nLab groupal model for universal principal infinity-bundles

Idea

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

For ordinary groups

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

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

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

It is also manifest then that this fits with the one-object delooping groupoid $\mathbf{B}G$ of $G$ into a sequence

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

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 $G$-bundle.

If $G$ 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 $G$-principal bundle, as discussed at ?LieGrpd – the universal G-principal bundle.

By applying the geometric realization functor

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

we obtain a sequence of topological spaces

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

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

For strict 2-groups

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

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

For $\infty$-Groups

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

This standard model $W G$ does happen to have the structure of a simplicial group itself, and this structure is compatible with that of $G$ in that the canonical inclusion $G \to W G$ 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 $(\infty,1)$-topos

Since the $W$ 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 of smooth ∞-groupoids we can obtain ∞-groups by Lie integration of $L_\infty$-algebras.

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

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

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

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

This is described at

The Lie-integrated universal principal ∞-bundle

Related concepts

• principal ∞-bundle

• universal principal ∞-bundle

• simplicial classifying space

• universal connection

References

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

• David Roberts, Urs Schreiber, The inner automorphism 3-group of a strict 2-group , Journal of Homotopy and Related Structures, vol. 3(1) (arXiv:0708.1741)

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

• David Roberts, The universal simplicial bundle is a simplicial group, New York Journal of Mathematics, Volume 19 (2013) 51-60, journal version, arXiv:1204.4886.

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

• David Roberts, Danny Stevenson, Simplicial principal bundles in parameterized spaces, (arXiv:1203.2460)

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

• Hisham Sati, U.S., Jim Stasheff,

  $L_\infty$-algebra connections in Fauser (eds.) Recent Developments in QFT, Birkhäuser (arXiv:0801.3480)