, F, A, \theta)$ where $\theta$ is an equivalence between $(v, F)$ and $(v, A)$ in $^p Alg\_{/O}(C)$. Or perhaps it's something else entirely.


What is, intuitively/philosophically, the role of the semi-inert morphisms, i.e. why do we really have to use them to make this definition?


I think I understand how an object in here can be viewed as generalization of the classical notion of a module over an algebra. In chapter 4 when Lurie defines the $\infty$-operad $\mathcal{LM}^\otimes$ and defines an $\infty$-category $LMod(C)$ (for $C$ a monoidal $\infty$-category) of left modules over the associative operad as $Alg\_{\mathcal{LM}/\mathcal{Ass}}(C)$, and then example 4.2.1.18 shows how to think of an object in this category as a generalization of the notion of a module. I assume a similar analysis holds for objects in $Mod^O(C)$ e.g given $(v, A)$, say $v\simeq (v\_1, ..., v\_n)$, where the $v\_i \in O$ then $A(v\_1), ..., A(v\_n)$ have various multiplications maps between them, controlled by the operad $O$. If this more explicitly spelled out in Higher Algebra let me know.



understanding the definition of $\infty$-operad of module objects
I know this question is a little old but I just came across it.


Roughly, as you say, from this data you get an object $v$ of $O^\otimes$ and an algebra $A$ of $Alg\_{/O}(C)$. However, you also get an *action* of $A$ on some object $M$ over $v$. The role of the semi-inert morphisms is as a "marking": a semi-inert morphism $v \to y$ in $O^\otimes$ marks some portion of the object $y$ (the image of $v$) as set aside for a *module* part, and the part outside the image describes an *algebra* part. Here is how this is expressed.


In the following, for the sake of convenience I am going to pretend that objects of $O^\otimes$ are literally $n$-tuples $(X\_1,\dots,X\_n)$ of objects of the underlying category $O$, and similarly for $C^\otimes$, so that the inert maps are given by projecting off factors. You can say the following without that assumption but it might obscure what's going on.


Your object $v$ is then a tuple $(X\_1,\dots,X\_k)$ of objects of $O$. Your module $M$ is going to actually be a tuple $(M\_1, \dots, M\_k)$, where $M\_i$ is a module living over the 1-object tuple $X\_i$. So for our purposes, it's simpler to just look at the case where $v = X$ is an object of $O$.


There's a special semi-inert morphism, the identity $v \to v$, which gets sent to an object $M$ in the fiber $C\_X$ over $X$. For any $w$ in $O$, there is also another special semi-inert morphism $v \to () \to w$: these together describe objects $A\_Y$ in $C\_Y$ for any $Y \in O$. Those are going to be the module and algebra objects. A general semi-inert morphism $v \to w$ is either isomorphic to one of the form $(X) \to (Y\_1, Y\_2, \dots, Y\_m, X)$ coming from a sequence of maps $() \to (Y\_i)$ in $O^\otimes$, or $(X) \to () \to (Y\_1,\dots,Y\_m)$. Your functor $F$ sends these to $(A\_{Y\_1},\dots,A\_{Y\_m},M)$ and $(A\_{Y\_1},\dots,A\_{Y\_m})$ respectively by the assumption that it preserves inert morphisms.


Now the rest of the structure says that we essentially get maps of spaces
$$
Map\_{O^\otimes}((Y\_1,\dots,Y\_m), Y) \to Map\_{C^\otimes}((A\_{Y\_1},\dots,A\_{Y\_m}), A\_Y)
$$
making $A$ into an $O$-algebra, and maps
$$
Map\_{O^\otimes}((Y\_1,\dots,Y\_m,X), X) \to Map\_{C^\otimes}((A\_{Y\_1},\dots,A\_{Y\_m},M), M)
$$
that give the action of $A$ on $M$. These satisfy an associativity constraint -- compatibility with composition. (You can also mix the order of the terms up but that's encapsulated by compatibility with an action of the symmetric group.)


If instead of having $v = (X)$ you had $v = (X\_1,\dots,X\_k)$, then you would have an algebra $A$ and a $k$-tuple of $A$-modules.


