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groupal model for universal principal infinity-bundles

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cohomology

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(,1)(\infty,1)-Topos theory

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Background

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Extra stuff, structure and property

Models

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structures in a cohesive (∞,1)-topos

Contents

Idea

For GG a model for an ∞-group, there is often a model for the universal principal ∞-bundle EG\mathbf{E}G that itself carries a group structure such that the canonical inclusion GEGG \to \mathbf{E}G is a homomorphism of group objects. This extra groupal structure is important for various constructions.

For ordinary groups

For GG an ordinary bare group, the action groupoid EG=G//G\mathbf{E}G = G//G of the right multiplcation action of GG on itself

G//G=(G×Gp 1G) G//G = \left( G \times G \stackrel{\overset{\cdot}{\to}}{\underset{p_1}{\to}} G \right)

is contractible. We may think of this as the groupoid INN(G)AUT(G)INN(G) \subset AUT(G) of inner automorphisms inside the automorphism 2-group of GG. But it is also simply isomorphic to the codiscrete groupoid on the set underlying GG. In this latter form the 2-group structure on EG\mathbf{E}G is manifest, which corresponds to the crossed module (GIdG)(G \stackrel{Id}{\to} G) .

Als manifest is then that this fits with the one-object delooping groupoid BG\mathbf{B}G of GG into a sequence

GEGBG, G \to \mathbf{E}G \to \mathbf{B}G \,,

where the first morphism is a homomorphism of strict 2-groups.

Regarded as a sequence of morphisms in the model KanCplx of the (∞,1)-topos ∞-Grpd this is already a model for the universal GG-bundle.

If GG here is refined to a Lie group or topological group then this is a sequence of ∞-Lie groupoids or topological ∞-groupoids, respectively, and also then, this is already a model for the universal GG-principal bundle, as discussed at ∞LieGrpd – the universal G-principal bundle.

By applying the geometric realization functor

:GrpdTop |-| : \infty Grpd \stackrel{\simeq}{\to} Top

we obtain a sequence of topological spaces

GGG, G \to \mathcal{E}G \to \mathcal{B}G \,,

where G\mathcal{E}G carries the structure of a topological group and the morphism GGG \to \mathcal{E}G is a topological group homomorphism. For bare groups GG and under mild assumptions also for general topological groups GG, this groupal topological model for the universal GG-bundle obtained from the realization of the groupoid G//GG//G was consider in Segal68.

For strict 2-groups

For GG \in ∞Grpd a strict 2-group a groupal model for EG\mathbf{E}G was given in RobertsSchr07 generalizing the INN(G)AUT(G)INN(G) \subset AUT(G) construction mentioned above. This yields a weak 3-group structure on EG\mathbf{E}G (A Gray-group).

In Roberts07 it is observed that there is also an analog of codisc(G)codisc(G) and that this yields a strict group structure on EG\mathbf{E}G. In fact, this strictly groupal model of EG\mathbf{E}G turns out to be isomorphic to the standard model for the universal simplicial principal bundle traditionally denoted WGW G. And this statement generalizes

For \infty-Groups

Every ∞-group may be modeled by a simplicial group GG. There is a standard Kan complex model for the universal GG-simplicial principal bundle EGBG\mathbf{E}G \to \mathbf{B}G denoted WGW¯GW G \to \bar W G.

This standard model WGW G does happen to have the structure of a simplicial group itself, and this structure is compatible with that of GG in that the canonical inclusion GWGG \to W G is a homomorphisms of simplicial groups.

This sounds like a statement that must be well known, but it was maybe not in the literature. The simplicial group structure is spelled out explicitly in Roberts07

For \infty-groups in an arbitrary (,1)(\infty,1)-topos

Since the WW construction discussed above is functorial, this generalizes to prresheaves of simplicial groups and hence gives models for group objects and groupal universal principal ∞-bundles in (∞,1)-toposes modeled by the model structure on simplicial presheaves.

(…)

For \infty-Lie groups

In the (∞,1)-topos ∞LieGrpd of ∞-Lie groupoids we can obtain ∞-groups by Lie integration of ∞-Lie algebras.

Corresponding to this is a construction of Lie-integrated groupal universal principal \infty-bundles:

for 𝔤\mathfrak{g} an L L_\infty-algebra, there is an L L_\infty-algebra inn(𝔤)inn(\mathfrak{g}), defined such that its Chevalley-Eilenberg algebra is the Weil algebra W(𝔤)W(\mathfrak{g}) of 𝔤\mathfrak{g}:

CE(inn(𝔤))=W(𝔤). CE(inn(\mathfrak{g})) = W(\mathfrak{g}) \,.

Under Lie integration this gives a groupal model for the universal principal \infty-bundle over the ∞-Lie group that integrates 𝔤\mathfrak{g}.

This is described at

The Lie-integrated universal principal ∞-bundle

References

The observation that for GG an ordinary group, its action groupoid sequence GG//GBGG \to G//G \to \mathbf{B}G – which is the strict 2-group coming from the crossed module (GIdG)(G \stackrel{Id}{\to} G) - maps under the nerve to the universal GG-bundle appeared in

  • G. B. Segal, Classifying spaces and spectral sequences , Publ. Math. IHES No. 34 (1968) pp. 105–112

A weak 3-group structure on GEGG \to \mathbf{E}G for GG a strict 2-group is descibed in

The simplicial group structure on GEGG \to \mathbf{E}G for GG a general simplicial group is stated explicitly in

A general abstract construction of this simplicial group structure is discussed in

The use of L-∞-algebras inn(𝔤)inn(\mathfrak{g}) as L L_\infty-algebraic models for universal 𝔤\mathfrak{g}-principal bundle (evident as it is) was considered as such in

Revised on August 30, 2013 11:10:14 by David Roberts (192.43.227.18)