ScalingStacks

The data of an oriented embedding $U\hookrightarrow[-1,1]$ from a finite disjoint union of oriented intervals, determines a linear ordering of the connected components of $U$. This is organized as a monoidal functor between $\infty$-operads

(4)

$$\disk^{\partial,{\sf or}}_{1/[-1,1]}\longrightarrow{\sf Assoc}_{\sf RL}$$

to the standard multi-category corepresenting the datum of an associative algebra $A$, together with a unital right module $T$ and a unital left module $S$. Because the space of oriented embeddings between two oriented intervals is contractible, this functor (4) is an equivalence of $\infty$-operads. In summary, there is an equivalence of $\infty$-categories

(5)

$$\Alg_{\sf RL}(\mathcal{V})\xrightarrow{~\simeq~}\Fun^{\otimes}\bigl(\disk_{1/[-1,1]}^{\partial,\sf or},\mathcal{V}\bigr)$$

where the lefthand $\infty$-category is that of algebras over ${\sf Assoc}_{\sf RL}$.