In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn inclusions . A right fibration is defined similarly with the condition . A Kan fibration is one with the right lifting property with respect to every horn inclusion; hence, a Kan fibration is exactly a map that is both a left and right fibration.
A right fibration is a cartesian fibration such that each fiber is a Kan complex.
In particular, a category fibered in groupoids over another category is a special case of a right fibration of simplicial sets in the âÂÂ-category setup.
A left anodyne extension is a map in the saturation of the set of the horn inclusions for in the category of simplicial sets, where the saturation of a class is the smallest class that contains the class and is stable under pushouts, retracts and transfinite compositions (compositions of infinitely many maps). A right anodyne extension is defined by replacing the condition with . The notions are originally due to GabrielâÂÂZisman and are used to study fibrations for simplicial sets.
A left (or right) anodyne extension is a monomorphism (since the class of monomorphisms is saturated, the saturation lies in the class of monomorphisms).
Given a class of maps, let denote the class of maps satisfying the right lifting property with respect to . Then for the saturation of . Thus, a map is a left (resp. right) fibration if and only if it has the right lifting property with respect to left (resp. right) anodyne extensions.
An inner anodyne extension is a map in the saturation of the horn inclusions for . The maps having the right lifting property with respect to inner anodyne extensions or equivalently with respect to the horn inclusions are called inner fibrations. Simplicial sets are then weak Kan complexes (âÂÂ-categories) if unique maps to the final object are inner fibrations.
An isofibration is an inner fibration such that for each object (0-simplex) in and an invertible map with in , there exists a map in such that . For example, a left (or right) fibration between weak Kan complexes is a conservative isofibration.
Given monomorphisms and , let denote the pushout of and . Then a theorem of Gabriel and Zisman says: if is a left (resp. right) anodyne extension, then the induced map
is a left (resp. right) anodyne extension. Similarly, if is an inner anodyne extension, then the above induced map is an inner anodyne extension.
A special case of the above is the covering homotopy extension property: a Kan fibration has the right lifting property with respect to for monomirphisms and .
As a corollary of the theorem, a map is an inner fibration if and only if for each monomirphism , the induced map
is an inner fibration. Similarly, if is a left (resp. right) fibration, then is a left (resp. right) fibration.
The category of simplicial sets sSet has the standard model category structure where
Because of the last property, an anodyne extension is also known as an acyclic cofibration (a cofibration that is a weak equivalence). Also, the weak equivalences between Kan complexes are the same as the simplicial homotopy equivalences between them.
Under the geometric realization | - | : sSet â Top, we have:
Let be the simplicial set where each n-simplex consists of
Now, a conjecture of Nichols-Barrer which is now a theorem says that U is the same thing as the âÂÂ-category of âÂÂ-groupoids (Kan complexes) together with some choices. In particular, there is a forgetful map
which is a left fibration. It is universal in the following sense: for each simplicial set X, there is a natural bijection
given by pulling-back , where means the simplicial homotopy classes of maps. In short, is the classifying space of left fibrations. Given a left fibration over X, a map corresponding to it is called the classifying map for that fibration.
In Cisinski's book, the hom-functor on an âÂÂ-category C is then simply defined to be the classifying map for the left fibration
where each n-simplex in is a map . In fact, is an âÂÂ-category called the twisted diagonal of C.
In his Higher Topos Theory, Lurie constructs an analogous universal cartesian fibration.