Internal principal $G$-bundles

Question

Let $(C, J)$ be a small site and let $\mathsf{Sh}_{(2, 1)}(C, J)$ be the $(2, 1)$-sheaf topos of sheaves of (small) groupoids on $(C, J)$. Let $G$ be a sheaf of groups on $(C, J)$, and let $\mathbf{Bun}_G$ be the hom-stack $[-, \mathbf{B}G]$, which is typically known as the moduli stack of principal $G$-bundles on $(C, J)$. On the nLab, it is stated that (up to homotopy equivalences), principal $G$-bundles over a given base space $X \in (C, J)$, (i.e. objects of $\mathbf{Bun}_G(X)$) are homotopy pullbacks of the following form:

$\require{AMScd}$ \begin{CD} P @>>> *\\ @V V V @VV V\\ X @>>> \mathbf{B}G \end{CD}

Would anyone mind explaining to me why this is the case, and moreover, how one might obtain "local trivialisations" out of the above pullback square ? The nLab article I linked does go into these topics, but their explanation is a bit too abstract non-sensical for me to be able to cut through.

Answer 1

The easiest way to see local trivializations is to compute the homotopy pullback using the local projective model structure.

For differential geometry, we can $C$ to be the category of cartesian spaces and smooth maps, whereas $J$ is the usual topology of open covers. (Other sites work in the same manner.)

Then maps $\def\B{{\bf B}} X→\B G$ in the homotopy category correspond to maps $Č(U)→\B G$ of presheaves of groupoids, where $U$ is a good cover of $X$ and $Č(U)$ denotes the Čech nerve of U.

Unfolding this construction, we see that a map $Č(U)→\B G$

• sends each $U_i$ to the only object of $\B G(U_i)$;
• sends each intersection $U_i ∩ U_j$ to a morphism in $\B G(U_i∩U_j)$, i.e., a smooth map $t_{i,j}\colon U_i∩U_j→G$;
• for each triple intersection $U_i∩U_j∩U_k$, it enforces a cocycle condition $t_{j,k}t_{i,j}=t_{i,k}$.

This data is precisely the traditional description of principal $G$-bundles in terms of cocycles.

Next, we can compute the homotopy pullback by replacing the map $*→\B G$ with a fibration, namely, $\def\E{{\bf E}} \E G→\B G$, where $\E G(V)$ is the nerve of the contractible groupoid with its objects being smooth maps $V→G$.

Let's now see what a smooth map $V→P$, i.e., an element of $P(V)$ is in concrete terms. By definition of a pullback, an element of $P(V)$ can be described as a pair $(b,f)$, where $b∈Č(U)(V)$, i.e., a smooth map $V→U_i$ for some $i$, whereas $f∈\E G(V)$, i.e., a smooth map $V→G$. The pullback compatibility condition for objects is trivial because $\B G(V)$ has a single object.

Thus, restricting our attention to a single $U_i⊂X$, we see that maps $V→P$ are pair of smooth maps $(b\colon V→U_i,f\colon V→G)$, or, equivalently, a smooth map $τ\colon V→U_i⨯G$. The manifold $U_i⨯G$ is precisely the total space of the trivial principal $G$-bundle $U_i⨯G→U_i$ over $U_i$.

Next, let's examine a generic morphism of the form $$(b\colon V→U_i,f\colon V→G)→(b'\colon V→U_i,f'\colon V→G).$$ According to the definition of a pullback, such a morphism is given by a compatible pair of morphisms $b→b'$ and $f→f'$. By definition of $\E G(V)$, there is a unique morphism $f→f'$, and its image in $\B G(V)$ is $f'f^{-1}$. By definition of $Č(U)$, a morphism $b→b'$ exists if and only if $b$ and $b'$ factor through the intersection $U_i∩U_j$, in which case such a morphism is unique and its image in $\B G(V)$ is $t_{i,j}$. Finally, the compatibility condition amounts to saying $f'f^{-1}=t_{i,j}$.

Collecting all the pieces together, we see that two maps $(b,f)\colon V→U_i⨯G$ and $(b',f')\colon V→U_j⨯G$ are identified if they satisfy the relation $f'f^{-1}=t_{i,j}$. This is precisely how the total space of a principal bundle is glued from individual pieces given by trivial principal bundles over $U_i$ and $U_j$.

Thus, a map $c\colon X→\B G$ picks out principal $G$-bundles, and for such a map, the homotopy pullback is precisely the total space of the principal $G$-bundle classified by $c$.