Spahn HTT, 2. fibrations of simplicial sets (Rev #4, changes)

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2.0 (overview and glossary of fibrations and anodyne morphisms)

Definition

A morphism of simplicial sets is called

  1. a Kan fibration if it has the right lifting property with respect to every horn inclusion.

  2. a trivial fibration if it has the right lifting property with respect to every boundary inclusion ∂Δ[n]↪Δ[n]\partial \Delta[n]\hookrightarrow \Delta[n].

  3. a left fibration if it has the right lifting property with respect to every horn inclusion except the right outer one.

  4. a left fibration if it has the right lifting property with respect to every horn inclusion except the left outer one.

  5. a left fibration if it has the right lifting property with respect to every inner horn inclusion.

  6. left anodyne if it has the left lifing property with respect to every left fibration.

  7. right anodyne if it has the left lifing property with respect to every right fibration.

  8. inner anodyne if it has the left lifing property with respect to every inner fibration.

  9. minimal fibration roughly said, when the morphism is determined by its values on the boundaries.

  10. cartesian fibration

  11. cocartesian fibration

  12. categorical fibration

In particular the following notions are important:
Remark

We have the following intuition in regard to these types of fibrations

  1. Right fibrations are the ∞\infty-categorical analog of fibrations in groupoids.

  2. Left fibrations are the ∞\infty-categorical analog of cofibrations in groupoids.

  3. Cartesian fibrations are the ∞\infty-categorical analog of fibrations (not necessarily in groupoids).

  • Left fibrations: These are the higher analogs of fibrations in groupoids.

  • Anodyne morphisms: These are morphisms possessing the left lifting property with respect to all Kan fibrations. Anodyne morphisms are acyclic cofibrations in the standard model structure on simplicial sets.

  • The covariant model structure: This is a model structure on the over category sSet/SsSet/S. It is a model structure for left fibrations. It is factorial in SS. It is called covariant since by the Grothendieck construction Fib l(S)Fib_l (S) is associated to the category of covariant functors Fun(S,∞Grpd)Fun(S,\infty Grpd). There is also a notion of contravariant model structure where right fibrations Fib r(S)Fib_r (S) is associated to Fun(S op,∞Grpd)Fun(S^{op},\infty Grpd).

  • The Joyal model structure: Precisely ∞\infty-categories are fibrant with respect to this model structure.

  • inner fibration and minimal fibration: These are used to develop a theory of n-categories.

  • cartesian fibration: These are higher analogs to Grothendieck fibrations (not necessarily in groupoids). These are defined with respect to cartesian morphisms.

  • categorical fibrations?
Remark

(anodyne morphisms)

  1. Every map of simplicial sets admits a factorization into an anodyne ((left anodyne, right anodyne, inner anodyne, a cofibration) followed by a Kan fibration (left fibration, right fibration, inner fibration, trivial fibration).

  2. The theory of left fibrations and left anodyne morphisms is dual to that of right fibrations and right anodyne morphisms; i.e. if S→TS\to T is a left fibration (left anodyne morphisms) iff the induced map S op→T opS^{op}\to T^{op} right fibrations and right anodyne morphisms.

For a general overview see model structure on simplicial sets.

2.1

Definition

A morphism of simplicial sets is called

  1. a Kan fibration if it has the right lifting property with respect to every horn inclusion.

  2. a trivial fibration if it has the right lifting property with respect to every boundary inclusion ∂Δ[n]↪Δ[n]\partial \Delta[n]\hookrightarrow \Delta[n].

  3. a left fibration if it has the right lifting property with respect to every horn inclusion except the right outer one.

  4. a left fibration if it has the right lifting property with respect to every horn inclusion except the left outer one.

  5. a left fibration if it has the right lifting property with respect to every inner horn inclusion.

  6. left anodyne if it has the left lifing property with respect to every left fibration.

  7. right anodyne if it has the left lifing property with respect to every right fibration.

  8. inner anodyne if it has the left lifing property with respect to every inner fibration.

  9. minimal fibration roughly said, when the morphism is determined by its values on the boundaries.

  10. cartesian fibration

  11. cocartesian fibration

  12. categorical fibration
Proposition Remark 2.1.1.3 +

Let We have the following intuition in regard to these types of fibrationsF:C→DF:C\to D be a functor between categories. The FF is a fibrations in groupoids iff the induced map N(F):N(C)→N(D)N(F):N(C)\to N(D) is a left fibration of simplicial sets.

  1. Right fibrations are the ∞\infty-categorical analog of fibrations in groupoids.

  2. Left fibrations are the ∞\infty-categorical analog of cofibrations in groupoids.

  3. Cartesian fibrations are the ∞\infty-categorical analog of fibrations (not necessarily in groupoids).
Lemma Remark 2.1.1.4 +

Let (anodyne morphisms)q:X→Sq:X\to S be a left fibration of simplicial sets. The assignment

{hS→H s↦X s f↦f 1\begin{cases} hS\to H \\ s\mapsto X_s \\f\mapsto f_1 \end{cases}

  1. Every map of simplicial sets admits a factorization into an anodyne ((left anodyne, right anodyne, inner anodyne, a cofibration) followed by a Kan fibration (left fibration, right fibration, inner fibration, trivial fibration).

  2. The theory of left fibrations and left anodyne morphisms is dual to that of right fibrations and right anodyne morphisms; i.e. if S→TS\to T is a left fibration (left anodyne morphisms) iff the induced map S op→T opS^{op}\to T^{op} right fibrations and right anodyne morphisms.
Remark (2.1.2.2, 2.1.2.9)

(left fibrations) 1. The projection from the over category is a left fibration.

1.The property of being a left fibration is stable under forming functor categories.

2.1 Left fibrations

2.3 Inner fibrations and minimal inner fibrations

2.1.1 Left fibrations in classical category theory

Corollary Left 2.3.2.2: fibrations are analogs to fibrations in groupoids.Fun(Δ[2],C)→Fun(Λ 1[2],C)Fun(\Delta[2],C)\to Fun(\Lambda_1[2],C) is a trivial fibration.

Every ∞\infty-category is categorial equivalenct to a minimal ∞\infty-category.

Proposition 2.1.1.3 +

Let F:C→DF:C\to D be a functor between categories. The FF is a fibrations in groupoids iff the induced map N(F):N(C)→N(D)N(F):N(C)\to N(D) is a left fibration of simplicial sets.

2.3.3. Minimal inner fibrations

Compare the following lemma with Proposition 2.1.3.1
Definition Lemma 2.3.3.1 2.1.1.4 +

Let p q:X→S p q:X\to : X \to S be an a inner left fibration fibration of simplicial sets. The assignmentpp is called minimal inner fibration if f=f ′f = f^\prime for every pair of maps f,f ′:Δ[n]→Xf , f ^\prime : \Delta[n] \to X which are homotopic relative to ∂Δ[n]\partial \Delta[n] over SS .

An ∞\infty-category CC is called minimal ∞\infty-category if C→*C\to * is minimal.

{hS→H s↦X s f↦f 1\begin{cases} hS\to H \\ s\mapsto X_s \\f\mapsto f_1 \end{cases}

(…)

2.1.2 Stability properties of left fibrations

Every The aim of this section is to show that left fibrations exist in abundance.∞\infty-category is equivalent to a minimal ∞\infty-category.

2.3.4 Theory of nn-categories

The following remark follows from a theorem of Joyal (not displayed here).
Proposition Remark 2.3.4.19 (2.1.2.2, 2.1.2.9)

Proposition (left 2.3.4.5: fibrations) For 1. The projection from the over category is a simplicial left set fibration.XX the following statements are equivalent:

  1. the unit u:X→N(hX)u:X\to N(hX) is an isomorphism of simplicial sets.

  2. There is small category CC and an isomorphism of simpliial sets X≃N(C)X\simeq N(C).

  3. SS is a 1-category.

1.The property of being a left fibration is stable under forming functor categories.
Proposition 2.3.4.19 Let CC be an ∞\infty-category. Let n≥−1n\ge -1. Then the following statements are equivalent:

  1. CC is an nn-category.

  2. For every simplicial set KK and every pair of maps f,g:K→Cf,g:K\to C such that f|sk nKf| sk^n K and g|sk nKg|sk^n K are homotopic relative to sk n−1Ksk^{n-1}K, we have f=gf=g.

2.1.3 A characterization of Kan fibrations
Corollary 2.3.4.8

Let CC be an nn-category and let XX be a simplicial set. Then Fun(X,C)Fun(X,C) is an nn-category.

Compare the following proposition with Lemma 2.1.1.4.
Proposition 2.3.4.12 2.1.3.1 +

Let C p:S→T C p:S\to T be an a left fibration of simplicial sets. Then the following statements are equivalent∞\infty-category. Let n≥1n\ge 1.

  1. There exists a simplicial set pph nCh_n C is a Kan fibration. with the following universal mapping property: Fun(K,h nC)=[K,C]/∼Fun(K,h_n C)=[K,C]/\sim.

  2. h nCh_n CFor every edge is an f:t→t ′f:t\to t^\primenn in -category.TT, the map f !:S t→S t ′f_!:S_t\to S_{t^\prime} is an isomorphism in the homotopy category of spaces.

  3. If CC is an nn-category, then the natural map Θ:C→h nC\Theta:C\to h_n C is an isomorphism.

    4.

  4. For every nn-category DD, composition with Θ\Theta is an isomorphism of simplicial sets Fun(h nC,D)→Fun(C,D)Fun(h_n C,D)\to Fun(C,D).

    5.
Corollary 2.3.4.19

Let XX be a Kan complex. Then is is equivalent to an nn-category iff it is nn-truncated.

2.1.3 (characterization of Kan fibrations by maps between their fibers.
Proposition 2.1.3.1

Let p:S→Tp:S\to T be a left fibration of simplicial sets. Then the following statements are equivalent

  1. pp is a Kan fibration.

  2. For every edge f:t→t ′f:t\to t^\prime in T,themap, the map f_!:S_t\to S_^{t^\prime}$ is an isomorphism in the homotopy category of spaces.

2.1.4 The covariant model structure

This section is a preparation for the Grothendieck consruction for ∞\infty-categories.

uses the model structure on simplicially enriched categories

Definition 2.1.4.2

left- and right cone of a morphism of simplicial sets

cone point

The covariant model structure is a ‘’relative model structure’‘

Definition 2.1.4.5

Let SS be a simplicial set . A morphism f:X→Yf:X\to Y in sSet /SsSet_{/S} is called a

(C) covariant cofibration if it is a monomorphism of simplicial sets.

(W) a covariant weak equivalence if the induced map

X ◃∐ XS→Y ◃∐ YSX^\triangleleft\coprod_X S\to Y^\triangleleft\coprod_Y S

is a categorical weak equivalence.

(F) covariant fibration if it has the right lifting property with respect to every map wich is both a covariant cofibration and a covariant equivalence.

Lemma 2.1.4.6

every left anodyne map is a covariant equivalence

Proposition 2.1.4.7

The covariant model structure determines a left proper, combinatorial model structure on sSet /SsSet_{/S}

Proposition 2.1.4.9

Every covariant fibration is a left fibration of simplicial sets

Proposition 2.1.4.10

The covariant model structure is functorial in SS.

Remark 2.1.4.12

There is also a contravariant model structure

2.2 Simplicial categories and ∞\infty-categories

2.2.5 The Joyal model structure

Requisite: Theorem 2.4.6.1: Let XX be a simplicial set. Then XX is fibrant in the Joyal model structure iff XX is an ∞\infty-category.

Theorem 2.2.5.1

The exists a left proper, combinatorial model structure on the category of simplicial sets such that

(C) Cofibrations are precisely monomorphisms

(W) A map pp is a categorical equivalence iff S(p)S(p) is an equivalence of simplicial categories. Where S:sSet→sCatS:sSet\to sCat denotes the functor induced via Kan extension by the cosimplicial object ℭ:Δ→sCat\mathfrak{C}:\Delta\to sCat, Definition 1.1.5.1, HTT.

here: give proof of Proposition 1.2.7.3

2.3 Inner fibrations
Corollary 2.3.2.2: Fun(Δ[2],C)→Fun(Λ 1[2],C)Fun(\Delta[2],C)\to Fun(\Lambda_1[2],C) is a trivial fibration.

Every ∞\infty-category is categorial equivalenct to a minimal ∞\infty-category.

2.3.3. Minimal inner fibrations
Definition 2.3.3.1

Let p:X→Sp : X \to S be an inner fibration of simplicial sets. pp is called minimal inner fibration if f=f ′f = f^\prime for every pair of maps f,f ′:Δ[n]→Xf , f ^\prime : \Delta[n] \to X which are homotopic relative to ∂Δ[n]\partial \Delta[n] over SS .

An ∞\infty-category CC is called minimal ∞\infty-category if C→*C\to * is minimal.

(…)

Every ∞\infty-category is equivalent to a minimal ∞\infty-category.

2.3.4 Theory of nn-categories
Proposition 2.3.4.19

Proposition 2.3.4.5: For a simplicial set XX the following statements are equivalent:

  1. the unit u:X→N(hX)u:X\to N(hX) is an isomorphism of simplicial sets.

  2. There is small category CC and an isomorphism of simpliial sets X≃N(C)X\simeq N(C).

  3. SS is a 1-category.
Proposition 2.3.4.19 Let CC be an ∞\infty-category. Let n≥−1n\ge -1. Then the following statements are equivalent:

  1. CC is an nn-category.

  2. For every simplicial set KK and every pair of maps f,g:K→Cf,g:K\to C such that f|sk nKf| sk^n K and g|sk nKg|sk^n K are homotopic relative to sk n−1Ksk^{n-1}K, we have f=gf=g.
Corollary 2.3.4.8

Let CC be an nn-category and let XX be a simplicial set. Then Fun(X,C)Fun(X,C) is an nn-category.
Proposition 2.3.4.12

Let CC be an ∞\infty-category. Let n≥1n\ge 1.

  1. There exists a simplicial set h nCh_n C with the following universal mapping property: Fun(K,h nC)=[K,C]/∼Fun(K,h_n C)=[K,C]/\sim.

  2. h nCh_n C is an nn-category.

  3. If CC is an nn-category, then the natural map Θ:C→h nC\Theta:C\to h_n C is an isomorphism.

  4. For every nn-category DD, composition with Θ\Theta is an isomorphism of simplicial sets Fun(h nC,D)→Fun(C,D)Fun(h_n C,D)\to Fun(C,D).
Corollary 2.3.4.19

Let XX be a Kan complex. Then is is equivalent to an nn-category iff it is nn-truncated.

2.4 Cartesian fibrations

Definiton 2.4.1.1

Let p:X→Sp:X\to S be an inner fibrations of simplicial sets. Let f:x→yf:x\to y be an edge in XX. Then ff is called pp-cartesian if the induced map

X /f→X /y× S p(y)S /p(f)X_{/f}\to X_{/y}\times_{S_{p(y)}} S_{/p(f)}

is a trivial Kan fibration.

2.4.1 Cartesian morphisms
Proposition Definiton 2.4.1.3 2.4.1.1

  1. Every edge of a simplicial set is pp cartesian for an isomorphism.

  2. Let pp be an inner fibration, let qq be the pullback of pp (which s then also an inner fibration). Then an edge is pp cartesian if ‘’its pullback’‘ is qq-cartesian.

  3. (…)

Let p:X→Sp:X\to S be an inner fibrations of simplicial sets. Let f:x→yf:x\to y be an edge in XX. Then ff is called pp-cartesian if the induced map
X /f→X /y× S p(y)S /p(f)X_{/f}\to X_{/y}\times_{S_{p(y)}} S_{/p(f)}

is a trivial Kan fibration.
Corollary Proposition 2.4.1.6 2.4.1.3

Let p:C→Dp:C\to D be an inner fibration between ∞\infty-categories. Every identity morphism of CC (i.e. every degenerate edge of CC) is pp-cartesian.

  1. Every edge of a simplicial set is pp cartesian for an isomorphism.

  2. Let pp be an inner fibration, let qq be the pullback of pp (which s then also an inner fibration). Then an edge is pp cartesian if ‘’its pullback’‘ is qq-cartesian.

  3. (…)
Proposition Corollary 2.4.1.7 2.4.1.6

(left cancellation) Letp:C→Dp:C\to D be an inner fibration between simplicial sets. Let∞\infty-categories. Every identity morphism of CC (i.e. every degenerate edge of CC) is pp-cartesian.

C 1 f↗ ↘ g C 0 →h C 2\array{ &C_1& \\ {}_f\nearrow&\searrow^g \\ C_0&\stackrel{h}{\to}&C_2 }

Let gg be pp-cartesian. Then ff is pp-cartesian iff hh is pp-cartesian.
Proposition 2.4.1.7

(left cancellation) Let p:C→Dp:C\to D be an inner fibration between simplicial sets. Let

C 1 f↗ ↘ g C 0 →h C 2\array{ &C_1& \\ {}_f\nearrow&\searrow^g \\ C_0&\stackrel{h}{\to}&C_2 }

Let gg be pp-cartesian. Then ff is pp-cartesian iff hh is pp-cartesian.
Proposition 2.4.1.10

(…)

Definition 2.4.2.1

Let p:X→Sp:X\to S be a map of simplicial sets. Then pp is called a cartesian fibration if the following coditions are satisfied.

  1. pp is an inner fibration.

  2. Every edge of has a pp-cartesian lift.

2.4.2 Cartesian fibrations
Definition 2.4.2.1

Let p:X→Sp:X\to S be a map of simplicial sets. Then pp is called a cartesian fibration if the following coditions are satisfied.

  1. pp is an inner fibration.

  2. Every edge of has a pp-cartesian lift.
Proposition 2.4.2.3
Proposition 2.4.2.4

2.4.4 Mapping spaces and cartesian fibrations
Corollary 2.4.4.4
Corollary 2.4.4.6
Corollary 2.4.4.7
Corollary 2.4.4.8

2.4.6 Application: Categorical fibrations
Corollary 2.4.6.1

2.4.7 Bifibrations
Corollary 2.4.7.11
Corollary 2.4.7.12

Revision on June 21, 2012 at 15:16:38 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

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