nLab associated infinity-bundle

Idea

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

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

Definition

Let $\mathbf{H}$ be an (∞,1)-topos.

Definition

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

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

Definition

Let $G \in Grp(\mathbf{H})$ be an ∞-group equipped with an ∞-action $\rho$ on $V$. Then for $P \to X$ a $G$-principal ∞-bundle over $X$, the $\rho$-associated $\infty$-bundle is

$$P \times_G V \to X \,,$$

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

Remark

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

Properties

General

Proposition

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

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

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

$$G Act \simeq \mathbf{H}_{/\mathbf{B}G}$$

it corresponds to a fiber sequence

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

in $\mathbf{H}$. This is the universal $\rho$-associated $V$-bundle in that for $P \to X$ any $G$-principal ∞-bundle modulated by $g \colon X \to \mathbf{B}G$ we have a natural equivalence

$$P \times_G V \simeq g^* \overline{\rho} \,.$$

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

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

Compare universal principal ∞-bundle.

Examples

Fibrations of topological spaces / simplicial sets

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

$\infty$-Gerbes

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

Related concepts

• action, ∞-action

• representation, ∞-representation

• principal bundle / torsor / associated bundle

• principal 2-bundle / gerbe / bundle gerbe

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

• principal ∞-bundle / ∞-gerbe / associated $\infty$-bundle

• vector bundle

• (∞,1)-vector bundle

References

Early work on associated $\infty$-bundles takes place in the $(\infty,1)$-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 $F$-fiber ∞-bundle in the pointed context is identified with $\mathbf{B}Aut_*(F)$ (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

• Matthias Wendt, Classifying spaces and fibrations of simplicial sheaves , Journal of Homotopy and Related Structures 6(1), 2011, pp. 1–38. (arXiv) (published version)

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

• Matthias Wendt, Fibre sequences and localization of simplicial sheaves (pdf)

Similar considerations and results are in

• Martin Blomgren, Wojciech Chacholski, On the classification of fibrations (arXiv:1206.4443)

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

• object classifier in ∞Grpd,

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

• Principal ∞-bundles – theory, presentations and applications

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

• Andrey Lazarev, Models for classifying spaces and derived deformation theory (arXiv:1209.3866)