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created
groupal model for universal principal infinity-bundles
in order to record and link David Roberts’s result.
to go with this, I also created universal principal infinity-bundle.
In case it rings bells for anyone, I point out that the functor $E:sGrp(C) \to sGrp(C)$, sending a simplicial group object $K$ (in a finitely complete category $C$, say) to a contractible simplicial group object $EK$ with $K$ a sub-simplicial group object, is a monad.
is a monad.
It’s maybe noteworthy that the functor underlying this functor after forgetting the group structure is just decalage, and that decalage is a comonad (as indicated at decalage).
What’s going on?
expanded the entry groupal models for universal principal infinity-bundles a bit more
I am sorry to talk as I did not read the background of the construction above, but it is often that the monadic and comonadic simplicial constructions for the same adjoint pair look confusingly similar. For example the bar construction for an action gives a simplicial set, but monads induce a cosimplicial object. I will once write about that in detail; the point is that objects in the category downstairs can be identified with free objects in the category upstairs. So the action can be described using endofunctors above or endofunctors below. Sometimes there is a confusion in identifying what is an appropriate choice here. Sorry for being vague, but I will write soon a discussion on this for the case of bar-like resolutions; the question of the augmentations is also relevant.
I am working my way up to including the following construction into groupal model for universal principal infinity-bundle:
Let $\mathfrak{g}$ be an $n$-truncated $L_\infty$-algebra, such that its Lie integration
$\tau_{\leq n}\exp(\mathfrak{g})$is equivalent to
$\mathbf{B}G := \tau_{\leq {n+1}} \exp(\mathfrak{g}) \,,$i.e. that we can truncate the Sullivan construction one degree “too high” and still not pick up a homotopy group. This is for instance the case for $\mathfrak{g}$ an ordinary Lie algebra: in that case $\tau_{\leq 1}\exp(\mathfrak{g})$ is $\mathbf{B}G$ for $G$ the simply connected Lie group integrating it, and since for every Lie algebra we have $\pi_2(G) = 0$ there is a trivial homotopy group in degree 2 and so $\mathbf{B}G \simeq \tau_{\leq 2}\exp(\mathfrak{g})$.
All right, so in that case, write $inn(\mathfrak{g})$ for the $L_\infty$-algebra whose Chevalley-Eilenberg algebra is the Weil algebra of $\mathfrak{g}$. This is a “$L_\infty$-algebroidal-model” for the universal $\mathfrak{g}$-bundle. (An observation going all the way back to Cartan and of course underlying the Cartan model of equivariant cohomologhy, which is nothing but a Lie algebroid model for the Borel construction $E G \times_G V$).
So we want to integrate this to get a groupal model of the universal $G$-principal $n$-bundle.
Since Lie integration is functorial, we simply have that
$(\mathbf{B}G \to \mathbf{B}\mathbf{E}G) := (\tau_{\leq {n+1}}\exp(\mathfrak{g}) \to \tau_{\leq {n+1}}\exp(inn(\mathfrak{g})))$So picking any explcit delooping functor gives an $\infty$-group homomorphism
$G \to \mathbf{E}G \,.$Since the Weil algebra has no cohomolog we have of course $\mathbf{E}G \simeq *$.
It remains to pass back to the base space object $\mathbf{B}G$ in a controled way.
okay, I added a paragraph with a pointer to a discussion along the above lines.
It’s maybe noteworthy that the functor underlying this functor after forgetting the group structure is just decalage,
..up to isomorphism, that is. The monad I give is not a strict lift of the decalage $Dec:sGrp \to sSet$ through the forgetful functor, but up to iso.
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