Contents

Idea

For $G$ a model for an ∞-group, there is often a model for the universal principal ∞-bundle $EG$ that itself carries a group structure such that the canonical inclusion $G\to EG$ 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 $EG=G//G$ of the right multiplcation action of $G$ on itself

is contractible. We may think of this as the groupoid $\mathrm{INN}(G)\subset \mathrm{AUT}(G)$ 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 $EG$ is manifest, which corresponds to the crossed module $(G\stackrel{\mathrm{Id}}{\to }G)$ .

Als manifest is then that this fits with the one-object delooping groupoid $BG$ of $G$ 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 $G$-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

we obtain a sequence of topological spaces

where $ℰG$ carries the structure of a topological group and the morphism $G\to ℰG$ 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 $EG$ was given in RobertsSchr07 generalizing the $\mathrm{INN}(G)\subset \mathrm{AUT}(G)$ construction mentioned above. This yields a weak 3-group structure on $EG$ (A Gray-group).

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

For $\mathrm{\infty }$-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 $EG\to BG$ denoted $WG\to \overline{W}G$.

This standard model $WG$ 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 $G\to WG$ 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 $\mathrm{\infty }$-groups in an arbitrary $(\mathrm{\infty },1)$-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 $\mathrm{\infty }$-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 $\mathrm{\infty }$-bundles:

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

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

This is described at

The Lie-integrated universal principal ∞-bundle

Related concepts

• principal ∞-bundle

• universal principal ∞-bundle , groupal model for universal principal ∞-bundle

• universal connection

References

The observation that for $G$ an ordinary group, its action groupoid sequence $G\to G//G\to BG$ – which is the strict 2-group coming from the crossed module $(G\stackrel{\mathrm{Id}}{\to }G)$ - 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 $G\to EG$ 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 $G\to EG$ for $G$ a general simplicial group is stated explicitly in

• David Roberts, The universal simplicial bundle is a simplicial group, (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 $\mathrm{inn}(𝔤)$ as $L_{\mathrm{\infty }}$-algebraic models for universal $𝔤$-principal bundle (evident as it is) was considered as such in

• Hisham Sati, U.S., Jim Stasheff, $L_{\mathrm{\infty }}$-algebra connections in Fauser (eds.) Recent Developments in QFT, Birkhäuser (arXiv:0801.3480)