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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?


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.

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    $\begingroup$ Here is my interpretation. An object of $\mathrm{Mod}^\mathscr{O}(\mathscr{C})^\otimes$ is a pair $(\mathcal{M},\mathscr{M})$, where $\mathcal{M}$ is an object of $\overline{\mathrm{Mod}}^\mathscr{O}(\mathscr{C})^\otimes$ and $\mathscr{M}\in\mathscr{O}^\otimes\times\mathrm{Alg}_{/\mathscr{O}}(\mathscr{C})$, such that if $F_1:\overline{\mathrm{Mod}}^\mathscr{O}(\mathscr{C})^\otimes \to^{\mathrm{P}} \mathrm{Alg}_{/\mathscr{O}}(\mathscr{C})$ and $F_2:\overline{\mathrm{Mod}}^\mathscr{O}(\mathscr{C})^\otimes \to \mathscr{O}^\otimes\times\mathrm{Alg}_{/\mathscr{O}}(\mathscr{C})$, ... $\endgroup$ – user62675 Sep 3 '14 at 15:33
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    $\begingroup$ ... then $F_1(\mathcal{M})=F_2(\mathscr{M})$. Indeed, we may write $\mathscr{M}$ itself as a pair $(O,\mathscr{A})$, where $\mathscr{A}$ and $O$ are objects of $\mathrm{Alg}_{/\mathscr{O}}(\mathscr{C})$ and $\mathscr{O}^\otimes$ respectively. $\endgroup$ – user62675 Sep 3 '14 at 15:34
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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.

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An object of $\mathrm{Mod}^\mathscr{O}(\mathscr{C})^\otimes$ is a pair $(\mathcal{M},\mathscr{M})$, where $\mathcal{M}$ is an object of $\overline{\mathrm{Mod}}^\mathscr{O}(\mathscr{C})^\otimes$ and $\mathscr{M}\in\mathscr{O}^\otimes\times\mathrm{Alg}_{/\mathscr{O}}(\mathscr{C})‌​$, such that if $F_1:\overline{\mathrm{Mod}}^\mathscr{O}(\mathscr{C})^\otimes\to ^{\mathrm{P}}{\mathrm{Alg}}_{/\mathscr{O}}(\mathscr{C})$ and $F_2:\overline{\mathrm{Mod}}^\mathscr{O}(\mathscr{C})^\otimes\to\mathscr{O}^{\otimes}\times\mathrm{Alg}_{/\mathscr{O}}(\mathscr{C})$, then $F_1(\mathcal{M})=F_2(\mathscr{M})$. Indeed, we may write $\mathscr{M}$ itself as a pair $(O,\mathscr{A})$, where $\mathscr{A}$ and $O$ are objects of $\mathrm{Alg}_{/\mathscr{O}}(C)$ and $\mathscr{O}^\otimes$ respectively.

Another formulation is given in this PhD thesis; see section $2.4$, especially Definition $2.4.3$. I will try to explain what the intuitive role of semi-inert morphisms are in a later edit.

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