Classes of bundles
Examples and Applications
Special and general types
An associated -bundle is a fiber bundle in an (∞,1)-topos with typical fiber that is classified by a cocycle with coefficients in the delooping of the internal automorphism ∞-group of . We say this is associated to the corresponding -principal ∞-bundle.
More generally there should be notions of associated -bundles whose fibers are objects in an (∞,n)-topos over for some .
Let be an (∞,1)-topos.
Let be an ∞-group equipped with an ∞-action on . Then for a -principal ∞-bundle over , the -associated -bundle is
where is the homotopy quotient of the diagonal -action.
For , write for the internal automorphism ∞-group of . This comes with a canonical action on . Then the operation of sending an -principal ∞-bundle to the associated establishes an equivalence
More specifically, if is an ∞-action of on some , then under the equivalence of (∞,1)-categories
it corresponds to a fiber sequence
in . This is the universal -associated -bundle in that for any -principal ∞-bundle modulated by we have a natural equivalence
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:
The universal --bundle is presented by the bar construction
Compare universal principal ∞-bundle.
Fibrations of topological spaces / simplicial sets
For the special case that ∞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).
In the case that the fiber is the delooping of an ∞-group object , the -associated -bundles are called -∞-gerbes. See there for more details.
Early work on associated -bundles takes place in the -topos ∞Grpd 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.
- Daniel Gottlieb, The total space of universal fibrations. Pacific J. Math. Volume 46, Number 2 (1973), 415-417.
the total space of the universal -fiber ∞-bundle in the pointed context is identified with (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 -fibrations, Topology and its applications 87 (1998) (pdf)
Consideration of associated -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 Top sSet this is discussed in
which reproduces the classical results (Stasheff, May).
For general (∞,1)-toposes the classification of associated -bundles is discussed in section I 4.1 of
Models in rational homotopy theory of classifying spaces for homotopy types go back to Sullivan’s remarks on the automorphism L-infinity algebra. Further developments are reviewed and developed in