Classes of bundles
Examples and Applications
Special and general types
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
For a model for an ∞-group, there is often a model for the universal principal ∞-bundle that itself carries a group structure such that the canonical inclusion is a homomorphism of group objects. This extra groupal structure is important for various constructions.
For ordinary groups
For an ordinary bare group, the action groupoid of the right multiplcation action of on itself
is contractible. We may think of this as the groupoid of inner automorphisms inside the automorphism 2-group of . But it is also simply isomorphic to the codiscrete groupoid on the set underlying . In this latter form the 2-group structure on is manifest, which corresponds to the crossed module .
Als manifest is then that this fits with the one-object delooping groupoid of into a sequence
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 -bundle.
If 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 -principal bundle, as discussed at ∞LieGrpd – the universal G-principal bundle.
By applying the geometric realization functor
we obtain a sequence of topological spaces
where carries the structure of a topological group and the morphism is a topological group homomorphism. For bare groups and under mild assumptions also for general topological groups , this groupal topological model for the universal -bundle obtained from the realization of the groupoid was consider in Segal68.
For strict 2-groups
For ∞Grpd a strict 2-group a groupal model for was given in RobertsSchr07 generalizing the construction mentioned above. This yields a weak 3-group structure on (A Gray-group).
In Roberts07 it is observed that there is also an analog of and that this yields a strict group structure on . In fact, this strictly groupal model of turns out to be isomorphic to the standard model for the universal simplicial principal bundle traditionally denoted . And this statement generalizes
Every ∞-group may be modeled by a simplicial group . There is a standard Kan complex model for the universal -simplicial principal bundle denoted .
This standard model does happen to have the structure of a simplicial group itself, and this structure is compatible with that of in that the canonical inclusion 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 -groups in an arbitrary -topos
Since the 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 -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 -bundles:
for an -algebra, there is an -algebra , defined such that its Chevalley-Eilenberg algebra is the Weil algebra of :
Under Lie integration this gives a groupal model for the universal principal -bundle over the ∞-Lie group that integrates .
This is described at
The Lie-integrated universal principal ∞-bundle
The observation that for an ordinary group, its action groupoid sequence – which is the strict 2-group coming from the crossed module - maps under the nerve to the universal -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 for a strict 2-group is descibed in
The simplicial group structure on for 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 as -algebraic models for universal -principal bundle (evident as it is) was considered as such in