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.