Boavida de Brito–Weiss non-discretized operadic model for Taylor towers

Let $M$ be a manifold, let $\mathcal{O}^s_\infty(M)$ and $\widetilde{\mathcal{O}}^s_k(\mathbb{R}^m)$ be the source categories of the continuous context-free functors under consideration, let $\operatorname{Top}$ denote topological spaces, let $\mathtt{B}_m$ be the topological little $m$-balls operad, and let $\mathtt{C}$ be the chain functor. Write $\operatorname{T}_k$ for the $k$th Taylor approximation, $\operatorname{hRmod}$ for the derived mapping space of right modules, and $\operatorname{hWBimod}_{\leq k}$ for the derived mapping space of truncated weak bimodules. Then the following conjectural operadic models hold. Non-discretized operadic Taylor-tower conjecture. If $F\colon\mathcal{O}^s_\infty(M)\longrightarrow\operatorname{Top}$ is a continuous context-free functor, then $F$ can be regarded as a right module over $\mathtt{B}_m$ and there are natural equivalences

$$\operatorname{T}_\infty F(M)\simeq\underset{\mathtt{B}_m}{\operatorname{hRmod}}\left(\operatorname{sEmb}(-,M),F(-)\right),$$ $$\operatorname{T}_\infty\mathtt{C}(F(M))\simeq\underset{\mathtt{C}(\mathtt{B}_m)}{\operatorname{hRmod}}\left(\mathtt{C}(\operatorname{sEmb}(-,M)),\mathtt{C}(F(-))\right).$$

If $F\colon\widetilde{\mathcal{O}}^s_k(\mathbb{R}^m)\longrightarrow\operatorname{Top}$ is context-free, then $F$ can be regarded as a weak bimodule over $\mathtt{B}_m$ and

$$\operatorname{T}_kF(\mathbb{R}^m)\simeq\underset{\mathtt{B}_m}{\operatorname{hWBimod}_{\leq k}}\left(\mathtt{B}_m,F(-)\right).$$

In particular, for $F(-)=\operatorname{sEmb}(-,\mathbb{R}^n)$, so that $F$ is equivalent to $\mathtt{B}_n$ as a weak bimodule over $\mathtt{B}_m$,

$$\operatorname{T}_k\overline{\operatorname{Emb}}_c(\mathbb{R}^m,\mathbb{R}^n)\simeq\underset{\mathtt{B}_m}{\operatorname{hWBimod}_{\leq k}}(\mathtt{B}_m,\mathtt{B}_n).$$

Moreover,

$$\operatorname{T}_k\mathtt{C}(F(\mathbb{R}^m))\simeq\underset{\mathtt{C}(\mathtt{B}_m)}{\operatorname{hWBimod}_{\leq k}}\left(\mathtt{C}(\mathtt{B}_m),\mathtt{C}(F(-))\right),$$

and hence

$$\operatorname{T}_k\mathtt{C}(\overline{\operatorname{Emb}}_c(\mathbb{R}^m,\mathbb{R}^n))\simeq\underset{\mathtt{C}(\mathtt{B}_m)}{\operatorname{hWBimod}_{\leq k}}\left(\mathtt{C}(\mathtt{B}_m),\mathtt{C}(\mathtt{B}_n)\right).$$

This conjectural non-discretized replacement for the discretized module models is significant because it connects manifold calculus with the topology and chain-level homotopy theory of little-balls operads. The source notes that the conjecture was subsequently proved by Pedro Boavida de Brito and Michael Weiss.