I asked this over on Math.SE but it remained completely silent for over a week so I've deleted it and am reposting it here (I'm not really sure which site it fits better). The question itself is somewhat vague since I don't know precisely what I'm looking for and I'm hoping someone can point me in the right direction. Let me also preface this by saying that while I am familiar with the basic definitions of groupoids and sheaves, that is basically where my knowledge stops, so I am very much feeling my way in the dark here. I think a concrete example would best illustrate the situation.

Consider the open sets in $\mathbb{C}$ (or any Riemann surface, really) as the objects of a category. Let $\mathrm{Hom}(U,V)=\{\text{holomorphic maps }U\longrightarrow V\}$; composition of morphisms is just composition of functions. If we keep only the isomorphisms in this category, we get the titular "conformal" groupoid $\mathrm{Conf}$.

Now consider some sheaf of functions $\mathcal{O}$ on $\mathbb{C}$, e.g. smooth or holomorphic or whatever. $\mathrm{Conf}$ "acts" on $\mathcal{O}$ the sense that any $\phi\in\mathrm{Hom}(U, V)$ can be used to transfer sections between $\mathcal{O}(U)$ and $\mathcal{O}(V)$ in a reasonable way: if $f\in\mathcal{O}(V)$, then we can define $f\circ\phi\in\mathcal{O}(U)$.

But note that this action doesn't really involve the whole structure of $\mathcal{O}$ since I haven't mentioned the restriction maps of $\mathcal{O}$. And indeed $\mathrm{Conf}$ has more structure than just a groupoid, there are also "restriction operations": if $W\subset U$ and $\phi\in\mathrm{Hom}(U, V)$, we can form $\phi\vert_W\in\mathrm{Hom}(W, \phi(W))$. The action from before behaves naturally with respect to this: if again $f\in\mathcal{O}(V)$, then $$ (f\circ\phi)\vert_W = f\vert_{\phi(W)}\circ\phi\vert_W $$ The reason for the CFT tag should be clearer now, but let me say some more on this. If we look at something with different symmetry, like a standard Poincaré-covariant field theory on Minkowski space, we could play a similar game, but it doesn't seem as interesting: once we know that a map $\phi:U\longrightarrow V$ between open subsets preserves the metric, it has (I believe) a unique extension to a global Poincaré transformation, so somehow the transformations are too rigid to admit an interesting local structure (but I may be glossing over something here, I don't know). On the other hand, the same is emphatically not the case for conformal transformations: here it is much more important to compare not just global field configurations but also individual subsystems, i.e. field configurations on open subsets. This post goes into more detail on this, also working with the groupoid of conformal maps and its corresponding algebroid.

As I have said, this is all somewhat vague, and I haven't even made my question explicit. I suppose it would be something like, how do we tie all of this together neatly? That is, what algebraic structure captures this kind of groupoid-with-restriction and how do we describe its action on a given sheaf more precisely? Whatever the answers are, given that the question originated in CFT, they would be particularly satisfying if it followed that $\mathrm{Conf}$ bore some strong relation to the Witt algebra.


1 Answer 1


That is, what algebraic structure captures this kind of groupoid-with-restriction and how do we describe its action on a given sheaf more precisely?

This structure is well known and has many equivalent incarnations: inductive groupoid, inverse semigroup, etale groupoid, etale stack.

Indeed, the bicategories of these objects turn out to be equivalent once one imposes certain reasonable conditions (such as requiring inverse semigroups to be complete and distributive) and equips them with an appropriate notion of 1-morphisms and 2-morphisms. This equivalence is the main result of the PhD thesis of Nilan Manoj Chathuranga (soon to be on arXiv).

These languages also provide a very satisfactory answer to your second question: the action of Conf on a given sheaf is now simply a sheaf of sets (appropriately defined) on the inductive groupoid or inverse semigroup associated to Conf. Equivalently, it is an etale map to the etale stack associated to Conf, i.e., an object in the slice category Et/Conf, where Et denotes the category of etale stacks and etale maps.

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    $\begingroup$ Thanks! I still have a lot of work to do before I understand these things, but at least now I have some concrete names and concepts to work with. $\endgroup$
    – J_P
    Commented Apr 21, 2022 at 18:30

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