Construction for algebras over little cubes operad

Question

Recently I came across the following construction:

Fix a dimension $k$. Let $C$ denote the space whose points are disjoint rectilinear embeddings $c\colon I^k\to \mathbb R^k$ of the (closed) $k$-dimensional cube into $\mathbb R^k$; let us call the points $c\in C$ cubes in $\mathbb R ^k$ (or just cubes) Given a cube $c$ and disjoint subcubes $c_1,\dots ,c_n\subset c$ this determines uniquely an $n$-ary operation $f\in O(n)$ in the little k-cubes operad (which we shall denote by $O$ here), given by the unique rectilinear factorization $\coprod_{i=1}^nc_i\colon \coprod^n I^k\xrightarrow{f} I^k \xrightarrow{c} \mathbb R$. Note that different such inclusions might give rise to the same operation in $O(n)$.

Now let us assume that some of these inclusions $\coprod c_i \subset c$ are marked and consider the functor $F$ which associates to an algebra $A$ (in spaces) over the little $k$-cubes operad the following space $FA$: Points in $FA$ are maps $g\colon C\to A$ such that for each marked inclusion $\coprod c_i\subset c$ and corresponding operation $f\in O(n)$ we have $f(g(c_1),\dots,g(c_n))= f(c)$.

Here are some issues

1. I am a bit worried about the equality constraint in the definition of $FA$. Maybe it might be more appropriate using a more homotopical construction which keeps track of (coherent) paths between $f(g(c_1),\dots,g(c_n))$ and $f(c)$.

2. I would expect that if we mark all inclusions $c\subset c'$ of a single cube into another (and only those), then $FA$ should recover the space $A$ itself; at least up to homotopy. Is there a way to write the construction such that this becomes obvious?

More generally it seems to me that this construction should be an instance of a very general procedure which I am only able to describe heuristically: We make the space $C$ into a topological colored operad which has $C$ as it's space of colors and which has operations $(c_1,\dots,c_n)\to c$ given by disjoint inclusions. Then $C$ comes equipped with a canonical map $C\to O$. Moreover, some of the operations in $C$ are marked. Now it seems that $FA$ should be some sort of space of "maps of marked topological (or infinity-) operads" from $C$ to $A^\otimes$ where $A^\otimes\to O$ is some sort of operadic cocartesian fibration which describes the $O$-algebra $A$.

So here are (some of) my questions: What is the relation between the colored operad $C$ describes above and the little cubes operad? Is there a different description of this construction? How exactly should I define $A^\otimes$? What is the best way to make this construction precise, preferably in the language of infinity-categories? Is there a good theory of operadic cocartesian fibrations analogous to the one for infinity-categories?

Answer 1

As you point out, it indeed seems that in order to get a well-behaved answer one should work in a suitable homotopical setting, for example, that of $\infty$-categories and $\infty$-operads. Using this language one may reformulate your construction as follows: identifying spaces with $\infty$-groupoids (i.e., $\infty$-categories in which all arrows are invertible) we may think of $A$ as an $\infty$-groupoid endowed with an $E_k$-monoidal product (for example, we may imagine for intuition purposes that $A$ is a braided monoidal groupoid). We may encode such an object by a left fibration $A^{\otimes} \to E_k$ of $\infty$-operads.

In the case when you consider all the operations as "marked", your $C$ should be thought of as an $\infty$-operad $C^{\otimes}$, whose space of colors are rectilinear embedding $I^k \to \textbf{R}^k$ of disks in $\textbf{R}^k$ and whose operations are given by compatible rectilinear embeddings between disks. When suitably interpreted, this is a particular case of a general construction, in which you can replace $\textbf{R}^k$ with any space $B$ equipped with a real vector bundle $\tau:E \to B$ (see section 5.4.2 of Lurie's book higher algebra, and in particular Definition 5.4.2.10). The resulting $\infty$-operad, let's call it $E_B$, will encode the algebraic theory of ($\tau$-twisted) $B$-families of $E_k$-algebras. In your case, since $B=\textbf{R}^k$ is contractible, you will just get back $C^{\otimes} = E_{\textbf{R}^k} \simeq E_k$. Your definition of $FA$ (when homotopically interpreted) then gives the space of $\infty$-operad maps $C^{\otimes} = E_k \to A$ over $E_k$, or, equivalently, the space of $E_k$-algebra objects in $A$. This space is however not very interesting: a formal consequence of the fact that $E_k$ is a unital $\infty$-operad is that the space of $E_k$-algebra objects in any $E_k$-monoidal $\infty$-groupoid is contractible (essentially because the unit map $1\to X$ provides a natural equivalence between any such $E_k$-algebra object and the trivial one).