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For $G$ a Lie group, we discuss $G$-principal bundles over a smooth manifold $X$ as a natural construction in the context of smooth groupoids.
Recall the discussion of Cech cohomology in degree 1 from geometry of physics -- smooth homotopy types – Pre-smooth groupoids
Let $G$ be a Lie group. Write $(\mathbf{B}G)_\bullet \in LieGrpd \hookrightarrow PreSmoothGrpd$ for the Lie groupoid
with composition induced from the product in $G$.
A useful schematic picture this groupoid is
where the $g_i \in G$ are elements in the group, and the bottom morphism is labeled by forming the product in the group. (The order of the factors here is a convention whose choice, once and for all, does not matter up to equivalence.)
The nerve of $(\mathbf{B}G)_\bullet$, def. 1, is a simplicial object of the form
with face maps of the form
where the outer face maps forget the corresponding outer copy in the Cartesian product of groups, and the inner face maps are given by group multiplication in two consecutive copies.
For $X$ a smooth manifold, we may regard it as a Lie groupoid with only identity morphisms. Its schematic depiction is simply
Let $\{U_i \to X\}_{i \in I}$ be an open cover of a smooth manifold $X$. The corresponding Cech groupoid $C(\{U_i\})_\bullet$ is
with the uniquely defined composition. The schematic depiction is
This indicates that the objects of this groupoid are pairs $(x,i)$ consisting of a point $x \in X$ and a patch $U_i \subset X$ that contains $x$, and a morphism is a triple $(x,i,j)$ consisting of a point and two patches, that both contain the point, in that $x \in U_i \cap U_j$. The triangle in the above cartoon symbolizes the evident way in which these morphisms compose. All this inherits a smooth structure from the fact that the $U_i$ are smooth manifolds and the inclusions $U_i \to X$ are smooth functions. hence also $C(U)$ becomes a Lie groupoid.
Given an open cover $\{U_i \to X\}$ there is a canonical morphism from its Cech groupoid to the manifold $X$ given by
A morphism
is given in components precisely by a collection of functions
such that on each $U_i \underset{X}{\times} U_k \cap U_j$ the equality $g_{j k} g_{i j} = g_{i k}$ of smooth functions holds:
This is precisely a cocycle in Cech cohomology on $X$ relative $\{U_i\}$ with coefficients in $G$.
For $G$ a Lie group (or any topological group), traditional literature highlights the universal principal bundle $E G \to B G$ over the classifying space of $G$, and the fact that under pullback of topological spaces this yields all isomorphism classes of smooth $G$-principal bundles. But an analogous construction exists in smooth groupoids which is both simpler as well as more powerful: it modulates the full groupoid of smooth $G$-principal bundles. We now discuss this smooth incarnation $(\mathbf{E}G)_\bullet$ of $E G$.
For $G$ a Lie group, write $\mathbf{E}G$ for the action groupoid of $G$ acting on itself from the right, hence for the Lie groupoid
whose manifold of objects is $G$, whose manifold of morphisms is $G \times G$, whose source-map is projection on the first factor, whose target map is multiplication in the group, whose identity-map is $g\mapsto (g,e)$ and whose composition operation is
The groupoid $(\mathbf{E}G)_\bullet$ of def. 4 has at most one morphism for every ordered pair of objects, hence the morphisms are uniquely identified by giving their source and target.
Schematically:
or simply
This means that it is isomorphic, as a pre-smooth groupoid, to the pair groupoid of $G$.
While therefore $(\mathbf{E}G)_\bullet$ is a rather simplistic object, it is nevertheless worthwhile to make its following properties explicit.
There is an evident morphism of smooth groupoids
given by
hence
There is an evident $G$-action
given by
The projection $p$ is the quotient projection of this action.
The nerve of $(\mathbf{E}G)_\bullet$ is a simplicial object of the form
with face maps of the form
If here $(\mathbf{E}G)_\bullet$ is though of isomorphically as $(G//G)_\bullet$, then
these face maps are such that all except one outermost (say the topmost) in each degree are given by group multiplication in two consecutive copies of $G$, with the remaining outermost one given by projection.
If on the other hand $(\mathbf{E}G)_\bullet$ is thought of isomorphically as the pair groupoid of $G$, via remark 3, then the $k$th face map in each degree is given simply by projecting out the $k$th factor (starting counting at 0).
A morphism $p\colon E_\bullet \to B_\bullet$ of pre-smooth groupoids is called a fibration if for each $n\in \mathbb{N}$ the functor $p(\mathbb{R}^n) \colon E(\mathbb{R}^n)_\bullet \to B(\mathbb{R}^n)_\bullet$ is an isofibration, hence if for each object $e \in E(\mathbb{R}^n)$, each morphism $p(e) \to b$ in $B(\mathbb{R}^n)$ has a lift through $p$ to a morphism $e \to e'$ in $E$:
The projection $p \colon (\mathbf{E}G)_\bullet \to (\mathbf{B}G)_\bullet$ of prop. 1 is a fibration of smooth groupoids, def. 6. Moreover, any point inclusion $\ast \longrightarrow \mathbf{E}G$ is over each $\mathbb{R}^n$ an equivalence of groupoids, hence is in particular a local weak equivalence of smooth groupoids (as defined here).
In summary, the morphisms $\ast \to (\mathbf{E}G)_\bullet \stackrel{p}{\to} (\mathbf{B}G)_\bullet$ constitute a factorization of the canonical $\ast \to (\mathbf{B}G)_\bullet$ into a local weak equivalence followed by a fibration.
The smooth groupoid $(\mathbf{E}G)_\bullet$ of def. 4 has the following equivalent incarnations as pre-smooth groupoids by isomorphic Lie groupoids
$(\mathbf{E}G)_\bullet \simeq (G//G)_\bullet$ is the action groupoid of $G$ acting on itself by right multiplication;
$(\mathbf{E}G)_\bullet \simeq ((\mathbf{B}G)^{I})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} \ast$ is the pullback of the point inclusion $\ast \to (\mathbf{B}G)_\bullet$ along one of the projection map $d_1 \colon ((\mathbf{B}G)^I)_\bullet \longrightarrow (\mathbf{B}G)_\bullet$ out of the path space object of $(\mathbf{B}G)_\bullet$.
The first statement is immediate from the definitions. The second is also fairly immediate, but worth making more explicit: the Lie groupoid $((\mathbf{B}G)^I)_\bullet$ has as objects the morphisms in $(\mathbf{B}G)_\bullet$, hence elements of $G$, and as morphisms $g_1 \to g_2$ commuting squares between these, schematically:
The morphism $d_1$ projects out the top horizontal morophisms:
The pullback then restricts this image to be constant and hence produces the groupoid whose objects are still the morphisms in $\mathbf{B}G$, hence elements of $G$, but whose morphisms are no longer all commuting squares, but just all commuting triangles between these, schematically:
Such a triangle exists precisely if $g_2 = g_1 h$, which gives $\mathbf{E}G$ as in def. 4, thought of as:
objects = $\left\{ \array{ && \bullet \\ & {}^g\swarrow \\ \bullet } \right\}$
morphisms = $\left\{ \array{ && \bullet \\ & {}^g\swarrow && \searrow^{g' = g h} \\ \bullet &&\stackrel{h}{\longrightarrow}&& \bullet } \right\} \,.$
The traditional construction of the $G$-principal bundle associated to a Cech cocycle is the following.
Let $X$ be a smooth manifold, $\{U_i \to X\}_I$ an open cover and $(g_{i j})_{i,j \in I}$ a Cech cocycle of degree 1 with values in $G$. Then the bundle $P \to X$ associated with this data is the quotient
of the Cartesian product of the cover (regarded as the disjoint union of its patches) with $G$, by the equivalence relation which identifies two elements in the product whenever they are related by the Cech cocycle:
Let $X$ be a smooth manifold, $\{U_i \to X\}_I$ an open cover and $(g_{i j})_{i,j \in I}$ a Cech cocycle of degree 1 with values in $G$. Then the associated $G$-bundle $P$, def. 7, is equivalent, regarded as a smooth groupoid with only identity morphisms, to the pullback of the morphism $(\mathbf{E}G)_\bullet \to (\mathbf{B}G)_\bullet$ of def. 4 along the cocycle regarded as a homomorphism of smooth groupoids $C(\{U_i\})_\bullet \stackrel{g}{\longrightarrow} (\mathbf{B}G)_\bullet$.
Pullbacks of pre-smooth groupoids are computed componentwise. Hence a morphism in $C(\{U_i\})_\bullet\underset{(\mathbf{B}G)_\bullet}{\times} (\mathbf{E}G)_\bullet$ is a pair consisting of a morphism $(x,i,j)$ in $C(\{U_i\})_\bullet$ and a morphism $(g_1, h)$ in $(\mathbf{E}G)_\bullet$ such that $h$ is the value of the cocycle on $(x,i,j)$.
With $(\mathbf{E}G)_\bullet$ thought of as in remark 3, then the pullback looks like:
objects =
morphisms =
This means that the morphisms in the pullback are of the form
and there is at most one for any ordered pair of objects. But this means that these morphisms represent precisely the equivalence relation of def. 7: the evident projection map from this pullback to $P$ (with $P$ regarded as a groupoid with only identity morphisms) is evidently essentially surjective and fully faithful, hence an equivalence.
By the pullback construction in prop. 4, $P$ inherits a $G$-action from that on $(\mathbf{E}G)_\bullet$ of def. 1:
via the pasting diagram of pullbacks
The morphism $\tilde P \times G \to \tilde P$ exhibits the principal $G$-action of $G$ on $\tilde P$.
We saw above that smooth $G$-principal bundles for $G$ a Lie group may naturally be understood in terms of pullbacks of the fibrant replacement $\mathbf{E}G\to \mathbf{B}G$ of the point inclusion $\ast \to \mathbf{B}G$ along a Cech cocycle, regarded as a homomorphism of smooth groupoids.
This is a special case of a very general construction of homotopy pullbacks which will also apply, below, to weakly principal simplicial bundles and then generally to principal infinity-bundles. We now discuss the general axiomatization of this construction via categories of fibrant objects.
A category of fibrant objects $\mathcal{C}$ is
a category with weak equivalences, i.e equipped with a subcategory $W$ that contains all isomorphisms
where $f \in Mor(W)$ is called a weak equivalence;
equipped with a further subcategory
where $f \in Mor(F)$ is called a fibration
Those morphisms which are both weak equivalences and fibrations are called acyclic fibrations .
This data has to satisfy the following properties:
$\mathcal{C}$ has finite products, and in particular a terminal object ${\ast}$;
the pullback of a fibration along an arbitrary morphism exists, and is again a fibration;
acyclic fibrations are preserved under pullback;
weak equivalences satisfy 2-out-of-3;
for every object there exists a path object
where $\sigma$ is a weak equivalence and $d_0 \times d_1$ is a fibration;
all objects are fibrant, i.e. all morphisms $B \to {\ast}$ to the terminal object are fibrations.
The category of pre-smooth groupoids (here) becomes a category of fibrant objects, def. 8 with fibrations as in def. 6 and weak equivalences the local weak equivalences (as defined here).
Let $C$ be a category of fibrant objects. The factorization lemma says the following.
Every morphism $f : X \to Y$ in $C$ factors as
where
$i$ is a weak equivalence (even a right inverse to an acyclic fibration);
$p$ is a fibration.
Let $Y^I$ with factorization $Y \stackrel{\simeq}{\to} Y^I \stackrel{(d_0,d_1)}{\longrightarrow} Y \times Y$ be a path space object for $Y$. Let $\hat X \coloneqq Y^I \times_Y X$ be the pullback of $f$ along one of its legs, to get the diagram
Take $p$ to be the composite vertical morphism in the above diagram, hence
To see that this is indeed a fibration, notice that, by the pasting law, the above pullback diagram can be refined to a double pullback diagram as follows
Both squares are pullback squares. Since pullbacks of fibrations are fibrations, the morphism $\hat X \to X \times Y$ is a fibration. Similarly, since $X$ is assumed to be fibrant (as all objects in a category of fibrant objects), also the projection map $X \times Y \to Y$ is a fibration (see here).
Since $p$ is thereby exhibited as the composite of two fibrations
(the first map being a pullback of a fibration as above, the composite of the second and the third map being the projection just menioned) it is itself a fibration.
Next, by the axioms of path space objects in a category of fibrant objects we have that $d_1 \;\colon\; Y^I \to Y$ is an acyclic fibration. Since these are stable under pullback, also $\hat X \to X$ is an acyclic fibration.
But, by the axioms, $Y^I \to Y$ has a right inverse $Y \to Y^I$. By the pullback property this induces a right inverse of $\hat X \to X$ fitting into a pasting diagram
This establishes the claim.
Let $\mathcal{C}$ be a category of fibrant objects.
Two morphism $f,g : A \to B$ in $\mathcal{C}(A,B)$ are
right homotopic, denoted $f \simeq g$, precisely if they fit into a diagram of the form
for some path space object $B^I$;
homotopic, denoted $f \sim g$, if they become right homotopic after pulled back to a weakly equivalent domain, i.e. precisely if they fit into a diagram of the form
for some object $\hat A$ and for some path space object $B^I$ of $I$
In view of this the following definition is natural.
A homotopy fiber product or homotopy pullback of two morphisms
in a category of fibrant objects is the object $A \times_C C^I \times_C B$ defined as the (ordinary) limit
The homotopy fiber product in def. 10 is isomorphic to the ordinary fiber product of either of the two morphisms with the fibration replacement of the other as given by the factorization lemma, def. 1.
By basic properties of limits the defining limit in def. 10 may be computed by two consecutive pullbacks.
Here the intermediate pullback is precisely the one appearing in the proof of the factorization lemma.
With $\mathcal{C}$ the category of fibrant objects given by pre-smooth groupoids, prop. 5, then for $G$ a Lie group, the factorization $\ast \to \mathbf{E}G \stackrel{p}{\to} \mathbf{B}G$ of prop. 2 is the one given by the factorization lemma. Hence a pullback of $p \colon \mathbf{E}G\to \mathbf{B}G$ as in prop. 4 is equivalently the homotopy pullback of $\ast \to \mathbf{B}G$.
(… under construction …)
It is no coincidence that the above statement looks akin to the maybe more familiar statement which says that equivalence classes of $G$-principal bundles are classified by homotopy-classes of morphisms of topological spaces
where $\mathbf{B}G \in$ Top is the topological classifying space of $G$. The category Top of topological spaces, regarded as an (∞,1)-category, is the archetypical (∞,1)-topos the way that Set is the archetypical topos. And it is equivalent to ∞Grpd, the $(\infty,1)$-category of bare ∞-groupoids. What we are seeing above is a first indication of how cohomology of bare $\infty$-groupoids is lifted to a richer $(\infty,1)$-topos to cohomology of $\infty$-groupoids with extra structure.
In fact, all of the statements that we have considered so far become conceptually simpler in the $(\infty,1)$-topos. We had already remarked that the anafunctor span $X \stackrel{\simeq}{\leftarrow} C(U) \stackrel{g}{\to} \mathbf{B}G$ is really a model for what is simply a direct morphism $X \to \mathbf{B}G$ in the $(\infty,1)$-topos. But more is true: that pullback of $\mathbf{E}G$ which we considered is just a model for the homotopy pullback of just the point
The principal ∞-bundles that we wish to model are already the main and simplest example of the application of these three items:
Consider an object $\mathbf{B}G \in [C^{op}, sSet]$ which is an $\infty$-groupoid with a single object, so that we may think of it as the delooping of an ∞-group $G$, let $*$ be the point and $* \to \mathbf{B}G$ the unique inclusion map. The good replacement of this inclusion morphism is the $G$-universal principal ∞-bundle $\mathbf{E}G \to \mathbf{B}G$ given by the pullback diagram
An ∞-anafunctor $X \stackrel{\simeq}{\leftarrow} \hat X \to \mathbf{B}G$ we call a cocycle on $X$ with coefficients in $G$, and the (∞,1)-pullback $P$ of the point along this cocycle, which by the above discussion is the ordinary limit
we call the principal ∞-bundle $P \to X$ classified by the cocycle.
It is now evident that our discussion of ordinary smooth principal bundles above is the special case of this for $\mathbf{B}G$ the nerve of the one-object groupoid associated with the ordinary Lie group $G$.
So we find the complete generalization of the situation that we already indicated there, which is summarized in the following diagram: