When in the context of a category of fibrant objects we use the following notation and conventions for common constructions.
For any morphisms of fibrant objects and for denoting a path object for , we write for the composite vertical morphism in the pullback diagram
This is always a fibration and is an acyclic fibration if is a weak equivalence. Also the morphism
is always an acyclic fibration. Moreover, this has a section that is necessarily a weak equivalence.
This yields a factorization of as
If is a given point of ( is the terminal object) then we write for the morphjism in the pullback diagram
This is the loop space object of at the point .
If for a given object there is an object with a unique point and such that we write . This is the delooping of .
The object in this case we denote by as a short form for . The lack of the subscript indicates the different use of the symbol in the case that we have a unique point.
Then in this case the fibration sequence reads
This is the universal -principal ∞-bundle in the context of the given category of fibrant objects.
Last revised on September 4, 2009 at 14:46:28. See the history of this page for a list of all contributions to it.