homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
An -fibration is the version of a Grothendieck fibration appropriate for -categories.
The idea is that a functor between -categories is an -fibration if the assignation of an object to its fiber can be made into a (contravariant) functor from to the -category .
(This definition is schematic, and needs to be adapted to be made precise for any particular definition of -category.)
Let be a functor between (weak) -categories.
A morphism in is cartesian (relative to ) if for any , the following square:
(which commutes, up to equivalence, by functoriality of ) is a (weak) pullback of -categories.
We say that is an -fibration (or just a fibration) if
is a morphism of -fibrations.
If and are -fibrations, a commutative square
is a morphism of -fibrations if
is a morphism of -fibrations.
The definition is recursive in , but if we unravel it, it makes perfect sense for . That is, saying that is a fibration requires some things about cartesian 1-cells, and also that its action on hom-categories be a fibration–which in turn requires some things about cartesian 2-cells, and also that its action on hom-categories be a fibration—which in turn which requires some things about cartesian 3-cells, and so on. After steps of unraveling, we are left with a list of conditions on cartesian -cells for every .
An equivalent, conciser way to say this is that we interpret the definition in the case as a coinductive definition.
When , this reduces to a Street fibration, a weakened version of a Grothendieck fibration. We recover Grothendieck’s original notion by requiring that for any and in , there exists a cartesian such that and are equal. This condition violates the principle of equivalence as stated, but not if it is rephrased to apply to displayed categories instead.
When , so that weak 2-categories are bicategories, this notion of fibration can be found in (Buckley). A strict version for strict 2-categories (though with one condition missing) was originally studied by (Hermida).
In general, given any notion of (semi)strict -category, we can expect to appropriately strictify the definition to make it correspond to stricter notions of pseudofunctor.
If is an -fibration, we define a functor (or ‘-pseudofunctor’) from to as follows. (Like the above definition, this is only a schematic sketch.)
Send to the essential fiber , whose objects are objects equipped with a equivalence .
For a morphism in , define by choosing, for each , a cartesian over and defining . The universal property of cartesian arrows makes a functor.
For a 2-cell in , define a transformation as follows. Given , we have a cartesian arrow over . Now choose a cartesian 2-cell over in . Since , factors essentially uniquely through the cartesian arrow , giving a morphism ; we define this to be the component of the transformation at .
and so on…
Note that the functor we obtain is “totally contravariant:” it is contravariant on -cells for all .
Conversely, if we have a totally contravariant ‘-pseudofunctor’ from to , we define by a generalization of the Grothendieck construction as follows:
The objects of over are those of .
The morphisms of over in from to are the morphisms from to in .
The 2-cells of over in from to are the 2-cells in from to the composite .
and so on…
One expects that in this way, the -category of fibered -categories over is equivalent to the -category of totally contravariant functors . These constructions are known precisely only for .
A notion of fibration of (∞,1)-categories exists in terms of Cartesian fibrations of simplicial sets. (See also fibration of quasi-categories, left fibration, and right fibration.)
The notion of 2-fibrations was introduced in:
In fact they also appeared earlier, in some form, in Gray's book.
However, Hermida’s definition was missing the stability of cartesian 2-cells under postcomposition, which is necessary for the “Grothendieck construction” turning a pseudofunctor into a fibration to have an inverse. This was rectified, and the definition generalized to bicategories, in
Igor Baković, Fibrations of bicategories (2010) [pdf, pdf]
Mitchell Buckley, Fibred 2-categories and bicategories, Journal of Pure and Applied Algebra 218 6 (2014) 1034-1074 [arXiv:1212.6283, doi:10.1016/j.jpaa.2013.11.002]
A definition for strict -categories due to Hermida is unpublished, but it is used and presented in another joint work with Marta Bunge, presented at CATS07 at Calais:
-pseudofunctors may be viewed (and defined) as anafunctors. For -groupoids such an approach to -pseudofunctors has been studied in
Last revised on May 5, 2023 at 15:13:14. See the history of this page for a list of all contributions to it.