Do you recognise this variant of the cubes operad?

Question

In the subject of operads one comes across many quirky variants of more or less the same operad. The cubes operad has many incarnations with various interesting properties. In a recent paper I came across one such variant. It's so simple I presume other people have come across it before, so it would be nice to have a reference if that's the case. I'll give a sketch of this rather simple construction, below.

Let me describe the operad, what I think it's good for, and how it relates to other operads. First, I'll set up some notation convention with the cubes operad.

Def'n: (cubes) An increasing affine-linear function $[-1,1] \to [-1,1]$ is a little interval. A product of little intervals $[-1,1]^n \to [-1,1]^n$ is a little $n$-cube. The space $\mathcal C_n(j)$ is the collection of $j$-tuples of little $n$-cubes whose images are required to have disjoint interiors, $\mathcal C_n(0)=\{*\}$ is the empty cube. The collection $\mathcal C_n = \sqcup_{j=0}^\infty \mathcal C_n(j)$ is the operad of little $n$-cubes, it is a $\Sigma$-operad with structure maps

$$\mathcal C_n(k) \times \left( \mathcal C_n(j_1) \times \cdots \times \mathcal C_n(j_k) \right) \to \mathcal C_n(j_1+\cdots+j_k)$$

defined by

$$(L,J_1,\cdots,J_k) \longmapsto (L_1 \circ J_1, \cdots, L_k \circ J_k)$$

and $\mathcal C_n(j) \times \Sigma_j \to \mathcal C_n(j)$ given by $(L, \sigma) \longmapsto L\circ \sigma$.

Def'n: (overlapping cubes) A collection of $j$ overlapping $n$-cubes is an equivalence class of pairs $(L, \sigma)$ where $L=(L_1,\cdots,L_j)$, each $L_i$ is a little $n$-cube and $\sigma \in \Sigma_j$. Two collections of $j$ overlapping $n$-cubes $(L,\sigma)$ and $(L',\sigma')$ are taken to be equivalent provided $L = L'$ and whenever the interiors of $L_i$ and $L_k$ intersect $\sigma^{-1}(i) < \sigma^{-1}(k) \Longleftrightarrow \sigma'^{-1}(i) < \sigma'^{-1}(k)$. Given $j$ overlapping $n$-cubes $(L_1,\cdots,L_j,\sigma)$ say the $i$-th cube $L_i$ is at height $\sigma^{-1}(i)$. $\sigma(1)$ is the index of the bottom cube, and $\sigma(j)$ is the index of the top cube. Let $\mathcal C_n'(j)$ be the space of all $j$ overlapping $n$-cubes, with the quotient topology induced by the equivalence relation.

The structure map $$\mathcal C_n'(k) \times \left( \mathcal C_n'(j_1) \times \cdots \times \mathcal C_n'(j_k) \right) \to \mathcal C_n'(j_1 + \cdots + j_k)$$

is defined by

$$\left((L,\sigma), (J_1,\alpha_1), \cdots, (J_k, \alpha_k)\right) \longmapsto ((L_1\circ J_1, \cdots, L_k\circ J_k), \beta)$$

the permutation $\beta$ is given for $1 \leq a \leq k$, $1 \leq b \leq j_a$ by

$$\beta^{-1}\left(\sum_{i<a} j_i + b\right) = \left( \sum_{i<\sigma^{-1}(a)} j_{\sigma(i)}\right) +\alpha^{-1}_a(b)$$

This permutation is obtained by taking the lexicographical order on the set $\{(a,b) : a \in \{1,\cdots,k\}, b \in \{1,\cdots,j_a\}\}$ and then identifying with $\{1, 2, \cdots,j_1+\cdots+j_k\}$ in the order-preserving way.

== The point ==

So there is a map of operads $\mathcal C_{n+1} \to \mathcal C'_n$ given by sending $(L_1, \cdots, L_j)$ to $(L_1^\pi, \cdots, L_j^\pi, \sigma)$ where we write $L_i = L_i^\pi \times L_i^\nu$ where $L_i^\pi$ is an $n$-cube and $L_i^\nu$ a $1$-cube. The permutation $\sigma$ is any element $\sigma \in \Sigma_j$ such that $L_{\sigma(j)}^\nu(1) \geq L_{\sigma(j-1)}^\nu(1) \geq \cdots \geq L_{\sigma(1)}^\nu(1)$.

Some of the nice things about this operad are:

• (1) it's a multiplicative operad, the inclusion of the associative operad is given by the elements $(Id_{\mathcal [-1,1]^n}, \cdots, Id_{\mathcal [-1,1]^n}, Id_{\{1,\cdots,j\}}) \in \mathcal C'_n(j)$.

• (2) The map above $\mathcal C_{n+1} \to \mathcal C'_n$ is an equivalence of operads.

• (3) While $\mathcal C_{n+1}$ acts on spaces such as $\Omega^{n+1} X$, $\mathcal C'_n$ does not. $\mathcal C'_n$ acts on spaces of the form $\Omega^n M$ where $M$ is a topological monoid.

The operad of overlapping intervals $\mathcal C'_1$ has a certain affinity to the cactus operad. For example, imagine $[-1/2,1/2]$ as an element of $\mathcal C'_1(1)$ as being represented by $[-1,1]$ with a $1$-cell attached at the points $-1/2$ and $1/2$.

And there are all kinds of variants of this idea -- overlapping discs, or overlapping framed discs, etc. So you can get cyclic multiplicative operads out of these types of constructions.

The criterion for getting the answer "right" is either showing me an occurance of this operad in the literature, or coming up with some convincing argument it's a new construction.

Answer 1

I just now noticed this question.

Your operad is a suboperad of the "surgery cylinder" operad described in arxiv 1009.5025 (more recent version available here). See Section 8 and figures therein. In the notation of that paper, your operad corresponds to case where all the manifolds $M_i$ and $N_i$ are $n$-cubes and all the homeomorphisms $f_i$ are the identity.

The surgery cylinder operad can be thought of as describing a sequence of (generalized) surgeries on an initial manifold $M_0$, yielding a final manifold $N_0$. At the $i$-th stage ($1\le i \le k$) we remove a codimension-0 submanifold $M_i$ replace it with $N_i$, where $m_i$ and $N_i$ have the same boundary. To get your overlapping $n$-cubes operad, let $M_0$ be the "big" $n$-cube and $M_i = N_i$ be the $i$-th little $n$-cube. In other words we have a sequence of pointless surgeries in which little $n$-cubes are removed and replaced with identical copies of themselves.