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principal infinity-bundle

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cohomology

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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

See also

Definition

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

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

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

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

( *× BG*× BG* *× BG* *)( G×G G *) \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 *BG{*} \to \mathbf{B}G.

GG-principal \infty-bundles

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

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

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

Definition

(principal GG-action)

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

P×G×G (p 2,p 3) G×G P×G p 2 G P * \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\mathbf{H};

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

Xlim (P//G:Δ opH) 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//GP//G as the action groupoid of the GG-action on PP. For us it defines this GG-action.

Proposition

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

P× XP× XP P×G×G P× XP P×G P P. \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//GP//G is effective. By Giraud's axioms characterizing (∞,1)-toposes, every groupoid object in H\mathbf{H} is effective.

Proposition

For XBGX \to \mathbf{B}G any morphism, its homotopy fiber PXP \to X is canonically equipped with a principal GG-action with base space XX.

Proof

First we show that we have a morphism of simplicial diagrams

P× XP× XP P×G×G G×G P× XP P×G p 2 G P = P * X = X g BG, \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× XPP \times_X P is

P× XP P P X \array{ P \times_X P &\to& P \\ \downarrow && \downarrow \\ P &\to& X }

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

P× XP P * P X BG. \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× XPP \times_X P is the pullback

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

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

P× XP * P * BG. \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

P× XP P×G G * P * BG. \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 GG and the remaining one on the left is just an (∞,1)-product.

Proceeding by induction from this case we find analogously that P × X n+1P×G × nP^{\times_X^{n+1}} \simeq P \times G^{\times_n}: suppose this has been shown for (n1)(n-1), then the defining pullback square for P × X n+1P^{\times_X^{n+1}} is

P× XP × X n P P × X n X. \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

P× XP × X n P * P × X n X BG \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 × X nXBG)(P × X n*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//GP//G is the Cech nerve of PXP \to X. It remains to show that indeed X=lim nP×G × nX = {\lim_\to}_n P \times G^{\times^n}. For this notice that *BG* \to \mathbf{B}G is an effective epimorphism in an (infinity,1)-category. Hence so is PXP \to X. This proves the claim, by definition of effective epimorphism.

using this we have

X BG BGX (lim nG × n) BGX lim n(G × n BGX) lim n(P×G × n) lim P//G. \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 XBGX \to \mathbf{B}G canonically induced a GG-principal action on the homotopy fiber PXP \to X. The following definition declares the GG-principal \infty-bundles to be those GG-principal actions that do arise this way.

Definition

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

P×G×G G×G P×G p 2 G P * X=lim (P×G ) g BG=lim (G ). \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 GG an infinity-group in H\mathbf{H} and XHX \in \mathbf{H} any object, write

GBund(X)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*//G on the GG-principal \infty-bundles as above.

Proposition

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

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

of GG-orincipal \infty-bundles over XX with cocycles XBGX \to \mathbf{B}G.

Proof

The arrow category H I\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(H I)(H I) eff I Grpd(\mathbf{H}^I) \simeq (\mathbf{H}^{I})^{I}_{eff}

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

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

Restricting this equivalence along the inclusion

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

given by sending a cocycle to its homotopy fiber diagram

(XBG)(P * X BG) (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

GBund(X) Grpd(H I) H(X,BG) hofib (H I) I. \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 XX with coefficients in GG classified GG-principal \infty-bundles, and in fact does so on the level of cocycles.

Connections on GG-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 H\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 CC be a small category and let

ρ P:CSSet \rho_P : C \to SSet

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

Consider first the case that CC has a single object, so that it is the delooping BG\mathbf{B}G of a monoid or group GG. Then

Let

P:=ρ P() P := \rho_P(\bullet)

be the simplicial set assigned to this single object and let

X:=P//G:=hocolimρ 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//GBGP//G \to \mathbf{B}G.

Theorem

Given the above, the diagram

P * X g BG \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 GG is group, in which case ρ C\rho_C necessarily takes values not just in weak equivalences but is isomorphisms of simplicial sets, this says that PXP \to X is a GG-principal \infty-bundle. In particular the principality of the action is manifestly exhibited by the fact that the base space XX is the (weak) quotient of PP by the action of GG.

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

More generally, when GG is just a monoid the above descibes something a bit more general than an ordinary GG-principal bundle (as then the action of GG 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 CC be a category and for

ρ P:CSSet \rho_P : C \to SSet

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

Let now for each object cCc \in C

P c:=ρ C(c) P_c := \rho_C(c)

be the “bundle of cc-fibers”.

Then for each cc the diagram

P c * *c X g C \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 (,1)(\infty,1)-topos

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

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 (,1)(\infty,1)-topos

For CC a site of test space, – for instance duals of algebras over a Lawvere theory as described at function algebras on infinity-stacks – let H=Sh (,1)(C)\mathbf{H} = Sh_{(\infty,1)}(C) be the (∞,1)-topos of ∞-stacks on CC. 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 GG an ordinary Lie group, a GG-principal bundle in the (,1)(\infty,1)-topos H=\mathbf{H} = ∞LieGrpd is an ordinary GG-principal bundle.

Circle nn-bundles

For G=B n1U(1)G = \mathbf{B}^{n-1} U(1) \in ∞LieGrpd, the circle Lie n-group, a GG-principal \infty-bundle is a circle nn-bundle.

See circle n-bundle with connection.

Classes of examples include

Bundle gerbes

Normal morphisms of \infty-groups

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

References

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 PXP \to X is axiomatized, but the \infty-action groupoid of the action of GG on it.

See the remarks at principal 2-bundle.

See also

on associated ∞-bundles.

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 H=Grpd\mathbf{H}= \infty Grpd the statement that homotopy types over BGB G are equivalently GG-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 B 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 BB on that fiber.

Revised on March 5, 2014 02:37:19 by Urs Schreiber (145.116.128.235)