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 ${\displaystyle {\mathrm{\Lambda }}_i^n\subset {\mathrm{\Delta }}^n,0\le i<n}$.^[1] A right fibration is defined similarly with the condition ${\displaystyle 0<i\le n}$.^[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]

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.

A left anodyne extension is a map in the saturation of the set of the horn inclusions ${\displaystyle {\mathrm{\Lambda }}_k^n\to {\mathrm{\Delta }}^n}$ for ${\displaystyle n\ge 1,0\le k<n}$ 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 ${\displaystyle 0\le k<n}$ with ${\displaystyle 0<k\le n}$. 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 ${\displaystyle F}$ of maps, let ${\displaystyle r(F)}$ denote the class of maps satisfying the right lifting property with respect to ${\displaystyle F}$. Then ${\displaystyle r(F)=r(\bar{F})}$ for the saturation ${\displaystyle \bar{F}}$ of ${\displaystyle F}$.^[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 ${\displaystyle {\mathrm{\Lambda }}_k^n\to {\mathrm{\Delta }}^n}$ for ${\displaystyle n\ge 1,0<k<n}$.^[6] The maps having the right lifting property with respect to inner anodyne extensions or equivalently with respect to the horn inclusions ${\displaystyle {\mathrm{\Lambda }}_k^n\to {\mathrm{\Delta }}^n,n\ge 1,0<k<n}$ 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 ${\displaystyle p:X\to Y}$ is an inner fibration such that for each object (0-simplex) ${\displaystyle x_0}$ in ${\displaystyle X}$ and an invertible map ${\displaystyle g:y_0\to y_1}$ with ${\displaystyle p(x_0)=y_0}$ in ${\displaystyle Y}$, there exists a map ${\displaystyle f}$ in ${\displaystyle X}$ such that ${\displaystyle p(f)=g}$.^[8] For example, a left (or right) fibration between weak Kan complexes is a conservative isofibration.^[9]

Given monomorphisms ${\displaystyle i:A\to B}$ and ${\displaystyle k:Y\to Z}$, let ${\displaystyle i{\bigsqcup }_{A\times Y}k}$ denote the pushout of ${\displaystyle i\times {\mathrm{id}}_Y}$ and ${\displaystyle {\mathrm{id}}_A\times k}$. Then a theorem of Gabriel and Zisman says:^[10][11] if ${\displaystyle i}$ is a left (resp. right) anodyne extension, then the induced map

${\displaystyle i{\bigsqcup }_{A\times Y}k\to B\times Z}$

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

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

${\displaystyle (i^{*},p_{*}):\underset{\_}{\mathrm{Hom}}(B,X)\to \underset{\_}{\mathrm{Hom}}(A,X){\times }_{\underset{\_}{\mathrm{Hom}}(A,Y)}\underset{\_}{\mathrm{Hom}}(B,Y)}$

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

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

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

• a map ${\displaystyle p:X\to {\mathrm{\Delta }}^n}$ from a (small) simplicial set X,
• a section ${\displaystyle s}$ of ${\displaystyle p}$,
• for each integer ${\displaystyle m\ge 0}$ and for each map ${\displaystyle f:{\mathrm{\Delta }}^m\to {\mathrm{\Delta }}^n}$, a choice of a pullback of ${\displaystyle p}$ along ${\displaystyle f}$.^[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

${\displaystyle p_{univ}:U\to \text{<mrow data-mjx-texclass="ORD">Kan</mrow>}}$ = 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

${\displaystyle [X,\text{<mrow data-mjx-texclass="ORD">Kan</mrow>}]\stackrel{\sim }{\to }}$ the set of the isomorphism classes of left fibrations over X

given by pulling-back ${\displaystyle p_{univ}}$, where ${\displaystyle [,]}$ means the simplicial homotopy classes of maps.^[23] In short, ${\displaystyle \text{<mrow data-mjx-texclass="ORD">Kan</mrow>}}$ is the classifying space of left fibrations. Given a left fibration over X, a map ${\displaystyle X\to \text{<mrow data-mjx-texclass="ORD">Kan</mrow>}}$ corresponding to it is called the classifying map for that fibration.

In Cisinski's book, the hom-functor ${\displaystyle \mathrm{Hom}:C^{op}\times C\to \text{<mrow data-mjx-texclass="ORD">Kan</mrow>}}$ on an ∞-category C is then simply defined to be the classifying map for the left fibration

${\displaystyle (s,t):S(C)\to C^{op}\times C}$

where each n-simplex in ${\displaystyle S(C)}$ is a map ${\displaystyle ({\mathrm{\Delta }}^n{)}^{op}*{\mathrm{\Delta }}^n\to C}$.^[24] In fact, ${\displaystyle S(C)}$ is an ∞-category called the twisted diagonal of C.^[25]

In his Higher Topos Theory, Lurie constructs an analogous universal cartesian fibration.^[26]

• small object argument

• 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. 

• 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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