nLab smooth groupoid

Idea

The concept of smooth groupoid is the first generalization of the concept of smooth sets (smooth spaces) to higher differential geometry, it is a joint generalization of smooth sets (smooth manifolds, diffeological spaces, …), orbifolds and Lie groupoids, hence it also subsumes deloopings of Lie groups and diffeological groups. It also subsumes smooth moduli stacks, for instance of gauge fields.

Technically, a smooth groupoid is a groupoid-valued stack on the site of smooth manifolds (sometimes called a “smooth stack”, but this is somewhat ambiguous), or equivalently on its dense subsite CartSp of just Cartesian spaces and smooth functions between them. This is equivalently an 1-truncated smooth ∞-groupoid.

For more see also at geometry of physics – smooth homotopy types.

Definition

Remark

on how not to define smooth groupoids

One could be tempted to define smooth groupoids to be internal groupoids in smooth sets. Restricting this definition to groupoids internal to diffeological spaces yields the concept of diffeological groupoid, restricting it further to groupoids internal to smooth manifolds yields the concept of Lie groupoid. While these are respectable definitions in themselves, care needs to be exercised in interpreting them correctly: they do not in themselves exhibit the correct homotopy theory of smooth groupoids. One may fix this by changing the concept of morphisms to generalized morphisms called Morita morphisms or bibundles. These are certainly useful tools for working with smooth groupoids, and they serve to present their correct homotopy theory, but they do not serve well as the definition of this homotopy theory. For instance without further insight it is impossible to guess from the concept of bibundles between groupoids its correct generalization to smooth 2-groupoids etc.

But just as the definition of smooth sets, contrasted with that of smooth manifolds, is not only more powerful but also simpler, so there is a definition of smooth groupoids which not only does give the correct homotopy theory, but it does so while being much more transparent than these more traditional presentations.

Now, this definition, given below, amounts to saying that smooth groupoids are stacks on the site of smooth manifolds, and that in turn may tend to not sound like a simple definition at all. But there is a further simplification at work. Traditional texts tend to define stacks in terms of the comparatively intricate structures of pseudofunctors or (dually under the Grothendieck construction) fibered categories satisfying descent, and the rich structure of these combinatorial objects tends to become unwieldy already for fairly simple examples. But the theory of localization of categories turns out to handle the same theory of stacks by much more tractable means (e.g. Hollander 01). Here a stack is presented by a plain functor with no descent condition imposed, something that is hence as simple as a pair of two presheaves. The nature of stacks is then instead just encoded in remembering that some of the morphisms of presheaves of groupoids are to be labeled as weak equivalences, namely those that locally restrict to equivalences of groupoids.

This style of definition combines the simplicity of the naive definition of Lie groupoids with the full power of homotopy theory, and it immediately generalizes to a definition of ∞-stacks which is just as simple, hence, in the present context, to a definition of smooth ∞-groupoid.

Pre-smooth groupoids

Definition

Write CartSp for the category of Cartesian spaces $\mathbb{R}^n$, $n \in \mathbb{N}$, with smooth functions between them. (The full subcategory of SmoothMfd on the Cartesian spaces.) Regard this as a site by equipping it with the coverage of (differentiably) good open covers.

For more on this see at geometry of physics – coordinate systems.

In view of the motivation for sheaves, cohomology and higher stacks and in direct analogy with the discussion at geometry of physics – smooth sets, just replacing sets by groupoids throughout, we set:

Definition

A pre-smooth groupoid $X$ is a presheaf of groupoids on CartSp, hence a functor

$$X \colon CartSp^{op} \longrightarrow Grpd$$

from CartSp to the 1-category Grpd.

See at geometry of physics – homotopy types – groupoids for more on bare groupoids. Here we will freely assume familiarity with these.

Remark

The intuition for def. is the following: a smooth structure on a groupoid is to be determined by which maps from abstract coordinate systems $\mathbb{R}^n$ into it are to be regarded as smooth maps. Since, by its groupoidal nature, two such maps may be related by a smooth homotopy/gauge transformation (in physics: “twisted sectors”), the collection of all smooth functions from $\mathbb{R}^n$ into the smooth groupoid $X$ is itself a bare groupoid. We here define a pre-smooth groupoid as something that assigns to each abstract coordinate systems a groupoid of would-be smooth maps into it

“$X(\mathbb{R}^n) = \left\{ \array{ & \nearrow \searrow^{\mathrlap{smooth}} \\ \mathbb{R}^n &\Downarrow& X \\ & \searrow \nearrow_{\mathrlap{smooth}} } \right\}$”

We sometimes therefore speak of $X(\mathbb{R}^n)$ as the groupoid of plots or of probes of $X$ (which here is only defined by these!) by $n$-dimensional coordinate systems.

The Yoneda lemma will turn this intuition into a theorem. For that we need to speak of homomorphisms of pre-smooth groupoids, hence of “smooth functors” betwen pre-smooth groupoids.

Definition

A homomorphism or smooth map between pre-smooth groupoids is a natural transformation between the presheaves that they are. We write

$$PreSmooth1Type \coloneqq Funct(CartSp^{op}, Grpd)$$

for the functor category, the category of pre-smooth groupoids, def. , regarded naturally as a Grpd-enriched category.

This means that for $X,Y \in PreSmooth1Type$ two pre-smooth groupoids, then the bare hom-groupoid

$$PreSmooth1Type(X,Y)_\bullet \in Grpd$$

between them has

• as objects the natural transformations $f \colon X \to Y$ between $X$ and $Y$ regarded as functors, hence collections $\{f(\mathbb{R}^n)\}_{n \in \mathbb{N}}$ of functors between groupoids of probes

  $$f(\mathbb{R}^n) \colon X(\mathbb{R}^n) \longrightarrow Y(\mathbb{R}^n)$$

  such that for every smooth function $\phi \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ between abstract coordinate charts, these form a (strictly) commuting diagram with the probe-pullback functors of $X$ and $Y$:

  $$\array{ X(\mathbb{R}^{n_1}) &\overset{f(\mathbb{R}^n)}{\longrightarrow}& Y(\mathbb{R}^{n_1}) \\ {}^{\mathllap{X(\phi)}}\downarrow && \downarrow^{\mathrlap{Y(\phi)}} \\ X(\mathbb{R}^{n_2}) &\overset{f(\mathbb{R}^n)}{\longrightarrow}& Y(\mathbb{R}^{n_2}) } \,;$$

• as morphisms $H \colon f \rightarrow g$ the “modifications” of these, hence collections $H(\mathbb{R}^n) \colon f(\mathbb{R}^n)\to g(\mathbb{R}^n)$ of natural transformations, such that for every smooth function $\phi \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ the whiskering of these satisfies

  $$\array{ X(\mathbb{R}^{n_1}) \\ \downarrow^{\mathrlap{X(\phi)}} \\ & \nearrow \searrow^{\mathrlap{f(\mathbb{R}^{n_2})}} \\ X(\mathbb{R}^{n_2}) &\Downarrow^{H(\mathbb{R}^{n_2})}& Y(\mathbb{R}^{n_2}) \\ & \searrow \nearrow_{\mathrlap{g(\mathbb{R}^{n_1})}} } \;\;\; = \;\;\; \array{ & \nearrow \searrow^{\mathrlap{f(\mathbb{R}^{n_1})}} \\ X(\mathbb{R}^{n_1}) &\Downarrow^{H(\mathbb{R}^{n_1})}& Y(\mathbb{R}^{n_1}) \\ & \searrow \nearrow_{\mathrlap{g(\mathbb{R}^{n_1})}} \\ && \downarrow^{\mathrlap{Y(\phi)}} \\ && Y(\mathbb{R}^{n_2}) }$$

Remark

Despite the size of the diagrams in def. , what they encode is immediate: this just says that smooth maps between pre-smooth groupoids take all probes of $X$ by abstract coordinate charts to probes of $Y$ by these charts, and take gauge transformations between these to gauge transformations between those, in a way that it compatible with changing probes along smooth maps.

The following is the Yoneda lemma in this context, and it says that this intuition in remark is fully correct:

Proposition

Every Cartesian space $\mathbb{R}^n$ defines a pre-smooth groupoid $\underline{\mathbb{R}}^n$, def. , by the assignment

$$\underline{\mathbb{R}}^n \colon \mathbb{R}^k \mapsto C^\infty(\mathbb{R}^k, \mathbb{R}^n) \in Set \hookrightarrow Grpd$$

where the set of smooth functions on the right is regarded as a groupoid with only identity morphisms. This construction constitutes a fully faithful functor

$$\underline{(-)} \colon CartSp \hookrightarrow PreSmooth1Type$$

making CartSp a full subcategory of that of pre-smooth groupoids (the Yoneda embedding). Under this embedding for any pre-smooth groupoid $X\in PreSmooth1Type$ and any Cartesian space $\mathbb{R}^n$, there is a natural equivalence of groupoids

$$PreSmooth1Type(\underline{\mathbb{R}}^n, X) \simeq X(\mathbb{R}^n)$$

(the Yoneda lemma proper).

Remark

The last statement of the Yoneda lemma in prop. expresses just the intuition of remark and justifies removing the quotation marks displayed there. It also justifies dropping the extra underline denoting the Yoneda embedding. We will freely identify from now on $\mathbb{R}^n$ with the pre-smooth groupoid that it represents.

Example

For $X$ a smooth set (e.g. a smooth manifold), hence in particular a functor

$$X \colon CartSp^{op} \to Set$$

then its embedding into groupoids

$$X \colon CartSp^{op} \to Set \hookrightarrow Grpd$$

is a pre-smooth groupoid.

More generally:

Example

Let

$$\mathcal{G}_\bullet = \left\{ \mathcal{G}_1 \stackrel{\overset{s}{\longrightarrow}}{\stackrel{\overset{i}{\longleftarrow}}{\underset{t}{\longrightarrow}}} \mathcal{G}_0 \right\}$$

be an internal groupoid in smooth sets, hence a pair $\mathcal{G}_0, \mathcal{G}_1 \in Smooth0Type$ of smooth sets, equipped with source, target, identity homomorphisms of smooth sets between them, and equipped with a compatibly unital, associative and invertible composition map $\mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 \to \mathcal{G}_1$. By definition of smooth sets, this means that for every abstract coordinate system $\mathbb{R}^n$ then

$$\mathcal{G}(\mathbb{R}^n)_\bullet = \left\{ \mathcal{G}(\mathbb{R}^n)_1 \stackrel{\overset{s(\mathbb{R}^n)}{\longrightarrow}}{\stackrel{\overset{i(\mathbb{R}^n)}{\longleftarrow}}{\underset{t(\mathbb{R}^n)}{\longrightarrow}}} \mathcal{G}(\mathbb{R}^n)_0 \right\}$$

is a bare groupoid. Moreover, this assignment is functorial and hence defines a pre-smooth groupoid

$$\mathbb{R}^n \mapsto \mathcal{G}(\mathbb{R}^n)_\bullet \,.$$

This exhibits sequence of full subcategory inclusions

$$Grp \hookrightarrow LieGrpd \hookrightarrow DiffGrpd \hookrightarrow Grpd(Smooth0Type) \hookrightarrow PreSmooth1Type$$

of internal groupoids in smooth sets into pre-smooth groupoids, and hence (by further restriction) also of diffeological groupoids, of Lie groupoids and of course of just bare groupoids, too.

Regarding a bare groupoid $\mathcal{G}_\bullet$ as a pre-smooth groupoid this way means to regard it as equipped with discrete smooth structure. It is given by the constant presheaf

$$\mathbb{R}^n \mapsto \mathcal{G}_\bullet$$

exhibiting the fact that smooth functions from an $\mathbb{R}^n$ into a geometrically discrete space are constant on one of the points of this space, a situation which here is only refined by the fact that every morphism in the groupoid thus gives a homotopy/gauge transformation between the two smooth functions that are constant on the two endpoints of this morphism.

A particular case of this of special importance is this:

Example

For $G$ a Lie group, we write $(\mathbf{B}G)_\bullet \in Grpd(Smooth0Type)$ for the Lie groupoid which

$$(\mathbf{B}G)_\bullet = \left( G \stackrel{\longrightarrow}{\stackrel{\overset{i}{\longleftarrow}}{\longrightarrow}} \ast \right)$$

whose composition is the product operation in the group (the groupoidal delooping of $G$). The pre-smooth groupoid that this corresponds to under the embedding of example has groupoids of probes of the form

$$\mathbb{R}^n \mapsto B \left(C^\infty(\mathbb{R}^n,G)_{disc}\right)$$

where on the right we have the homotopy 1-type whose fundamental group is that of smooth $G$-valued functions on $\mathbb{R}^n$, under pointwise mulitplication.

More specifically, the following class of examples plays a special role in the theory, as the encode what it takes for a pre-smooth groupoid to be a genuinely smooth groupoid.

Example

For $X$ a smooth manifold, let $\{U_i \to X\}_{i \in I}$ be an open cover of $X$. Its Cech groupoid is the Lie groupoid (diffeological groupoid) $C(\{U_i\})_\bullet$ whose

• manifold of objects is $C(\{U_i\})_0 \coloneqq \underset{i \in I}{\coprod} U_i$ is the disjoint union of all the charts of the cover;

• manifold of morphisms is $C(\{U_i\})_1 \coloneqq \underset{i,j \in I}{\coprod}U_i \underset{X}{\times} U_j$ is the disjoint union of all intersections of charts.

and whose source, target and identity maps are the evident inclusions. There is then a unique composition operation.

So a global point $\ast \to C(\{U_i\})_\bullet$ may be thought of as a pair $(x,i)$ of a point in the manifold $X$ and a label $i$ of a chart $U_i$ that contains it, and there is is precisely one morphism between two such global point $(x,i)\to (y,j)$ whenever $x = y$ in $X$ and both $U_i$ as well as $U_j$ contain $x$, hence one morphism for each point in an intersection of two patches. Composition of morphism is just re-remembering which intersections they sit in, the schematic picture of the Cech groupoid is this this:

$$C(\{U_i\})_\bullet = \left\{ \array{ && (x,j) \\ & \nearrow && \searrow \\ (x,i) && \longrightarrow && (x,k) } \right\} \,.$$

Under the embedding of example there is an evident morphism of pre-smooth groupoids

$$C(\{U_i\}) \longrightarrow X$$

from this Cech groupoid of the cover to the manifold that is being covered. This morphism simply forgets the information of which chart or intersection of charts a point is regarded to be in, and just remembers it as a point of $X$.

$$((x,i) \to (x,j)) \mapsto (x \stackrel{id_x}{\to} x) \,.$$

Proposition

Let $X$ a smooth manifold, $\{U_i \to X\}$ an open cover and $C(\{U_i\})$ the corresponding Cech groupoid, def. . Let $G$ be a Lie group and $\mathbf{B}G$ its groupoidal delooping according to example .

Then the hom-groupoid $PreSmooth1Type(C(\{U_i\}), \mathbf{B}G)$ of maps from $C(\{U_i\})$ to $\mathbf{B}G$, def. , has

• as objects the Cech cocycles of degree 1 on $X$ relative to the cover and with values in $G$; i.e. collections of smooth functions

  $$g_{ i j} \;\colon\; U_i \underset{X}{\times} U_j \longrightarrow G$$

  satisfying on each triple intersection the cocycle condition

  $$g_{i j} g_{j k} = g_{i k}$$

• as morphisms the coboundaries between such cocycles $\{g_{i j}\}$ and $\{\tilde g_{j k}\}$, hence collections of smooth functions

  $$h_{i} \colon U_i \longrightarrow G$$

  such that on each intersection of charts

  $$g_{i j} h_j = h_i \tilde g_{i j} \,.$$

In particular the connected components of the Hom-groupoid is hence the Cech cohomology itself:

$$\pi_0 PreSmooth1Type(C(\{U_i\}), \mathbf{B}G) \simeq H^1_{Cech}(X,\{U_i\}; G) \,.$$

Proof

It is a matter of unwinding the definitions that the statement holds disregarding smooth structure everywhere, hence after evaluating the morphisms of pre-smooth groupoids on $\mathbb{R}^0$. That the component functions that one finds this way all have to be smooth then follows componentwise with the Yoneda lemma for presheaves of sets on $CartSp$ (as discussed at geometry of physics – smooth sets).

Gluing

The “bootstrap”-definition of pre-smooth groupoids above works as intended, by prop. , it just needs to be restricted now to something a little less general. The issue is that while this definition consistently identifies a smooth-structure-to-be by what its possible smooth probes are, it does not enforce yet the consistent gluing of probes:

for $X$ a pre-smooth groupoid, then given a probe of it of the form $\sigma \colon \mathbb{R}^n \to X$ and given a covering $\{U_i \to \mathbb{R}^n\}$ of the probe space by other probe spaces $U_i \simeq \mathbb{R}^n$, then it should be possible to reconstruct $\sigma$ from knowing its restrictions $\sigma_{|U_i}\colon U_i \to X$ to these charts, and the information of how these are identified by gauge transformations on double overlaps.

Such a system of local data and of gauge identification on double overlaps is just what maps out of the Cech groupoid $C(\{U_i\})$, def. , encode. This is shown by prop. for the case that $X$ is of the form $(\mathbf{B}G)_\bullet$, example , but this is already the archetypical case.

In other words, the condition that smooth probes of $X$ by coordinate charts $\mathbb{R}^n$ glue along covers $\{U_i \to \mathbb{R}^n\}$ of these charts is the condition that the groupoid of smooth maps out of $\mathbb{R}^n$ itself

$$\mathbb{R}^n \longrightarrow X$$

is equivalent to the groupoid of maps out of the Cech groupoid of any cover

$$C(\{U_i\}) \longrightarrow X$$

and is so via the “restriction” map that takes the former and precomposes it with the canonical map $C(\{U_i\}) \to \mathbb{R}^n$.

$$\array{ C(\{U_i\}) &\longrightarrow& X \\ \downarrow & \nearrow \\ \mathbb{R}^n } \,.$$

Definition

A pre-smooth groupoid $X \in PreSmooth1Type$, def. , satisfies descent if for all $n \in \mathbb{N}$ and for all differentiably good open covers $\{U_i \to \mathbb{R}^n\}$ of the $n$-dimensional abstract coordinate chart, the functor

$$X(\mathbb{R}^n) \simeq PreSmooth1Type(\mathbb{R}^n, X) \longrightarrow PreSmooth1Type(C(\{U_i\}),X)$$

given by pre-composition with $C(\{U_i\}) \to \mathbb{R}^n$, is an equivalence of groupoids.

Remark

The condition in def. is called the stack condition, or the condition of descent, alluding to the fact that it says that $X$ “descends” down along the cover projection. So a smooth groupoid is a stack on the site CartSp. This is a higher analog of the sheaf condition (see the next example) and hence a more systematic terminology would be to say that such $X$ is a 2-sheaf or rather a (2,1)-sheaf (since it takes values in groupoids as opposed to in more general categories).

Proposition

Let $X \in PreSmooth0Type \hookrightarrow PreSmooth1Type$ be a pre-smooth groupoid which is really just a pre-smooth set, hence a presheaf on $CartSp$ that takes values in groupoids with only identity morphisms

$$X \colon CartSp^{op} \longrightarrow Set \hookrightarrow Grp \,.$$

Then $X$ is a smooth groupoid in the sense of def. precisely if it is a smooth set, hence precisely if, as a presheaf, it satisfies the sheaf condition.

Example

In particular, for $X \in SmoothMfd \hookrightarrow PreSmooth0Type \hookrightarrow PreSmooth1Type$ a smooth manifold, it satisfies descent as a pre-smooth groupoid.

Remark

There is an alternative formulation of the whole theory where instead of the site CartSp one uses the site SmoothMfd of all smooth manifolds. Everything discussed so far goes through verbatim for that site, too, but then the descent condition in def. is a much stronger condition.

For instance the presheaves of the form $(\mathbf{B}G)_\bullet = (G \stackrel{\longrightarrow}{\longrightarrow} \ast)$ from example satisfy descent on $CartSp$, but not all $SmoothMfd$. Still, once we have defined the higher category of smooth groupoids, the definition will be equivalent for both choices of sites.

The choice of the smaller site is the one that is easier to work with, and therefore we stick with that. In fact, most every example of a pre-smooth groupoid that one runs into satisfies descent on $CartSp$.

For example:

Proposition

For $G$ a Lie group, then the pre-smooth delooping groupoid $(\mathbf{B}G)_\bullet$ of example satisfies descent on $CartSp$, def. .

Proof

For $\{U_i \to \mathbb{R}^n\}$ a differentiably good open cover, then by prop. the groupoid $PreSmooth1Type(C(\{U_i\}),(\mathbf{B}G)_\bullet)$ is the groupoid of Cech 1-cocycles and coboundaries with coeffcients in $G$ on $\mathbb{R}^n$ relative to the cover. But this is equivalently the groupoid of $G$-principal bundles on $\mathbb{R}^n$. Now because the underlying topological space of $\mathbb{R}^n$ is contractible, all $G$-principal bundles on it are equivalent to the trivial one. But this is evidently represented by the image of point under the map

$$B (C^\infty(G)_{disc}) \simeq PreSmooth1Type(\mathbb{R}^n, (\mathbf{B}G)_\bullet) \longrightarrow PreSmooth1Type(C(\{U_i\}), (\mathbf{B}G)_\bullet) \,.$$

Therefore this is an essentially surjective functor of groupoids. Moreover, the automorphisms of the trivial $G$-principal bundles are precisely the smooth $G$-functions, hence this is also a fully faithful functor of groupoids. Accordingly it is an equivalence of groupoids.

The same argument however shows that on the larger site SmoothMfd this object does not satisfy descent. Put positively, this is the content of prop. . below.

Weak equivalences

While the morphisms of pre-smooth groupoids defined above correctly encode morphisms of smooth structures (by taking smooth probes compatibly to smooth probes), they are not sensitive enough yet to the required concept of equivalence. This is because smooth structure, being about existence of differentiation, is to be detected entirely locally, namely stalk-wise. If for instance $X$ is a smooth manifold, then its smooth structure is determined, around any of its points, by the smooth structure of an arbitrarily small open ball around that point.

Definition

For $n \in \mathbb{N}$, and $0 \lt r \lt 1 \in \mathbb{R}$ let

$$\mathbb{R}^n \simeq D^n_r \hookrightarrow \mathbb{R}^n$$

be the smooth function that regards the Cartesian space $\mathbb{R}^n$ as the standard $n$-disk of radius $r$ around the origin in $\mathbb{R}^n$.

For $X \in PreSmooth1Type$ we write

$$(D^n)^* X \coloneqq {\underset{\longrightarrow_r}{\lim}} X(D^n_r) \in Grpd$$

for the colimit (in the 1-category of groupoids) of the restrictions of its groupoids of plots along the inclusion of these open balls – the $n$-stalk of $X$. This extends to a functor

$$(D^n)^* \colon PreSmooth1Type \longrightarrow Grpd$$

This means that objects in $(D^n)^\ast X$ are equivalence classes of pairs $(r,x_r)$ where $0 \lt r \lt 1$ and where $x_r \in X(D^n_r)$, with two such pairs being equivalent $(r_1, x_{r_1})\sim (r_2, x_{r_2})$ precisely if there is $r_0 \lt r_1,r_2$ such that $x_{r_1}$ becomes equal to $x_{r_2}$ after restriction to $D^n_{r_0}$.

Definition

A morphism $f \colon X \longrightarrow Y$ of pre-smooth groupoids, def. , is called a local weak equivalence if for every $n \in \mathbb{N}$ the $n$-stalk, def. , is an equivalence of groupoids

$$((D^n)^* f) \colon (D^n)^* X \stackrel{}{\longrightarrow} (D^n)^* Y \,.$$

We write $X\stackrel{\simeq}{\longrightarrow} Y$ for local weak equivalences of pre-smooth groupoids. We will mostly just say weak equivalence for short.

Proposition

For $X$ a smooth manifold and $\{U_i \to X\}$ an open cover for it, then the canonical morphism from the corresponding Cech groupoid to $X$, def. , is a local weak equivalence in the sense of def. .

Proof

The $n$-stalk of the smooth manifold $X$ regarded as a presheaf is the set of equivalence classes of maps from open pointed $n$-disks into it, where two such are identified if they coincide on some small joint sub-disk of their domain. We may call this the set of germs of $X$ (but beware that this terminology is typically used for something a little bit more restrictive, namely for the case that $n$ is the dimension of $X$ and that all maps from the disks into $X$ are required to be embeddings).

On the other hand the $n$-stalk of $C(\{U_i\})$ is the groupoid whose set of objects is the set of germs, in this sense, of the disjoint union $\underset{i}{\coprod} U_i$, and whose set of morphisms is the set of germs of the disjoint union $\underset{i,j}{\coprod} U_i \underset{X}{\times} U_j$.

But now since the cover is by open subsets, it follows that for every $(x,i,j) \in \underset{i,j}{\coprod} U_i \underset{X}{\times} U_j$, then every germ of objects $[g]$ around $(x,i)$ has a representative $g$ that factors through this double intersection charts: $g \colon D^n_r \to U_i \underset{X}{\times} U_j \to \underset{i}{\coprod} U_u$. And similarly for $(x,j)$.

This means that the groupoid of $n$-stalks is a disjoint union of groupoids, one for each germ of $X$, all whose components are groupoids in which there is a unique morphism between any two objects, which are copies of this germ regarded as sitting in one of the charts of the cover. This means that each of these connected components is equivalent to the point.

Now the canonical cover projection sends each of these connected components to the germ that it corresponds to. Hence this is a an equivalence of groupoids.

Hypercovers

Definition

A morphism $p \colon Y \longrightarrow X$ of pre-smooth groupoids is called a split hypercover if

1. $Y$ is

   1. degreewise a coproduct of Cartesian spaces;

   2. such that the degenerate elements split off as a dijoint summand.

2. $p$ is a weak equivalence, def. .

Proposition

For $X$ a smooth manifold and $\{U_i \to X\}$ an open cover, then the canonical projection $C(\{U_i\}) \to X$ from the corresponding Cech groupoid, def. , is a split hypercover precisely if the cover is differentiably good.

Proof

For every cover the map is a weak equivalence, by prop. .

For the Cech groupoid the condition of cofibrancy in def. means that every non-empty finite intersection of patches is diffeomorphic to a Cartesian space, hence to an open ball. This is precisely the definition of differentialbly good open cover.

The $(2,1)$-category of smooth groupoids

The Grpd-enriched category of genuine smooth groupoids is that obtained from that of pre-smooth groupoids, def. by “universally turning local weak equivalences, def. , into actual homotopy equivalences”. This is stated formally by def. below, but for many applications in practice certain concrete presentations of what this means concretely are well sufficient, one of these we state below in prop. .

Definition

Write

$$Smooth1Type \coloneqq L_{lwe} PreSmooth1Type$$

for the (2,1)-category which is the simplicial localization of groupoid-valued presheaves at the local weak equivalences, def. .

An object $X \in Smooth1Type$ we call a smooth groupoid or smooth homotopy 1-type.

Proposition

Let $X,A \in PreSmooth1Type$ such that $A$ satisfies descent, def. . Let $Y \to X$ be a split hypercover of $X$, def. .

Then there is an equivalence of groupoids

$$Smooth1Type(X,A) \simeq PreSmooth1Type(Y,A)$$

between the hom-groupoid of smooth groupoids from $X$ to $A$, and that of pre-smooth groupoids, def. , from $Y$ to $A$.

Such statements follow with model structures on simplicial presheaves after embedding the present situation in the more general context of smooth infinity-groupoids. See there for more.

Remark

There is a canonical localization functor

$$PreSmooth1Type \longrightarrow L_{lwe} PreSmooth1Type = Smooth1Type \,.$$

which is really just the identity as a functor. Instead of doing anything to the objects, passing along this functor just means to change the definition of the hom-groupoids from the direct definition of def. to the localized definition.

When a pre-smooth groupoid is given by an internal groupoid $\mathcal{G}_\bullet$ in smooth sets via example , then we indicate its image under this functor by removing the subscript index, writing just $\mathcal{G}$. This reflects the fact that in $Smooth1Type$ it no longer makes sense to ask what the space of 0-cells and of 1-cells of an object is, as these are concepts not invariant under local weak equivalence.

In particular this means that we write $\mathbf{B}G$ for the image of $(\mathbf{B}G)_\bullet$ in $Smooth1Type$.

Proposition

Let $X$ be a smooth manifold, regarded as a smooth 0-groupoid, and let $G$ be a Lie group, with smooth delooping groupoid $\mathbf{B}G$ (example , remark ).

Then

$$Smooth1Type(X,\mathbf{B}G) \simeq G Bund(X)$$

is equivalently the groupoid of $G$-principal bundles on $X$.

Proof

By prop. the object $(\mathbf{B}G)_\bullet$ satisfies descent on CartSp. Choose $\{U_i \to X\}$ a differentiably good open cover. By prop. the correspoding Cech groupoid projection $C(\{U_i\}) \to X$ is a split hypercover, def. . Hence, by prop. , there is an equivalence of groupoids

$$Smooth1Type(X,\mathbf{B}G) \simeq PreSmooth1Type(C(\{U_i\}), (\mathbf{B}G)_\bullet) \,.$$

The groupoid on the right is, by prop. , the groupoid of Cech 1-cocycles and coboundaries with values in $G$ relative to a good open cover. This is equivalently the groupoid of $G$-principal bundles.

Remark

The content of prop. is in common jargon that: $\mathbf{B}G \in SmoothGrpd$ is the moduli stack of $G$-principal bundles“.

Properties

Embedding into smooth $\infty$-groupoids

By generalizing here groupoids to general Kan complexes and equivalences of groupoids to homotopy equivalences of Kan complexes, one obtains smooth ∞-stacks or smooth ∞-groupoids, which we write

$\;\;\;$Smooth∞Grpd $\coloneqq Sh_{(\infty,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, KanCplx)$.

We often write $\mathbf{H} \coloneqq$ Smooth∞Grpd for short.

By the (∞,1)-Yoneda lemma there is a sequence of faithful inclusions

$\;\;\;$ SmoothMfd $\hookrightarrow$ SmoothGrpd $\hookrightarrow$ Smooth∞Grpd.

This induces a corresponding inclusion of simplicial objects and hence of groupoid objects

$$LieGrpd \hookrightarrow Grpd_\infty(SmoothMfd) \hookrightarrow Grpd_\infty(Smooth\infty Grpd) \,.$$

For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a groupoid object we write

$$\mathcal{G}_0 \to \mathcal{G} \coloneqq \underset{\longrightarrow}{\lim}_{n} \mathcal{G}_n$$

for its (∞,1)-colimiting cocone, hence $\mathcal{G} \in \mathbf{H}$ (without subscript decoration) denotes the realization of $\mathcal{G}_\bullet$ as a single object in $\mathbf{H}$.

By the Giraud-Rezk-Lurie axioms of the (∞,1)-topos $\mathbf{H}$ this morphism $\mathcal{G}_0 \to \mathcal{G}$ is a 1-epimorphism – an atlas – and its construction establishes is an equivalence of (∞,1)-categories $Grpd_\infty(\mathbf{H}) \simeq \mathbf{H}^{\Delta^1}_{1epi}$, hence morphisms $\mathcal{G}_\bullet \to \mathcal{K}_\bullet$ in $Grpd_\infty(\mathbf{H})$ are equivalently diagrams in $\mathbf{H}$ of the form

$$\array{ \mathcal{G}_0 &\to& \mathcal{K}_0 \\ \downarrow &\swArrow& \downarrow \\ \mathcal{G} &\to& \mathcal{K} } \,.$$

This is evidently more constrained that just morphisms

$$\mathcal{G} \to \mathcal{K}$$

by themselves. The latter are the “generalized” or Morita morphisms between the groupoid objects $\mathcal{G}_\bullet$, $\mathcal{K}_\bullet$. These can be modeled as $\mathcal{G}_\bullet$-$\mathcal{K}_\bullet$-bibundles.

Examples

Zoo of concepts subsumed by smooth groupoids

Example

Every smooth space is canonically a smooth groupoid with only identity morphisms.

Proposition

The canonical identification yields a full subcategory

$$Smooth0Type \hookrightarrow Smooth1Type \,.$$

Every Lie groupoid presents a smooth groupoids. Those of this form are also called differentiable stacks.

A 0-truncated smooth groupoid is equivalently a smooth space.

For $G$ a smooth group, its delooping $\mathbf{B}G$ is a smooth groupoid, the moduli stack of smooth $G$-principal bundles.

Action groupoids

Example

For $X$ a smooth space and $G$ a smooth group and

$$\rho : X \times G \to X$$

an action then the action groupoid

$$X // G \in Smooth1Type$$

$$X // G = \left( X \times G \stackrel{\overset{\rho}{\longrightarrow}}{\underset{p_1}{\longrightarrow}} X \right)$$

is a smooth groupoid.

Moduli spaces of gauge fields

Groupoid of electromagnetic field configurations

The mathematical concept of smooth groupoid is well familiar, in slight disguise, in the physics of basic gauge theory. Here we make this explicit for basic electromagnetism. For more exposition and details along these lines see (Eggersson 14).

We have seen in The electromagnetic field strength that a configuration of the electromagnetic field on $\mathbb{R}^4$ is given by a differential 1-form $A \in \Omega^1(\mathbb{R}^4)$, the “electromagnetic potential”. It describes a field configuration with field strength Lorentz tensor (with respect to the canonical coordinates $(t, x^1, x^2, x^3) \coloneqq (x^i)_{i = 1}^4 =$ on $\mathbb{R}^4$)

$$\begin{aligned} F & = \mathbf{d}A \\ & = E_1 \mathbf{d}x^1 \wedge \mathbf{d}t + E_1 \mathbf{d}x^1 \wedge \mathbf{d}t + E_1 \mathbf{d}x^1 \wedge \mathbf{d}t + \\ & + B_1 \mathbf{d}x^2 \wedge \mathbf{d}x^3 + B_2 \mathbf{d}x^3 \wedge \mathbf{d}x^1 + B_3 \mathbf{d}x^1 \wedge \mathbf{d}x^2 \end{aligned}$$

with electric field strength $(E_i \in C^\infty(\mathbb{R}^4))_{i = 1}^3$ and magnetic field strength $(B_i \in C^\infty(\mathbb{R}^4))_{i = 1}^3$ given that is given by the de Rham differential of $A$:

$$\begin{aligned} F = \mathbf{d}A \end{aligned} \,.$$

After Maxwell it was thought that $F \in \Omega^2(\mathbb{R}^4)$ alone genuinely reflects the configuration of the electromagnetic field. But with the discovery of quantum mechanics it became clear that it is indeed the potential $A$ itself that reflects the configuration of the electromagnetic field: in the presence of magnetic flux or other topoligical constraints, there can be different $A, A'$ with the same $F = \mathbf{d}A = \mathbf{d}A'$ which nevertheless describe experimentally distinguishable electromagnetic field configurations. (Distinguishable by the Aharonov-Bohm effect and also to some extent by Dirac charge quantization; this is discussed at Circle-principal connections below.)

However, not all different gauge potentials describe different physics. The actual configuration space of electromagnetism on a spacetime $X$ is finer than $\mathbf{\Omega}^2_{cl}(X)$ but coarser than $\mathbf{\Omega}^1(X)$. And it is not quite a smooth space itself, but a smooth groupoid:

one finds that two electromagnetic potentials $A, A' \in \Omega^1(\mathbb{R}^4)$ for which there is a function $\lambda \in C^\infty(\mathbb{R}^4)$ such that

$$A' = A + \mathbf{d}\lambda$$

represent different but equivalent field configurations. One says that $\lambda$ induces a gauge transformation from $A$ to $A'$. We write $\lambda \colon A \stackrel{\simeq}{\to} A'$ to reflect this.

So the configuration space of electromagnetism does not just have points and coordinate systems. But it is also equipped with the information of a space of gauge transformations between any two coordinate systems laid out in it (which may be empty).

To see what the structure of such a smooth gauge groupoid should be, notice that the above defines an action of smooth functions $\lambda$ on smooth $1$-forms $A$,

Definition

For any $U \in$ CartSp, Write $\Omega^1_{vert}(X \times U)$ for the set of differential 1-forms on $X \times U$ with no components along $U$, and write $C^\infty(X \times U , U(1))$ for the group of circle group valued smooth functions. There is an action of this group on the 1-forms

$$\rho_U \colon \Omega^1_{vert}(X \times U) \times C^\infty(X \times U, U(1)) \to \Omega^1_{vert}(X \times U)$$

given by

$$\rho_U \colon (A,\lambda) \mapsto A + \mathbf{d}_X \lambda \,.$$

Given such an action of a discrete group on a set, we might be demoted to form the quotient set $\Omega^1_{vert}(X\times U)/C^\infty(X \times U, U(1))$. This set contains the gauge equivalence classes of $U$-parameterized collections of electromagnetic gauge fields on $X$. But it turns out that this is too little information to correctly capture physics. For that instead we need to remember not just that two gauge fields are equivalent, but how they are equivalent. That is, we for $\lambda$ a gauge transformation from $A_1$ to $A_2$, we should have an equivalence $\lambda \colon A_1 \stackrel{\simeq}{\to} A_2$.

Definition

The action groupoid

$$\Omega^1_{vert}(X\times U)// C^\infty(X \times U, U(1)) \coloneqq ( \Omega^1_{vert}(X\times U) \times C^\infty(X \times U, U(1)) \stackrel{\overset{t = \rho}{\longrightarrow}}{\underset{s = p_1}{\longrightarrow}} \Omega^1_{vert}(X\times U) )$$

is the groupoid whose

• objects are $U$-parameterized collections of gauge potentials $A \in \Omega^1_{vert}(X \times U)$;

• morphisms are pairs $(A,\lambda)$ with $A$ an object and $\lambda \in C^\infty(X \times U, U(1))$, with domain $A$ and codomain $A + \mathbf{d}_X \lambda$;

• composition is given by multiplication of the $\lambda$-labels in the group $C^\infty(X \times U, U(1))$.

This is the discrete gauge groupoid for $U$-parameterized collections of fields. It refines the gauge group, which is recoverd as its fundamental group:

Proposition

Let $A_0 \coloneqq 0 \in \Omega^1_{vert}(X \times U)$ be the trivial gauge field. Then its automorphism group in the gauge groupoid of def. is the group of circle-group valued functions on $U$:

$$\pi_1(\Omega^1_{vert}(X\times U)// C^\infty(X \times U, U(1)), A_0) = Aut(A_0) \simeq C^\infty(U,U(1)) \,.$$

Proof

By definition, an automorphism of $A_0$ is given by a function $\lambda \in C^\infty(X \times U, U(1))$ such that $A_0 = A_0 + \mathbf{d}_X \lambda$. This is the case precisely if $\mathbf{d}_X \lambda = 0$, hence if $\lambda$ is contant along $X$. But a function on $X \times U$ which is constant along $X$ is canonically identified with a function on just $U$.

All this data in in fact natural in $U$. Recall that $\Omega^1_{vert}(X \times U) = \mathbf{\Omega}^1(X)(U)$ is the set of $U$-charts of the smooth moduli space $\mathbf{\Omega}^1(X)$ of smooth 1-forms on $X$. Similarly $C^\infty(X \times U) = \mathbf{C}^\infty(X)(U)$.

Proposition

There is a homomorphism of smooth spaces (def. )

$$\rho \colon \mathbf{\Omega}^1(X)\times \mathbf{C}^\infty(X) \to \mathbf{\Omega}^1(X)$$

from the product smooth space, def. , of the smooth moduli spaces of 1-forms and 0-forms on $X$, def. , to that of smooth functions, def. , whose component over $U \in$ CartSp is the action

$$\rho_U \colon (A,\lambda) \mapsto A + \mathbf{d}\lambda$$

of def. .

We then also want to consider a smooth action groupoid.

Definition

Write

$$\mathbf{\Omega}^1(X) // \mathbf{C}^\infty(X, U(1)) : CartSp^{op} \to Grpd$$

for the contravariant functor from coordinate systems to the category of groupoids, which assigns to $U \in CartSp$ the discrete action groupoid of $U$-collections of gauge fields of def. .

$$U \mapsto \Omega^1_{vert}(X \times U) // C^\infty(X\times U, U(1)) \,.$$

Such a structure presheaf of groupoids is a common joint generalization of the notion of discrete groupoids and smooth spaces. We call them smooth groupoids. This is what we turn to in Smooth groupoids

Related concepts

• moduli stack

• centrally extended groupoid

References

Discussion of diffeological groups (such as diffeomorphism groups and quantomorphism groups) goes back to

• Jean-Marie Souriau, Groupes différentiels, in Differential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128. (MathScinet)

Exposition of the concept of smooth groupoids motivated from basic gauge theory is in

• Ragnar Eggertsson?, Stacks in gauge theory, 2014-

Discussion of smooth stacks as target spaces for sigma-model quantum field theories is in

• Tony Pantev, Eric Sharpe, String compactifications on Calabi-Yau stacks, Nucl.Phys. B733 (2006) 233-296, (arXiv:hep-th/0502044)

• Tony Pantev, Eric Sharpe, Gauged linear sigma-models for gerbes (and other toric stacks), (arXiv:hep-th/0502053)

• S. Hellerman, Eric Sharpe, Sums over topological sectors and quantization of Fayet-Iliopoulos parameters, (arXiv:1012.5999)

Discussion of geometric Langlands duality in terms of 2d sigma-models on stacks (moduli stacks of Higgs bundles over a given algebraic curve) is in

• Edward Witten, section 6 of Mirror Symmetry, Hitchin’s Equations, And Langlands Duality (arXiv:0802.0999)