category of fibrant objects (changes) in nLab

Context

Model category theory

Homotopy theory

• Idea
• Definition
• Examples
  • Full subcategories of model categories
  • Right proper model categories
  • ∞\infty-Groupoids
  • Simplicial sheaves
    • Example
  • Slice categories
• Properties
  • Simple consequences of the definition
  • Generalized universal bundles and the factorization lemma
  • More sophisticated consequences of the definition
  • Homotopy fiber product
  • Homotopies
  • The homotopy category
  • Pointed category of fibrant objects
  • Fibers
  • Fibration Sequences
  • Derived hom-spaces
• Application in cohomology theory
• Related concept
• References

Definition

A category of fibrant objects C\mathbf{C} is

• a category with weak equivalences, i.e equipped with a subcategory

  Core(C)↪W↪C Core(\mathbf{C}) \hookrightarrow W \hookrightarrow C

  where f∈Mor(W)f \in Mor(W) is called a weak equivalence;

• equipped with a further subcategory

  Core(C)↪F↪C, Core(\mathbf{C}) \hookrightarrow F \hookrightarrow C \,,

  where f∈Mor(F)f \in Mor(F) is called a fibration

  Those morphisms which are both weak equivalences and fibrations are called acyclic fibrations .

Those morphisms which are both weak equivalences and fibrations are called acyclic fibrations.

This data has to satisfy the following properties:

• CC has finite products, and in particular a terminal object *{*};

• the pullback of a fibration along an arbitrary morphism exists, and is again a fibration;

• acyclic fibrations are preserved under pullback;

• weak equivalences satisfy 2-out-of-3

• for every object there exists a path object

  • this means: for every object BB there exists at least one object denoted B IB^I (not necessarily but possibly the internal hom with an interval object) that fits into a diagram

    (B→Id×IdB×B)=(B→σB I→d 0×d 1B×B) (B \stackrel{Id \times Id}{\to} B \times B) = (B \stackrel{\sigma}{\to} B^I \stackrel{d_0 \times d_1}{\to} B \times B)

    where σ\sigma is a weak equivalence and d 0×d 1d_0 \times d_1 is a fibration;

• all objects are fibrant, i.e. all morphisms B→*B \to {*} to the terminal object are fibrations.

Proof

The path object of any XX can be chosen to be the internal hom

X I=[Δ 1,X] X^I = [\Delta^1, X]

in with respect to the closed monoidal structure on SSet with the simplicial 1-simplex Δ 1\Delta^1.

The morphism X→X IX \to X^I is given by the degeneracy map σ 0:Δ 0→Δ 1\sigma_0 : \Delta^0 \to \Delta^1 as

X→≃[Δ 0,X]→[σ 0,X][Δ 1,X]. X \stackrel{\simeq}{\to} [\Delta^0, X] \stackrel{[\sigma_0, X]}{\to} [\Delta^1, X] \,.

This is indeed a weak equivalence, since by the simplicial identities it is a section (a right inverse) for the morphism

[Δ 1,X]→[δ 0,X][Δ 0,X]. [\Delta^1, X] \stackrel{[\delta_0,X]}{\to} [\Delta^0, X] \,.

This map, one checks, has the right lifting property with respect to all boundary of a simplex-inclusions ∂Δ n→Δ n\partial \Delta^n \to \Delta^n. By a lemma discussed at Kan fibration this means that [δ 0,X][\delta_0,X] is an acyclic fibration. Hence [σ 0,X][\sigma_0, X], being its right inverse, is a weak equivalence.

The remaining morphism of the path space object X I→X×XX^I \to X \times X is

[Δ 1,X]→[δ 0⊔δ 1,X][Δ 0⊔Δ 0,X]→≃X×X. [\Delta^1, X] \stackrel{[\delta_0 \sqcup \delta_1, X]}{\to} [\Delta^0 \sqcup \Delta^0, X] \stackrel{\simeq}{\to} X \times X \,.

One checks that this is indeed a Kan fibration.

The stability of fibrations and acyclic fibrations follows from the above fact that both are characterized by a right lifting property (as described a model structure on simplicial sets).

See for instance section 1 of

• Goerss, Jardine, Simplicial homotopy theory .

Definition

(∞\infty-groupoid valued sheaves)

For CC be a site such that the sheaf topos Sh(C)Sh(C) has enough points, i.e. so that a morphism f:A→Bf : A \to B in Sh(X)Sh(X) is an isomorphism precisely if its image

x *f:x *A→x *B x^* f : x^* A \to x^* B

is a bijection of sets for all points (geometric morphisms from Sh(*)≃SetSh({*}) \simeq Set)

x:Set≃Sh(*)→x *←x *Sh(C). x : Set \simeq Sh({*}) \stackrel{\stackrel{x^*}{\leftarrow}} {\stackrel{x_*}{\to}} Sh(C) \,.

Then let

C=SSh(C) \mathbf{C} = SSh(C)

be the full subcategory of

• sheaves on CC with values in the category SSet of simplicial sets

• equivalently: simplicial objects in the category of sheaves on CC

on those sheaves AA for which each stalk x *A∈SSetx^* A \in SSet is a Kan complex.

Define a morphism f:A→Bf : A \to B to be a fibration or a weak equivalence, if on each stalk x *f:x *A→x *Bx^* f : x^* A \to x^* B is a fibration or weak equivalence, respectively, of Kan complexes (in terms of the standard model structure on simplicial sets).

Lemma

SSh(X)SSh(X) with this structure is a category of fibrant objects.

Proof

The terminal object *=X{*} = X is the sheaf constant on the 0-simplex Δ 0\Delta^0, which represents the space XX itself as a sheaf.

For every simplicial sheaf AA and every point x∈Xx \in X the stalk of the unique morphism A→*A \to {*} is x *A→x *Xx^* A \to x^* {X}, which is the unique morphism from the Kan complex x *Ax^* A to Δ 0\Delta^0. Since Kan complexes are fibrant, this is a Kan fibration for every x∈Xx \in X. So every AA is a fibrant object by the above definition.

The fact that fibrations and acyclic fibrations are preserved under pullback follows from the fact that the stalk operation

x *:SSh(X)→SSh(pt)≃SSet x^* : SSh(X) \to SSh(pt) \simeq SSet

is the inverse image of a geometric morphism and hence preserves finite limits and in particular pullbacks. So if f:A→Bf : A \to B is a fibration or acyclic fibration in SSh(X)SSh(X) and

A× BC → B ↓ h *f ↓ f C →h B \array{ A \times_B C &\to& B \\ \downarrow^{\mathrlap{h^* f}} && \downarrow^\mathrlap{f} \\ C &\stackrel{h}{\to}& B }

is a pullback diagram in SSh(X)SSh(X), then for x∈Xx \in X any point of XX also

x *(A× BC) → x *B ↓ x *(h *f) ↓ x *f x *C →x * B \array{ x^*(A \times_{B} C) &\to& x^*B \\ \downarrow^{\mathrlap{x^* (h^* f)}} && \downarrow^{\mathrlap{x^* f}} \\ x^*C &\stackrel{x^*}{\to}& B }

is a pullback diagram, now of Kan complexes. Since Kan complexes form a category of fibrant objects, by the above, it follows that x *(h *f)x^* (h^* f) is a fibration or acyclic fibration of Kan complexes, respectively. Since this holds for every xx, it follows that h *fh^* f is a fibration or acyclic fibration, respectively, in SSh(X)SSh(X).

Recall that a functorial choice of path object for a Kan complexe KK is the internal hom [Δ 1,K][\Delta^1, K] with respect to the closed monoidal structure on simplicial sets:

K→=[Δ 0,K]→s 0[Δ 1,K]→d 0×d 1[Δ 0,K]×[Δ 0,K]→=K×K, K \stackrel{=}{\to} [\Delta^0, K] \stackrel{s_0}{\to} [\Delta^1, K] \stackrel{d_0 \times d_1}{\to} [\Delta^0, K] \times [\Delta^0, K] \stackrel{=}{\to} K \times K \,,

where s is_i and d id_i denote the degeneracy and face maps, respectively.

For A∈SSh(X)A \in SSh(X) let [Δ 1,A][\Delta^1,A] denote the sheaf

[Δ 1,A]:U↦[Δ 1,A(U)], [\Delta^1,A] : U \mapsto [\Delta^1,A(U)] \,,

where on the left we have new notation and on the right we have the internal hom in SSet.

(The notation on the left defines the way in which SSh(X)SSh(X) is copowerered over SSet).

We want to claim that [Δ 1,A][\Delta^1,A] is a path object for AA.

To check that [Δ 1,A][\Delta^1,A] is fibrant, let x∈Xx \in X be any point and consider the stalk x *[Δ 1,A]∈SSetx^* [\Delta^1,A] \in SSet. We compute laboriously

x *[Δ 1,A] ≃colim U∋x[Δ 1,A(U)] ≃colim U∋xSSet(Δ 1×Δ •,A(U)) ≃([n]↦colim U∋xSSet(Δ 1×Δ n,A(U)) ≃([n]↦colim U∋x∫ [k]∈ΔSet(Δ([k],[1])×Δ([k],[n]),A(U) k) ≃([n]↦∫ [k≤n+1]∈Δ(colim U∋xSet(Δ([k],[1])×Δ([k],[n]),A(U) k) ≃([n]↦∫ [k]∈Δ| n+1(Set(Δ([k],[1])×Δ([k],[n]),colim U∋xA(U) k) ≃([n]↦∫ [k]∈Δ| n+1(Set(Δ([k],[1])×Δ([k],[n]),(colim U∋xA(U)) k) ≃[Δ 1,colim U∋xA(U)] ≃[Δ 1,x *A]) \begin{aligned} x^* [\Delta^1,A] &\simeq colim_{U \ni x} [\Delta^1,A(U)] \\ &\simeq colim_{U \ni x} SSet(\Delta^1 \times \Delta^\bullet, A(U)) \\ &\simeq ([n] \mapsto colim_{U \ni x} SSet(\Delta^1 \times \Delta^\n, A(U)) \\ & \simeq ([n] \mapsto colim_{U \ni x} \int_{[k] \in \Delta} Set(\Delta([k],[1])\times\Delta([k],[n]), A(U)_k) \\ & \simeq ([n] \mapsto \int_{[k \leq n+1] \in \Delta}( colim_{U \ni x} Set(\Delta([k],[1])\times\Delta([k],[n]), A(U)_k ) \\ &\simeq ([n] \mapsto \int_{[k] \in \Delta|_{n+1}}( Set(\Delta([k],[1])\times\Delta([k],[n]), colim_{U \ni x} A(U)_k ) \\ &\simeq ([n] \mapsto \int_{[k] \in \Delta|_{n+1}}( Set(\Delta([k],[1])\times\Delta([k],[n]), (colim_{U \ni x} A(U))_k ) \\ &\simeq [\Delta^1, colim_{U \ni x} A(U)] \\ & \simeq [\Delta^1, x^* A] ) \end{aligned}

Where the

• first step is the general formula for the stalk;

• second step is the formula for the internal hom in the closed monoidal structure on simplicial sets;

• third step is the fact that colimits of presheaves are computed objectwise (see examples at colimit);

• the fourth step is the definition of the SSet-enriched functor category by an end

• the fifth step uses that

  • the end truncates to a finite limit with k≤n+1k \leq n+1 since Δ 1×Δ n\Delta^1 \times \Delta^n is (n+1)(n+1)-skeletal

  • and that the colimit is over a filtered category

  • and that filtered colimits commute with finite limits;

• the sixth step uses that the set Δ([k],[1])×Δ([k],[n])\Delta([k],[1])\times \Delta([k],[n]) is finite, hence a compact object so that the colimit can be taken into the hom;

• the seventh step uses again that colimits of presheaves are computed objectwise

• the remaining steps then just rewind the first ones, only that now A(U)A(U) has been replaced by colim U∋xA(U)colim_{U \ni x} A(U).

That the morphism A→[Δ 1,A]A \to [\Delta^1,A] is a weak equivalence and that [Δ 1,A]→d 0×d 1A×A[\Delta^1,A] \stackrel{d_0 \times d_1}{\to} A \times A is a fibration follows similarly by taking the stalk colimit inside to reduce to the statement that x *A→[Δ 1,x *A]x^* A \to [\Delta^1,x^* A] is a weak equivalence and [Δ 1,x *A]→d 0×d 1x *A×x *A[\Delta^1,x^* A] \stackrel{d_0 \times d_1}{\to} x^*A \times x^* A is a fibration, using that [Δ 1,x *A][\Delta^1,x^*A] is a path object for the Kan complex x *Ax^* A.

Lemma

With this structure, C B F\mathbf{C}_B^F becomes a category of fibrant objects.

Proof

Below is proven the factorization lemma that holds in any category of fibrant objects. This implies in particular that every morphism

A →Id×Id A× BA ∈F↘ ↙ ∈F B \array{ A &&\stackrel{Id \times Id}{\to}&& A \times_B A \\ & {}_{\in F}\searrow && \swarrow_{\in F} \\ && B }

may be factored as

A →∈W A I →∈F A× BA ∈F↘ ↙ ∈F B. \array{ A &\stackrel{\in W}{\to}& A^I &\stackrel{\in F}{\to}& A \times_B A \\ & {}_{\in F}\searrow && \swarrow_{\in F} \\ && B } \,.

This provides the path space objects in C B F\mathbf{C}^F_B.

Lemma

For every two objects A 1,A 2∈CA_1, A_2 \in \mathbf{C}, the two projection maps

p i:A 1×A 2→∈FA i p_i : A_1 \times A_2 \stackrel{\in F}{\to} A_i

out of their product are fibrations.

Proof

Because by assumption both morphisms A i→*A_i \to {*} are fibrations and fibrations are preserved under pullback

A 1×A 2 → A 2 p 1↓ ⇒∈F ↓ ∈F A 1 → *. \array{ A_1 \times A_2 &\to& A_2 \\ \;{}^{p_1}\downarrow^{\mathrlap{\Rightarrow \in F}} && \downarrow^{\mathrlap{\in F}} \\ A_1 &\to& {*} } \,.

Lemma

For every object B∈CB \in \mathbf{C} and everey path object B IB^I of BB, the two morphisms

d i:B I→∈W∩FB d_i : B^I \stackrel{\in W \cap F}{\to} B

(whose product d 0×d 1d_0 \times d_1, recall, is required to be a fibration) are each separately acyclic fibrations.

Proof

By the above lemma d i:B I→d 0×d 1B×B→p iBd_i : B^I \stackrel{d_0 \times d_1}{\to} B \times B \stackrel{p_i}{\to} B is the composite of two fibrations and hence itself a fibration.

Moreover, from the diagram

B →≃ B I →d 0×d 1 B×B ↘ d i ↓ p i ↘ Id B \array{ B &\stackrel{\simeq}{\to}& B^I &\stackrel{d_0 \times d_1}{\to}& B \times B \\ &&&\searrow^{d_i}& \downarrow^{\mathrlap{p_i}} \\ & \searrow^{Id}&&& B }

one reads off that the 2-out-of-3 property for weak equivalences implies that d id_i is also a weak equivalence.

Lemma (factorization lemma)

For every morphism f:C→Df : C \to D in a category C\mathbf{C} of fibrant objects, there is an object E fB\mathbf{E}_f B such that ff factors as

E fB σ f∈W↗ ↘ p f∈F C →f B \array{ && \mathbf{E}_f B \\ & {}^{\sigma_f \in W}\nearrow && \searrow^{p_f \in F} \\ C &&\stackrel{f}{\to}&& B }

with

• p fp_f a fibration

• σ f\sigma_f a weak equivalence that is a section ( a right inverse) of an acyclic fibration:

  Id E fB=(C→σ fE fB→≃C). Id_{\mathbf{E}_f B} = ( C \stackrel{\sigma_f}{\to} \mathbf{E}_f B \stackrel{\simeq}{\to} C ) \,.

Definition

For f:C→Bf : C \to B a morphism in C\mathbf{C}, we say that the morphism p f:E fB→Bp_f : \mathbf{E}_f B \to B defined as the composite vertical morphism in the pullback diagram

E fB →≃ C ↓ ↓ f B I →d 0 B ↓ d 1 B \array{ \mathbf{E}_f B &\stackrel{\simeq}{\to}& C \\ \downarrow && \downarrow^{\mathrlap{f}} \\ B^I &\stackrel{d_0}{\to}& B \\ \downarrow^{\mathrlap{d_1}} \\ B }

for some path space object B IB^I is the generalized universal bundle over BB relative to ff.

Lemma

The morphism p f:E fB→Bp_f : \mathbf{E}_f B \to B is a fibration.

Proof

The defining pullback diagram for E fB\mathbf{E}_f B can be refined to a double pullback diagram as follows

E fB →∈F C×B →p 1 C ↓ ↓ f×Id ↓ f B I →d 0×d 1∈F B×B →p 1 B ↓ d 1 ↙ p 2∈F B. \array{ \mathbf{E}_f B &\stackrel{\in F}{\to}& C \times B &\stackrel{p_1}{\to}& C \\ \downarrow && \downarrow^{\mathrlap{f \times Id}} && \downarrow^{\mathrlap{f}} \\ B^I &\stackrel{d_0 \times d_1 \in F}{\to}& B \times B &\stackrel{p_1}{\to}& B \\ \downarrow^{\mathrlap{d_1}} & \swarrow_{p_2 \in F} \\ B } \,.

Both squares are pullback squares. Since pullbacks of fibrations are fibrations, the morphism E fB→C×B\mathbf{E}_f B \to C \times B is a fibration.

By one of the lemmas above, also the projection map p i:B×B→Bp_i : B \times B \to B is a fibration.

The above diagram exibits p fp_f as the the composite

p f :E fB→C×B→f×IdB×B→p 2B =E fB→C×B→p 2B \begin{aligned} p_f &: \mathbf{E}_f B \to C \times B \stackrel{f \times Id}{\to} B \times B \stackrel{p_2}{\to} B \\ & = \mathbf{E}_f B \to C \times B \stackrel{p_2}{\to} B \end{aligned}

of two fibrations. Therefore it is itself a fibration.

Lemma

The morphism E fB→≃C\mathbf{E}_f B \stackrel{\simeq}{\to} C has a section (a right inverse) σ f:C→≃E fB\sigma_f : C \stackrel{\simeq}{\to} \mathbf{E}_f B and its composite with p fp_f is ff:

E fB ←σ f C ↓ p f ↙ f B \array{ \mathbf{E}_f B &\stackrel{\sigma_f}{\leftarrow}&& C \\ \downarrow^{\mathrlap{p_f}} && \swarrow_{f} \\ B }

Proof

The section

σ f=Id×σ∘f \sigma_f = Id \times \sigma \circ f

is the morphism induced via the universal property of the pullback by the section σ:B→B I\sigma : B \to B^I of d 0:B I→Bd_0 : B^I \to B:

C →σ f∈W E fB →∈W∩F C ↓ f ↓ ↓ f B →σ B I →d 1∈W∩F B Id↘ ↓ d 0 B=C →Id C ↓ f ↓ f B →Id B. \array{ C &\stackrel{\sigma_f \in W}{\to}& \mathbf{E}_f B &\stackrel{\in W \cap F}{\to}& C \\ \downarrow^{\mathrlap{f}} && \downarrow && \downarrow^{\mathrlap{f}} \\ B &\stackrel{\sigma}{\to}& B^I &\stackrel{d_1 \in W \cap F}{\to}& B \\ & {}_{Id}\searrow & \downarrow^{\mathrlap{d_0}} \\ && B } \;\;\;\; = \;\;\;\; \array{ C &\stackrel{Id}{\to}& C \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f}} \\ B &\stackrel{Id}{\to}& B } \,.

Lemma

Let

A 1 →f A 2 ∈F↘ ↙ ∈F B \array{ A_1 &&\stackrel{f}{\to}&& A_2 \\ & {}_{\in F}\searrow && \swarrow_{\in F} \\ && B }

be a morphism of fibrations over some object BB in C\mathbf{C} and let u:B′→Bu : B' \to B be any morphism in C\mathbf{C}. Let

u *A 1 →u *f u *A 2 ∈F↘ ↙ ∈F B′ \array{ u^*A_1 &&\stackrel{u^* f}{\to}&& u^* A_2 \\ & {}_{\in F}\searrow && \swarrow_{\in F} \\ && B' }

be the corresponding morphism pulled back along uu.

Then

• if f∈Ff \in F then also u *f∈Fu^* f \in F;

• if f∈Wf \in W then also u *f∈Wu^* f \in W.

Proof

For f∈Ff \in F the statement follows from the fact that in the diagram

B′× BA 1 → A 1 ↓ u *f∈F ↓ f∈F B′× BA 2 → A 2 ↓ ∈F ↓ ∈F B′ →u B \array{ B' \times_B A_1 &\to& A_1 \\ \;\;\downarrow^{\mathrlap{u^* f \in F}} && \;\;\downarrow^{\mathrlap{f \in F}} \\ B' \times_B A_2 &\to& A_2 \\ \;\downarrow^{\mathrlap{\in F}} && \;\downarrow^{\mathrlap{\in F}} \\ B' &\stackrel{u}{\to}& B }

all squares (the two inner ones as well as the outer one) are pullback squares, since pullback squares compose under pasting.

The same reasoning applies for f∈W∩Ff \in W \cap F.

To apply this reasoning to the case where f∈Wf \in W, we first make use of the factorization lemma to decompose ff as a right inverse to an acyclic fibration followed by an acyclic fibration.

f:A 1→∈WE fA 2→∈W∩FA 2. f : A_1 \stackrel{\in W}{\to} \mathbf{E}_f A_2 \stackrel{\in W \cap F}{\to} A_2 \,.

(Compare the definition of the category of fibrant objects C B F\mathbf{C}_B^F of fibrations over BB, discussed in the example section above.)

Using the above this reduces the proof to showing that the pullback of the top horizontal morphism of

A 1 → E fA 2 ∈F↘ ↙ ∈F B \array{ A_1 &&\stackrel{}{\to}&& \mathbf{E}_f A_2 \\ & {}_{\in F}\searrow && \swarrow_{\in F} \\ && B }

(here the fibration on the right is the composite of the fibration E fA 2→A 2\mathbf{E}_f A_2 \to A_2 with A 2→BA_2 \to B)

along uu is a weak equivalence. For that consider the diagram

B′× BA 1 → A 1 ↓ ↓ B′× BE fA 2 → E fA 2 ↓ ∈W∩F ↓ ∈W∩F B′× BA 1 → A 1 ↓ ∈F ↓ ∈F B′ → B \array{ B' \times_B A_1 &\to& A_1 \\ \downarrow && \downarrow \\ B' \times_B \mathbf{E}_f A_2 &\to& \mathbf{E}_f A_2 \\ \;\;\downarrow^{\mathrlap{\in W \cap F}} && \;\;\downarrow^{\mathrlap{\in W \cap F}} \\ B' \times_B A_1 &\to& A_1 \\ \;\downarrow^{\mathrlap{\in F}} && \;\downarrow^{\mathrlap{\in F}} \\ B' &\to& B }

where again all squares are pullback squares. The top two vertical composite morphisms are identities. Hence by 2-out-of-3 the morphism B′× BE 1→B′× BEB' \times_B E_1 \to B' \times_B \mathbf{E} is a weak equivalence.

Lemma

The pullback of a weak equivalence along a fibration is again a weak equivalence.

Definition

A homotopy fiber product or homotopy pullback of two morphisms

A→uC←vB A \stackrel{u}{\to} C \stackrel{v}{\leftarrow} B

in a category of fibrant objects is the object A× CC I× CBA \times_C C^I \times_C B defined as the (ordinary) limit

A× CC I× CB → B ↓ v C I →d 0 C ↓ ↓ d 1 A →u C. \array{ A \times_C C^I \times_C B &&&\to & B \\ &&&& \downarrow^v \\ & &C^I & \stackrel{d_0}{\to}& C \\ \downarrow && \downarrow^{\mathrlap{d_1}} \\ A &\stackrel{u}{\to} & C } \,.

Lemma

The projection

A× CC I× CB→A A \times_C C^I \times_C B \to A

out of a homotopy fiber product is a fibration. If v:B→Cv : B \to C is a weak equivalence, then this is an acyclic fibration.

Proof

One may compute this limit in terms of two consecutive pullbacks in two different ways.

On the one hand we have

A× CC I× CB → E vC → B ↓ ↓ v C I →d 0 C ↓ ↓ d 1 A →u C \array{ A \times_C C^I \times_C B &\to& \mathbf{E}_v C &\to & B \\ && \downarrow && \downarrow^{\mathrlap{v}} \\ & &C^I & \stackrel{d_0}{\to}& C \\ \downarrow && \downarrow^{\mathrlap{d_1}} \\ A &\stackrel{u}{\to}& C }

where both squares are pullback squares.

By the above lemma on generalized universal bundles, the map E vC→C\mathbf{E}_v C \to C is a fibration. The first claim follows then since fibrations are stable under pullback.

On the other hand we can rewrite the limit diagram also as

A× CC I× CB → B ↓ ↓ v E uC →∈W∩F C I →d 0∈W∩F C ↓ ∈W∩F ↓ d 1∈W∩F A →u C \array{ A \times_C C^I \times_C B &\to& && B \\ \downarrow && && \downarrow^{\mathrlap{v}} \\ \mathbf{E}_u C & \stackrel{\in W \cap F}{\to} &C^I & \stackrel{d_0 \in W \cap F}{\to}& C \\ \downarrow^{\mathrlap{\in W \cap F}} && \;\;\downarrow^{\mathrlap{d_1\in W \cap F}} \\ A &\stackrel{u}{\to} & C }

where again both inner squares are pullback squares.

Again by the above statement on generalized universal bundles, we have that the morphism E uC→C\mathbf{E}_u C \to C is a fibration. By one of the above propositions, weak equivalences are stable under pullback along fibrations, hence the pullback A× CC I× CB→E uCA \times_C C^I \times_C B \to \mathbf{E}_u C of vv is a weak equivalence. Since also E uC→A\mathbf{E}_u C \to A is a weak equivalence (being the pullback of an acyclic fibration) the entire morphism A× CC I× CB→AA \times_C C^I \times_C B \to A is.

Definition

Two morphism f,g:A→Bf,g : A \to B in C(A,B)C(A,B) are

• right homotopic, denoted f≃gf \simeq g, precisely if they fit into a diagram

  B f↗ ↑ d 0 A →η B I g↘ ↓ d 1 B \array{ && B \\ & {}^f\nearrow & \uparrow^{d_0} \\ A &\stackrel{\eta}{\to}& B^I \\ & {}_g\searrow & \downarrow^{\mathrlap{d_1}} \\ && B }

  for some path space object B IB^I;

• homotopic, denoted f∼gf \sim g, if they become right homotopic after pulled back to a weakly equivalent domain, i.e. precisely if they fit into a diagram

A →f B w∈W↗ ↑ d 0 A^ →η B I w∈W↘ ↓ d 1 A →g B \array{ && A &\stackrel{f}{\to}& B \\ &{}^{w \in W}\nearrow&&& \uparrow^{d_0} \\ \hat A && \stackrel{\eta}{\to} && B^I \\ &{}_{w\in W}\searrow & && \downarrow^{d_1} \\ && A &\stackrel{g}{\to}& B }

for some object A^\hat A and for some path space object B IB^I of II

Lemma

For A,B∈CA,B \in \mathbf{C}, right homotopy is an equivalence relation on the hom-set C(A,B)\mathbf{C}(A,B).

Proof

This follows by “piecing path spaces together”:

Let B I 1B^{I_1} and B I 2B^{I_2} be two path space objects of BB. Then the pullback

B I 1∨I 2 → B I 2 ↓ ↓ d 0 B I 1 →d 1 B \array{ B^{I_1 \vee I_2} &\to& B^{I_2} \\ \downarrow && \downarrow^{d_0} \\ B^{I_1} &\stackrel{d_1}{\to}& B }

defines a new path object, with structure maps

B→σ 1×σ 2B I 1∨I 2→(d 0∘p 1)×(d 1∘p 2)B×B. B \stackrel{\sigma_1 \times \sigma_2}{\to} B^{I_1 \vee I_2} \stackrel{(d_0 \circ p_1) \times (d_1\circ p_2)}{\to} B \times B \,.

So given two right homotopies with respect to B I 1B^{I_1} and B i 2B^{i_2} we can paste them next to each other and deduce a homotopy through B I 1∨I 2B^{I_1 \vee I_2}

B f↗ ↑ d 0 1 A →η 1 B I 1 g↘ ↓ d 1 1 ↖ B B I 1∨I 2 g↗ ↓ d 0 2 ↙ A →η 2 B I 2 h↘ ↓ d 1 2 B \array{ && B \\ & {}^f\nearrow & \uparrow^{d_0^1} \\ A &\stackrel{\eta_1}{\to}& B^{I_1} \\ & {}_{g}\searrow& \downarrow^{\mathrlap{d_1^1}} & \nwarrow \\ && B && B^{I_1 \vee I_2} \\ & {}^{g}\nearrow & \downarrow^{\mathrlap{d_0^2}} & \swarrow \\ A &\stackrel{\eta_2}{\to}& B^{I_2} \\ & {}_h\searrow & \downarrow^{\mathrlap{d_1^2}} \\ &&B }

Lemma

Every diagram

A → E ↓ i∈W ↓ p∈F X → B \array{ A &\to& E \\ \;\;\downarrow^{\mathrlap{i \in W}} && \;\;\downarrow^{\mathrlap{p \in F}} \\ X &\to& B }

may be refined to a diagram

A → X′ → E i↘ ↓ t∈W∩F ↓ p∈F X → B \array{ A &\to & X' &\to& E \\ & {}_{i}\searrow & \;\;\downarrow^{\mathrlap{t \in W \cap F}} && \;\;\downarrow^{\mathrlap{p \in F}} \\ && X &\to& B }

Proof

Consider the pullback square

A → X× BE → E i∈W↘ ↓ ∈F ↓ ∈F X → B \array{ A &\to& X \times_B E &\to& E \\ &{}_{i \in W}\searrow& \;\; \downarrow^{\mathrlap{\in F}} && \;\; \downarrow^{\mathrlap{\in F}} \\ && X &\to& B }

and apply the factorization lemma, lemma 5, to factor the universal morphism A→X× BE→EA \to X \times_B E \to E into the pullback as

A→∈WEE→∈FE A \stackrel{\in W}{\to} \mathbf{E} E \stackrel{\in F}{\to} E

to obtain the diagram

A →≃ EE → E i∈W↘ ↓ ∈F ↓ ∈F X → B, \array{ A &\stackrel{\simeq}{\to}& \mathbf{E} E &\to& E \\ &{}_{i \in W}\searrow& \;\; \downarrow^{\mathrlap{\in F}} && \;\; \downarrow^{\mathrlap{\in F}} \\ && X &\to& B } \,,

where the middle vertical morphism is still a fibration, being the composite of two fibrations. By 2-out-of-3 it follows that it is also a weak equivalence.

Lemma

For u:B→C u : B \to C a morphism and B IB^I, C IC^I choices of path objects, there is always another path object B I′B^{I'} with an acyclic fibration B I←∈W∩FB I′B^I \stackrel{\in W \cap F}{\leftarrow} B^{I'} and a span of morphisms of path space objects

B ←= B →u C ↓ σ ↓ σ′ ↓ σ C B I ←∈W∩F B I′ →u¯ C I ↓ d 0×d 1 ↓ d′ 0×d′ 1 ↓ d 0 C×d 1 C B×B ←= B×B →u×u C×C \array{ B &\stackrel{=}{\leftarrow}& B &\stackrel{u}{\to}& C \\ \downarrow^{\mathrlap{\sigma}} && \downarrow^{\sigma'} && \downarrow^{\mathrlap{\sigma_C}} \\ B^I &\stackrel{\in W \cap F}{\leftarrow}& B^{I'} &\stackrel{\bar u}{\to}& C^I \\ \;\;\downarrow^{d_0 \times d_1} && \;\;\downarrow^{\mathrlap{d'_0 \times d'_1}} && \;\;\downarrow^{\mathrlap{d_0^C \times d_1^C}} \\ B \times B &\stackrel{=}{\leftarrow}& B \times B &\stackrel{u \times u}{\to}& C \times C }

Proof

Apply the lemma above to the square

B →u C →σ C C I ↓ σ ↓ d 0×d 1 B I →d 0×d 1 B×B →u×u C×C. \array{ B &\stackrel{u}{\to}& C &\stackrel{\sigma_C}{\to}& C^I \\ \downarrow^{\mathrlap{\sigma}} &&&& \downarrow^{\mathrlap{d_0 \times d_1}} \\ B^I &\stackrel{d_0 \times d_1}{\to}& B \times B &\stackrel{u \times u}{\to}& C \times C } \,.

Proposition

Right homotopy f≃gf \simeq g between morphisms is preserved under pre- and postcomposition with a given morphism.

More precisely, let f,g:B→Cf, g : B \to C be two homotopic morphisms. Then

• for all morphisms A→BA \to B and C→DC \to D the composites A→B→fC→DA \to B \stackrel{f}{\to} C \to D and A→B→gC→DA \to B \stackrel{g}{\to} C \to D are still right homotopic.

• moreover, the right homotopy may be realized with every given choice of
  path space object D ID^I for DD.

Proof

We decompose this into two statements:

1. for any A→BA \to B the morphisms A→B→f,gBA \to B \stackrel{f,g}{\to} B are right homotopic.

2. for any u:C→Du : C \to D and choice D ID^I of path object there is an acyclic fibration B′→B B' \to B such that B′→B→fC→DB' \to B \stackrel{f}{\to} C \to D is right homotopic to B′→B→gC→DB' \to B \stackrel{g}{\to} C \to D by a right homotopy η:B′→D I\eta : B' \to D^I.

The first of these follows trivially.

The second one follows by using the weak functoriality property of path objects from above: let B′:=B× C IC I′B' := B \times_{C^I} C^{I'} be the pullback in the following diagram

B′ → C I′ →u¯ D I ↓ ∈W∩F ↓ ∈W∩F ↓ B →η C I f×g↘ ↓ ↓ C×C →u×u D×D \array{ B' &\to& C^{I'} &\stackrel{\bar u}{\to}& D^I \\ \;\;\;\downarrow^{\mathrlap{\in W \cap F}} && \;\;\;\downarrow^{\mathrlap{\in W \cap F}} && \downarrow \\ B &\stackrel{\eta}{\to}& C^I \\ &{}_{f \times g}\searrow & \downarrow && \downarrow \\ && C \times C &\stackrel{u \times u}{\to}& D \times D }

Lemma

• Every diagram

  B ↓ w∈W A → C \array{ && B \\ && \downarrow^{\mathrlap{w \in W}} \\ A &\to& C }

  in C\mathbf{C} extends to a (right) homtopy-commutative diagram

  A′ → B ↓ w′∈W ↓ w∈W A → C. \array{ A' &\to & B \\ \downarrow^{\mathrlap{w' \in W}} && \downarrow^{\mathrlap{w \in W}} \\ A &\to& C } \,.

• For every pair of morphisms

  f,g,A→→B f, g, A \stackrel{\to}{\to} B

  and weak equivalence t:B→∈WCt : B \stackrel{\in W}{\to} C such that there is a right homotopy t∘f≃t∘gt \circ f \simeq t \circ g, there exists a weak equivalence t′:A′→At' : A' \to A such that f∘t′≃g∘t′f \circ t' \simeq g \circ t'.

Proof

• The first point we accomplish this by letting A′:=A× CC I× CBA' := A \times_C C^I \times_C B be the homotopy fiber product in CC of a representative of the pullback diagram. The lemma about morphisms out of the homotopy fiber product says that A′→AA' \to A is a weak equivalence.

• The second point is more work. Let η:A→C I\eta : A \to C^I the right homotopy in question. We start by considering the homotopy fiber product

  D:=B× CC I× CB → →∈W B ↓ ∈W ↓ t∈W C I →d 0 C ↓ ↓ d 1 B →t∈W C, \array{ D := B \times_C C^I \times_C B &\to&&\stackrel{\in W}{\to}& B \\ \downarrow^{\mathrlap{\in W}} &&&& \downarrow^{\mathrlap{t \in W}} \\ && C^I &\stackrel{d_0}{\to}& C \\ \downarrow && \downarrow^{d_1} \\ B &\stackrel{t \in W}{\to}& C } \,,

  where the long morphisms are weak equivalences by the lemma on morphisms out of homotopy fiber products.

Then consider the two universal morphisms

(f,η,g):A→B× CC I× CB (f,\eta,g) : A \to B \times_C C^I \times_C B

and

(Id,σ∘t,Id):B→∈WB× CC I× CB (Id, \sigma \circ t, Id) : B \stackrel{\in W}{\to} B \times_C C^I \times_C B

into that. It follows by 2-out-of-3 that the latter is a weak equivalence. Factoring this using the factorization lemma produces hence

B→∈WD′→∈W∩FD. B \stackrel{\in W}{\to} D' \stackrel{\in W \cap F}{\to} D \,.

We know moreover that the product map D→∈FB×BD \stackrel{\in F}{\to} B \times B is a fibration, as we can rewrite the homotopy limit as the pullback

D → C I ↓ ↓ ∈F B×B →f×g C×C. \array{ D &\to& C^I \\ \downarrow && \downarrow^{\mathrlap{\in F}} \\ B \times B &\stackrel{f \times g}{\to}& C \times C } \,.

It follows that the composite D′→D→B×BD' \to D \to B \times B is a fibration and hence D′D' a path space object for BB.

Finally, by setting A′=A× DD′A' = A \times_D D' we obtaine the desired right homotopy f∘t′≃g∘t′f \circ t' \simeq g \circ t'.

A′ → D′ ↓ t′ ↓ A → D → C I f×g↘ ↓ ↓ B×B →t×t C×C. \array{ A' &\to& D' \\ \downarrow^{\mathrlap{t'}} && \downarrow \\ A &\to & D &\to & C^I \\ & {}_{f \times g}\searrow & \downarrow && \downarrow \\ && B \times B &\stackrel{t \times t}{\to}& C \times C } \,.

Lemma

The relation “f,g∈C(A,B)f, g \in C(A,B) are homotopic”, f∼gf \sim g, is an equivalence relation on C(A,B)C(A,B).

Proof

The nontrivial part is to show transitivity. This now follows with the above lemma about homtopy commutative composition of pullback diagrams and then using the “piecing together of path objects” used above to show that right homotopy is an equivalence relation.

Definition

For CC a category of fibrant objects the category πC\pi C is defined to be the category

• with the same objects as CC;

• with hom-sets the set of equivalence classes

  πC(A,B):=C(A,B)/ ∼ \pi C(A,B) := C(A,B)/_\sim

  under the above equivalence relation.

• Composition in πC\pi C is given by composition of representatives in CC.

Definition

The obvious functor

C→πC C \to \pi C

is the identity on objects and the projection to equivalence classes on hom-set.

Let πW⊂Mor(πC)\pi W \subset Mor(\pi C) be the image of the weak equivalences of CC in πC\pi C under this functor, and πF\pi F the image of the fibrations.

Theorem

The weak equivalences in πC\pi C form a left multiplicative system.

Proof

This is now a direct consequence of the above lemma on homotopy-commutative completions of diagrams.

Theorem

For CC a category of fibrant objects, its homotopy category is (equivalent to) the category Ho CHo_C with

• the same objects as CC;

• the hom-set Ho C(A,B)Ho_C(A,B) for all A,B∈Obj(C)A, B \in Obj(C) given naturally by

  Ho C(A,B) ≃colim A^→w∈πWAπC(A^,B) =colim A^→f∈πW∩FAπC(A^,B). \begin{aligned} Ho_C(A,B) & \simeq colim_{\hat A \stackrel{w\in \pi W}{\to} A} \pi C (\hat A,B) \\ & = colim_{\hat A \stackrel{f\in \pi W\cap F}{\to} A} \pi C (\hat A,B) \end{aligned} \,.

Lemma

The homotopy categories of CC and πC\pi C coincide:

Ho C≃Ho πC. Ho_C \simeq Ho_{\pi C} \,.

Proof

By one of the lemmas above, the morphisms d i:B I→Bd_i : B^I \to B are weak equivalences and become isomorphisms in Ho CHo_C. The section σ:B→B I\sigma : B \to B^I then becomes an inverse for both of them, hence the images of d 0d_0 and d 1d_1 in Ho CHo_C coincide. Therefore the above diagram says that homotopic morphisms in CC become equal in Ho CHo_C.

But this means that the localization morphism

Q C:C→Ho C Q_C : C \to Ho_C

factors through πC\pi C as

Q C:C→πC→Q πCHo C Q_C : C \to \pi C \stackrel{Q_{\pi C}}{\to} Ho_C

where Q πCQ_{\pi C} sends weak equivalences in πC\pi C to isomorphisms in Ho CHo_C.

The universal property of QQ then implies the universal property for Q πCQ_{\pi C}

C → πC → A ↓ Q C ↙ Q πC ↙ Ho C. \array{ C &\to& \pi C &\to & A \\ \downarrow^{\mathrlap{Q_C}} & \swarrow^{Q_{\pi C}} && \swarrow \\ Ho_C } \,.

Definition

For 𝒞\mathcal{C} a category of fibrant objects, write for any X,A∈Obj(𝒞)X, A \in Obj(\mathcal{C})

Cocycle(X,A),wCocycle(X,A)∈Cat Cocycle(X,A) , wCocycle(X,A) \in Cat

for the categories (“categories of cocycles on XX with coefficients in AA”) whose objects are correspondences

X⟵≃X^⟶A X \stackrel{\simeq}{\longleftarrow} \hat X \stackrel{}{\longrightarrow} A

with the left leg an acyclic fibration (for Cocycle(X,A)Cocycle(X,A)) or just a weak equivalence (for wCocycle(X,A)wCocycle(X,A)); and whose morphisms are morphisms of spans

X^ ↙ ↘ X ↓ A ↖ ↗ X^′ \array{ && \hat X \\ & \swarrow && \searrow \\ X && \downarrow && A \\ & \nwarrow && \nearrow \\ && \hat X' }

Proposition

Write L we H𝒞L^H_{we} \mathcal{C} for the simplicial localization of the category of fibrant objects 𝒞\mathcal{C} at its weak equivalences (hence essentially the (infinity,1)-category that it presents). Then for all objects X,A∈Obj(𝒞)X,A \in Obj(\mathcal{C}) the canonical maps

NCocycle(X,A)→NwCocycle(X,A)→L we H𝒞(X,A) N Cocycle(X,A) \to N wCocycle(X,A) \to L^H_{we} \mathcal{C}(X,A)

of simplicial sets (on the left the nerves of the cocycle categories of def. 8, on the right the derived hom space given by the simplicial localization) are weak homotopy equivalences.

Proof

Observe that for each object XX the 2-functor NCocycle(X,−):𝒞→sSetN Cocycle(X,-) \colon \mathcal{C} \to sSet of def. 8 sends fibrations to Kan fibrations of simplicial sets (the horn-filling condition comes down to factoring maps through the given fibration, which is possible by pullback along the fibration). Moreover, it is evident that NCocycle(X,−)N Cocycle(X,-) preserves ordinary pullbacks. This means that NCocycle(X,−)N Cocycle(X,-) takes pullbacks along a fibration in 𝒞\mathcal{C} to pullbacks in sSet one of whose maps is a Kan fibration. Since the standard model structure on simplicial sets sSet QuillensSet_{Quillen} is a right proper model category, this means that these are homotopy pullbacks (as discussed there) in sSet QuillensSet_{Quillen}. Finally by prop. 3 this means that the derived hom-space functor ℝHom(X,−)\mathbb{R}Hom(X,-) sends pullbacks along fibrations to homotopy pullbacks of the correct derived hom-spaces. This means (as discussed for instance at homotopy Kan extension) that the original pullbacks in 𝒞\mathcal{C} are the correct homotopy pullbacks.