Classes of bundles
Examples and Applications
Special and general types
In higher category theory
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
A universal principal ∞-bundle over an ∞-group-object in an (∞,1)-topos is a morphism in a 1-categorical model for (a homotopical category) such that every -principal ∞-bundle in is modeled in by an (ordinary) pullback of .
Notice that in the proper (∞,1)-topos-context the universal -principal ∞-bundle for an ∞-group is nothing but the point inclusion into the delooping of : every -principal -bundle is the (∞,1)-pullback
of the point in , namely the homotopy kernel of its classifying map . In other words, in a full -categorical context the notion of universal bundle disappears. It is a notion genuinely associated with 1-categorical models for .
Assume that we have a homotopical category model for that has the structure of a category of fibrant objects. Notably this can be the full subcategory on fibrant objects of a model structure on simplicial presheaves.
By standard results on homotopy pullbacks every morphism that
is a fibration
fits into a diagram
with the horizontal morphisms being weak equivalences;
is a model for the universal -principal -bundle.
By path fibrations
A standard construction of a fibration is above is obtained as follows:
by standard results on homotopy pullbacks, we have that the bundle classified by a morphism is given by the limit
where is a path space object for .
This limit may be computed as two consecutive pullbacks
The intermediate pullback
is the path fibration over . By the factorization lemma we have that the projection is indeed a fibration and by the fact that the acyclic fibration is preserved under pullback that indeed is a weak equivalence.
For a Kan complex with a single vertex, the decalage construction is a Kan fibration that fits into a diagram
For a simplicial group the standard simplicial model for the delooping of in ∞Grpd is denoted . This is a Kan complex with a single vertex and is the standard model for the universal simplicial principal bundle, traditionally written .
These constructions are functorial and hence extend to models for (∞,1)-toposes by a model structure on simplicial presheaves.
The model for the universal -principal bundle has the special property that it is a groupal model for universal principal ∞-bundles.