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associated infinity-bundle

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Contents

Idea

An associated \infty-bundle EXE \to X is a fiber bundle in an (∞,1)-topos H\mathbf{H} with typical fiber FHF \in \mathbf{H} that is classified by a cocycle XBAut̲(F)X \to \mathbf{B}\underline{Aut}(F) with coefficients in the delooping of the internal automorphism ∞-group of FF. We say this is associated to the corresponding Aut̲(F)\underline{Aut}(F)-principal ∞-bundle.

More generally there should be notions of associated \infty-bundles whose fibers are objects in an (∞,n)-topos over H\mathbf{H} for some n>1n \gt 1.

Definition

Let H\mathbf{H} be an (∞,1)-topos.

Definition

For V,XHV,X \in \mathbf{H} two objects, say a VV-fiber ∞-bundle over XX is a morphism EXE \to X (an object in the slice (∞,1)-topos H /X\mathbf{H}_{/X}) such that there exists an effective epimorphism UXU \to X and an (∞,1)-pullback square

U×V E U X. \array{ U \times V &\to& E \\ \downarrow && \downarrow \\ U &\to& X } \,.
Definition

Let GGrp(H)G \in Grp(\mathbf{H}) be an ∞-group equipped with an ∞-action ρ\rho on VV. Then for PXP \to X a GG-principal ∞-bundle over XX, the ρ\rho-associated \infty-bundle is

P× GVX, P \times_G V \to X \,,

where P× GV:=(P×V)//GP \times_G V := (P \times V)//G is the homotopy quotient of the diagonal GG-action.

Remark

Below in Properties we see that every ρ\rho-associated \infty-bundle is a VV-fiber \infty-bundle and that every VV-fiber \infty-bundle arises as associated to an Aut(V)\mathbf{Aut}(V)-principal ∞-bundle

Properties

General

Proposition

For VHV \in \mathbf{H}, write Aut(V)Grp(H)\mathbf{Aut}(V) \in Grp(\mathbf{H}) for the internal automorphism ∞-group of VV. This comes with a canonical action on VV. Then the operation of sending an Aut\mathbf{Aut}-principal ∞-bundle PXP \to X to the associated P× GVXP \times_G V \to X establishes an equivalence

H 1(X,Aut(V)){Vfiberbundles}. H^1(X, \mathbf{Aut}(V)) \simeq \{V-fiber\;\infty-bundles\} \,.

More specifically, if ρ\rho is an ∞-action of GG on some VHV \in \mathbf{H}, then under the equivalence of (∞,1)-categories

GActH /BG G Act \simeq \mathbf{H}_{/\mathbf{B}G}

it corresponds to a fiber sequence

V V//G ρ¯ BG \array{ V &\to& V//G \\ && \downarrow^{\mathrlap{\overline{\rho}}} \\ && \mathbf{B}G }

in H\mathbf{H}. This is the universal ρ\rho-associated VV-bundle in that for PXP \to X any GG-principal ∞-bundle modulated by g:XBGg \colon X \to \mathbf{B}G we have a natural equivalence

P× GVg *ρ¯. P \times_G V \simeq g^* \overline{\rho} \,.

This is discussed in (NSS, section I 4.1).

Presentation in simplicial presheaves

In (Wendt), section 5.5, a presentation of the general situation for 1-localic (∞,1)-toposes is given in terms of the model structure on simplicial presheaves (as discussed at models for ∞-stack (∞,1)-toposes) .

Under this presentation we have:

Proposition

The universal FF-\infty-bundle EFBAut(F)\mathbf{E} F \to \mathbf{B}Aut(F) is presented by the bar construction

FB(*,Aut(F),F)B(*,Aut(F),*). F \to B(*, Aut(F), F) \to B(*, Aut(F), *) \,.

Compare universal principal ∞-bundle.

Examples

Fibrations of topological spaces / simplicial sets

For the special case that H=\mathbf{H} = ∞Grpd and using the presentation by the model structure on topological spaces/model structure on simplicial sets the classification theorem 1 reduces to the classical statement of (Stasheff, May).

\infty-Gerbes

In the case that the fiber FF is the delooping F=BGF = \mathbf{B}G of an ∞-group object GG, the Aut̲(BG)\underline{Aut}(\mathbf{B}G)-associated \infty-bundles are called GG-∞-gerbes. See there for more details.

References

Early work on associated \infty-bundles takes place in the (,1)(\infty,1)-topos ∞Grpd \simeq Top. In

  • Jim Stasheff, A classification theorem for fiber spaces , Topology 2 (1963) 239-246
  • Jim Stasheff, H-spaces and classifying spaces: foundations and recent developments. Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), pp. 247–272. MR0321079 (47 #9612)

a classification of fibrations of CW-complexes with given CW-complex fiber in terms of maps into a classifying CW-complex is given.

In

  • Daniel Gottlieb, The total space of universal fibrations. Pacific J. Math. Volume 46, Number 2 (1973), 415-417.

the total space of the universal FF-fiber ∞-bundle in the pointed context is identified with BAut *(F)\mathbf{B}Aut_*(F) (the pointed automorphism ∞-group).

A generalization or more systematic account of the classification theory is then given in

  • Peter May, Classifying Spaces and Fibrations Mem. Amer. Math. Soc. 155 (1975) (pdf)

This has been reproven in various guises, such as the statement of univalence in the model sSet for homotopy type theory. See the references at univalence for more on this.

Generalizations with extra structure on the fibers are discussed in

  • Claudio Pacati, Petar Pavesic, Renzo Piccinini, On the classification of \mathcal{F}-fibrations, Topology and its applications 87 (1998) (pdf)

Consideration of associated \infty-bundles / fiber sequences in general 1-localic (∞,1)-toposes presented by a model structure on simplicial presheaves (which subsumes the above case for the trivial site) is discussed in

Related discussion on the behaviour of fiber sequences under left Bousfield localization of model categories is in

Similar considerations and results are in

With the advent of (∞,1)-topos theory all these statements and their generalizations follow from the existence of object classifiers in an (∞,1)-topos. For the classical case in ∞Grpd \simeq Top {}^\circ sSet {}^\circ this is discussed in

which reproduces the classical results (Stasheff, May).

For general (∞,1)-toposes the classification of associated \infty-bundles is discussed in section I 4.1 of

Models in rational homotopy theory of classifying spaces for homotopy types Aut(F)Aut(F) go back to Sullivan’s remarks on the automorphism L-infinity algebra. Further developments are reviewed and developed in

Revised on December 8, 2013 08:29:38 by Tim Porter (2.26.35.180)