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 ${\mathrm{\Lambda }}_i^n\subset {\mathrm{\Delta }}^n,0\le i<n$.Template:Sfn A right fibration is defined similarly with the condition $0<i\le n$.Template:Sfn 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.

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 ${\mathrm{\Lambda }}_k^n\to {\mathrm{\Delta }}^n$ for $n\ge 1,0\le k<n$ 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). A right anodyne extension is defined by replacing the condition $0\le k<n$ with $0<k\le n$. 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, 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 $r(F)=r(\bar{F})$ for the saturation $\bar{F}$ of $F$. 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.

An inner anodyne extension is a map in the saturation of the horn inclusions ${\mathrm{\Lambda }}_k^n\to {\mathrm{\Delta }}^n$ for $n\ge 1,0<k<n$.Template:Sfn The maps having the right lifting property with respect to inner anodyne extensions or equivalently with respect to the horn inclusions ${\mathrm{\Lambda }}_k^n\to {\mathrm{\Delta }}^n,n\ge 1,0<k<n$ are called inner fibrations.Template:Sfn Simplicial sets are then weak Kan complexes (∞-categories) if unique maps to the final object are inner fibrations.

An isofibration $p:X\to Y$ is an inner fibration such that for each object (0-simplex) $x_0$ in $X$ and an invertible map $g:y_0\to y_1$ with $p(x_0)=y_0$ in $Y$, there exists a map $f$ in $X$ such that $p(f)=g$.Template:Sfn For example, a left (or right) fibration between weak Kan complexes is a conservative isofibration.Template:Sfn

Theorem of Gabriel and Zisman

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

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

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.Template:Sfn

A special case of the above is the covering homotopy extension property:Template:Sfn a Kan fibration has the right lifting property with respect to $(Y\times I)\bigsqcup (Z\times 0)\to Z\times I$ for monomorphisms $Y\to Z$ and $0\to I={\mathrm{\Delta }}^1$.

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

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

is an inner fibration.Template:Sfn Similarly, if $p$ is a left (resp. right) fibration, then $(i^{\ast },p_{\ast })$ is a left (resp. right) fibration.Template:Sfn

Model category structure

Template:Main The category of simplicial sets sSet has the standard model category structure where Template:Sfn

• The cofibrations are the monomorphisms,
• The fibrations are the Kan fibrations,
• The weak equivalences are the maps $f$ such that $f^{\ast }$ 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.Template:Sfn
• A map $f$ is a fibration if and only if $|f|$ is a (usual) fibration in the sense of Hurewicz or of Serre.Template:Sfn
• For an anodyne extension $i$, $|i|$ admits a strong deformation retract.Template:Sfn

Universal left fibration

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

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

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. In particular, there is a forgetful map

$p_{univ}:U\to \mathbf{\text{𝐊𝐚𝐧}}$ = 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

$[X,\mathbf{\text{𝐊𝐚𝐧}}]\stackrel{\sim }{\to }$ the set of the isomorphism classes of left fibrations over X

given by pulling-back $p_{univ}$, where $[,]$ means the simplicial homotopy classes of maps. In short, $\mathbf{\text{𝐊𝐚𝐧}}$ is the classifying space of left fibrations. Given a left fibration over X, a map $X\to \mathbf{\text{𝐊𝐚𝐧}}$ corresponding to it is called the classifying map for that fibration.

In Cisinski's book, the hom-functor $\mathrm{Hom}:C^{op}\times C\to \mathbf{\text{𝐊𝐚𝐧}}$ on an ∞-category C is then simply defined to be the classifying map for the left fibration

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

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

In his Higher Topos Theory, Lurie constructs an analogous universal cartesian fibration.

See also

• small object argument

References

• Template:Citation
• Template:Cite web
• Template:Cite book
• 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
• Template:Cite web

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