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I have sitting in front of me two 2-categories. On the left, I have the 2-category GPOID, whose:

  • objects are groupoids;
  • 1-morphisms are (left-principal?) bibundles;
  • 2-morphisms are bibundle homomorphisms.

On the right, I have the 2-category ALG, whose:

  • objects are algebras (over $\mathbb C$, say);
  • 1-morphisms are (adjectives?) bimodules;
  • 2-morphisms are bimodule homomorphisms.

And probably I should go through and add "in TOP" to every word on the left and "C-star" to every word on the right.

(I have the impression that ALG is two-equivalent to another category, which I will put on the far right, whose:

  • objects are cocomplete VECT-enriched categories (with extra adjectives?);
  • 1-morphisms are cocontinuous VECT-enriched functors;
  • 2-morphisms are VECT-enriched natural transformations.

In one direction, the functor takes an algebra to its full category of modules. In the other direction, there might be extra adjectives needed, and I mean to appeal to the Mitchell embedding theorem; but I'm pretty sure an equivalence exists if I insist that every object on the far right comes with a cocontinuous faithful VECT-enriched functor to VECT, and my idea is that Mitchell says that every category admits such a functor.

So anyway, the point is that either category on the right or far right is a sort of "algebraic" category, as opposed to the more "geometric" category on the left.)

Then I've been told on numerous occasions that there is a close relationship between GPOID and ALG. See, for example, the discussion at Geometric interpretation of group rings? — in fact, it's reasonable to think of the present question as a follow-up on that one.

The relationship is something like the following. To each (locally compact Hausdorff) topological, say, groupoid, we can associate a C-star, say, algebra — the construction restricts in various special cases to: a (locally compact Hausdorff) topological space $X$ going to its algebra of continuous vanishing-at-infinity functions $C_0(X)$; a finite group $G$ going to its group ring $\mathbb C G$; etc. The construction extends to 1- and 2-morphisms to build a (contra)functor. At least if I get all the adjectives right, the functor should be a two-equivalence.

Question: What's the precisification of what I have said above? What exactly is the two-functor from groupoids to algebras, and which adjectives make it into an equivalence of two-categories? Groupoids have natural "disjoint union" and "product" constructions; these presumably correspond to Cartesian product and tensor product (?!? that's not the coproduct in the category of algebras, but maybe in this two-category it is?) on the algebraic side?

Let me end with an example to illustrate my confusion, which I brought up in Op. cit.. Let $G$ be a finite abelian group; then it has a Pontryagin dual $\hat G$. Now, there is a canonical way to think of $G$ as a groupoid: it is the groupoid $\{\text{pt}\}//G$ with only one object and with $G$ many morphisms. If I'm not mistaken, the corresponding algebra should be the group algebra $\mathbb C G$. But $\mathbb C G$ is also the algebra $C_0[\hat G]$ of functions on the space $\hat G$. And if there is one thing I am certain of, it is that the underlying space of $\hat G$ (a groupoid with no non-identity morphisms) and the one-object groupoid $\{\text{pt}\}//G$ are not equivalent as groupoids. And yet their "function algebras" are the same. So clearly I am confused.

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3 Answers 3

up vote 8 down vote accepted

I think the problem is that the left hand side is part of commutative geometry, while the right hand side is part of noncommutative geometry (regardless of whatever vague claims I may have made in my previous response). Groupoids (or stacks) certainly have interesting noncommutative aspects that are captured by the construction you discuss, but it's not an equivalence. More precisely, (an algebro-geometric version of) your construction attaches to a stack the category of (quasi)coherent sheaves (which is given by replacing algebras by their categories of modules as you suggest). However, one cannot hope to recover the stack from this category, as your finite group example shows: representations of a finite group are the same as (algebraic) vector bundles on BG, or on $\widehat G$, and these are certainly nonisomorphic.

In order to get something that can be an equivalence with appropriate adjectives, we need to change the right hand side into its commutative version: replace categories by tensor (symmetric monoidal) categories - ie remember that vector bundles/coherent sheaves have a tensor product. Now we're in a good situation: Tannakian theory tells us that in a variety of settings this construction is an equivalence. For example tensor of representations corresponds to tensor of vector bundles on BG, but has nothing to do with tensor of vector bundles on $\widehat G$. This is of course anathema to noncommutative geometers - vector bundles on a noncommutative space (modules for noncommutative rings) don't have tensor products. It's in this sense that the LHS is part of commutative geometry.

A nice formal way to say this is by looking at the adjoint functor to the functor from stacks to categories, or to symmetric monoidal categories, given by considering bundles/sheaves with or without tensor product. The first adjoint takes a category to the moduli stack of objects in it; the second is the Tannakian reconstruction functor, sending a tensor category to its "spectrum" (fiber functors over various rings). The first functor is very far from recovering a stack - the moduli stack of objects in a category of modules for a ring is much much bigger than the spectrum of the ring!

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Maybe what you're saying is the following, or maybe not. The "algebra of functions on a stack" carries some extra data, namely a (symmetric?) monoidal structure on its representation theory. When the stack is a (classical, commutative) space, the algebra is commutative, and this monoidal structure is the tensor over the algebra. When the structure is a group (a one-object groupoid), the monoidal structure is the tensor structure or group representations. And there is a construction for arbitrary groupoids that generalizes both of these. Is what I said correct? –  Theo Johnson-Freyd Jul 10 '10 at 2:38
    
That's right. You only get an honest commutative algebra representing functions when the space is affine - in algebraic geometry this means spec of a ring, though in the differentiable setting any manifold is affine. Otherwise you can represent functions by a noncommutative algebra (up to Morita equivalence), but this doesn't help you see the symmetric monoidal structure. But I believe (comm and NC) geometry is better captured categories of sheaves, not about algebras of functions - in the former, these categories can be made symm monoidal, in the latter not. –  David Ben-Zvi Jul 10 '10 at 2:48
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There is a notion, I think due to Alan Weinstein, of "Hopfish algebra", which is more or less an algebra and a tensor structure on its representation theory (and an antipode). More precisely, since any cocomplete functor between representation categories "is" tensor with a bimodule, a Hopfish algebra is an algebra $A$ and an $A^{\otimes 2}$-$A$ bimodule satisfying an associativity constraint up to a pentagon. I should probably go read more about them --- I only met them recently. For "affine groupoids", if I knew the correct definitions, how hands-on are the constructions? –  Theo Johnson-Freyd Jul 10 '10 at 4:03
    
The answer to my question at the end of these comments is mostly "yes". I will describe it in an "answer" to my original question below. –  Theo Johnson-Freyd Jul 13 '10 at 3:36

Both David Ben-Zvi and Jeffrey Giansiracusa have given excellent answers, so I will only fill in some bits that with Douglas Rizzolo we worked through based on their suggestions. Moreover, for the purposes of this answer, I will avoid almost all analysis questions. So although I think I know how to say some of what I want to say more generally, I will take as my "geometric" category the category $\text{FinSet}$ of finite sets. The final disclaimer is that I will not mention the "inverse" map: rather than groupoids, I will describe the construction for category objects; rather than Hopf algebras, I will have bialgebras; there will not be a C-star structure. Nevertheless, I will use the shorthand of "groupoid"/"Hopf"/... in place of "category object"/"bialgebra"/...

Groupoid algebras

So, abusing language, a groupoid is a finite set $C_0$, a finite set $C_1$, maps $i: C_0 \to C_1$ and $s,t: C_1 \to C_0$ such that $s\circ i = t\circ i = \operatorname{id}$, and a composition map $m: C_1 \underset{C_0}\times C_1 \to C_1$ satisfying a number of equations. The short-hand way to say all this is to say "algebra in the category of spans from $C_0$ to $C_0$". I will denote a generic groupoid by $C = \{C_1 \rightrightarrows C_0\}$. I have worked through precisely three examples:

  1. Any set $M$ is a groupoid $M = \{M \rightrightarrows M\}$, where all maps are identities.
  2. Any group $G$ is a groupoid $\text{pt}/G = \{G \rightrightarrows \text{pt}\}$, where the maps are the only thing they can be.
  3. The "equivalence relation" groupoid $n/n = \{n^2 \rightrightarrows n\}$ has $n$ objects and for each pair of objects a unique morphism. It is equivalent to $\text{pt}$.

Fix a field $\mathbb K$, and let $\text{FinVect}$ denote the category of finite-dimensional $\mathbb K$-vector spaces. Recall the "linearization" functor $\mathbb K: \text{FinSet} \to \text{FinVect}$, which takes a set $S$ to the vector space $\mathbb KS$, which has $S$ as a (distinguished) basis. This functor preserves colimits (being adjoint to $\text{forget}$) and takes products to tensor products.

For any groupoid $C = \{C_1 \rightrightarrows C_0\}$, we define the algebra of functions on $C$ to be the vector space $\mathbb K C_1$, with the multiplication determined by the multiplication in $C$. More precisely, if $a,b \in C_1$ are basis elements, we set their multiplication in $\mathbb K C_1$ to be: $$ab = \begin{cases} ab \in C_1, & s(a) = t(b), \\ 0 ,& \text{otherwise.} \end{cases}$$ In the three examples above:

  1. The "algebra of functions on $M$" is the algebra of functions $M \to \mathbb K$ with pointwise multiplication.
  2. The "algebra of functions on $\text{pt}/G$" is the "group algebra" $\mathbb K G$.
  3. The "algebra of functions on $n/n$" is the matrix algebra $\operatorname{Mat}(n,\mathbb K)$. It is Morita equivalent to $\mathbb K$.

A "vector bundle" or "sheaf" on a groupoid $C$ is precisely a representation of $\mathbb KC_1$. My earlier objection was that as algebras, if $G$ is abelian with Pontryagin dual $\hat G$, then as algebras the algebras of functions on $\text{pt}/G$ and on $\hat G$ are the same, when these are different as sheaves.

Hopfish structure

David Ben-Zvi hinted that what distinguishes the algebras of functions over $\text{pt}/G$ and $\hat G$ is the tensor structure on their categories of sheaves. The following notion is due to Xiang Tang, Alan Weinstein, and Chenchang Zhu, Hopfish algebras, 2008, arXiv:math/0510421v2. (Actually, they decategorify, considering only isomorphism classes of bimodules; I will describe the correct version, which is presumably the version they secretly wanted but were afraid to write about. Also, I will continue to ignore antipodes.) A hopfish algebra is:

  • A $\mathbb K$-algebra $A$,
  • a bimodule ${_A \Delta _{A\otimes A}}$,
  • and an "associativity" isomorphism ${_A \Delta_{A\otimes A}} \underset{A\otimes A}\otimes ({_A A _A} \otimes {_A \Delta_{A\otimes A}}) \overset{\phi}\to {_A \Delta_{A\otimes A}} \underset{A\otimes A}\otimes ( {_A \Delta_{A\otimes A}}\otimes{_A A _A} )$,
  • such that $\phi$ satisfies a pentagon.

Coining some words, a hopfish algebra is commutatish if there is an isomorphism ${_A \Delta _{A\otimes A}} \to {_A \Delta^{\rm flip} _{A\otimes A}}$, where ${_A \Delta^{\rm flip} _{A\otimes A}}$ is the module $\Delta$ with the action by the two $A$s on the right flipped, which is an involution and satisfies (with $\phi$) two hexagons.

The idea is the following. If $A$ is an algebra so that the category $_A\operatorname{mod}$ of left $A$-modules has a $\mathbb K$-linear cocontinuous monoidal structure, then this structure makes $A$ hopfish. Given a hopfish structure, the corresponding monoidal structure is: $$_A\bigl((_A X) \underset{\Delta}\otimes (_AY) \bigr) \overset{\rm def}= {_A \Delta _{A\otimes A}} \underset{A\otimes A}\otimes ({_AX}\underset{\mathbb K}\otimes {_AY})$$

The three examples above are naturally hopfish.

  1. Any commutative ring $R$ is hopfish, with the bimodule given as the "comodulation" of the multiplication map $R \otimes R \to R$ (an algebra homomorphism iff $R$ is commutative), i.e. the bimodule is $_R R _{R\otimes R}$, where the right action is obvious and the left action is "multiply and act". When $R$ is the algebra of functions on a set $M$, then the corresponding monoidal structure on sheaves is the "pointwise" or "fiber-by-fiber" tensor product.
  2. A group algebra $\mathbb K G$ is hopfish with the bimodule given by the "modulation" of the "duplication" algebra homomorphism $\mathbb K G \to \mathbb K G \otimes \mathbb K G$ given on a basis by $g \mapsto g\otimes g$. I.e. the bimodule is $_{\mathbb K G} {\mathbb K G \otimes \mathbb K G} _{\mathbb K G \otimes \mathbb K G}$, where the left action is obvious and the right action is given by the duplication homomorphism. The corresponding monoidal structure on sheaves is the usual tensor structure on $G$-representations.
  3. Hopfish structures are designed to be well-behaved under Morita equivalence. The Morita equivalence between $\operatorname{Mat}(n,\mathbb K)$ and $\mathbb K$ is given by $\mathbb K^{\oplus n}$ with the obvious actions. Pushing the hopfish structure on $\mathbb K$ through this Morita equivalence gives the bimodule: $$_{\operatorname{Mat}(n)} {\operatorname{Mat}(n\times n^2)} _{\operatorname{Mat}(n^2) = \operatorname{Mat}(n) \otimes \operatorname{Mat}(n)}$$

In fact, all three examples follow from a general construction that I will now describe.

Let $C = \{C_1 \rightrightarrows C_0\}$ be a groupoid and $A = \mathbb K C_1$ the corresponding algebra of functions. As a vector space, $\Delta = \mathbb K ( C_1 { \underset{^t{C_0}^t}\times} C_1)$ is spanned by pairs $(g,h)$ of morphisms with the same target. Identifying the basis of $A\otimes A$ as all pairs of morphisms $(x,y)$, the bimodule structure is: $$ a \cdot (g,h) \cdot (x,y) = (agx,ahy) $$ with the convention that if any multiplication is non-composable, the whole pair is $0$. This determines an action. The associativity isomorphism comes from identifying both sides with the space spanned by triples of arrows with the same target. This hopfish algebra is always commutatish, by switching $(g,h) \mapsto (h,g)$.

In cartoons, $\mathbb K C_0 = \{\bullet\}$, $\mathbb K C_1 = \{\leftarrow\}$, and $\Delta = \{ \rightarrow \bullet \leftarrow\}$.

Extending to a 2-functor

Let $C = \{ C_1 \rightrightarrows C_0\}$ be a groupoid. A left $C$-set is a functor $C \to \text{FinSet}$, or more precisely an arrow $C_0 \leftarrow S$ and an action $C_1 \underset{C_0}\times S \to S$ satisfying an obvious square. If $C,D$ are two groupoids, a bibundle is a diagram $C_0 \leftarrow S \rightarrow D_0$ with commuting actions $C_1 \underset{C_0}\times S \underset{D_0}\times D \to S$.

It should be pretty much clear that if $S$ is a left $C$-set, then $\mathbb K S$ is a left $\mathbb K C_1$-module, and similar on the right, so that linearization takes bibundles to bimodules. Moreover, it should be clear that $\mathbb K : \underset{C_0}\times \to \underset{\mathbb K C_1}\otimes$, and that morphisms of bibundles go to morphisms of bimodules. Thus, linearization is a 2-functor from $\text{Gpoid}$ to $\text{Alg}$. In fact, it lands within the sub-2-category of commutatish hopfish algebras.

In particular, the hopfish algebra corresponding to a groupoid, considered up to Morita equivalence, gives an invariant of the stack represented by said groupoid.

Further questions

There are largely two issues that I would like to understand in the construction.

  • If our groupoids consist of topological spaces, then taking "linear combinations of points" may not be the right thing, as it ignores the topology. But it is no longer correct to identify, as vector spaces, $\mathbb K S$ with the algebra of, say, continuous functions on $S$. In particular, if $G$ is a compact Lie group, then the algebra of functions on $\text{pt}/G$ should be some "convolution algebra" for $G$, and in particular the elements of the convolution algebra are not functions, but rather distributions. I don't know how to say such words when I move far away from manifolds, but presumably the C-star algebraists do.
  • Nowhere did I use the antipode. In Op. Cit., defining what I have called a "Hopfish algebra" (what they call "sesquialgebra") takes a page, whereas the antipode takes most of the paper. The antipode must also come into play to define the $*$-involution in the C-star approach.

Finally, the question arises:

  • Is the functor from groupoids to commutative Hopfish algebras the right 2-notion of "full and faithful"? I.e. it is a complete invariant of stacks? I.e. is a stack recoverable from its commutatish Hopfish algebra up to Morita? Is a morphism of stacks recoverable from a morphism of algebras? Etc.
  • What is the essential image of the functor described above? Is every commutative Hopfish algebra Morita-equivalent to one coming from a groupoid? Does every bimodule thereof come from a bibundle? The second is doubtful; the first is more promising.
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In the algebraic setting the functor is indeed fully faithful.. however I have no idea about the essential image (nor does my source, <a href="arxiv.org/abs/math/0412266">Lurie's paper</a>).. though I would love to learn some useful properties satisfied by the image. –  David Ben-Zvi Jul 13 '10 at 19:34
    
Oh, awesome. Lurie's paper is the thing missing from my library, I think. –  Theo Johnson-Freyd Jul 13 '10 at 21:57
    
Proper link to Lurie's paper Tannaka Duality for Geometric Stacks: arxiv.org/abs/math/0412266 –  David Roberts Nov 3 at 7:57

The construction that goes from groupoids to C* algebras is given by forming the reduced C-star algebra, which is a generalisation of the algebra of functions on a space and the convolution algebra of functions on a compact Lie group.

This construction can probably be made into a nice 2-functor, but it will almost certainly not be an equivalence of 2-categories. Passing from a Lie groupoid to its reduced C-star algebra preserves K-theoretic information, but it throws other information away. Ulrich Bunke once told me a simple example of two Lie groupoids that are not Morita equivalent but have Morita equivalent reduced C-star algebras. Unfortunately I can't remember what the example is, but it was sufficiently straightforward that I doubt adding in some extra adjectives could reasonably exclude it or similar examples.

The fact of the matter is that the groupoids really contain more information than the C-star algebras. You should think of the C-star algebra as representing a K-theory homotopy type (something like the spectrum $X\wedge KU$ if $X$ is an ordinary topological space), whereas the groupoid represents the full homotopy type of a 'noncommutative space'. I remember Graeme Segal explaining in some talks about how Morita equivalence in C-star algebras automatically imposes a kind of periodicity that can be seen as Bott periodicity. This certainly indicates that the passage from groupoids to C-star algebras is not generally an equivalence.

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Going from a group to its reduced $C^*$-algebra is not functorial -- that would be the full group $C^*$-algebra construction -- but perhaps I'm missing something which works better in the groupoid case? –  Yemon Choi Jul 9 '10 at 21:57
    
So, "reduced C-star algebra" didn't turn up any useful hits on Google. Would you mind giving the details of the construction? –  Theo Johnson-Freyd Jul 10 '10 at 2:33
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For details on the constructions that go from groupoids to C* algebras (reduced and full C* algebras) see the book: J. Renault, A groupoid approach to C^*-algebras. Lecture Notes in Mathematics, 793. Springer, Berlin, 1980. –  Jeffrey Giansiracusa Jul 10 '10 at 7:16
    
@Yemon: I agree that the reduced C* algebra construction is not strictly functorial, but we have 2-categories here and I think it can be made to define a 2-functor. –  Jeffrey Giansiracusa Jul 10 '10 at 7:41
    
@Jeffrey: That sounds plausible: presumably the reason is loosely that, in a 2-cat setup, we're now remembering the Hilbert space being acted on? algebras as 0-cells, (bi-)modules as 1-cells, etc? –  Yemon Choi Jul 10 '10 at 9:24

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