nLab
Atiyah Lie groupoid

Idea

The Atiyah Lie groupoid $\mathrm{At}(P)$ of a $G$-principal bundle $P\to X$ is the Lie groupoid whose objects are the fibers of the bundle, and whose morphisms are the $G$-equivariant morphisms between the fibers. Schematically:

$$\mathrm{At}(P)=\{P_x\stackrel{\alpha }{\to }{P}_y\mid x,y\in X\}.$$At(P) = \left\{ P_x \stackrel{\alpha}{\to} P_y | x,y \in X \right\} \,.

Its Lie algebroid is the Atiyah Lie algebroid $\mathrm{at}(P)$ of $P$.

Both the Atiyah Lie groupoid and its Lie algebroid are used to characterize and are characterized by connections on $P$.

Definition

As generally for every Lie algebroid, there are different Lie groupoids integrating the Atiyah Lie algebroid. We describe two of them.

The Aityah Lie algebroid $\mathrm{at}(P)$ of the principal bundle $P\to X$ comes canonically with a morphism $\mathrm{at}(X)\to TX$ to the tangent Lie algebroid.

The simplest Lie integration of the tangent Lie algebroid is the codiscrete groupoid $X\times X$ of $X$. On the other hand, the universal integration is the fundamental groupoid $\Pi (X)$ (both coincide precisey if $X$ is a simply connected space).

Accordingly, there is a version of the Atiyah Lie groupoid over $X\times X$, and a richer version over $\Pi (X)$.

Over the pair groupoid

For $G$ a Lie group and $P\to X$ a $G$-principal bundle, the Atiyah groupoid $\mathrm{At}(P)$ – also called the gauge groupoid or transport groupoid – of $P$ is the Lie groupoid with

• $\mathrm{Obj}(\mathrm{At}(P))=X$;
• $\mathrm{Mor}(\mathrm{At}(P))=(P\times P)/G$, where the quotient is taken with respect to the diagonal action? of $G$ on $P\times P$.

The Atiyah groupoid sits in an sequence of groupoids

$$\mathrm{Ad}(P)\to \mathrm{At}(P)\to \mathrm{Codisc}(X)$$Ad(P) \to At(P) \to Codisc(X)

where

• $\mathrm{Ad}(P)=P{\times }_{G}G$ is the adjoint bundle of groups associated via the adjoint action of $G$ on itself; regarded as a smooth union ${\coprod }_{x\in X}B{P}_x{\times }_{G}G$ of one-object groupoids coming from groups;

• $\mathrm{Codisc}(X)=(X\times X\rightrightarrows X)$ is the codiscrete groupoid of $X$

• the functor $\mathrm{Ad}(P)\to \mathrm{At}(P)$ is the identity on objects and on morphisms given by the canonical identification $P_x{\times }_{G}G\stackrel{\simeq }{\to }(P_x\times P_x)G$, where again we use the diagonal action of $G$ on $P_x\times P_x$.

• the functor $\mathrm{At}(P)\to \mathrm{Codisc}(X)$ is the unique one that is the identity on objects.

What is all of this $\mathrm{diag}$ stuff? I don't understand either $(P\times P){/}_{\mathrm{diag}}G$ or $(P_x\times P_x{)}_{\mathrm{diag}}G$. —Toby

David Roberts: It’s to do with the diagonal action of $G$ on $P\times P$ as opposed to the antidiagonal (if $G$ is abelian) or the action on only one factor. I agree that it’s a bad notation.

Toby: How well do you think it works now, with the notation suppressed and a note added in words? (For what it's worth, the diagonal action seems to me the only obvious thing to do here, although admittedly the others that you mention do exist.)

Todd: I personally believe it works well. A small note is that this construction can also be regarded as a tensor product, regarding the first factor $P$ as a right $G$-module and the second a left module, where the actions are related by $gp=p{g}^{-1}$.

Toby: H'm, maybe we should write diagonal action? if there's something interesting to say about it.

Notice that a splitting (a section)

$$\mathrm{Codisc}(X)\to \mathrm{At}(P)$$Codisc(X) \to At(P)

of the Atiyah groupoid is a trivialization of $P$. On the other hand, locally on contractible $U\subset X$ we have $\mathrm{Codisc}(U)\simeq {\Pi }_1(U)$ with $U$ the fundamental groupoid of $U$, and a splitting $\mathrm{Codisc}(U)\simeq {\Pi }_1(U)\to \mathrm{At}(P){\mid }_U$ is still a trivialization over $U$ but indicates now that one may want to interpret it as giving rise to a flat connection.

Indeed, we have the sequence of of surjective and full functors of path categories

$$P_1(X)\to {\Pi }_1(X)\to \mathrm{Codisc}(X)$$P_1(X) \to \Pi_1(X) \to Codisc(X)

with ${\Pi }_1(X)$ the fundamental groupoid and $P_1(X)$ the smooth path groupoid and may refine the Atiyah groupoid by pulling back along these.

Write therefore $\mathrm{At}\prime (P):=\mathrm{At}(P){\times }_{\mathrm{Codisc}(X)}{\Pi }_1(X)$ for the pullback

$$\begin{array}{ccc}\mathrm{At}\prime (P)& \to & {\Pi }_1(X)\\ \downarrow & & \downarrow \\ \mathrm{At}(P)& \to & \mathrm{Codisc}(X)\end{array}.$$\array{ At'(P) &\to& \Pi_1(X) \\ \downarrow && \downarrow \\ At(P) &\to& Codisc(X) } \,.

A splitting ${\Pi }_1(X)\to \mathrm{At}\prime (P)$ of the top row is now precisely a flat connection on $P$.

If we pull back further to $A″$

$$\begin{array}{ccc}\mathrm{At}″(P)& \to & P_1(X)\\ \downarrow & & \downarrow \\ \mathrm{At}\prime (P)& \to & {\Pi }_1(X)\\ \downarrow & & \downarrow \\ \mathrm{At}(P)& \to & \mathrm{Codisc}(X)\end{array}.$$\array{ At''(P) &\to& P_1(X) \\ \downarrow && \downarrow \\ At'(P) &\to& \Pi_1(X) \\ \downarrow && \downarrow \\ At(P) &\to& Codisc(X) } \,.

then splittings of $P_1(X)\to \mathrm{At}″(X)$ are precisely (not necessarily flat) connections on $P$.

All this is more well known in terms of the Lie algebroid underlying the Atiyah Lie groupoid, i.e. the Atiyah Lie algebroid sequence

$$\mathrm{ad}(P)\to \mathrm{at}(P)\to TX,$$ad(P) \to at(P) \to T X \,,

where

• $\mathrm{ad}(P)=P{\times }_g\mathrm{Lie}(G)$ is the adjoint bundle of Lie algebras, associated via the adjoint action of $G$ on its Lie algebra;

• $\mathrm{at}(P)=(TP)/G$ is the Atiyah Lie algebroid

• $TX$ is the tangent Lie algebroid.

Indeed, a splitting ${\nabla }_{\mathrm{flat}}:TX\to \mathrm{at}(P)$ of this sequence in the category of Lie algebroids is precisely again a flat connection on $P$ and integrates under Lie integration to the splitting of $\mathrm{At}\prime (P)$ discussed above.

To get non-flat connections in the literature one often sees discussed splittings of the Atiyah Lie algebroid sequence in the category just of vector bundles. In that case one finds the curvature of the connection precisely as the obstruction to having a splitting even in Lie algebroids.

One can describe non-flat connections without leaving the context of Lie algebroids by passing to higher Lie algebroids, namely $L_{\mathrm{\infty }}$-algebroids.

Over the path $\mathrm{\infty }$-groupoid

…

Relation to differential nonabelian cohomology

The following is not in the literature. – Urs

We discuss the Atiyah Lie groupoid along the lines of nonabelian groupoid cohomology of the path n-groupoid of the base manifold $X$.

Write $BG$ for the delooping of $G$. Write $\Pi (X)$ for the path ∞-groupoid of $X$. For the present purpose of Atiyah 1-groupoids this may be taken to be simply the path 2-groupoid.

Remark

Much of what makes the following discussion somewhat less than straightforward is the fact already manifest in and discussed at nonabelian group cohomology: nonabelian degree 2-cocycles with coefficients in the (delooping of the) automorphism 2-group $\mathrm{AUT}(G)$ classify really action groupoids of actions on bundles with typical fiber $F=\mathrm{Aut}(G)$, while the object that one wants to see classified (there: a group extension; here: the Atiyah Lie groupoid) is obtained from this only after forgetting the elements in these fibers, and just remembering the action.

Claim

A $G$-principal bundle $P\to X$ is classified by a morphism $g:X\to BG$.

Putting a connection on $P$ give a parallel transport morphism $\Pi (X)\to B\mathrm{INN}(G)$ into the inner automorphism 2-group of $G$, which fits into a diagram

$$\begin{array}{cccccc}X& \stackrel{g}{\to }& BG& & & \mathrm{underlying}\mathrm{cocycle}\\ \downarrow & & \downarrow \\ \Pi (X)& \stackrel{(A,F)}{\to }& B\mathrm{INN}(G)& & & \mathrm{connection}\\ {\downarrow }^{\mathrm{Id}}& & \downarrow \\ \Pi (X)& \stackrel{F}{\to }& B\mathrm{AUT}(G)& & & \mathrm{nonabelian}\mathrm{curvature}\end{array},$$\array{ X &\stackrel{g}{\to}& \mathbf{B}G &&& {underlying\;cocycle} \\ \downarrow && \downarrow \\ \Pi(X) &\stackrel{(A,F)}{\to}& \mathbf{B} INN(G) &&& {connection} \\ \downarrow^{\mathrlap{Id}} && \downarrow \\ \Pi(X) &\stackrel{F}{\to}& \mathbf{B} AUT(G) &&& {nonabelian\; curvature} } \,,

where we have added the morphism $\mathrm{INN}(G)\to \mathrm{AUT}(G)$ into the automorphism 2-group.

The action groupoid $(P{\times }_G\mathrm{Aut}(G))//\mathrm{At}(P)$ of the Atiyah Lie groupoid acting on the associated bundle $(P{\times }_G\mathrm{Aut}(G))$, regarded as a groupoid over $\Pi (X)$, is classified by $F$: it is the homotopy fiber of $F$, in that there is a homotopy pullback diagram

$$\begin{array}{c}\begin{array}{ccc}(P{\times }_G\mathrm{Aut}(G))//\mathrm{At}(P)& \to & *\\ \downarrow & & \downarrow \\ \Pi (X)& \stackrel{F}{\to }& B\mathrm{AUT}(G)\end{array}\end{array}.$$\array{ \array{ (P \times_{G} Aut(G) )//At(P) &\to& {*} \\ \downarrow && \downarrow \\ \Pi(X) &\stackrel{F}{\to}& \mathbf{B} AUT(G) } } \,.

The Atiyah Lie groupoid itself is the corresponding quotient

$$(P{\times }_G\mathrm{Aut}(G))\to (P{\times }_G\mathrm{Aut}(G))//\mathrm{At}(P)\to \mathrm{At}(P).$$(P \times_G Aut(G)) \to (P \times_{G} Aut(G))//At(P) \to At(P) \,.

Proof

Setup

Choose to model all ∞-Lie groupoids in terms of the local projective model structure on simplicial presheaves on the site CartSp of cartesian spaces.

All the Lie $2$-groupoids appearing here are represented by Kan complex valued sheaves. They satisfy descent on contractible domains, hence on all representables, hence are fibrant in the local projective model structure.

As discussed in the subsection on cofibrant objects, a cofibrant replacement for the manifold $X$ is given by the Cech nerve $Y:=C(\coprod U_i)$ of any good cover $\{U_i\to X\}$. So fix any such good cover of $X$.

Accordingly, a cofibrant version of the path ∞-groupoid is given by the coend $\Pi (Y)={\int }^{U\in C}\Pi (U)\cdot Y(U)$. This is the $n$-groupoid generated in degree $k$ from

• points in $(k+1)$-fold intersections;

• paths in $k$-fold intersections

• surfaces in $(k-1)$-fold intersections,

etc.

The bundle $P{\times }_G\mathrm{Aut}(G)$

The bundle $P{\times }_G\mathrm{Aut}(G)$ is, by definition, the one classified by the morphism

$$\mathrm{Ad}(g):X\stackrel{g}{\to }BG\stackrel{\mathrm{Ad}}{\to }B\mathrm{Aut}(G)$$Ad(g) : X \stackrel{g}{\to} \mathbf{B}G \stackrel{Ad}{\to} \mathbf{B}Aut(G)

represented by a morphism

$$X\stackrel{\simeq }{\leftarrow }Y\to B\mathrm{Aut}(G)$$X \stackrel{\simeq}{\leftarrow}Y \to \mathbf{B}Aut(G)

in $\mathrm{SPSh}(\mathrm{CartSp}{)}_{\mathrm{proj}}^{\mathrm{loc}}$.

We compute the homotopy pullback

$$\begin{array}{ccc}P{\times }_G\mathrm{Aut}(G)& \to & *\\ \downarrow & & \downarrow \\ X& \stackrel{\mathrm{Ad}(g)}{\to }& B\mathrm{Aut}(G)\end{array}$$\array{ P \times_G Aut(G) &\to& {*} \\ \downarrow && \downarrow \\ X &\stackrel{Ad(g)}{\to}& \mathbf{B}Aut(G) }

as usual in terms of an ordinary pullback

$$\begin{array}{ccc}P{\times }_G\mathrm{Aut}(G)& \to & EG\\ \downarrow & & \downarrow \\ Y& \stackrel{\mathrm{Ad}(g)}{\to }& B\mathrm{Aut}(G)\end{array}$$\array{ P \times_G Aut(G) &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ Y &\stackrel{Ad(g)}{\to}& \mathbf{B}Aut(G) }

along the representing cocycle of the universal bundle $E\mathrm{Aut}(G)\to B\mathrm{Aut}(G)$, which is the groupoid

$$E\mathrm{Aut}(G)=\left\{\begin{array}{ccc}& & •\\ & {}^{\alpha }\swarrow & & {\searrow }^{\beta }\\ •& & \stackrel{\gamma }{\to }& & •\end{array}\right\}$$\mathbf{E}Aut(G) = \left\{ \array{ && \bullet \\ & {}^{\mathllap{\alpha}}\swarrow && \searrow^{\mathrlap{\beta}} \\ \bullet &&\stackrel{\gamma}{\to}&& \bullet } \right\}

whose morphisms are commuting trianfles in $B\mathrm{Aut}(G)$, with the projection to $mathbB\mathrm{Aut}(G)$ being the projection onto the horizontal component.

So the objects of $P{\times }_G\mathrm{Aut}(G)$ are pairs $((x,i),\alpha )$ consisting of a point $x\in X$ in a patch $U_i$, together with an element $\alpha \in \mathrm{Aut}(G)$, and a morphism $((x,i),\alpha )\to ((y,j)\beta )$ exists iff $x=y$ and is then given by

$$\left\{\begin{array}{ccc}& & •\\ & {}^{\alpha }\swarrow & & {\searrow }^{\beta }\\ •& & \stackrel{\mathrm{Ad}(g_{ij}(x))}{\to }& & •\\ \\ (x,i)& & \to & & (x,j)\end{array}\right\}.$$\left\{ \array{ && \bullet \\ & {}^{\mathllap{\alpha}}\swarrow && \searrow^{\mathrlap{\beta}} \\ \bullet &&\stackrel{Ad(g_{i j}(x))}{\to}&& \bullet \\ \\ (x,i) &&\to&& (x,j) } \right\} \,.

The homotopy fiber of $\Pi (X)\to B\mathrm{AUT}(G)$

We compute now the homotopy pullback

$$\begin{array}{ccccc}Q& \to & & \to & *\\ \downarrow & & & & \downarrow \\ \Pi (X)& \to & B\mathrm{INN}(G)& \to & B\mathrm{AUT}(G)\end{array}$$\array{ Q &\to& &\to& {*} \\ \downarrow &&&& \downarrow \\ \Pi(X) &\to& \mathbf{B}INN(G) &\to& \mathbf{B}AUT(G) }

for $\Pi (X)\to B\mathrm{INN}(G)$ coming from the choice of any connection on $P$, in terms of the ordinary pullback

$$\begin{array}{ccc}Q& \to & E\mathrm{AUT}(G)\\ \downarrow & & \downarrow \\ \Pi (Y)& \to & B\mathrm{AUT}(G)\end{array}$$\array{ Q &\to& \mathbf{E}AUT(G) \\ \downarrow && \downarrow \\ \Pi(Y) &\to& \mathbf{B}AUT(G) }

along the representing cocycle $\Pi (Y)\to B\mathrm{AUT}(G)$ of the generalized universal bundle $E\mathrm{AUT}(G)$ of the automorphism 2-group $\mathrm{AUT}(G)$.

The computation is (apart from the fact that it takes place in a smooth context which is taken care of by the abstract nonsense) entirely analogous as that of the 2-cocycles discussed at nonabelian group cohomology, the only difference being that the domain is here not a one-object groupoid that is the delooping of a group, but the genuine many-object groupoid $\Pi (Y)$.

As there, we ventually make use of the translation between strict 2-groups and crossed modules - as described at strict 2-group – in terms of crossed modules – in order to translate some of the diagrams to follow into formulas. We follow the convention LB, as described there, in order to do so.

In $E\mathrm{AUT}(G)$

• an object is an automorphism $\alpha \in \mathrm{Aut}(G)$

• a morphism $(\gamma ,b):\alpha \to \beta$ is a triangle

  $$\begin{array}{ccc}& & •\\ & {}^{\alpha }\swarrow & {}^b⇙& {\searrow }^{\beta }\\ •& & \stackrel{\gamma }{\to }& & •\end{array}$$\array{ && \bullet \\ & {}^{\mathllap{\alpha}}\swarrow & {}^{\mathllap{b}}\swArrow& \searrow^{\mathrlap{\beta}} \\ \bullet &&\stackrel{\gamma}{\to}&& \bullet }

• a 2-cell $k:({\gamma }_1,b_1)\cdots ({\gamma }_2,b_2)\Rightarrow ({\gamma }_3,b_3)$

  is a 2-cell

  $$\begin{array}{ccc}& & •\\ & {}^{{\gamma }_1}\swarrow & {}^k⇙& {\searrow }^{{\gamma }_2}\\ •& & \stackrel{{\gamma }_3}{\to }& & •\end{array}$$\array{ && \bullet \\ & {}^{\mathllap{\gamma_1}}\swarrow & {}^{\mathllap{k}}\swArrow& \searrow^{\mathrlap{\gamma_2}} \\ \bullet &&\stackrel{\gamma_3}{\to}&& \bullet }

  in $B\mathrm{AUT}(G)$ such that

  $$\begin{array}{ccc}& & •\\ & \swarrow & {}^{b_3}⇙& \searrow \\ •& & \to & & •\end{array}=\begin{array}{ccc}& & •\\ & \swarrow & \downarrow & \searrow \\ \downarrow & {}^{b_1}⇙& •& {}^{b_2}⇙& \downarrow \\ \downarrow & \nearrow & {\Downarrow }^k& \searrow & \downarrow \\ •& & \stackrel{}{\to }& & •\end{array}.$$\array{ && \bullet \\ &\swarrow &{}^{\mathllap{b_3}}\swArrow& \searrow \\ \bullet &&\to&& \bullet } \;\;\; = \;\;\; \array{ && \bullet \\ &\swarrow& \downarrow & \searrow \\ \downarrow & {}^{\mathllap{b_1}}\swArrow & \bullet & {}^{\mathllap{b_2}}\swArrow& \downarrow \\ \downarrow & \nearrow &\Downarrow^{\mathrlap{k}} & \searrow & \downarrow \\ \bullet && \stackrel{}{\to} && \bullet } \,.

Using this, we find the pullback 2-groupoid $\Pi (Y){\times }_{B\mathrm{AUT}(K)}EG$ to be given as follows:

• an object is pair $((x,i),\alpha )$ consisting of a point $x\in X$ and a patch $U_i$ containing it, and an automorphism $\alpha \in \mathrm{Aut}(G)$

• a morphism is, over a path $(p,i):(x,i)\to (y,i)$ in $U_i$ a diagram

  $$\begin{array}{ccc}& & •\\ & {}^{\alpha }\swarrow & {}^b⇙& {\searrow }^{\beta }\\ •& & \stackrel{\mathrm{Ad}({\mathrm{tra}}_{\nabla }(p))}{\to }& & •\\ \\ (x,i)& & \stackrel{(p,i)}{\to }& & (y,i)\end{array},$$\array{ && \bullet \\ & {}^{\mathllap{\alpha}}\swarrow & {}^{\mathllap{b}}\swArrow& \searrow^{\mathrlap{\beta}} \\ \bullet &&\stackrel{Ad(tra_{\nabla}(p))}{\to}&& \bullet \\ \\ (x,i) &&\stackrel{(p,i)}{\to}&& (y,i) } \,,

  where ${\mathrm{tra}}_{\nabla }(p)$ is the parallel transport of the chosen connection on $P$ along the path $p$;

  and over a Cech morphism $(x,i)\to (x,j)$ a diagram

  $$\begin{array}{ccc}& & •\\ & {}^{\alpha }\swarrow & {}^b⇙& {\searrow }^{\beta }\\ •& & \stackrel{\mathrm{Ad}(g_{ij}(x))}{\to }& & •\\ \\ (x,i)& & \stackrel{}{\to }& & (x,j)\end{array},$$\array{ && \bullet \\ & {}^{\mathllap{\alpha}}\swarrow & {}^{\mathllap{b}}\swArrow& \searrow^{\mathrlap{\beta}} \\ \bullet &&\stackrel{Ad(g_{i j}(x))}{\to}&& \bullet \\ \\ (x,i) &&\stackrel{}{\to}&& (x,j) } \,,

• a 2-cell over a surface $\Sigma :p_1\cdot p_2\Rightarrow p_3$ is

  $$\begin{array}{ccc}& & •\\ & \swarrow & {}^{b_3}⇙& \searrow \\ •& & \stackrel{\mathrm{Ad}({\mathrm{tra}}_{\nabla }(p_3))}{\to }& & •\end{array}=\begin{array}{ccc}& & •\\ & \swarrow & \downarrow & \searrow \\ \downarrow & {}^{b_1}⇙& •& {}^{b_2}⇙& \downarrow \\ \downarrow & {}^{\mathrm{Ad}({\mathrm{tra}}_{\nabla }(p_1))}\nearrow & {\Downarrow }^{{\mathrm{tra}}_{\nabla }(\Sigma )}& {\searrow }^{\mathrm{Ad}({\mathrm{tra}}_{\nabla }(p_2))}& \downarrow \\ •& & \stackrel{}{\to }& & •\end{array}$$\array{ && \bullet \\ &\swarrow &{}^{\mathllap{b_3}}\swArrow& \searrow \\ \bullet &&\stackrel{Ad(tra_\nabla(p_3))}{\to}&& \bullet } \;\;\; = \;\;\; \array{ && \bullet \\ &\swarrow& \downarrow & \searrow \\ \downarrow & {}^{\mathllap{b_1}}\swArrow & \bullet & {}^{\mathllap{b_2}}\swArrow& \downarrow \\ \downarrow & {}^{\mathclap{Ad(tra_{\nabla}(p_1))}} \nearrow &\Downarrow^{\mathrlap{tra_\nabla(\Sigma)}} & \searrow^{\mathclap{Ad(tra_\nabla(p_2))}} & \downarrow \\ \bullet && \stackrel{}{\to} && \bullet }

  and over a Cech 2-cell is

  $$\begin{array}{ccc}& & •\\ & \swarrow & {}^{b_3}⇙& \searrow \\ •& & \stackrel{\mathrm{Ad}(g_{ik}(x))}{\to }& & •\end{array}=\begin{array}{ccc}& & •\\ & \swarrow & \downarrow & \searrow \\ \downarrow & {}^{b_1}⇙& •& {}^{b_2}⇙& \downarrow \\ \downarrow & {}^{\mathrm{Ad}(g_{ij}(x))}\nearrow & {\Downarrow }^{\mathrm{Id}}& {\searrow }^{\mathrm{Ad}(g_{jk}(x))}& \downarrow \\ •& & \stackrel{}{\to }& & •\end{array}.$$\array{ && \bullet \\ &\swarrow &{}^{\mathllap{b_3}}\swArrow& \searrow \\ \bullet &&\stackrel{Ad(g_{i k}(x))}{\to}&& \bullet } \;\;\; = \;\;\; \array{ && \bullet \\ &\swarrow& \downarrow & \searrow \\ \downarrow & {}^{\mathclap{b_1}}\swArrow & \bullet & {}^{\mathclap{b_2}}\swArrow& \downarrow \\ \downarrow & {}^{\mathclap{Ad(g_{i j}(x))}} \nearrow &\Downarrow^{\mathrlap{Id}} & \searrow^{\mathclap{Ad(g_{j k}(x))}} & \downarrow \\ \bullet && \stackrel{}{\to} && \bullet } \,.

The identification with the groupoid $(P{\times }_G\mathrm{Aut}(G))//\mathrm{At}(P)$

We now show that the homotopy fiber computed above is indeed the groupoid $(P{\times }_G\mathrm{Aut}(G))//\mathrm{At}(P)$.

The main point is the realizaiton of the Atiyah Lie groupoid $\mathrm{At}(P)$ as a groupoid over $\Pi (Y)$.

A morphism in $\mathrm{At}(P)$ over a path $p:x\to y$ is a fiber homomorphism $f:P_x\to P_y$. Since we have chose a connection $\nabla$ on $P$, we may always express this relative to the parallel transport of the connection along $p$, which provides a reference frame in which to measure morpisms between fibers, and defines for given $f$ a group element $b$ defined by

$$\begin{array}{ccc}& & P_y\\ & {}^{{\mathrm{tra}}_{\nabla }(p)}\nearrow & {\downarrow }^b\\ P_x& \stackrel{f}{\to }& P_y\end{array}.$$\array{ && P_y \\ & {}^{\mathllap{tra_\nabla(p)}}\nearrow & \downarrow^{b} \\ P_x &\stackrel{f}{\to}& P_y } \,.

When we express composition of fiber morphisms in terms of the compositiion of the group elements associated to them this way, we find an adjoint action of the parallel transport on the group elements:

$$\begin{array}{ccccc}& & & & P_z\\ & & & {}^{{\mathrm{tra}}_{\nabla }(p_2)}\nearrow & {\downarrow }^{\mathrm{Ad}({\mathrm{tra}}_{\nabla })(p_2)(b_1)}\\ & & P_y& & P_z\\ & {}^{{\mathrm{tra}}_{\nabla }(p_1)}\nearrow & {\downarrow }^{b_1}& {}^{{\mathrm{tra}}_{\nabla }(p_2)}\nearrow & {\downarrow }^{b_2}\\ P_x& \stackrel{f_1}{\to }& P_y& \stackrel{f_2}{\to }& P_z\end{array}.$$\array{ &&&& P_z \\ && & {}^{\mathllap{tra_\nabla(p_2)}}\nearrow & \downarrow^{\mathrlap{Ad(tra_\nabla)(p_2)(b_1)}} \\ && P_y && P_z \\ & {}^{\mathclap{tra_\nabla(p_1)}}\nearrow & \downarrow^{b_1} & {}^{\mathclap{tra_\nabla(p_2)}}\nearrow & \downarrow^{b_2} \\ P_x &\stackrel{f_1}{\to}& P_y &\stackrel{f_2}{\to}& P_z } \,.

We see that this is precisely the composition rule satisfied by the group elements labeled $b_i$ in the above description of the homotopy pullback:

$$\begin{array}{cccc}& & & •\\ & \swarrow & {⇙}_{b_1}& \downarrow & {⇙}_{b_2}& & \searrow \\ •& & \stackrel{\mathrm{Ad}(\mathrm{tra}(p_1))}{\to }& •& \stackrel{\mathrm{Ad}(\mathrm{tra}(p_2))}{\to }& & \end{array}=\begin{array}{ccc}& & •\\ & \swarrow & {⇙}_{b_2\mathrm{Ad}({\mathrm{tra}}_{\nabla }(p_2)(b_1))}& \searrow \\ •& & \stackrel{\mathrm{Ad}({\mathrm{tra}}_{\nabla }(p_2\cdot p_1))}{\to }& & \end{array}$$\array{ &&& \bullet \\ & \swarrow &\swArrow_{b_1}& \downarrow &\swArrow_{b_2} & & \searrow \\ \bullet &&\stackrel{Ad(tra(p_1))}{\to}& \bullet &\stackrel{Ad(tra(p_2))}{\to}&& } \;\;\; = \;\;\; \array{ && \bullet \\ & \swarrow &\swArrow_{b_2 Ad(tra_\nabla(p_2)(b_1))}& \searrow \\ \bullet &&\stackrel{Ad(tra_\nabla(p_2 \cdot p_1 ))}{\to}&& }

(with labels in terms of the L B-convention-group – in terms of crossed modules](http://ncatlab.org/nlab/show/strict+2-group#InTermsOfCrossedModules))).

If $p_1\cdot p_2$ and $p_3$ are two homotopica but different paths from $x$ to $y$, then the parallel transport between them differs by the integrated curvature

$$\begin{array}{ccc}& & y\\ & {}^{p_1}\nearrow & {\Downarrow }^{\Sigma }& {\searrow }^{p_2}\\ x& & \stackrel{p_3}{\to }& & z\end{array}\mapsto \begin{array}{ccc}& & •\\ & {}^{\mathrm{Ad}(\mathrm{tra}(p_1))}\nearrow & {\Downarrow }^{\mathrm{tra}(\partial \Sigma )}& {\searrow }^{\mathrm{Ad}(\mathrm{tra}(p_2))}\\ •& & \stackrel{\mathrm{Ad}(\mathrm{tra}(p_3))}{\to }& & •\end{array}.$$\array{ && y \\ & {}^{\mathllap{p_1}}\nearrow &\Downarrow^{\Sigma} & \searrow^{\mathrlap{p_2}} \\ x &&\stackrel{p_3}{\to}&& z } \;\;\; \mapsto \;\;\; \array{ && \bullet \\ & {}^{\mathllap{Ad(tra(p_1))}}\nearrow &\Downarrow^{tra(\partial \Sigma)} & \searrow^{\mathrlap{Ad(tra(p_2))}} \\ \bullet &&\stackrel{Ad(tra(p_3))}{\to}&& \bullet } \,.

The same morphism $f:P_x\to P_z$ corresponds to two different group elements $b_{12}$ and $b_3$ depending on whether it is expressed with respect to the path $p_1\cdot p_2$ or $p_3$. The difference is precisely the integrated curvature $\mathrm{tra}(\partial \Sigma )$

$$\begin{array}{ccc}& & •\\ & {}^{\mathrm{tra}(p_1\cdot p_2)}\nearrow & {\downarrow }^{b_{12}}\\ •& \stackrel{f}{\to }& •\\ & {}_{\mathrm{tra}(p_3)}\searrow & {\uparrow }^{b_3}\\ & & •\end{array}$$\array{ && \bullet \\ & {}^{\mathllap{tra(p_1 \cdot p_2)}} \nearrow & \downarrow^{\mathrlap{b_{1 2}}} \\ \bullet &\stackrel{f}{\to}& \bullet \\ & {}_{\mathllap{tra(p_3)}}\searrow & \uparrow^{\mathrlap{b_3}} \\ && \bullet }

i.e.

$$b_3=\mathrm{tra}(\partial \Sigma )b_{12}.$$b_3 = tra(\partial \Sigma) b_{1 2} \,.

This is precisely the relation imposed by the 2-cells in the homotopy pullback, as read off from the above diagrams.

In summary, this shows that in the homotopy pullback

• if we forget the labels $\alpha ,\beta ,\cdots \in \mathrm{Aut}(K)$ on the diagonal morphism, the resulting groupoid is the Atiyah groupoid over $\Pi (X)$;

• the action of this on the labels, taken into account, is the action of the Atiyah Lie groupoid on $P{\times }_g\mathrm{Aut}(G)$.