nLab
principal infinity-bundle

Idea

The notion of principal $\mathrm{\infty }$-bundle is a categorification of principal bundle from groups and groupoids to ∞-groupoids, or rather from parameterized groupoids (generalized spaces called stacks) to parameterized $\mathrm{\infty }$-groupoids (generalized spaces called ∞-stacks).

For motivation, background and further details see

• motivation for sheaves, cohomology and higher stacks
• principal bundle
• gerbe
• principal 2-bundle.

Definition in a general $(\mathrm{\infty }\mathrm{,1})$-topos

We define $G$-principal $\mathrm{\infty }$-bundles in the general context of an ∞-stack (∞,1)-topos $H$, with $G$ a group object in the (∞,1)-topos.

Recall that for $A\in H$ an object equipped with a point ${\mathrm{pt}}_A:*\to A$, its corresponding loop space object $\Omega A$ is the homotopy pullback

$$\begin{array}{ccc}\Omega A& \to & *\\ \downarrow & & \downarrow \\ *& \to & A\end{array}.$$\array{ \Omega A &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& A } \,.

Conversely, for $G\in H$ we say an object $BG$ is a delooping of $H$ if it has an essentially unique point and if $G\simeq \Omega BG$. We call $G$ an $\mathrm{\infty }$-group. More in detail, its structure as a group object in an (∞,1)-category is exhibited by the Čech nerve

$$\left(\begin{array}{ccccccc}& \cdots & *{\times }_{BG}*{\times }_{BG}*& \overrightarrow{\overrightarrow{\to }}& *{\times }_{BG}*& \overrightarrow{\to }& *\end{array}\right)\simeq \left(\begin{array}{ccccccc}& \cdots & G\times G& \overrightarrow{\overrightarrow{\to }}& G& \overrightarrow{\to }& *\end{array}\right)$$\left( \array{ &\cdots& {*} \times_{\mathbf{B}G} {*} \times_{\mathbf{B}G} {*} &\stackrel{\to}{\stackrel{\to}{\to}}& {*} \times_{\mathbf{B}G} {*} &\stackrel{\to}{\to}& {*} } \right) \simeq \left( \array{ &\cdots& G \times G &\stackrel{\to}{\stackrel{\to}{\to}}& G &\stackrel{\to}{\to}& {*} } \right)

of $*\to BG$.

$G$-principal $\mathrm{\infty }$-bundles

By the general reasoning discussed at cohomology, a cocycle for $G$-nonabelian cohomology on $X\in H$ is just a morphism $g:X\to BG$ in $H$.

To this is canonically associated its homotopy fiber

$$\begin{array}{ccc}P& \to & *\\ \downarrow & & \downarrow \\ X& \to & BG.\end{array}$$\array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\to& \mathbf{B}G \,. }

and we claim that $P\to X$ canonically extends to the structure of a groupoid object in an (∞,1)-topos that exhibits the action of $G$ on $P$ in that it is a groupoid object over $G$: it fits naturally into a diagram

$$\begin{array}{ccccc}⋮& & ⋮& & ⋮\\ P{\times }_{X}P{\times }_{X}P& \stackrel{\simeq }{\to }& P\times G\times G& \to & G\times G\\ \downarrow \downarrow \downarrow & & \downarrow \downarrow \downarrow & & \downarrow \downarrow \downarrow \\ P{\times }_{X}P& \stackrel{\simeq }{\to }& P\times G& \stackrel{p_2}{\to }& G\\ \downarrow \downarrow & & \downarrow \downarrow & & \downarrow \downarrow \\ P& \stackrel{=}{\to }& P& \stackrel{}{\to }& *\\ \downarrow & & \downarrow & & \downarrow \\ X& \stackrel{=}{\to }& X& \stackrel{g}{\to }& BG\end{array}$$\array{ \vdots && \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\simeq}{\to}& P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{=}{\to}& P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{=}{\to}& X &\stackrel{g}{\to}& \mathbf{B}G }

Proposition

Here the horizontal morphisms on the left are indeed equivalences, as indicated.

Proof

The defining homotopy pullback square for $P{\times }_{X}P$ is

$$\begin{array}{ccc}P{\times }_{X}P& \to & P\\ \downarrow & & \downarrow \\ P& \to & X\end{array}$$\array{ P \times_X P &\to& P \\ \downarrow && \downarrow \\ P &\to& X }

To compute this, we may attach the defining homotopy pullback square of $P$ to obtain

$$\begin{array}{ccccc}P{\times }_{X}P& \to & P& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ P& \to & X& \to & BG.\end{array}$$\array{ P \times_X P &\to& P &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ P &\to& X &\to& \mathbf{B}G \,. }

Since homotopy pullback squares paste to homotopy pullback squares, this says that $P{\times }_{X}P$ is the homotopy limit

$$\begin{array}{ccccc}P{\times }_{X}P& \to & & \to & *\\ \downarrow & & & & \downarrow \\ P& \to & X& \to & BG.\end{array}$$\array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& X &\to& \mathbf{B}G \,. }

But again by the defnition of $P$, the lower horizontal morphism is homotopic to $P\to *\to BG$, so that $P{\times }_{X}P$ is also (equivalent to) the homotopy pullback

$$\begin{array}{ccccc}P{\times }_{X}P& \to & & \to & *\\ \downarrow & & & & \downarrow \\ P& \to & *& \to & BG.\end{array}$$\array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& {*} &\to& \mathbf{B}G \,. }

This finally may be computed in turn as the pasting of two homotopy pullbacks

$$\begin{array}{ccccc}P{\times }_{X}P\simeq P\times G& \to & G& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ P& \to & *& \to & BG.\end{array}$$\array{ P \times_X P\simeq P \times G &\to& G &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ P &\to& {*} &\to& \mathbf{B}G \,. }

of which the one on the right is the defining one of $G$ and the remaining one on the left is just a product.

Remark

(terminology)

For ordinary principal bundles the following terminology is standard:

• the morphism $P{\times }_{X}P\stackrel{}{\to }P\times G$ is the division map;

• the fact that the division map is an equivalence is the principality condition on the action;

• the image $\rho :P\times G\to P$ of the projection to the second factor $p_2:P{\times }_{X}P\to P$ under this equaivelence is the action of $G$ on $P$.

The above shows how every cocycle $X\to BG$ induces a map $P\to X$ equipped with a $G$-action. Conversely, we may define a $G$-action on an object $V$ to be a groupoid object in an (∞,1)-topos sitting over $G$.

Definition

($G$-action)

Let $G$ be a group object in the (∞,1)-topos $H$. An action of $G$ on another object $V\in H$ is a group object in the (∞,1)-topos $V//G\to G$ over $G$ in that we have a morphism of simplicial diagrams

$$\begin{array}{ccc}⋮& & ⋮\\ V\times G\times G& \to & G\times G\\ \downarrow \downarrow \downarrow & & \downarrow \downarrow \downarrow \\ V\times G& \stackrel{p_2}{\to }& G\\ \downarrow \downarrow & & \downarrow \downarrow \\ V& \stackrel{}{\to }& *\end{array}$$\array{ \vdots && \vdots \\ V \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ V \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow \\ V &\stackrel{}{\to}& {*} }

such that the corresponding diagram of homotopy colimits

$$\begin{array}{cccc}& V& \to & *\\ & \downarrow & & \downarrow \\ V//G:=& \underrightarrow{\lim }V\times G^{\times n}& \to & BG\end{array}$$\array{ & V &\to& {*} \\ & \downarrow && \downarrow \\ V//G := & \lim_\to V\times G^{\times n} &\to& \mathbf{B}G }

is a homotopy pullback.

We call the object $V//G$ with this structure of a group object in the (∞,1)-topos the action groupoid of $G$ acting on $V$.

Definition

($G$-$\mathrm{\infty }$-torsor / $G$-principal $\mathrm{\infty }$-bundle)

The above shows that every $G$-cocycle induces

1. an $\mathrm{\infty }$-bundle $P\to X$

2. equipped with a $G$-action on $P$

3. which is principal in that $X\simeq P//G$

Conversely, any such collection we call a $G$-principal $\mathrm{\infty }$-bundle or a $G$-$\mathrm{\infty }$-torsor over $X$ in the given (∞,1)-topos $H$.

Remark

(classification of $G$-principal $\mathrm{\infty }$-bundles)

The ambient ∞-stack (∞,1)-topos $H$ satisfies (as described there) the analog of Giraud’s axioms. These serve to show that $G$-principal $\mathrm{\infty }$-bundles are indeed classified by their corresponding cocycles:

one of the axioms says that every groupoid object in $H$ is effective, in that the morphism to its homotopy colimit is an effective epimorphism.

But this means that starting with the $G$-action on $P$, passing to the cocycle $g$ obtained from the quotient

$$\begin{array}{ccc}P& \to & *\\ \downarrow & & \downarrow \\ X\stackrel{\simeq }{\to }P//G& \stackrel{g}{\to }& BG\end{array}$$\array{ P &\to& {*} \\ \downarrow && \downarrow \\ X \stackrel{\simeq}{\to} P//G &\stackrel{g}{\to}& \mathbf{B}G }

and then reconstructing from this cocycle its homotopy fiber with the induced $G$-action as above, we do reproduce, up to equivalence the $G$-principal bundle that we started with.

Morphisms of $G$-principal $\mathrm{\infty }$-bundles

The morphisms of $G$-principal $\mathrm{\infty }$-bundles are to be such that they respect the $G$-action, so that the ∞-groupoid $G\mathrm{Bund}(X)$ of $G$-principal $\mathrm{\infty }$-bundles on $X$ is naturally equivalent to the cocycle $\mathrm{\infty }$-groupoid ${\mathrm{Hom}}_C(X,BG)$

$$G\mathrm{Bund}(X)\simeq {\mathrm{Hom}}_C(X,BG).$$G Bund(X) \simeq Hom_C(X, \mathbf{B} G) \,.

Then this says that $G$-principal bundles are classified by $G$-cohomology

$$G\mathrm{Bund}(X){/}_{\simeq }=H(X,BG):={\mathrm{Ho}}_C(X,BG)$$G Bund(X)/_\simeq = H(X,\mathbf{B}G) := Ho_C(X, \mathbf{B}G)

given by the hom-set in the homotopy category of the (∞,1)-category $C$.

Connections on $G$-principal $\mathrm{\infty }$-bundles

By refining the classifying cocycles $g:X\to BG$ to cocycles $\Pi (X)\to BG$ on higher path groupoids of $X$, one obtains higher versions of the notion of connection on a bundle. This is described in more detail at Differential Nonabelian Cohomology.

Concrete realizations

We discuss realizations of the general idea in various (∞,1)-toposes.

In topological spaces

The following general construction was originally due to Quillen and defines principal groupoid $\mathrm{\infty }$-bundles in the (∞,1)-topos Top in its presentation by the model structure on simplicial sets.

Let $C$ be a small category and let

$${\rho }_P:C\to \mathrm{SSet}$$\rho_P : C \to SSet

be a functor with values in SSet such that it sends all morphisms in $C$ to weak equivalences in SSet (weak homotopy equivalences of simplicial sets).

Consider first the case that $C$ has a single object, so that it is the delooping $BG$ of a monoid or group $G$. Then

Let

$$P:={\rho }_P(•)$$P := \rho_P(\bullet)

be the simplicial set assigned to this single object and let

$$X:=P//G:=\mathrm{hocolim}{\rho }_P$$X := P//G := hocolim \rho_P

be the corresponding action groupoid (see there for the description as a weak colimit).

Notice that, as every action group, this comes with a canonical map $P//G\to BG$.

Theorem

Given the above, the diagram

$$\begin{array}{ccc}P& \to & *\\ \downarrow & & \downarrow \\ X& \stackrel{g}{\to }& BG\end{array}$$\array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G }

is a homotopy pullback (i.e. defines a fibration sequence).

Proof

This is originally due to

• D. Quillen, Higher algebraic K-theory I, Springer Lecture notes in Math. 341 (1973) 85–147.

The statement is reproduced in section IV of

• P. G. Goerss and J. F. Jardine, 1999, Simplicial Homotopy Theory, number 174 in Progress in Mathematics, Birkhauser. (ps)

Remark

For the simple case that $G$ is group, in which case ${\rho }_C$ necessarily takes values not just inweak equivalences but is isomorphisms of simplicial sets, this says that $P\to X$ is a $G$-principal $\mathrm{\infty }$-bundle. In particular the principality of the action is manifestly exhibited by the fact that the base space $X$ is the (weak) quotient of $P$ by the action of $G$.

The above reproduces manifest the description of ordinary $G$-principal topological bundles in the incarnation as groupoids as described in detail at generalized universal bundle.

More generally, when $G$ is just a monoid the above descibes something a bit more general than an ordinary $G$-principal bundle (as then the action of $G$ on the total space may be by weak equivalences that are not isomorphisms).

Quillen’s original construction is more general than this, concerning in fact 1-groupoid-principal $\mathrm{\infty }$-bundles:

Theorem

Let now $C$ be a category and for

$${\rho }_P:C\to \mathrm{SSet}$$\rho_P : C \to SSet

a functor that sends all morphisms to weak equivalences of simplicial sets.

David Roberts: Do you mean ordinary functor or something weaker? Unless you have something weaker, then all those weak equivalences are actually isomorphisms. Also cf simplicial localization.

Urs: right, $C$ is allowed to be a category, not required to be a groupoid. I have corrected that now. See the beginning of Jardine’s article.

Let now for each object $c\in C$

$$P_c:={\rho }_C(c)$$P_c := \rho_C(c)

be the “bundle of $c$-fibers”.

Then for each $c$ the diagram

$$\begin{array}{ccc}{P}_c& \to & *\\ \downarrow & & {\downarrow }^{*\mapsto c}\\ X& \stackrel{g}{\to }& C\end{array}$$\array{ P_c &\to& {*} \\ \downarrow && \downarrow^{{*} \mapsto c} \\ X &\stackrel{g}{\to}& C }

is a homotopy pullback (i.e. defines a fibration sequence).

This classical construction is recalled in the introduction of

• Jardine, Diagrams and torsors (pdf)

In simplicial presheaves

For a small site $C$, let $H$ be the (∞,1)-topos of ∞-stacks on $C$, in its presentation by the model structure on simplicial presheaves.

In

• Jardine, Diagrams and torsors (pdf)

the discussion of $\mathrm{\infty }$-principal bundles is discussed in this context.

Notice however that this means placing oneself always in the petit topos of sheaves on $C$, which will describe always $\mathrm{\infty }$-bundles over $C$. So if $C=\mathrm{Op}(X)$ is the category of open subsets of some topological space $X$, then this is $\mathrm{\infty }$-bundles over $X$.

Notably in this (∞,1)-topos $(\mathrm{\infty }\mathrm{,1})\mathrm{Sh}(X)$ the terminal object is not the expected abstract point, but rather the space $X$ itself.

As a consequence, the above simple picture of a principal $\mathrm{\infty }$-bundle being just the homotopy pullback of the point is no longer directly available. To circumvent this Jardine introduces the notion of diagrams (not meant in the simple conventional sense).

…

Alternatively, one can consider simplicial presheaves on a gros topos, say on some version of the site of Diff of manifolds as discussed for instance in some detail in

• Daniel Dugger, Sheaves and homotop theory (web, pdf).

That gives a notion of smooth $\mathrm{\infty }$-bundles. And in that case the the desrption of principal $\mathrm{\infty }$-bundles as homotopy pullbacks of the point continues to be valid.

In low degree this is then, more or less explicitly, the description of higher principal bundles by Bartels, Baković, Wockel, etc., as referenced at principal 2-bundle.

References

The notion of principal $\mathrm{\infty }$-bundle (often addressed in the relevant literature in the language of torsors) appears in the context of the simplicial presheaf model for generalized spaces in

• Jardine, Luo, Higher order principal bundles (pdf).

• Jardine, Cocycle categories (pdf).

An earlier description in terms of simplicial objects is

• P. Glenn, Realization of cohomology classes in arbitrary exact categories, J. Pure Appl. Algebra 25, 1982, no. 1, 33–105.

In that article not the total space of the bundle $P\to X$ is axiomatized, but the $\mathrm{\infty }$-action groupoid of the action of $G$ on it.

See the remarks at principal 2-bundle.