ct.category theory - Is a groupoid determined by its Hopfish algebra?

This is a follow up to my question What is the precise relationship between groupoid language and noncommutative algebra language?. I will briefly review some definitions; for details, a good place to look is Christian Blohmann, Alan Weinstein. Group-like objects in Poisson geometry and algebra. 2007. arXiv:math/0701499v1. And actually, there are two versions of my question, one for groupoids and the other for categories. So that I can avoid all analysis, I will restrict my attention to finite things; if you know the answer in, say, topological spaces, or smooth manifolds, or..., then I'm also interested.

A category is a span of sets $C = { C_0 overset l leftarrow C_1 overset r rightarrow C_0}$ which is an algebra object in the category of $C_0,C_0$ spans. I.e. there are maps of spans $i: {C_0 = C_0 = C_0} to C$ and $m: C underset{C_0}times C to C$ making the usual diagrams commute. A category is a groupoid if additionally there is an involution ${^{-1}} : { C_0 overset l leftarrow C_1 overset r rightarrow C_0} to { C_0 overset r leftarrow C_1 overset l rightarrow C_0}$ satisfying some condition. A category $C$ is finite if both $C_0$ and $C_1$ are finite.

A finite-dimensional algebra $A$ (over a fixed field $mathbb K$) is sesqui if it is equipped with a bimodule ${_A Delta _{Aotimes A}}$ and an "associativity isomorphism" $$varphi: {_A Delta _{Aotimes A}} underset{Aotimes A}otimes bigl( {_A A _A} underset{mathbb K}otimes {_A Delta _{Aotimes A}}bigr) oversetsimto {_A Delta _{Aotimes A}} underset{Aotimes A}otimes bigl( {_A Delta _{Aotimes A}} underset{mathbb K}otimes {_A A _A} bigr) $$
of $A, A^{otimes 3}$ bimodules, which satisfies a pentagon. There should also be a "counit" bimodule $_A epsilon _{mathbb K}$, some triangle isomorphisms, and some more equations.
A sesquialgebra is hopfish if a hard-to-write-down condition is satisfied; see Xiang Tang, Alan Weinstein, Chenchang Zhu. Hopfish algebras. 2006. arXiv:math/0510421v2. Let ${_A {Delta^{rm flip}} _{Aotimes A}}$ denote the bimodule $Delta$ with the two right $A$-actions flipped. A sesquialgebra is symmetric if it comes equipped with a bimodule isomorphism $psi: {_A Delta _{Aotimes A}} oversetsimto {_A {Delta^{rm flip}} _{Aotimes A}}$ so that $varphi,psi$ satisfy two hexagons. A sesquialgebra is finite if $A,Delta, dots$ are finite-dimensional over $mathbb K$.

Let $C = { C_0 overset l leftarrow C_1 overset r rightarrow C_0}$ be a finite category. Then it gives rise to a finite symmetric sesquialgebra as follows. The algebra $A$ is given by the vector space $mathbb K C_1$ with the convolution product (given on the basis by $aotimes b mapsto ab$ if $(a,b)$ is a composable pair of morphisms, and $aotimes b mapsto 0$ otherwise). The bimodule $Delta$ is given as the vector space with basis all pairs $(a,b) in C_1 times C_1$ with $l(a) = l(b)$. I will let you work out the rest: the actions, the associator $varphi$ and symmetrizer $psi$, etc. If $C$ is actually a groupoid, then $mathbb K C_1$ is hopfish.
This construction extends to a 2-functor, and so sends equivalences of categories to Morita equivalences of sesquialgebras.

Question: It is well known that a groupoid $C$ cannot be recovered from the algebra $mathbb K C_1$; compare for example the group with two elements, thought of as a groupoid with one object, and the set with two elements, thought of as a groupoid with only identity morphisms. But the examples I know can be distinguished by remembering the hopfish structure.

• Can a finite category be recovered from its symmetric sesquialgebra?


• If not, can a finite groupoid be recovered from its symmetric hopfish structure?


• Can an equivalence class of finite categories be recovered from the Morita-equivalence class of finite symmetric sesquialgebras?


• If not, do we at least have the corresponding statement for groupoids/hopfish algebras?

In a footnote, Blohmann and Weinstein suggest that they do not know the answers to the above questions. But that was three years ago; perhaps there has been more recent work?