7
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.