Simplicial groups model all ∞-groups in ∞Grpd. Accordingly principal ∞-bundles in ∞Grpd (a discrete -bundle) should be modeled by Kan complexes 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 .
is called principal if it is degreewise principal, i.e. if for all the only elements that have any fixed point in that are the neutral elements.
The canonical action
of any simplicial group on itself is principal.
(simplicial principal bundle)
the action is principal;
the base is isomorphic to the quotient by the action:
A simplicial -principal bundle is necessarly a Kan fibration.
This appears as (May, Lemma 18.2).
This is (May, prop. 18.4).
Recall from generalized universal bundle that a universal -principal simplicial bundle should be a principal bundle such that every other -principal simplicial bundle arises up to equivalence as the pullback of along a morphism .
This is described at simplicial group - delooping. The following establishes a model for the universal simplicial bundle over this model of .
i.e. as the homotopy fiber of the cocycle .
We call the simplicial -principal bundle corresponding to .
Let be the twisting function corresponding to by the above discussion.
Then the simplicial set is explicitly given by the formula called the twisted Cartesian product :
its cells are
with face and degeneracy maps
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 defined by a twisting function arises as the pullback of along a morphism of simplicial sets can be found there by combining
the last sentence on p. 81 which asserts that pullbacks of PTCPs along morphisms of simplicial sets yield PTCPs corresponding to the composite of with ;
section 21 which establishes that is the PTCP for some universal twisting function .
lemma 21.9 states in the language of composites of twisting functions that every PTCP comes from composing a cocycle with the universal twisting function . In view of the relation to pullbacks in item 1, this yields the statement in the form we stated it above.
On page 239 there it is mentioned that
One place in the literature where the observation that is the decalage of is mentioned fairly explicitly is page 85 of
Discussion of topological simplicial principal bundles is in