Fibration of simplicial sets

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 Λin⊂Δn,0≤i<n.[1] A right fibration is defined similarly with the condition 0<i≤n.[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]

## Contents

* 1 Examples
* 2 Anodyne extensions
* 3 Theorem of Gabriel and Zisman
* 4 Model category structure
* 5 Universal left fibration
* 6 See also
* 7 Footnotes
* 8 References
* 9 Further reading

## 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

A left anodyne extension is a map in the saturation of the set of the horn inclusions Λkn->Δn for n≥1,0≤k<n 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 0≤k<n with 0<k≤n. 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 F of maps, let r(F) denote the class of maps satisfying the right lifting property with respect to F. Then r(F)=r(F‾) for the saturation F‾ of F.[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 Λkn->Δn for n≥1,0<k<n.[6] The maps having the right lifting property with respect to inner anodyne extensions or equivalently with respect to the horn inclusions Λkn->Δn,n≥1,0<k<n 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 p:X->Y is an inner fibration such that for each object (0-simplex) x0 in X and an invertible map g:y0->y1 with p(x0)=y0 in Y, there exists a map f in X such that p(f)=g.[8] For example, a left (or right) fibration between weak Kan complexes is a conservative isofibration.[9]

## Theorem of Gabriel and Zisman

Given monomorphisms i:A->B and k:Y->Z, let i⊔A×Yk denote the pushout of i×idY and idA×k. Then a theorem of Gabriel and Zisman says:[10][11] if i is a left (resp. right) anodyne extension, then the induced map

i⊔A×Yk->B×Z

is a left (resp. right) anodyne extension. Similarly, if i 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 (Y×I)⊔(Z×0)->Z×I for monomirphisms Y->Z and 0->I=Δ1.

As a corollary of the theorem, a map p:X->Y is an inner fibration if and only if for each monomirphism i:A->B, the induced map

(i*,p*):Hom_(B,X)->Hom_(A,X)×Hom_(A,Y)Hom_(B,Y)

is an inner fibration.[14][15] Similarly, if p is a left (resp. right) fibration, then (i*,p*) is a left (resp. right) fibration.[16]

## Model category structure

Main article: Kan–Quillen model structure

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 f such that f* 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 f is a weak equivalence if and only if |f| is a homotopy equivalence.[18]
* A map f is a fibration if and only if |f| is a (usual) fibration in the sense of Hurewicz or of Serre.[19]
* For an anodyne extension i, |i| admits a strong deformation retract.[20]

## Universal left fibration

Let U be the simplicial set where each n-simplex consists of

* a map p:X->Δn from a (small) simplicial set X,
* a section s of p,
* for each integer m≥0 and for each map f:Δm->Δn, a choice of a pullback of p along f.[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

puniv:U->Kan = 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

[X,Kan]->∼ the set of the isomorphism classes of left fibrations over X

given by pulling-back puniv, where [,] means the simplicial homotopy classes of maps.[23] In short, Kan is the classifying space of left fibrations. Given a left fibration over X, a map X->Kan corresponding to it is called the classifying map for that fibration.

In Cisinski's book, the hom-functor Hom⁡:Cop×C->Kan on an ∞-category C is then simply defined to be the classifying map for the left fibration

(s,t):S(C)->Cop×C

where each n-simplex in S(C) is a map (Δn)op*Δn->C.[24] In fact, S(C) 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

* small object argument

## Footnotes

1. ↑ 1.0 1.1 Lurie 2009a, Definition 2.0.0.3.
2. ↑ Beke, Tibor (2008). "Fibrations of simplicial sets". arXiv:0810.4960 [math.CT].
3. ↑ 3.0 3.1 Cisinski 2023, Definition 3.4.1.
4. ↑ Proof: Let l(F) = the class of maps having the left lifting property with respect to a class F of maps. Then l(F) can be shown to be saturated. By the axiom of choice, if F is the class of surjective maps, then l(F) is the class of injective maps. This implies the same is true for monomorphisms between preshaves.
5. ↑ Proof: Since l(G), l for the left lifting property, is saturated and F⊂l(r(F)), we have: F‾⊂l(r(F)) and so r(F)=r(l(r(F))⊂r(F‾)⊂r(F).
6. ↑ Cisinski 2023, Definition 3.2.1.
7. ↑ Cisinski 2023, Definition 3.2.5.
8. ↑ Cisinski 2023, Definition 3.3.15.
9. ↑ Cisinski 2023, Proposition 3.4.8.
10. ↑ Joyal & Tierney 2008, Theorem 3.2.2.
11. ↑ Cisinski 2023, Proposition 3.4.3.
12. ↑ Cisinski 2023, Corollary 3.2.4.
13. ↑ Joyal & Tierney 2008, Proposition 3.2.2.
14. ↑ Cisinski 2023, Corollary 3.2.8.
15. ↑ Proposition 4.1.4.1. in https://kerodon.net/tag/01BS
16. ↑ Cisinski 2023, Proposition 3.4.4.
17. ↑ Joyal & Tierney 2008, Theorem 3.4.1, Proposition 3.4.2, Proposition 3.4.3.
18. ↑ Joyal & Tierney 2008, Proposition 4.6.3.
19. ↑ Joyal & Tierney 2008, § 2.1.
20. ↑ Joyal & Tierney 2008, Proposition 4.6.1.
21. ↑ Cisinski 2023, Definition 5.2.3.
22. ↑ Cisinski 2023, Theorem 5.2.10.
23. ↑ Cisinski 2023, Corollary 5.3.21.
24. ↑ Cisinski 2023, § 5.6.1. and § 5.8.1.
25. ↑ Cisinski 2023, Proposition 5.6.2.
26. ↑ Lurie 2009a, § 3.3.2.

## References

* Lurie, Jacob (2009a), Higher topos theory, Annals of Mathematics Studies, 170, Princeton University Press, ISBN 978-0-691-14049-0
* Lurie, J. (2009b). "Lecture 9 of Algebraic K-Theory and Manifold Topology (Math 281)". https://www.math.ias.edu/~lurie/281notes/Lecture9-Fibrations2.pdf.
* Cisinski, Denis-Charles (2023) (in en). Higher Categories and Homotopical Algebra. Cambridge University Press. ISBN 978-1108473200. https://cisinski.app.uni-regensburg.de/CatLR.pdf.
* Pierre Gabriel, Michel Zisman, chapter IV.2 of Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) [1]
* Lurie, Kerodon
* Joyal, André; Tierney, Myles (2008). "Notes on simplicial homotopy theory". https://ncatlab.org/nlab/files/JoyalTierneyNotesOnSimplicialHomotopyTheory.pdf.

## Further reading

* https://ncatlab.org/nlab/show/anodyne+morphism
* https://math.stackexchange.com/questions/1061303/history-of-the-term-anodyne-in-homotopy-theory
* https://mathoverflow.net/questions/313635/cellularity-of-anodyne-extensions
* nlab, https://ncatlab.org/nlab/show/inner+fibration

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