$\Omega X$-modules are functors from $X$

Question

Let $X$ be a connected (nice) space, $x\in X$ and $\Omega X$ the loopspace at $x$.

Then $\Omega X$ has an $E_1$-structure, and so we may consider left $\Omega X$-modules. There are a few ways to do so :

1) one is to use Lurie's general formalism of left modules over an associative algebra in an $\infty$-category (here, the $\infty$-category of pointed spaces is $\mathbb E_1$-monoidal, and we may view $\Omega X\in Alg_{/\mathbb E_1}(\mathcal S_*)$, and then consider $LMod_{\Omega X}(\mathcal S_*)$. .

2) Do it "by hand" : take for instance $\mathbb E_1$ to be explicitly the little $1$-disks operad $D_1$, define the morphism $D_1\to End(\Omega X)$ the usual way, and define a left module to be a pointed space $M$ with pointed maps $D_1(n)\times (\Omega X)^{n-1}\times M\to M$ such that all the appropriate diagrams commute.

3) Replace $\Omega X$ by a strictly associative model : take $\Omega^+X = \{(p,t)\mid t\in \mathbb R_+, p\in \hom([0,t],X) , p(0)=p(t)=x\}$ with the obvious concatenation. It is strictly associative, and there's a map $\Omega^+X\to \Omega X$ which respects multiplication and is a weak equivalence. Then consider "strict" $\Omega^+X$-modules : that is, spaces $M$ with a multiplication $\Omega X\times M\to M$ such that the two maps $\Omega X\times \Omega X\times M\to M$ coincide.

4) Say that $\Omega X$-modules are just $\infty$-functors $X\to \mathcal S_*$.

For 2) and 3), you can also equip the categories of modules you obtain with projective model structures , and then take the associated $\infty$-category.

This gives us four $\infty$-categories that could be sensibly named "the category of $\Omega X$-modules". I'm wondering whether they're all equivalent - I think they are. I'm now going to explain the connections I already see, and then point to what I'm missing. A great answer would either fill the gaps or point to the relevant literature that can fill the gaps, or explain (/give references) why the gaps can't be filled.

I'm not going to use 1) or relate it to the others, but if it's somewhere in the literature (e.g. in Lurie, I haven't read all of it), I'd be glad to know.

Relation between 3) and 2).

The map $\Omega^+X\to \Omega X$ is a weak equivalence of $D_1$-algebras, therefore general results ensure that it induces a Quillen equivalence between ($D_1-$) left modules on $\Omega X$ and left modules on $\Omega^+X$. Therefore the associated $\infty$-categories of such are equivalent.

Now it suffices to relate $D_1$-$\Omega^+X$-modules and "strict" $\Omega^+X$-modules. Of course I have a fully faithful inclusion of strict modules into $D_1$-modules, and I suspect that every $D_1$-module $Y$ has a strict module $Z$ with a weak equivalence of $D_1$-modules $Z\to Y$, and perhaps a universal one. This would amount to saying that the inclusion is left Quillen or something in that area, and that suggests that the two associated $\infty$-categories are equivalent. I'm not entirely sure here, and this is the first gap.

I think this is perhaps related to something called rectification, but I'm not entirely sure what I'm looking for. It might be related to this result, but I can't convince myself that it's enough : one would have to have operads $P,Q$ with a weak equivalence between them such that one of the operads encodes strict modules and the other one encodes $D_1$-modules over $\Omega^+X$ (or more generally a topological monoid). I know how to encode strict modules : take an operad concentrated in degree $1$ with the obvious thing in degree $1$; I'm less certain about $D_1$-modules. So that's something that could be enough to conclude.

Now for 3) and 4) I have essentially the same issue here. I know how to relate $Fun(X,\mathcal S_*)$ and strict $\Omega^+ X$-modules, but I'm not sure how to relate those to general ones.

Indeed, a teacher of mine told me that I could find somewhere in HTT the following result (or something along those lines - by the way, if you know where that result is in HTT, it would be great to include it in your answer !) :

If $C$ is a small simplicial category, and $M$ a combinatorial simplicial model category, then the category of simplicial functors $C\to M$ together with the projective model structure has an associated $\infty$-category which is equivalent to the $\infty$-category of $\infty$-functors $C\to \mathcal M$ (what I'm denoting $\mathcal M$ is either the homotopy-coherent nerve of the fibrant-cofibrant objects of $M$, or the localization of the homotopy-coherent nerve of $M$ at weak equivalences, if I'm not mistaken it's known that both are equivalent).

Using this, and the fact that, as an $\infty$-category, $X$ is equivalent to the homotopy-coherent nerve of the (small) simplicial category with one object and as mapping space $Sing(\Omega^+ X)$, and given that $\mathbf{sSet}_*$ is combinatorial, we only have to examine what simplicial functors between these two are to get a model for $Fun(X,\mathcal S_*)$.

Now these will be Quillen-equivalent to simplicial functors to $\mathbf{Top}_*$ (here again I'm not so sure, you can count that as a gap : $\mathbf{Top}$ is not accessible, so it's not combinatorial, so I'm not sure it has a projective model structure, but I think this existence theorem is enough), and those simplicial functors are easy to identify to strict modules : indeed a simplicial functor amounts to a pointed space $Y$ with a map of simplicial sets $Sing(\Omega^+X)\to \hom(Y,Y)$ compatible with composition, which amounts to a map of spaces $|Sing(\Omega^+X)|\times Y\to Y$ which presents a strict $|Sing(\Omega^+X)|$-module (strict because we're asking it to be compatible with composition on the nose), and we have a weak equivalence of topological monoids $|Sing(\Omega^+X)|\to \Omega^+X$ which induces a Quillen-equivalence of the associated categories of modules.

So I essentially have the same gap between 2) and 3) and between 3) and 4) (so maybe 2) and 4) can be related without having to fill this gap ? )

My question is thus

How can I fill this gap ? Are there any mistakes in what I claimed ? What is the relevant literature ? Are there any easier ways to relate the different points of view ?

Answer 1

Here is an answer for the first part of the question, i.e. the relation between $D_1$-modules and strict modules.

As you know there is a weak equivalence of operads $D_1 \to \mathrm{Ass}$, which induces a Quillen equivalence between the category of $D_1$-algebras and the category of associative algebras. This essentially comes from the fact that path components of $D_1(m)$ are contractible and that $\pi_0 D_1 = \mathrm{Ass}$, thus the quotient map $D_1 \to \pi_0 D_1 = \mathrm{Ass}$ is a weak equivalence.

This equivalence can be upgraded to become an equivalence between categories of modules. Let us build two-colored operads whose "algebras" will be pairs of the type (algebra, left module). Let's say that the two colors are respectively called "closed" and "open". These colored operads are going to be built from $D_1$ and $\mathrm{Ass}$. In both cases, if the output is closed, then if there are open inputs the operation space is empty, otherwise it's just the corresponding components of $D_1$ or $\mathrm{Ass}$. Now if the output is closed:

• for $\mathrm{Ass}$: the space of operations with open output, $m$ closed inputs and $n$ open inputs is denoted $\mathrm{LMod}(m,n)$. It is empty if $n \neq 1$, and $\mathrm{LMod}(m,1) \subset \mathrm{Ass}(m+1) = \Sigma_{m+1}$ is given by permutations which fix the last element (in some sense we are always acting from the left).
• for $D_1$, the procedure is similar expect you consider $\mathrm{LMod}_{D_1}(m,1) \subset D_1(m+1)$ to be configurations of intervals where the $(m+1)$st interval is always to the right of all the other intervals.

An algebra over $\mathrm{LMod}$ is a pair $(A,M)$ where $A$ is an associative algebra and $M$ is a left $A$-module. An algebra over $\mathrm{LMod}_{D_1}$ is a similar pair but with $D_1$ everywhere.

Now it's pretty clear that $\mathrm{LMod} = \pi_0(\mathrm{LMod}_{D_1})$, and the connected components of $\mathrm{LMod}_{D_1}(m,1)$ are contractible, so the quotient map $\mathrm{LMod}_{D_1} \to \mathrm{LMod}$ is a weak equivalence and we are done.

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Remarks: what I've built above is sometimes called a moperad (contraction of module and operad), see e.g. "The Homotopy Braces Formality Morphism" by Willwacher. They can equivalently be described as algebras in the category of right modules over an operad, see "Modules over Operads and Functors" by Fresse. These can be used to build notions of modules over algebras over an operad, see also "Operads, Modules and Topological Fields Theories" by Horel.

Canonically, the moperad associated to an operad $P$ is the shift $P[1]$ given by $P[1](m,1) = P(m+1)$, which is part of the universal enveloping operad of $P$. Algebras over this colored operad are pairs $(A,M)$ where $A$ is a $P$-algebra and $M$ is what is usually called a module over $A$. For associative algebras, what you get in this case are $(A,A)$-bimodule (as there's no reason left or right should be singled out a priori). The moperads built above are sub-moperads of $P[1]$.