group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Simplicial groups model all ∞-groups in ∞Grpd. Accordingly principal ∞-bundles in ∞Grpd (a discrete $\infty$-bundle) should be modeled by Kan complexes $E \to X$ equipped with a principal action by a simplicial group. It is suficient to assume the action to be strict. This yields the notion of simplicial principal bundles .
(strict principal simplicial bundle)
A strict simplicial principal bundle is…
(weakly principal simplicial bundle)
For $G$ a simplicial group, a Kan fibration $P \to X$ of Kan complexes equipped with a $G$-action on $P$ over $X$
is a weakly $G$-principal simplicial bundle if the shear map
(to the fiber product of $P$ with itself over $X$) is a simplicial weak equivalence.
(e.g. NSS 12b, Def. 3.79)
A simplicial $G$-principal bundle $P \to X$ is necessarily a Kan fibration.
This appears as (May, Lemma 18.2).
Let $E \to B$ be a twisted cartesian product of the simplicial set $B$ with a simplicial group $G$. Then with respect to the canonical $G$-action this is a simplicial principal bundle.
This is (May, prop. 18.4).
It is simplicial principal bundles of this form that one is mainly interested in. These are the objects that are classified by the evident classifying space $\overline{W} G$. This is discussed below.
Recall from generalized universal bundle that a universal $G$-principal simplicial bundle should be a principal bundle $\mathbf{E}G \to \mathbf{B}G$ such that every other $G$-principal simplicial bundle $P \to X$ arises up to equivalence as the pullback of $\mathbf{E}G$ along a morphism $X \to \mathbf{B}G$.
A standard model for the delooping Kan complex $\mathbf{B}G$ for $G$ a simplicial group goes by the simplicial set, $\overline{W} G \,.$
This is described at simplicial group - delooping. The following establishes a model for the universal simplicial bundle over this model of $\mathbf{B}G$.
For $G$ a simplicial group, define the simplicial set $W G$ to be the decalage of $\overline{W}G$
By the discussion at homotopy pullback this means that for $X_\bullet$ any Kan complex, an ordinary pullback diagram
in sSet exhibits $P_\bullet$ as the homotopy pullback in $sSet_{Quillen}$ / (∞,1)-pullback in ∞Grpd
i.e. as the homotopy fiber of the cocycle $g$.
We call $P_\bullet := X_\bullet \times^g W G$ the simplicial $G$-principal bundle corresponding to $g$.
Let $\{\phi : X_n \to G_{(n-1)}\}$ be the twisting function corresponding to $g : X_\bullet \to \overline{W}G$ by the above discussion.
Then the simplicial set $P_\bullet := X_\bullet \times_{g} W G$ is explicitly given by the formula called the twisted Cartesian product $X_\bullet \times^\phi G_\bullet$:
its cells are
with face and degeneracy maps
$d_i (x,g) = (d_i x , d_i g)$ if $i \gt 0$
$d_0 (x,g) = (d_0 x, \phi(x) d_0 g)$
$s_i (x,g) = (s_i x, s_i g)$.
Here are some pointers on where precisely in the literature the above statements can be found.
One useful reference is
There the abbreviation PCTP ( principal twisted cartesian product ) is used for what above we called just twisted Cartesian products.
The fact that every PTCP $X \times_\phi G \to X$ defined by a twisting function $\phi$ arises as the pullback of $W G \to \overline{W}G$ along a morphism of simplicial sets $X \to \overline{W}G$ can be found there by combining
the last sentence on p. 81 which asserts that pullbacks of PTCPs $X \times_\phi G \to X$ along morphisms of simplicial sets $f : Y \to X$ yield PTCPs corresponding to the composite of $f$ with $\phi$;
section 21 which establishes that $W G \to \bar W G$ is the PTCP for some universal twisting function $r(G)$.
lemma 21.9 states in the language of composites of twisting functions that every PTCP comes from composing a cocycle $Y \to \bar W G$ with the universal twisting function $r(G)$. In view of the relation to pullbacks in item 1, this yields the statement in the form we stated it above.
An explicit version of the statement that twisted Cartesian products are nothing but pullbacks of a generalized universal bundle is on page 148 of
On page 239 there it is mentioned that
is a model for the loop space object fiber sequence
One place in the literature where the observation that $W G$ is the decalage of $\overline{W}G$ is mentioned fairly explicitly is page 85 of
Discussion of topological simplicial principal bundles is in
Discussion of principal simplicial bundles in more general categories of simplicial presheaves (presenting (infinity,1)-toposes):
Last revised on November 28, 2023 at 14:04:20. See the history of this page for a list of all contributions to it.