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.

A model for principal $\mathrm{\infty }$-bundles is given by

• simplicial principal bundles.

See also

• universal principal ∞-bundle

• groupal model for universal principal ∞-bundles.

Definition

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 ∞-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

To every cocycle $g:X\to BG$ is canonically associated its homotopy fiber $P\to X$, the (∞,1)-pullback

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

We discuss now that $P$ canonically has the structure of a $G$-principal ∞-bundle and that $BG$ is the fine moduli space for $G$-principal $\mathrm{\infty }$-bundles.

Definition

(principal $G$-action)

Let $G$ be a group object in the (∞,1)-topos $H$. A principal action of $G$ on an object $P\in H$ is a groupoid object $P//G$ that sits over $*//G$ in that we have a morphism of simplicial diagrams ${\Delta }^{\mathrm{op}}\to H$

$$\begin{array}{ccc}⋮& & ⋮\\ P\times G\times G& \stackrel{(p_2,p_3)}{\to }& G\times G\\ \downarrow \downarrow \downarrow & & \downarrow \downarrow \downarrow \\ P\times G& \stackrel{p_2}{\to }& G\\ \downarrow \downarrow & & \downarrow \downarrow \\ P& \stackrel{}{\to }& *\end{array}$$\array{ \vdots && \vdots \\ P \times G \times G &\stackrel{(p_2, p_3)}{\to}& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{}{\to}& {*} }

in $H$.

We say that the (∞,1)-colimit

$$X:\underrightarrow{\lim }(P//G:{\Delta }^{\mathrm{op}}\to H)$$X : \lim_\to (P//G : \Delta^{op} \to \mathbf{H})

is the base space defined by this action.

We may think of $P//G$ as the action groupoid of the $G$-action on $P$. For us it defines this $G$-action.

Proposition

The $G$-principal action as defined above satisfies the principality condition in that we have an equivalence of groupoid objects

$$\begin{array}{ccc}⋮& & ⋮\\ P{\times }_{X}P{\times }_{X}P& \stackrel{\simeq }{\to }& P\times G\times G\\ \downarrow \downarrow \downarrow & & \downarrow \downarrow \downarrow \\ P{\times }_{X}P& \stackrel{\simeq }{\to }& P\times G\\ \downarrow \downarrow & & \downarrow \downarrow \\ P& \stackrel{\simeq }{\to }& P\end{array}.$$\array{ \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\simeq}{\to}& P \times G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{\simeq}{\to}& P } \,.

Proof

This principality condition asserts that the groupoid object $P//G$ is effective. By Giraud's axioms characterizing (∞,1)-toposes, every groupoid object in $H$ is effective.

Proposition

For $X\to BG$ any morphism, its homotopy fiber $P\to X$ is canonically equipped with a principal $G$-action with base space $X$.

Proof

First we show that we have a morphism of simplicial diagrams

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

with the right square swhere the left horizontal morphisms are equivalences, as indicated. We proceed by induction through the height of this diagram.

The defining (∞,1)-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 $(\mathrm{\infty },1)$-pullback square of $P$ to obtain the pasting diagram

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

and use the pasting law for pullbacks, to conclude that $P{\times }_{X}P$ is the pullback

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

By definition of $P$ as the homotopy fiber of $X\to BG$, the lower horizontal morphism is equivbalent to $P\to *\to BG$, so that $P{\times }_{X}P$ is also equivalent to the 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 as the pasting of two pullbacks

$$\begin{array}{ccccccc}P{\times }_{X}P& \simeq & P\times G& \to & G& \to & *\\ & & \downarrow & & \downarrow & & \downarrow \\ & & P& \to & *& \to & BG.\end{array}$$\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 for $G$ and the remaining one on the left is just an (∞,1)-product.

Proceeding by induction from this case we find analogously that $P^{{\times }_X^{n+1}}\simeq P\times G^{{\times }_n}$: suppose this has been shown for $(n-1)$, then the defining pullback square for $P^{{\times }_X^{n+1}}$ is

$$\begin{array}{ccc}P{\times }_X{P}^{{\times }_X^n}& \to & P\\ \downarrow & & \downarrow \\ P^{{\times }_X^n}& \to & X\end{array}.$$\array{ P \times_X P^{\times_X^n} &\to& P \\ \downarrow && \downarrow \\ P^{\times_X^n}&\to& X } \,.

We may again paste this to obtain

$$\begin{array}{ccccc}P{\times }_X{P}^{{\times }_X^n}& \to & P& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ P^{{\times }_X^n}& \to & X& \to & BG\end{array}$$\array{ P \times_X P^{\times_X^n} &\to& P &\to& * \\ \downarrow && \downarrow && \downarrow \\ P^{\times_X^n}&\to& X &\to& \mathbf{B}G }

and use from the previous induction step that

$$(P^{{\times }_X^n}\to X\to BG)\simeq (P^{{\times }_X^n}\to *\to BG)$$(P^{\times_X^n} \to X \to \mathbf{B}G) \simeq (P^{\times_X^n} \to * \to \mathbf{B}G)

to conclude the induction step with the same arguments as before.

This shows that $P//G$ is the Cech nerve of $P\to X$. It remains to show that indeed $X={{\lim }_{\to }}_{n}P\times G^{{\times }^n}$. For this notice that $*\to BG$ is an effective epimorphism in an (infinity,1)-category. Hence so is $P\to X$. This proves the claim, by definition of effective epimorphism.

using this we have

$$\begin{array}{rl}X& \simeq BG\prod _{BG}X\\ & \simeq \left({\underrightarrow{\lim }}_n{G}^{{\times }^n}\right)\prod _{BG}X\\ & \simeq {\underrightarrow{\lim }}_n(G^{{\times }^n}\prod _{BG}X)\\ & \simeq {\underrightarrow{\lim }}_n(P\times G^{{\times }^n})\\ & \simeq \underrightarrow{\lim }P//G\end{array}.$$\begin{aligned} X & \simeq \mathbf{B}G \prod_{\mathbf{B}G} X \\ & \simeq \left({\lim_{\to}}_n G^{\times^n}\right) \prod_{\mathbf{B}G} X \\ & \simeq {\lim_{\to}}_n ( G^{\times^n} \prod_{\mathbf{B}G} X ) \\ & \simeq {\lim_\to}_n ( P\times G^{\times^n} ) \\ & \simeq {\lim_\to} P//G \end{aligned} \,.

We have established that every cocycle $X\to BG$ canonically induced a $G$-principal action on the homotopy fiber $P\to X$. The following definition declares the $G$-principal $\mathrm{\infty }$-bundles to be those $G$-principal actions that do arise this way.

Definition

We say a $G$-principal action of $G$ on $P$ over $X$ is a $G$-principal ∞-bundle if the colimit over $P//G\to *//G$ produces a pullback square: the bottom square in

$$\begin{array}{ccc}⋮& & ⋮\\ P\times G\times G& \to & G\times G\\ \downarrow \downarrow \downarrow & & \downarrow \downarrow \downarrow \\ P\times G& \stackrel{p_2}{\to }& G\\ \downarrow \downarrow & & \downarrow \downarrow \\ P& \stackrel{}{\to }& *\\ \downarrow & & \downarrow \\ X=\underrightarrow{\lim }(P\times G^{•})& \stackrel{g}{\to }& BG=\underrightarrow{\lim }(G^{•})\end{array}.$$\array{ \vdots && \vdots \\ P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow \\ X = \lim_\to (P \times G^\bullet) &\stackrel{g}{\to}& \mathbf{B}G = \lim_\to( G^\bullet) } \,.

Definition

For $G$ an infinity-group in $H$ and $X\in H$ any object, write

$$G\mathrm{Bund}(X)\subset \mathrm{Grpd}(H)/*//G$$G Bund(X) \subset Grpd(\mathbf{H})/{*//G}

for the sub-(infinity,1)-category on the over-(infinity,1)-category of the groupoid objects over $*//G$ on the $G$-principal $\mathrm{\infty }$-bundles as above.

Proposition

We have an equivalence of (∞,1)-categories

$$G\mathrm{Bund}(X)\simeq H(X,BG)$$G Bund(X) \simeq \mathbf{H}(X, \mathbf{B}G)

of $G$-orincipal $\mathrm{\infty }$-bundles over $X$ with cocycles $X\to BG$.

Proof

The arrow category $H^I$ is still an (infinity,1)-topos and hence the Griraud-Lurie axioms still hold. This means that by the discussion at groupoid object in an (infinity,1)-category (using the statement below HTT, cor. 6.2.3.5) we have an equivalence

$$\mathrm{Grpd}(H^I)\simeq (H^I{)}_{\mathrm{eff}}^I$$Grpd(\mathbf{H}^I) \simeq (\mathbf{H}^{I})^{I}_{eff}

between groupoid objects in $H^I$ and effective epimorphisms in the arrow category.

Notice that groupoid objects and effective epis in $H^I$ are given objectwise over the two objects of the inerval $I=\Delta [1]$.

Restricting this equivalence along the inclusion

$$H(X,BG)\hookrightarrow (H^I{)}^I$$\mathbf{H}(X, \mathbf{B}G) \hookrightarrow (\mathbf{H}^I)^I

given by sending a cocycle to its homotopy fiber diagram

$$(X\to BG)\mapsto \left(\begin{array}{ccc}P& \to & *\\ \downarrow & & \downarrow \\ X& \to & BG\end{array}\right)$$(X \to \mathbf{B}G) \mapsto \left( \array{ P &\to& * \\ \downarrow && \downarrow \\ X &\to& \mathbf{B}G } \right)

therefore yields precisely the equivalence in question

$$\begin{array}{ccc}G\mathrm{Bund}(X)& \hookrightarrow & \mathrm{Grpd}(H^I)\\ {\downarrow }^{\simeq }& & {\downarrow }^{\simeq }\\ H(X,BG)& \stackrel{\mathrm{hofib}}{\hookrightarrow }& (H^I{)}^I\end{array}.$$\array{ G Bund(X) &\hookrightarrow& Grpd(\mathbf{H}^I) \\ \downarrow^\simeq && \downarrow^\simeq \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{hofib}{\hookrightarrow}& (\mathbf{H}^I)^I } \,.

In words this says that the cohomology on $X$ with coefficients in $G$ classified $G$-principal $\mathrm{\infty }$-bundles, and in fact does so on the level of cocycles.

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

For some comments on the generalization of the notion of connection on a bundle to principal $\mathrm{\infty }$-bundles see differential cohomology in an (∞,1)-topos -- survey.

Concrete realizations

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

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 in weak 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.

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 sets / Kan complexes

See simplicial principal bundle.

In a petit $(\mathrm{\infty },1)$-topos

For $X$ a topological space $C=\mathrm{Op}(X)$ the category of open subsets of $X$, let $H={\mathrm{Sh}}_{(\mathrm{\infty },1)}(X)$ be the (∞,1)-topos of ∞-stacks on $C$. This is the petit topos incarnation of $X$.

In its presentation by the model structure on simplicial presheaves this is the context in which princpal $\mathrm{\infty }$-bundles are discussed in

• Jardine, Diagrams and torsors (pdf)

In a gros $(\mathrm{\infty },1)$-topos

For $C$ a site of test space, – for instance duals of algebras over a Lawvere theory as described at function algebras on infinity-stacks – let $H={\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)$ be the (∞,1)-topos of ∞-stacks on $C$. This is a gros topos.

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

Smooth principal $\mathrm{\infty }$-bundles are realized in the $\mathrm{\infty }$-Cahiers topos as described in some detail at ∞-Lie groupoid.

In this context there is a notion of connection on a principal ∞-bundle.

Examples

Ordinary principal bundles

For $G$ an ordinary Lie group, a $G$-principal bundle in the $(\mathrm{\infty },1)$-topos $H=$ ∞LieGrpd is an ordinary $G$-principal bundle.

Circle $n$-bundles

For $G=B^{n-1}U(1)\in$ ∞LieGrpd, the circle Lie n-group, a $G$-principal $\mathrm{\infty }$-bundle is a circle $n$-bundle.

See circle n-bundle with connection.

Classes of examples include

• the Chern-Simons circle 3-bundle.

Bundle gerbes

• A bundle gerbe is a concrete model for the total space groupoid of the total space of a $BU(1)$-principal 2-bundle.

  More generally, a nonabelian bundle gerbe is a concrete model for the groupoid of the total space of a general principal 2-bundle.

• A bundle 2-gerbe is a concrete model for the total space 2-groupoid of the total space of a $B^{2}U(1)$-principal 3-bundle.

  More generally, a nonabelian bundle 2-gerbe is a concrete model for the 2-groupoid of the total space of a general principal 3-bundle.

  Classes of examples include

  • the Chern-Simons bundle 2-gerbe.

Normal morphisms of $\mathrm{\infty }$-groups

A principal $\mathrm{\infty }$-bundle over a 0-connected object / delooping object $mathfBK$ is a normal morphism of ∞-groups. See there for more details.

Related concepts

• principal bundle / torsor / associated bundle

• principal 2-bundle / gerbe / bundle gerbe

• principal 3-bundle / 2-gerbe / bundle 2-gerbe

• principal $\mathrm{\infty }$-bundle / associated ∞-bundle / ∞-gerbe

• vector bundle

• (∞,1)-vector bundle / $(\mathrm{\infty },n)$-vector 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

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

• Rick 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.

See also

• Matthias Wendt, Classifying spaces and fibrations of simplicial sheaves (arXiv)

on associated ∞-bundles.

The fully general abstract formalization in (∞,1)-topos theory as discussed here was first indicated in

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted Differential String and Fivebrane Structures

A more comprehensive conceptual account is in

• Urs Schreiber, differential cohomology in a cohesive topos

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal ∞-bundles -- models and general theory