Fibration of simplicial sets
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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 .[1] A right fibration is defined similarly with the condition .[1] 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.[2]
Examples
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 ∞-category setup.
Anodyne extensions
Summarize
Perspective
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).[3] A right anodyne extension is defined by replacing the condition with . The notions are originally due to Gabriel–Zisman 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,[4] 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 .[5] 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.[3]
An inner anodyne extension is a map in the saturation of the horn inclusions for .[6] 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.[7] Simplicial sets are then weak Kan complexes (∞-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 .[8] For example, a left (or right) fibration between weak Kan complexes is a conservative isofibration.[9]
Theorem of Gabriel and Zisman
Given monomorphisms and , let denote the pushout of and . Then a theorem of Gabriel and Zisman says:[10][11] 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.[12]
A special case of the above is the covering homotopy extension property:[13] 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.[14][15] Similarly, if is a left (resp. right) fibration, then is a left (resp. right) fibration.[16]
Model category structure
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Perspective
The category of simplicial sets sSet has the standard model category structure where [17]
- The cofibrations are the monomorphisms,
- The fibrations are the Kan fibrations,
- The weak equivalences are the maps such that is bijective on simplicial homotopy classes for each Kan complex (fibrant object),
- A fibration is trivial (i.e., has the right lifting property with respect to monomorphisms) if and only if it is a weak equivalence,
- A cofibration is an anodyne extension if and only if it is a weak equivalence.
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 → Top, we have:
- A map is a weak equivalence if and only if is a homotopy equivalence.[18]
- A map is a fibration if and only if is a (usual) fibration in the sense of Hurewicz or of Serre.[19]
- For an anodyne extension , admits a strong deformation retract.[20]
Universal left fibration
Summarize
Perspective
Let be the simplicial set where each n-simplex consists of
- a map from a (small) simplicial set X,
- a section of ,
- for each integer and for each map , a choice of a pullback of along .[21]
Now, a conjecture of Nichols-Barrer which is now a theorem says that U is the same thing as the ∞-category of ∞-groupoids (Kan complexes) together with some choices.[22] In particular, there is a forgetful map
- = the ∞-category of Kan complexes,
which is a left fibration. It is universal in the following sense: for each simplicial set X, there is a natural bijection
- the set of the isomorphism classes of left fibrations over X
given by pulling-back , where means the simplicial homotopy classes of maps.[23] 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 ∞-category C is then simply defined to be the classifying map for the left fibration
where each n-simplex in is a map .[24] In fact, is an ∞-category called the twisted diagonal of C.[25]
In his Higher Topos Theory, Lurie constructs an analogous universal cartesian fibration.[26]
See also
Footnotes
References
Further reading
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