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 <math>\Lambda^n_i \subset \Delta^n, 0 \le i < n</math>.{{sfn|Lurie|2009a|loc=Definition 2.0.0.3}} A '''right fibration''' is defined similarly with the condition <math>0 < i \le n</math>.{{sfn|Lurie|2009a|loc=Definition 2.0.0.3}} 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.<ref>{{Cite arXiv |last=Beke |first=Tibor |date=2008 |title=Fibrations of simplicial sets |class=math.CT |eprint=0810.4960 }}</ref>

== 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 <math>\Lambda^n_k \to \Delta^n</math> for <math>n \ge 1, 0 \le k < n</math> 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).<ref name="def">{{harvnb|Cisinski|2023|loc=Definition 3.4.1.}}</ref> A '''right anodyne extension''' is defined by replacing the condition <math>0 \le k < n</math> with <math>0 < k \le n</math>. 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,<ref>''Proof'': Let <math>l(F)</math> = the class of maps having the left lifting property with respect to a class <math>F</math> of maps. Then <math>l(F)</math> can be shown to be saturated. By the axiom of choice, if <math>F</math> is the class of surjective maps, then <math>l(F)</math> is the class of injective maps. This implies the same is true for monomorphisms between preshaves.</ref> the saturation lies in the class of monomorphisms).

Given a class <math>F</math> of maps, let <math>r(F)</math> denote the class of maps satisfying the right lifting property with respect to <math>F</math>. Then <math>r(F) = r(\overline{F})</math> for the saturation <math>\overline{F}</math> of <math>F</math>.<ref>''Proof'': Since <math>l(G)</math>, ''l'' for the left lifting property, is saturated and <math>F \subset l(r(F))</math>, we have: <math>\overline{F} \subset l(r(F))</math> and so <math>r(F) = r(l(r(F)) \subset r(\overline{F}) \subset r(F)</math>.</ref> 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.<ref name="def" />

An '''inner anodyne extension''' is a map in the saturation of the horn inclusions <math>\Lambda^n_k \to \Delta^n</math> for <math>n \ge 1, 0 < k < n</math>.{{sfn|Cisinski|2023|loc=Definition 3.2.1}} The maps having the right lifting property with respect to inner anodyne extensions or equivalently with respect to the horn inclusions <math>\Lambda^n_k \to \Delta^n, \, n \ge 1, 0 < k < n</math> are called '''inner fibrations'''.{{sfn|Cisinski|2023|loc=Definition 3.2.5}} Simplicial sets are then weak Kan complexes (∞-categories) if unique maps to the final object are inner fibrations.

An '''isofibration''' <math>p : X \to Y</math> is an inner fibration such that for each object (0-simplex) <math>x_0</math> in <math>X</math> and an invertible map <math>g : y_0 \to y_1</math> with <math>p(x_0) = y_0</math> in <math>Y</math>, there exists a map <math>f</math> in <math>X</math> such that <math>p(f) = g</math>.{{sfn|Cisinski|2023|loc=Definition 3.3.15}} For example, a left (or right) fibration between weak Kan complexes is a conservative isofibration.{{sfn|Cisinski|2023|loc=Proposition 3.4.8}}

== Theorem of Gabriel and Zisman == Given monomorphisms <math>i : A \to B</math> and <math>k : Y \to Z</math>, let <math>i \sqcup_{A \times Y} k</math> denote the pushout of <math>i \times \operatorname{id}_Y</math> and <math>\operatorname{id}_A \times k</math>. Then a theorem of Gabriel and Zisman says:{{sfn|Joyal|Tierney|2008|loc=Theorem 3.2.2}}{{sfn|Cisinski|2023|loc=Proposition 3.4.3}} if <math>i</math> is a left (resp. right) anodyne extension, then the induced map :<math>i \sqcup_{A \times Y} k \to B \times Z</math> is a left (resp. right) anodyne extension. Similarly, if <math>i</math> is an inner anodyne extension, then the above induced map is an inner anodyne extension.{{sfn|Cisinski|2023|loc=Corollary 3.2.4}}

A special case of the above is the covering homotopy extension property:{{sfn|Joyal|Tierney|2008|loc=Proposition 3.2.2}} a Kan fibration has the right lifting property with respect to <math>(Y \times I) \sqcup (Z \times 0) \to Z \times I</math> for monomorphisms <math>Y \to Z</math> and <math>0 \to I = \Delta^1</math>.

As a corollary of the theorem, a map <math>p : X \to Y</math> is an inner fibration if and only if for each monomorphism <math>i : A \to B</math>, the induced map :<math>(i^*, p_*) : \underline{\operatorname{Hom}}(B, X) \to \underline{\operatorname{Hom}}(A, X) \times_{\underline{\operatorname{Hom}}(A, Y)} \underline{\operatorname{Hom}}(B, Y)</math> is an inner fibration.{{sfn|Cisinski|2023|loc=Corollary 3.2.8}}<ref>Proposition 4.1.4.1. in https://kerodon.net/tag/01BS</ref> Similarly, if <math>p</math> is a left (resp. right) fibration, then <math>(i^*, p_*)</math> is a left (resp. right) fibration.{{sfn|Cisinski|2023|loc=Proposition 3.4.4}}

== Model category structure == {{main|Kan–Quillen model structure}} The category of simplicial sets '''sSet''' has the standard model category structure where {{sfn|Joyal|Tierney|2008|loc=Theorem 3.4.1, Proposition 3.4.2, Proposition 3.4.3}} *The cofibrations are the monomorphisms, *The fibrations are the Kan fibrations, *The weak equivalences are the maps <math>f</math> such that <math>f^*</math> 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 <math>f</math> is a weak equivalence if and only if <math>|f|</math> is a homotopy equivalence.{{sfn|Joyal|Tierney|2008|loc=Proposition 4.6.3}} * A map <math>f</math> is a fibration if and only if <math>|f|</math> is a (usual) fibration in the sense of Hurewicz or of Serre.{{sfn|Joyal|Tierney|2008|loc=§ 2.1}} * For an anodyne extension <math>i</math>, <math>|i|</math> admits a strong deformation retract.{{sfn|Joyal|Tierney|2008|loc=Proposition 4.6.1}}

== Universal left fibration == Let <math>U</math> be the simplicial set where each ''n''-simplex consists of *a map <math>p : X \to \Delta^n</math> from a (small) simplicial set ''X'', *a section <math>s</math> of <math>p</math>, *for each integer <math>m \ge 0</math> and for each map <math>f : \Delta^m \to \Delta^n</math>, a choice of a pullback of <math>p</math> along <math>f</math>.<ref>{{harvnb|Cisinski|2023|loc=Definition 5.2.3.}}</ref> 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.<ref>{{harvnb|Cisinski|2023|loc=Theorem 5.2.10.}}</ref> In particular, there is a forgetful map :<math>p_{univ} : U \to \textbf{Kan}</math> = 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 :<math>[X, \textbf{Kan}] \, \overset{\sim}\to </math> the set of the isomorphism classes of left fibrations over ''X'' given by pulling-back <math>p_{univ}</math>, where <math>[, ]</math> means the simplicial homotopy classes of maps.<ref>{{harvnb|Cisinski|2023|loc=Corollary 5.3.21.}}</ref> In short, <math>\textbf{Kan}</math> is the classifying space of left fibrations. Given a left fibration over ''X'', a map <math>X \to \textbf{Kan}</math> corresponding to it is called the classifying map for that fibration.

In Cisinski's book, the hom-functor <math>\operatorname{Hom} : C^{op} \times C \to \textbf{Kan}</math> on an ∞-category ''C'' is then simply defined to be the classifying map for the left fibration :<math>(s, t) : S(C) \to C^{op} \times C</math> where each ''n''-simplex in <math>S(C)</math> is a map <math>(\Delta^n)^{op} * \Delta^n \to C</math>.<ref>{{harvnb|Cisinski|2023|loc=§ 5.6.1. and § 5.8.1.}}</ref> In fact, <math>S(C)</math> is an ∞-category called the twisted diagonal of ''C''.<ref>{{harvnb|Cisinski|2023|loc=Proposition 5.6.2.}}</ref>

In his ''Higher Topos Theory'', Lurie constructs an analogous universal cartesian fibration.<ref>{{harvnb|Lurie|2009a|loc=§ 3.3.2.}}</ref>

== See also == *small object argument

== Footnotes == {{reflist}}

== References == *{{Citation | last1=Lurie | first1=Jacob | title=Higher topos theory | arxiv=math.CT/0608040 | publisher=Princeton University Press | series=Annals of Mathematics Studies | isbn=978-0-691-14049-0| mr=2522659 | year=2009a | volume=170}} *{{cite web |first=J. |last=Lurie |url=https://www.math.ias.edu/~lurie/281notes/Lecture9-Fibrations2.pdf |title=Lecture 9 of Algebraic K-Theory and Manifold Topology (Math 281) |year=2009b}} * {{cite book |last=Cisinski |first=Denis-Charles |author-link=Denis-Charles Cisinski |url=https://cisinski.app.uni-regensburg.de/CatLR.pdf |title=Higher Categories and Homotopical Algebra |date=2023|publisher=Cambridge University Press |isbn=978-1108473200 |location= |language=en |authorlink=}} * Pierre Gabriel, Michel Zisman, chapter IV.2 of Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) [https://people.math.rochester.edu/faculty/doug/otherpapers/GZ.pdf] * Lurie, [https://kerodon.net/ Kerodon] * {{cite web|first1=André|last1=Joyal|author1-link=André Joyal|first2= Myles|last2= Tierney|author2-link=Myles Tierney|title= Notes on simplicial homotopy theory|url=https://ncatlab.org/nlab/files/JoyalTierneyNotesOnSimplicialHomotopyTheory.pdf|year=2008}}

== 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

Category:Simplicial sets