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  Λ i n ⊂ Δ n , 0 ≤ i < n {\displaystyle \Lambda _{i}^{n}\subset \Delta ^{n},0\leq i<n} .[1] A right fibration is defined similarly with the condition  0 < i ≤ n {\displaystyle 0<i\leq 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]

## Examples

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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

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A left anodyne extension is a map in the saturation of the set of the horn inclusions  Λ k n -> Δ n {\displaystyle \Lambda _{k}^{n}\to \Delta ^{n}} for  n ≥ 1 , 0 ≤ k < n {\displaystyle n\geq 1,0\leq 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 {\displaystyle 0\leq k<n} with  0 < k ≤ n {\displaystyle 0<k\leq 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 {\displaystyle F} of maps, let  r ( F ) {\displaystyle r(F)} denote the class of maps satisfying the right lifting property with respect to  F {\displaystyle F} . Then  r ( F ) = r ( F ¯ ) {\displaystyle r(F)=r({\overline {F}})} for the saturation  F ¯ {\displaystyle {\overline {F}}} of  F {\displaystyle 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  Λ k n -> Δ n {\displaystyle \Lambda _{k}^{n}\to \Delta ^{n}} for  n ≥ 1 , 0 < k < n {\displaystyle n\geq 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  Λ k n -> Δ n , n ≥ 1 , 0 < k < n {\displaystyle \Lambda _{k}^{n}\to \Delta ^{n},\,n\geq 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 {\displaystyle p:X\to Y} is an inner fibration such that for each object (0-simplex)  x 0 {\displaystyle x_{0}} in  X {\displaystyle X} and an invertible map  g : y 0 -> y 1 {\displaystyle g:y_{0}\to y_{1}} with  p ( x 0 ) = y 0 {\displaystyle p(x_{0})=y_{0}} in  Y {\displaystyle Y} , there exists a map  f {\displaystyle f} in  X {\displaystyle X} such that  p ( f ) = g {\displaystyle 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

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Given monomorphisms  i : A -> B {\displaystyle i:A\to B} and  k : Y -> Z {\displaystyle k:Y\to Z} , let  i ⊔ A × Y k {\displaystyle i\sqcup _{A\times Y}k} denote the pushout of  i × id Y {\displaystyle i\times \operatorname {id} _{Y}} and  id A × k {\displaystyle \operatorname {id} _{A}\times k} . Then a theorem of Gabriel and Zisman says:[10][11] if  i {\displaystyle i} is a left (resp. right) anodyne extension, then the induced map

i ⊔ A × Y k -> B × Z {\displaystyle i\sqcup _{A\times Y}k\to B\times Z}

is a left (resp. right) anodyne extension. Similarly, if  i {\displaystyle 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 {\displaystyle (Y\times I)\sqcup (Z\times 0)\to Z\times I} for monomirphisms  Y -> Z {\displaystyle Y\to Z} and  0 -> I = Δ 1 {\displaystyle 0\to I=\Delta ^{1}} .

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

( i ∗ , p ∗ ) : Hom _ ( B , X ) -> Hom _ ( A , X ) × Hom _ ( A , Y ) Hom _ ( B , Y ) {\displaystyle (i^{*},p_{*}):{\underline {\operatorname {Hom} }}(B,X)\to {\underline {\operatorname {Hom} }}(A,X)\times _{{\underline {\operatorname {Hom} }}(A,Y)}{\underline {\operatorname {Hom} }}(B,Y)}

is an inner fibration.[14][15] Similarly, if  p {\displaystyle p} is a left (resp. right) fibration, then  ( i ∗ , p ∗ ) {\displaystyle (i^{*},p_{*})} is a left (resp. right) fibration.[16]

## Model category structure

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

## Universal left fibration

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Let  U {\displaystyle U} be the simplicial set where each n-simplex consists of

* a map  p : X -> Δ n {\displaystyle p:X\to \Delta ^{n}} from a (small) simplicial set X,
* a section  s {\displaystyle s} of  p {\displaystyle p} ,
* for each integer  m ≥ 0 {\displaystyle m\geq 0} and for each map  f : Δ m -> Δ n {\displaystyle f:\Delta ^{m}\to \Delta ^{n}} , a choice of a pullback of  p {\displaystyle p} along  f {\displaystyle 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

p u n i v : U -> Kan {\displaystyle p_{univ}:U\to {\textbf {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 ] -> ∼ {\displaystyle [X,{\textbf {Kan}}]\,{\overset {\sim }{\to }}} the set of the isomorphism classes of left fibrations over X

given by pulling-back  p u n i v {\displaystyle p_{univ}} , where  [ , ] {\displaystyle [,]} means the simplicial homotopy classes of maps.[23] In short,  Kan {\displaystyle {\textbf {Kan}}} is the classifying space of left fibrations. Given a left fibration over X, a map  X -> Kan {\displaystyle X\to {\textbf {Kan}}} corresponding to it is called the classifying map for that fibration.

In Cisinski's book, the hom-functor  Hom : C o p × C -> Kan {\displaystyle \operatorname {Hom} :C^{op}\times C\to {\textbf {Kan}}} on an ∞-category C is then simply defined to be the classifying map for the left fibration

( s , t ) : S ( C ) -> C o p × C {\displaystyle (s,t):S(C)\to C^{op}\times C}

where each n-simplex in  S ( C ) {\displaystyle S(C)} is a map  ( Δ n ) o p ∗ Δ n -> C {\displaystyle (\Delta ^{n})^{op}*\Delta ^{n}\to C} .[24] In fact,  S ( C ) {\displaystyle 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

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* small object argument

## Footnotes

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1. ^ a b Lurie 2009a, Definition 2.0.0.3.
2. ^ Beke, Tibor (2008). "Fibrations of simplicial sets". arXiv:0810.4960 [math.CT].
3. ^ a b Cisinski 2023, Definition 3.4.1.
4. ^ Proof: Let  l ( F ) {\displaystyle l(F)} = the class of maps having the left lifting property with respect to a class  F {\displaystyle F} of maps. Then  l ( F ) {\displaystyle l(F)} can be shown to be saturated. By the axiom of choice, if  F {\displaystyle F} is the class of surjective maps, then  l ( F ) {\displaystyle l(F)} is the class of injective maps. This implies the same is true for monomorphisms between preshaves.
5. ^ Proof: Since  l ( G ) {\displaystyle l(G)} , l for the left lifting property, is saturated and  F ⊂ l ( r ( F ) ) {\displaystyle F\subset l(r(F))} , we have:  F ¯ ⊂ l ( r ( F ) ) {\displaystyle {\overline {F}}\subset l(r(F))} and so  r ( F ) = r ( l ( r ( F ) ) ⊂ r ( F ¯ ) ⊂ r ( F ) {\displaystyle r(F)=r(l(r(F))\subset r({\overline {F}})\subset 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

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* Lurie, Jacob (2009a), Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, arXiv:math.CT/0608040, ISBN 978-0-691-14049-0, MR 2522659
* Lurie, J. (2009b). "Lecture 9 of Algebraic K-Theory and Manifold Topology (Math 281)" (PDF).
* Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.
* 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" (PDF).

## Further reading

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* 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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