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}. A right fibration is defined similarly with the condition 0 < i ≤ n {\displaystyle 0<i\leq n}. 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 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). 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, 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}. 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 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}. 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. 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}. For example, a left (or right) fibration between weak Kan complexes is a conservative isofibration.

Theorem of Gabriel and Zisman

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: 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.

A special case of the above is the covering homotopy extension property: 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. Similarly, if p {\displaystyle p} is a left (resp. right) fibration, then ( i ∗ , p ∗ ) {\displaystyle (i^{*},p_{*})} is a left (resp. right) fibration.

Model category structure

The category of simplicial sets sSet has the standard model category structure where

  • 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.
  • 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.
  • For an anodyne extension i {\displaystyle i}, | i | {\displaystyle |i|} admits a strong deformation retract.

Universal left fibration

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}.

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 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. 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}. In fact, S ( C ) {\displaystyle 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

Footnotes

Further reading

  • nlab,