Contents

Contents

Idea

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

For motivation, background and further details see

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

Definition

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

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

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

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

$\left( \array{ &\cdots& {*} \times_{\mathbf{B}G} {*} \times_{\mathbf{B}G} {*} &\stackrel{\to}{\stackrel{\to}{\to}}& {*} \times_{\mathbf{B}G} {*} &\stackrel{\to}{\to}& {*} } \right) \simeq \left( \array{ &\cdots& G \times G &\stackrel{\to}{\stackrel{\to}{\to}}& G &\stackrel{\to}{\to}& {*} } \right)$

of ${*} \to \mathbf{B}G$.

$G$-principal $\infty$-bundles

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

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

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

Definition

(principal $G$-action)

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

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

in $\mathbf{H}$;

and such that $P \to X$ exhibits the (∞,1)-colimit

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

called the base space over which the action takes place.

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

$\array{ \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\simeq}{\to}& P \times G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{\simeq}{\to}& P } \,.$
Proof

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

Proposition

For $X \to \mathbf{B}G$ 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

$\array{ \vdots && \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\simeq}{\to}& P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{=}{\to}& P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{=}{\to}& X &\stackrel{g}{\to}& \mathbf{B}G } \,,$

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

$\array{ P \times_X P &\to& P \\ \downarrow && \downarrow \\ P &\to& X }$

To compute this, we may attach the defining $(\infty,1)$-pullback square of $P$ to obtain the pasting diagram

$\array{ P \times_X P &\to& P &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ P &\to& X &\to& \mathbf{B}G \,. }$

and use the pasting law for pullbacks, to conclude that $P \times_X P$ is the pullback

$\array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& X &\to& \mathbf{B}G \,. }$

By definition of $P$ as the homotopy fiber of $X \to \mathbf{B}G$, the lower horizontal morphism is equivbalent to $P \to {*} \to \mathbf{B}G$, so that $P \times_X P$ is also equivalent to the pullback

$\array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& {*} &\to& \mathbf{B}G \,. }$

This finally may be computed as the pasting of two pullbacks

$\array{ P \times_X P &\simeq& P \times G &\to& G &\to& {*} \\ &&\downarrow && \downarrow && \downarrow \\ &&P &\to& {*} &\to& \mathbf{B}G \,. }$

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

$\array{ P \times_X P^{\times_X^n} &\to& P \\ \downarrow && \downarrow \\ P^{\times_X^n}&\to& X } \,.$

We may again paste this to obtain

$\array{ P \times_X P^{\times_X^n} &\to& P &\to& * \\ \downarrow && \downarrow && \downarrow \\ P^{\times_X^n}&\to& X &\to& \mathbf{B}G }$

and use from the previous induction step that

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

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 $* \to \mathbf{B}G$ 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{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 $X \to \mathbf{B}G$ canonically induced a $G$-principal action on the homotopy fiber $P \to X$. The following definition declares the $G$-principal $\infty$-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

$\array{ \vdots && \vdots \\ P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow \\ X = \lim_\to (P \times G^\bullet) &\stackrel{g}{\to}& \mathbf{B}G = \lim_\to( G^\bullet) } \,.$
Definition

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

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

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

Proposition

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

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

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

Proof

The arrow category $\mathbf{H}^I$ 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

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

between groupoid objects in $\mathbf{H}^I$ and effective epimorphisms in the arrow category.

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

Restricting this equivalence along the inclusion

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

given by sending a cocycle to its homotopy fiber diagram

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

therefore yields precisely the equivalence in question

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

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

Connections on $G$-principal $\infty$-bundles

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

Concrete realizations

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

In topological spaces

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

Let $C$ be a small category and let

$\rho_P : C \to SSet$

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 $\mathbf{B}G$ of a monoid or group $G$. Then

Let

$P := \rho_P(\bullet)$

be the simplicial set assigned to this single object and let

$X := P//G := hocolim \rho_P$

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 $P//G \to \mathbf{B}G$.

Theorem

Given the above, the diagram

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

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 $\infty$-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 $\infty$-bundles:

Theorem

Let now $C$ be a category and for

$\rho_P : C \to SSet$

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

Let now for each object $c \in C$

$P_c := \rho_C(c)$

be the “bundle of $c$-fibers”.

Then for each $c$ the diagram

$\array{ P_c &\to& {*} \\ \downarrow && \downarrow^{{*} \mapsto c} \\ X &\stackrel{g}{\to}& C }$

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 a petit $(\infty,1)$-topos

For $X$ a topological space $C = Op(X)$ the category of open subsets of $X$, let $\mathbf{H} = Sh_{(\infty,1)}(X)$ 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 $\infty$-bundles are discussed in

• Jardine, Diagrams and torsors (pdf)

In a gros $(\infty,1)$-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 $\mathbf{H} = Sh_{(\infty,1)}(C)$ be the (∞,1)-topos of ∞-stacks on $C$. This is a gros topos.

Smooth principal $\infty$-bundles

Smooth principal $\infty$-bundles are realized in the $\infty$-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 $(\infty,1)$-topos $\mathbf{H} =$ ?LieGrpd? is an ordinary $G$-principal bundle.

Circle $n$-bundles

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

Classes of examples include

Normal morphisms of $\infty$-groups

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

The notion of principal $\infty$-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

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 $\infty$-action groupoid of the action of $G$ on it.

See the remarks at principal 2-bundle.

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

A more comprehensive conceptual account is in

The classifying spaces for a large class of principal $\infty$-bundles are discussed in

A fairly comprehensive account of the literature is also in the introduction of NSS 12, “Presentations”.

For $\mathbf{H}= \infty Grpd$ the statement that homotopy types over $B G$ are equivalently $G$-infinity-actions is maybe due to

• E. Dror, William Dwyer, and Daniel Kan, Equivariant maps which are self homotopy equivalences, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 (JSTOR)

This is mentioned for instance as exercise 4.2in

• William Dwyer, Homotopy theory of classifying spaces, Lecture notes Copenhagen (June, 2008) pdf

Closely related discussion of homotopy fiber sequences and homotopy action but in terms of Segal spaces is in section 5 of

There, conditions are given for a morphism $A_\bullet \to B_\bullet$ to a reduced Segal space to have a fixed homotopy fiber, and hence encode an action of the loop group of $B$ on that fiber.