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 ${\displaystyle \Lambda _{i}^{n}\subset \Delta ^{n},0\leq i<n}$.¹ A right fibration is defined similarly with the condition ${\displaystyle 0<i\leq n}$.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

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 ${\displaystyle \Lambda _{k}^{n}\to \Delta ^{n}}$ for ${\displaystyle n\geq 1,0\leq k<n}$ 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 ${\displaystyle 0\leq k<n}$ with ${\displaystyle 0<k\leq n}$. 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 ${\displaystyle F}$ of maps, let ${\displaystyle r(F)}$ denote the class of maps satisfying the right lifting property with respect to ${\displaystyle F}$. Then ${\displaystyle r(F)=r({\overline {F}})}$ for the saturation ${\displaystyle {\overline {F}}}$ 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 ${\displaystyle \Lambda _{k}^{n}\to \Delta ^{n}}$ for ${\displaystyle n\geq 1,0<k<n}$.⁶ The maps having the right lifting property with respect to inner anodyne extensions or equivalently with respect to the horn inclusions ${\displaystyle \Lambda _{k}^{n}\to \Delta ^{n},\,n\geq 1,0<k<n}$ 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 ${\displaystyle p:X\to Y}$ 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 ${\displaystyle i\sqcup _{A\times Y}k}$ denote the pushout of ${\displaystyle i\times \operatorname {id} _{Y}}$ and ${\displaystyle \operatorname {id} _{A}\times k}$. Then a theorem of Gabriel and Zisman says:¹⁰¹¹ if ${\displaystyle i}$ is a left (resp. right) anodyne extension, then the induced map

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

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 ${\displaystyle (Y\times I)\sqcup (Z\times 0)\to Z\times I}$ for monomorphisms ${\displaystyle Y\to Z}$ and ${\displaystyle 0\to I=\Delta ^{1}}$.

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

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

is an inner fibration.¹⁴¹⁵ 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 ${\displaystyle p:X\to \Delta ^{n}}$ from a (small) simplicial set X,
• a section ${\displaystyle s}$ of ${\displaystyle p}$,
• for each integer ${\displaystyle m\geq 0}$ and for each map ${\displaystyle f:\Delta ^{m}\to \Delta ^{n}}$, 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

${\displaystyle p_{univ}:U\to {\textbf {Kan}}}$ = 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

${\displaystyle [X,{\textbf {Kan}}]\,{\overset {\sim }{\to }}}$ 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, ${\displaystyle {\textbf {Kan}}}$ is the classifying space of left fibrations. Given a left fibration over X, a map ${\displaystyle X\to {\textbf {Kan}}}$ corresponding to it is called the classifying map for that fibration.

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

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

where each n-simplex in ${\displaystyle S(C)}$ is a map ${\displaystyle (\Delta ^{n})^{op}*\Delta ^{n}\to C}$.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