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

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 $\Lambda _{k}^{n}\to \Delta ^{n}$ for $n\geq 1,0\leq k<n$ 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 $0\leq k<n$ with $0<k\leq n$ . 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 $F$ of maps, let $r(F)$ denote the class of maps satisfying the right lifting property with respect to $F$ . Then $r(F)=r({\overline {F}})$ for the saturation ${\overline {F}}$ of $F$ .^[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 $\Lambda _{k}^{n}\to \Delta ^{n}$ for $n\geq 1,0<k<n$ .^[6] The maps having the right lifting property with respect to inner anodyne extensions or equivalently with respect to the horn inclusions $\Lambda _{k}^{n}\to \Delta ^{n},\,n\geq 1,0<k<n$ 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 $p:X\to Y$ 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 $p(f)=g$ .^[8] For example, a left (or right) fibration between weak Kan complexes is a conservative isofibration.^[9]

Theorem of Gabriel and Zisman

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

i\sqcup _{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.^[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 $(Y\times I)\sqcup (Z\times 0)\to Z\times I$ for monomirphisms $Y\to Z$ and $0\to I=\Delta ^{1}$ .

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

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

Model category structure

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

Universal left fibration

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

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

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,{\textbf {Kan}}]\,{\overset {\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.^[23] In short, ${\textbf {Kan}}$ is the classifying space of left fibrations. Given a left fibration over X, a map $X\to {\textbf {Kan}}$ corresponding to it is called the classifying map for that fibration.

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

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

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

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

Footnotes

^ ^a ^b Lurie 2009a, Definition 2.0.0.3.
^ Beke, Tibor (2008). "Fibrations of simplicial sets". arXiv:0810.4960 [math.CT].
^ ^a ^b Cisinski 2023, Definition 3.4.1.
^ Proof: Let $l(F)$ = the class of maps having the left lifting property with respect to a class $F$ of maps. Then $l(F)$ can be shown to be saturated. By the axiom of choice, if $F$ is the class of surjective maps, then $l(F)$ is the class of injective maps. This implies the same is true for monomorphisms between preshaves.
^ Proof: Since $l(G)$ , l for the left lifting property, is saturated and $F\subset l(r(F))$ , we have: ${\overline {F}}\subset l(r(F))$ and so $r(F)=r(l(r(F))\subset r({\overline {F}})\subset r(F)$ .
^ Cisinski 2023, Definition 3.2.1.
^ Cisinski 2023, Definition 3.2.5.
^ Cisinski 2023, Definition 3.3.15.
^ Cisinski 2023, Proposition 3.4.8.
^ Joyal & Tierney 2008, Theorem 3.2.2.
^ Cisinski 2023, Proposition 3.4.3.
^ Cisinski 2023, Corollary 3.2.4.
^ Joyal & Tierney 2008, Proposition 3.2.2.
^ Cisinski 2023, Corollary 3.2.8.
^ Proposition 4.1.4.1. in https://kerodon.net/tag/01BS
^ Cisinski 2023, Proposition 3.4.4.
^ Joyal & Tierney 2008, Theorem 3.4.1, Proposition 3.4.2, Proposition 3.4.3.
^ Joyal & Tierney 2008, Proposition 4.6.3.
^ Joyal & Tierney 2008, § 2.1.
^ Joyal & Tierney 2008, Proposition 4.6.1.
^ Cisinski 2023, Definition 5.2.3.
^ Cisinski 2023, Theorem 5.2.10.
^ Cisinski 2023, Corollary 5.3.21.
^ Cisinski 2023, § 5.6.1. and § 5.8.1.
^ Cisinski 2023, Proposition 5.6.2.
^ Lurie 2009a, § 3.3.2.