If $\mathcal{O}$ is a (classical) topological operad with unit $1\in \mathcal{O}(1)$, $\mathcal{O}(0)=\{0\}$ and multiplications $(m;a_1,\dotsc,a_r)\mapsto m(a_1,\dotsc,a_r)$. Let $X$ be an algebra over $\mathcal{O}$. Then each choice of $m\in \mathcal{O}(2)$ gives us a binary product $$X\times X\to X, (x,x')\mapsto x\cdot x' := m\cdot (x,x').$$ It is well-known that the topological structure of $\mathcal{O}$ characterises the product:

- If $1,m(0,1),m(1,0)\in \mathcal{O}(1)$ lie in the same component, then $e:=0\cdot ()\in X$ is an $H$-unit.
- If $m(m,1),m(1,m)\in \mathcal{O}(3)$ lie in the same component, then the product is $H$-associative.

I thought about an analogous construction for coloured operads $\mathcal{O}\binom{n}{k_1,\dotsc,k_r}$ and algebras $(X_n)$ over it. Each choice $m^n_{k_1,k_2}\in \mathcal{O}\binom{n}{k_1,k_2}$ gives a binary product
$$X_{k_1}\times X_{k_2}\to X_n, (x,x')\mapsto x\cdot_n x' := m^n_{k_1,k_2}\cdot (x,x').$$
A system $(m^n_{k_1,k_2})$ may be called *multiplication system*. We see:

- If all $\mathcal{O}\binom{n}{n}$ are connected, then $e_k:=0_k\cdot ()\in X_k$ is an $H$-unit in the sense that $$(-\cdot_n e_k)|_{n}\simeq (e_k\cdot_n -)|_{n}\simeq \mathrm{id}_n$$
- If all $\mathcal{O}\binom{n}{k_1,k_2,k_3}$ are connected, then the restricted product is $H$-associative.

My hope was that if we fix such a multiplication system, we get an interesting $H$-something. As a coloured operad is the natural “horizontal categorification” of an operad, my first idea was that we obtain an $H$-category with colours as objects, and the above graded product as compositions and the $e_k$ as identities, but apparently, this is not the kind of structure we have.

(If you prefer algebraic categories, apply singular homology $H_*(-;R)$ to the above system and look at the Pontryagin structure. I first thought that the result is an $R$-algebroid, but again, the structure looks different.)

Does someone see what this is? A “monoid” where we can choose which product we want to use dependind on where we want to land?