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 ${_A \Delta _{A\otimes A}}$ and an "associativity isomorphism" $$\varphi: {_A \Delta _{A\otimes A}} \underset{A\otimes A}\otimes \bigl( {_A A _A} \underset{\mathbb K}\otimes {_A \Delta _{A\otimes A}}\bigr) \overset\sim\to {_A \Delta _{A\otimes A}} \underset{A\otimes A}\otimes \bigl( {_A \Delta _{A\otimes A}} \underset{\mathbb K}\otimes {_A A _A} \bigr) $$ 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 ${_A {\Delta^{\rm flip}} _{A\otimes A}}$ denote the bimodule $\Delta$ with the two right $A$-actions flipped. A sesquialgebra is symmetric if it comes equipped with a bimodule isomorphism $\psi: {_A \Delta _{A\otimes A}} \overset\sim\to {_A {\Delta^{\rm flip}} _{A\otimes A}}$ 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 $a\otimes b \mapsto ab$ if $(a,b)$ is a composable pair of morphisms, and $a\otimes b \mapsto 0$ 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?

  • $\begingroup$ Incidentally, I think that the correct language just adds "ish" to everything. Rather than calling something a "sesquialgebra", I would call it an "algebrish algebra"; rather than "symmetric", I would say "commutatish". Then the idea is that in general in a 2-category, and "algebrish object" is the correctly weakened version of an "algebra object". So a monoidal category is an "algebrish object in Cat", etc. $\endgroup$ – Theo Johnson-Freyd Jul 13 '10 at 18:31
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    $\begingroup$ Huh? "Commutatish" instead of the perfectly natural "symmetric"? Does "skew-symmetric" become "commutatishish"? And a symmetric sesquilinear algebra is a "commutatish algebrish algebra"? This does not seem like a good idea. $\endgroup$ – BCnrd Jul 13 '10 at 19:25
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    $\begingroup$ Well, ok, so it doesn't exactly roll of the tongue. But "symmetric" and "skew-symmetric" are overloaded words. And I like the consonance of "algebrish/gibberish". $\endgroup$ – Theo Johnson-Freyd Jul 13 '10 at 20:27

I believe this is the subject of Tannakian reconstruction (as in this question)? i.e. if I understand correctly the Hopfish algebra attached to a groupoid is built so that its category of modules as a tensor category is the category of vector bundles on the groupoid, in the discrete case, or of (quasi)coherent sheaves in the algebraic case, or some topological substitute? If so one can try to reconstruct the groupoid as the groupoid of tensor functors from this tensor category to vector spaces (more generally you can construct a functor of points out of the groupoid of tensor functors to R-modules for varying rings R, or to vector bundles of the appropriate type on general "test" spaces). For discrete groupoids, or for quasicompact stacks with affine diagonal, this reconstruction works to reconstruct the space from the tensor category, i.e. from the Hopfish algebra.

  • $\begingroup$ So, this certainly won't reconstruct any particular groupoid I might have started with, since the reconstruction you propose does not see the particular Hopfish algebra, just its Morita class. But it's possible that this reconstructs the equivalence class of the original groupoid. $\endgroup$ – Theo Johnson-Freyd Jul 13 '10 at 20:26
  • $\begingroup$ If it did, then it would presumably give for each stack a "universal" presentation. I'm dubious. $\endgroup$ – Theo Johnson-Freyd Jul 13 '10 at 20:32
  • $\begingroup$ Perhaps I misunderstood your goal, but I thought you were looking at a functor from a 2-category of (reasonable) groupoids or stacks to a Morita 2-category of algebras, or equivalently a 2-category of [compactly generated] linear categories. We have now enhanced this to a functor to a 2-category of Hopfish algebras, which I had understood you to say was a standin for a 2-category of symmetric monoidal categories. The Spec/Tannakian functor goes back - it's a one-sided quasiinverse. What stronger would we want - looking at groupoids up to isom (rather than equivalence) seems unnatural no? $\endgroup$ – David Ben-Zvi Jul 13 '10 at 20:55
  • $\begingroup$ Sorry, yes, this exactly does the trick of recovering a stack from its Morita class of Hopfish algebras. My dubiousness in the second sentence was that there are set-theoretic problems with implementing your proposal in any sort of Bourbakian framework. But set theory has never troubled me before, so it shouldn't now. $\endgroup$ – Theo Johnson-Freyd Jul 13 '10 at 21:52
  • $\begingroup$ All I meant in the first comment was that this reconstruction does not answer the question in the title, but that's OK --- of course the real question is to see the stack, not the particular presentation. The question I myself really want to know is the third on my list in the blockquote: namely, whether a category (up to equivalence) is recoverable from the Morita theory of its sesquialgebra. The answer should still be "yes" by the same construction, but the proof I gave myself in the case of groupoids in SET uses too much to transfer. $\endgroup$ – Theo Johnson-Freyd Jul 13 '10 at 21:55

Let $A=KC_0$ be the direct sum of fields indexed by the base of the groupoid. Then the algebra $H=KC_1$ is an $A|K$-Hopf algebra in terminilogy going back to Sweedler. The groupoid can be recovered from it as the set of grouplike elements, in the same way as a group can be recovered from its Hopf group algebra.

  • $\begingroup$ If I only know KC_1 as a hopfish algebra, then I don't see immediately how to reconstruct KC_0, at least not without some argument like David Ben-Zvi's. $\endgroup$ – Theo Johnson-Freyd Jul 14 '10 at 17:46

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