Floer homology associates a vector space $HF^\ast(L_1,L_2)$ to any pair of Lagrangian submanifolds $L_1,L_2$ inside a symplectic manifold $X$. By a **categorification** of Floer homology, I mean a category $\mathcal H\mathcal F(L_1,L_2)$ whose Hochschild homology is isomorphic to $HF^\ast(L_1,L_2)$.

**Example:** Let $M$ be a closed symplectic manifold, and consider the diagonal $\Delta\subseteq M\times M^-$, which is Lagrangian. It self-Floer homology $HF^\bullet(\Delta,\Delta)$ is (conjectured to be) isomorphic to the Hochschild homology of the Fukaya category of $M$. Thus $\mathcal F(M)$ is a categorification $\mathcal H\mathcal F(\Delta,\Delta)$ of $HF^\ast(\Delta,\Delta)$. Note $HF^\ast(\Delta,\Delta)$ is a unital algebra, and expressing this unit as a Hochschild cycle of $\mathcal F(M)$ is of fundamental importance in the study of the Fukaya category.

Are there any other known circumstances under which a categorification of Floer homology exists?

Floer homology also has a product $HF^\ast(L_1,L_2)\otimes HF^\ast(L_2,L_3)\to HF^\ast(L_1,L_3)$, so it is natural to further ask that a categorification $\mathcal H\mathcal F$ have natural functor $\mathcal H\mathcal F(L_1,L_2)\times\mathcal H\mathcal F(L_2,L_3)\to\mathcal H\mathcal F(L_1,L_3)$ which upon applying Hochschild homology recovers the product on Floer homology.

What is a natural geometric description of a functor $\mathcal F(M)\times\mathcal F(M)\to\mathcal F(M)$ corresponding to the product on $HF^\ast(\Delta,\Delta)$?

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