nLab
(infinity,1)-Grothendieck construction

Contents

Idea

The (,1)(\infty,1)-Grothendieck construction is a generalization of the Grothendieck construction – which establishes an equivalence

Fib(C)2Func(C op,Cat) Fib(C) \simeq 2Func(C^{op}, Cat)

and

Fib Grpd(C)2Func(C op,Grpd) Fib_{Grpd}(C) \simeq 2Func(C^{op}, Grpd)

between fibered categories/categories fibered in groupoids and pseudofunctors to Cat/to Grpd – from category theory to (∞,1)-category-theory.

The Grothendieck construction for ∞-groupoids constitutes an equivalence of (∞,1)-categories

RFib(C)Func(C op,Grpd) RFib(C) \simeq \infty Func(C^{op}, \infty Grpd)

between right fibrations of quasi-categories and (∞,1)-functors to ∞ Grpd, while the full Grothendieck construction for (∞,1)-categories constitutes an equivalence

CartFib(C)Func(C op,(,1)Cat) CartFib(C) \simeq \infty Func(C^{op}, (\infty,1)Cat)

between Cartesian fibrations of quasi-categories and (∞,1)-functors to (∞,1)Cat.

This correspondence may be modeled

For fibrations in \infty-groupoids

The generalization of a category fibered in groupoids to quasi-category theory is a right fibration of quasi-categories.

Theorem

((,0)(\infty,0)-Grothendieck construction)

Let CC be an (∞,1)-category. There is an equivalence of (∞,1)-categories

RFib(C)Func(C op,Grpd) RFib(C) \simeq Func(C^{op}, \infty Grpd)

where

In the next section we discuss how this statement is presented in terms of model categories.

Model category presentation

We discuss a presentation of the (,0)(\infty,0)-Grothendieck construction by a simplicial Quillen adjunction between simplicial model categories. (HTT, section 2.2.1).

Definition

(extracting a simplicial presheaf from a fibration)

Let

In particular we will be interested in the case that ϕ\phi is the identity, or at least an equivalence, identifying CC with τ hc(S)\tau_{hc}(S).

For any object (p:XS)(p : X\to S) in sSet/SsSet/S consider the sSet-category K(ϕ,p)K(\phi,p) obtained as the (ordinary) pushout in SSet Cat

τ hc(X) τ hc(X ) ϕ(p) C K(ϕ,p), \array{ \tau_{hc}(X) &\stackrel{}{\to}& \tau_{hc}(X^{\triangleright}) \\ {}^{\mathllap{\phi(p)}}\downarrow && \downarrow \\ C &\to& K(\phi,p) } \,,

where X =X{v}X^{\triangleright} = X \star \{v\} is the join of simplicial sets of XX with a single vertex vv.

Using this construction, define a functor, the straightening functor,

St ϕ:sSet/S[C op,sSet] St_\phi : sSet/S \to [C^{op}, sSet]

from the overcategory of sSet over SS to the enriched functor category of sSet-enriched functors from C opC^{op} to sSetsSet by defining it on objects (p:XS)(p : X \to S) to act as

St ϕ(X):=K(ϕ,p)(,v):C opSSet. St_\phi(X) := K(\phi,p)(-,v) : C^{op} \to SSet \,.
Example

The straightening functor effectively computes the fibers of a Cartesian fibration (p:XC)(p : X \to C) over every point xCx \in C. As an illustration for how this is expressed in terms of morphisms in that pushout, consider the simple situation where

  • C=*C = * only has a single point;

  • X={abc}X = \left\{ a \to b \;\;\; c\right\} is a category with three objects, two of them connected by a morphism

  • p:XCp : X\to C is the only possible functor, sending everything to the point.

Then

  • C ={a b c v}C^{\triangleright} = \left\{ \array{ a &\to& b && c \\ & \searrow \Leftarrow& \downarrow & \swarrow \\ && v } \right\}

and

  • X XC={ v}X^{\triangleright} \coprod_{X} C = \left\{ \array{ && \bullet \\ & \swarrow & \downarrow & \searrow \\ \downarrow& \Leftarrow & \downarrow \\ & \searrow & \downarrow & \swarrow \\ && v } \right\}

Therefore the category of morphisms in this pushout from ** to vv is indeed again the category {abc}\{a \to b \;\;\; c\}.

More on this is at Grothendieck construction in the section of adjoints to the Grothendieck construction.

Proposition

With the definitions as above, let π:CC\pi : C \to C' be an sSet-enriched functor between sSet-categories. Write

π !:[C op,sSet][C op,sSet] \pi_! : [C^{op}, sSet] \to [{C'}^{op}, sSet]

for the left sSet-Kan extension along π\pi.

There is a natural isomorphism of the straightening functor for the composite πϕ\pi \circ \phi and the original straightening functor for ϕ\phi followed by left Kan extension along π\pi:

St πϕπ !St ϕ. St_{\pi \circ \phi} \simeq \pi_! \circ St_\phi \,.

This is HTT, prop. 2.2.1.1.. The following proof has kindly been spelled out by Harry Gindi.

Proof

We unwind what the sSet-categories with a single object adjoined to them look like:

let

F:C opsSet F : C^{op} \to sSet

be an sSet-enriched functor. Define from this a new sSet-category C FC_F by setting

  • Obj(C F)=Obj(C){ν}Obj(C_F) = Obj(C) \coprod \{\nu\}

  • C F(c,d)={C(c,d) forc,dObj(C) F(c) forcObj(c)andd=ν forc=νanddObj(C) * forc=d=νC_F(c,d) = \left\{ \array{ C(c,d) & for c,d \in Obj(C) \\ F(c) & for c \in Obj(c) and d = \nu \\ \emptyset & for c = \nu and d \in Obj(C) \\ * & for c = d = \nu } \right.

The composition operation is that induced from the composition in CC and the enriched functoriality of FF.

Observe that the sSet-category K(ϕ,p)K(\phi,p) appearing in the definition of the straightening functor is

K(ϕ,p)C St ϕ(X) K(\phi,p) \simeq C_{St_\phi(X)}

(because K(ϕ,p)K(\phi,p) is CC with a single object ν\nu and some morphisms to ν\nu adjoined, such that there are no non-degenerate morphisms originating at ν\nu, we have that K(ϕ,p)K(\phi,p) is of form C FC_F for some FF; and St ϕ(X)St_\phi(X) is that FF by definition).

To prove the proposition, we need to compute the pushout

τ hc(X) τ hc(X ) C K(ϕ,p)=C St ϕ(X) π C Q \array{ \tau_{hc}(X) &\to& \tau_{hc}(X^{\triangleright}) \\ \downarrow && \downarrow \\ C &\to& K(\phi,p) = C_{St_\phi(X)} \\ {}^{\mathllap{\pi}}\downarrow && \downarrow \\ C' &\to& Q }

and show that indeed QC π !St ϕ(X)Q \simeq C'_{\pi_! St_\phi(X)}.

Using the pasting law for pushouts (see pullback) we just have to compute the lower square pushout. Here the statement is a special case of the following statement: for every sSet-category of the form C FC_F, the pushout of the canonical inclusion CC FC\to C_F along any sSetsSet-functor π:CC\pi : C \to C' is C π !FC'_{\pi_! F}.

This follows by inspection of what a cocone

C ι C F π d C r Q \array{ C &\stackrel{\iota}{\to}& C_F \\ {}^{\mathllap{\pi}}\downarrow && \downarrow^{\mathrlap{d}} \\ C' &\underset{r}{\to}& Q }

is like: by the nature of C FC_F the functor dd is characterized by a functor d| C:CQd|_C : C \to Q, an object d(ν)Qd(\nu) \in Q together with a natural transformation

F(c)Q(d| C(c),d(ν)) F(c) \to Q(d|_C(c), d(\nu))

being the component F c,ν:C F(c,ν)Q(d(c),d(ν))F_{c,\nu} : C_F(c,\nu) \to Q(d(c), d(\nu)) of the sSetsSet-functor.

We may write this natural transformation as

F(d| C) *Q(,d(ν))=ι *d *νQ(,d(ν)), F \to (d|_C)^* Q(-,d(\nu)) = \iota^* d^* \nu Q(-,d(\nu)) \,,

where d *d^* etc. means precomposition with the functor dd.

By commutativity of the diagram this is

π *r *Q(,d(ν)). \cdots \simeq \pi^* r^* Q(-,d(\nu)) \,.

Now by the definition of left Kan extension π !\pi_! as the left adjoint to prescomposition with a functor, this is bijectively a transformation

η:π !Fr *Q(,d(ν)). \eta : \pi_! F \to r^* Q(-,d(\nu)) \,.

Using this we see that we may find a universal cocone by setting Q:=C π !FQ := C'_{\pi_! F} with r:CQr : C' \to Q the canonical inclusion and C FC π !FC_{F} \to C'_{\pi_! F} given by π\pi on the restriction to CC and by the unit Fπ *π !FF \to \pi^* \pi_! F on C F(c,ν)C_F(c,\nu). For this the adjunct transformation η\eta is the identity, which makes this universal among all cocones.

Proposition

This functor has a right adjoint

Un ϕ:[C op,sSet]sSet/S, Un_\phi : [C^{op}, sSet] \to sSet/S \,,

that takes a simplicial presheaf on CC to a simplicial set over SS – this is the unstraightening functor.

Proof

One checks that St ϕSt_\phi preserves colimits. The claim then follows with the adjoint functor theorem.

Theorem

(presentation of the (,0)(\infty,0)-Grothendieck construction)

The straightening and the unstraightening functor constitute a Quillen adjunction

(St ϕUn ϕ):sSet/SSt ϕUn ϕ[C op,sSet] (St_\phi \dashv Un_\phi) : sSet/S \stackrel{\overset{Un_{\phi}}{\leftarrow}}{\underset{St_\phi}{\to}} [C^{op}, sSet]

between the model structure for right fibrations and the global projective model structure on simplicial presheaves on SS.

If ϕ\phi is a weak equivalence in the model structure on simplicial categories then this Quillen adjunction is a Quillen equivalence.

This is HTT, theorem 2.2.1.2.

This models the Grothendieck construction for ∞-groupoids in the following way:

Hence the unstraightening functor is what models the Grothendieck construction proper, in the sense of a construction that generalizes the construction of a fibered category from a pseudofunctor.

Remark: (,0)(\infty,0)-fibrations over an \infty-groupoid

Observation

Let CC itself be an \infty-groupoid. Then RFib(C)Grpd/CRFib(C) \simeq \infty Grpd/C and hence

Grpd/C[C op,Grpd]. \infty Grpd/C \simeq [C^{op}, \infty Grpd] \,.
Proof

By the fact that there is the standard model structure on simplicial sets we have that every morphism of \infty-groupoids XCX \to C factors as

X X^ fib C, \array{ X &&\stackrel{\simeq}{\to}&& \hat X \\ & \searrow && \swarrow_{\mathrlap{fib}} \\ && C } \,,

where the top morphism is an equivalence and the right morphism a Kan fibration. Moreover, as discussed at right fibration, over an \infty-groupoid the notions of left/right fibrations and Kan fibrations coincide. This shows that the full sub-(∞,1)-category of Grpd/X\infty Grpd/X on the right fibrations is equivalent to all of Grpd/X\infty Grpd/X.

For general fibered (,1)(\infty,1)-categories

The generalization of a fibered category to quasi-category theory is a Cartesian fibration of quasi-categories.

Theorem

((,1)(\infty,1)-Grothendieck construction)

Let CC be an (∞,1)-category. There is an equivalence

Cart(C)Func(C op,(,1)Cat) Cart(C) \simeq Func(C^{op}, (\infty,1) Cat)

where

In the next section we discuss how this statement is presented in terms of model categories.

Model category presentation

Regard the (∞,1)-category CC in its incarnation as a simplicially enriched category.

Let SS be a simplicial set, τ hc(S)\tau_{hc}(S) the corresponding simplicially enriched category (where τ hc\tau_{hc} is the left adjoint of the homotopy coherent nerve) and let ϕ:τ hc(S)C\phi : \tau_{hc}(S) \to C be an sSet-enriched functor.

Definition

(extracting a marked simplicial presheaf from a marked fibration) (HTT, section 3.2.1)

The straightening functor

St ϕ:sSet +/S[C op,sSet +] St_\phi : sSet^+/S \to [C^{op}, sSet^+]

from marked simplicial sets over SS to marked simplicial presheaves on C opC^{op} is on the underlying simplicial sets (forgetting the marking) the same straightening functor as above.

On the markings the functor acts as follows.

Each edge f:def: d \rightarrow e of XsSet/SX \in sSet/S gives rise to an edge f˜St ϕ(X)(d)=K(ϕ,p)(d,v)\tilde f \in St_\phi (X)(d) = K(\phi,p)(d,v): the join 2-simplex fvf \star v of X X^{\triangleright}

d f e d˜ f˜ e˜ v \array{ d && \stackrel{f}{\to} && e \\ & {}_{\mathllap{\tilde d}}\searrow & \stackrel{\tilde f}{\Rightarrow} & \swarrow_{\mathrlap{\tilde e}} \\ && v }

with image f˜:d˜f *e˜\tilde f : \tilde d \to f^* \tilde e in the pushout K(ϕ,p)(d,v)=St ϕX(d)K(\phi,p)(d,v)=St_\phi X(d).

We define the straightening functor to assign that marking of edges which is the minimal one such that all such morphisms f˜\tilde f are marked in St ϕX(d)St_\phi X(d), for all marked f:def : d \to e in XX: this means that this marking is being completed under the constraint that St ϕ(X)St_\phi(X) be sSet-enriched functorial.

For that, recall that the hom simplicial sets of sSet +sSet^+ are the spaces Map (X,Y)Map^\sharp(X,Y), which consist of those simplices of the internal hom Map(X,Y):=Y XMap(X,Y) := Y^X whose edges are all marked:

Map(X,Y) n=Hom sSet +(X×Δ[n] #,Y). Map(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^#, Y) \,.

So we need to find a marking on the St ϕ(X)()St_\phi(X)(-) such that for all g:Δ[1]C(c,d)g : \Delta[1] \to C(c,d) the composite

Δ[1]gC(c,d)St ϕ(X)(c,d)Map(St ϕ(X)(d),St ϕ(X)(c)) \Delta[1] \stackrel{g}{\to} C(c,d) \stackrel{St_\phi(X)(c,d)}{\to} Map(St_\phi(X)(d), St_\phi(X)(c))

is a marked edge of the mapping complex. By the internal hom-adjunction this edge corresponds to a morphism

St ϕ(X)(g):St ϕ(X)(d)×Δ[1]St ϕ(X)(c) St_\phi(X)(g) : St_\phi(X)(d) \times \Delta[1] \rightarrow St_\phi(X)(c)

and to be marked needs to carry edges of the form f˜×{01}\tilde f \times \{0 \to 1\} i.e. 1-cells (f˜,Id):Δ[1]St ϕ(X)(d)×Δ[1](\tilde f , Id) : \Delta[1] \to St_\phi(X)(d) \times \Delta[1] to marked edges

g *f˜:Δ[1](f˜,Id)St ϕ(X)(d)×Δ[1]St ϕ(X)(g)St ϕ(X)(c) g^* \tilde f : \Delta[1] \stackrel{(\tilde f,Id)}{\to} St_\phi(X)(d)\times \Delta[1] \stackrel{St_\phi(X)(g)}{\to} St_{\phi}(X)(c)

in St ϕ(X)(c)St_\phi(X)(c). So we need to ensure that the edges of this form are marked:

we define that the straightening functor marks an edge in St ϕ(X)(c)St_\phi(X)(c) iff it is of this form g *f˜g^* \tilde f, for f:def : d \to e a marked edge of XX and gC(c,d) 1g \in C(c,d)_1.

As in the unmarked cae, the straightening functor has an sSet-right adjoint, the unstraightening functor

n ϕ:[C op,sSet +]sSet +/S. n_\phi : [C^{op}, sSet^+] \to sSet^+/S \,.

This functor Un ϕUn_\phi exhibits the (,1)(\infty,1)-Grothendieck-construction proper, in that it constructs a Cartesian fibration from a given (,1)(\infty,1)-functor:

Theorem

(presentation of (,1)(\infty,1)-Grothendieck construction)

This induces a Quillen adjunction

(St ϕUn ϕ):SSet +/SSt ϕUn ϕ[C op,SSet +] (St_\phi \dashv Un_\phi) : SSet^+/S \stackrel{\overset{Un_{\phi}}{\leftarrow}}{\underset{St_\phi}{\to}} [C^{op}, SSet^+]

between the model structure for Cartesian fibrations and the projective global model structure on functors with values in the model structure on marked simplicial sets.

If ϕ\phi is an equivalence in the model structure on simplicial categories then this Quillen adjunction is a Quillen equivalence.

Proof

This is HTT, theorem 3.2.0.1.

Over an ordinary category

In the case that CC happens to be an ordinary category, the (,1)(\infty,1)-Grothendieck construction can be simplified and given more explicitly.

This is HTT, section 3.2.5.

Definition

(relative nerve functor)

Let CC be a small category and let f:CsSetf : C \to sSet be a functor. The simplicial set N f(C)N_f(C), the relative nerve of CC under ff is defined as follows:

an nn-cell of N f(C)N_f(C) is

  1. a functor σ:[n]C\sigma : [n] \to C;

  2. for every [k][n][k] \subset [n] a morphism τ(k):Δ[k]f(σ(k))\tau(k) : \Delta[k] \to f(\sigma(k));

  3. such that for all [j][k][n][j] \subset [k] \subset [n] the diagram

    Δ[j] τ(j) f(σ(j)) f(σ(jk)) Δ[k] τ(k) f(σ(k)) \array{ \Delta[j] &\stackrel{\tau(j)}{\to}& f(\sigma(j)) \\ \downarrow && \downarrow^{\mathrlap{f(\sigma(j\to k))}} \\ \Delta[k] &\stackrel{\tau(k)}{\to}& f(\sigma(k)) }

    commutes.

There is a canonical morphism

N f(C)N(C) N_f(C) \to N(C)

to the ordinary nerve of CC, obtained by forgetting the τ\taus.

This is HTT, def. 3.2.5.2.

Remark

When ff is constant on the point, then N f(C)N(C)N_f(C) \to N(C) is an isomorphism of simplicial sets, so N f(C)N_f(C) this is the ordinary nerve of CC.

The fiber of N f(C)N(C)N_f(C) \to N(C) over an object cCc \in C is given by taking σ\sigma to be constant on CC. Then all the τ\taus are fixed by the maximal τ(n):Δ[n]f(c)\tau(n) : \Delta[n] \to f(c). So the fiber of N f(C)N_f(C) over cc is f(c)f(c).

Definition

(marked relative nerve functor)

Let CC be a small category. Define a functor

sSet +/N(C)[C,sSet +]:N + sSet^+/N(C) \leftarrow [C, sSet^+] : N^+

by

(CFsSet +)(N f(C),E F), (C \stackrel{F}{\to} sSet^+) \mapsto (N_f(C), E_F) \,,

where f:C opFsSet +sSetf : C^{op} \stackrel{F}{\to} sSet^+ \to sSet is FF with the marking forgotten, where N f(C)N_f(C) is the relative nerve of CC under ff, and where the marking E FE_F is given by …

This is HTT, def. 3.2.5.12.

This functor has a left adjoint +\mathcal{F}^+. The value of +(F)\mathcal{F}^+(F) on some cCc \in C is equivalent to the value of St(F)St(F).

This is HTT, Lemma 3.2.5.17.

Proposition

((,1)(\infty,1)-Grothendieck construction over a category)

The adjunction

( +N +):sSet /N(C) +N + +[C,sSet +]. (\mathcal{F}^+ \dashv N^+) : sSet^+_{/N(C)} \stackrel{\overset{\mathcal{F}^+}{\to}}{\underset{N^+}{\leftarrow}} [C,sSet^+] \,.

is a Quillen equivalence between the model structure for coCartesian fibrations and the projective global model structure on functors.

Proof

This is HTT, prop. 3.2.5.18.

Relation beween the model structures

Theorem (HTT, section 3.1.5)

Let SS be a simplicial set.

There is a sequence of Quillen adjunctions

(sSet/S) JoyalsSet +/S(sSet +/S) loc(sSet/S) rfib(sSet/S) Quillen. (sSet/S)_{Joyal} \stackrel{\overset{}{\to}}{\overset{}{\leftarrow}} sSet^+/S \stackrel{\overset{}{\to}}{\overset{}{\leftarrow}} (sSet^+/S)^{loc} \stackrel{\overset{}{\to}}{\overset{}{\leftarrow}} (sSet/S)_{rfib} \stackrel{\overset{}{\to}}{\overset{}{\leftarrow}} (sSet/S)_{Quillen} \,.

Where from left to right we have

  1. the model structure on an overcategory for the Joyal model structure for quasi-categories;

  2. the model structure for Cartesian fibrations;

  3. some localizaton of the model structure for Cartesian fibrations;

  4. the model structure for right fibrations

  5. the model structure on an overcategory for the Quillen model structure on simplicial sets;

The first and third Quillen adjunction here is a Quillen equivalence if SS is a Kan complex.

Examples

Cartesian fibrations over the point

For the base category SS being the point S=*S = {*}, the (,1)(\infty,1)-Grothendieck construction naturally becomes essentially trivial. However, its model by the Quillen functor

St ϕ:sSet/*sSet[*,sSet]sSet St_\phi : sSet/* \simeq sSet \to [*,sSet] \simeq sSet

is not entirely trivial and in fact produces a Quillen auto-equivalence of sSet QuillensSet_{Quillen} with itself that plays a central role in the proof of the corresponding Quillen equivalence over general SS.

Definition

Let Q:ΔsSetQ : \Delta \to sSet be the cosimplicial simplicial set given by

Q[n]:=|J n|(x,v), Q[n] := |J^n|(x,v) \,,

where

J n=C (Δ[n]{x}). J^n = C^{\triangleleft}(\Delta[n] \to \{x\}) \,.

Then: nerve and realization associated to QQ yield a Quillen equivalence of sSet QuillensSet_{Quillen} with itself.

HTT, section 2.2.2.

Cartesian fibrations over the interval

A Cartesian fibration p:KΔ[1]p : K \to \Delta[1] over the 1-simplex corresponds to a morphism Δ[1] op\Delta[1]^{op} \to (∞,1)Cat, hence to an (∞,1)-functor F:DCF : D \to C.

By the above procedure we can express FF as the image of pp under the straightening functor. However, there is a more immediate way to extract this functor, which we now describe.

First recall the situation for the ordinary Grothendieck construction: given a Grothendieck fibration K{01}K \to \{0 \to 1\}, we obtain a functor f:K 1K 0f : K_1 \to K_0 between the fibers, by choosing for each object dK 1d \in K_1 a Cartesian morphism e dde_d \to d. Then the universal property of Cartesian morphism yields for every morphism d 1d 2d_1 \to d_2 in K 1K_1 the unique left vertical filler in

e d 1 d 1 e d 2 d 2. \array{ e_{d_1} &\to& d_1 \\ \downarrow && \downarrow \\ e_{d_2} &\to& d_2 } \,.

And again by universality, this assignment is functorial: K 1K 0K_1 \to K_0.

Diagrammatically, the choice of Cartesian morphisms here is a lift ee in the diagram

K 1 K e K 1×{01} {01}. \array{ K_1 &\hookrightarrow& K \\ \downarrow &\nearrow_e& \downarrow \\ K_1 \times \{0 \to 1\} &\to& \{0 \to 1\} } \,.

This diagrammatic way of encoding the functor associated to a Grothendieck fibration over the interval generalizes straightforwardly to the quasi-category context.

Definition

Given a Cartesian fibration p:KΔ[1]p : K \to \Delta[1] with fibers the quasi-categories C:=K 0C := K_{0} and D:=K 1D := K_{1}, an (,1)(\infty,1)-functor associated to the Cartesian fibration pp is a functor f:DCf : D \to C such that there exists a commuting diagram in sSet

D×Δ[1] F K p Δ[1] \array{ D \times \Delta[1] &&\stackrel{F}{\to}&& K \\ & \searrow && \swarrow_{\mathrlap{p}} \\ && \Delta[1] }

such that

  • F| 1=Id DF|_{1} = Id_D;

  • F| 0=fF|_{0} = f;

  • and for all dDd \in D, F({d}×{01})F(\{d\}\times \{0 \to 1\}) is a Cartesian morphism in KK.

More generally, if we also specify possibly nontrivial equivalences of quasi-categories h 0:CK 0h_0 : C \stackrel{\simeq}{\to} K_{0} and h 1:DK 1h_1 : D \stackrel{\simeq}{\to} K_{1}, then a functor is associated to KK and this choice of equivalences if the first twoo conditions above are generalized to

  • F| 1=h 1F|_{1} = h_1;

  • F| 0=h 0fF|_{0} = h_0 \circ f;

This is HTT, def. 5.2.1.1.

Proposition

For p:KΔ[1]p : K \to \Delta[1] a Cartesian fibration, the associated functor exists and is unique up to equivalence in the (∞,1)-category of (∞,1)-functors Func(K 0,K 1)Func(K_{0}, K_{1}).

Proof

This is HTT, prop 5.2.1.5.

Set C:=K 0C := K_{0} and D:=K 1D := K_{1}.

With the notation described at model structure for Cartesian fibrations, consider the commuting diagram

D ×{1} K p D ×Δ[1] # Δ[1] # \array{ D^\flat \times \{1\} &\hookrightarrow& K^{\sharp} \\ \downarrow && \downarrow^{\mathrlap{p}} \\ D^{\flat} \times \Delta[1]^{#} &\to& \Delta[1]^# }

in the category sSet +sSet^+ of marked simplicial sets.

Here the left vertical morphism is marked anodyne: it is the smash product of the marked cofibration (monomorphism) Id:D D Id : D^\flat \to D^\flat with the marked anodyne morphism Δ[1] #Δ[0]\Delta[1]^# \to \Delta[0]. By the stability properties discussed at Marked anodyne morphisms, this implies that the morphism itself is marked anodyne.

As discussed there, this means that a lift d:D ×Δ[1] #K d : D^\flat \times \Delta[1]^# \to K^{\sharp} against the Cartesian fibration in

D ×{1} K s p D ×Δ[1] # Δ[1] # \array{ D^\flat \times \{1\} &\hookrightarrow& K^{\sharp} \\ \downarrow &\nearrow_{s}& \downarrow^{\mathrlap{p}} \\ D^{\flat} \times \Delta[1]^{#} &\to& \Delta[1]^# }

exists. This exhibits an associated functor f:=s 0f := s_0.

Suppose now that another associated functor ff' is given. It will correspondingly come with its diagram

D ×{1} K s p D ×Δ[1] # Δ[1] #. \array{ D^\flat \times \{1\} &\hookrightarrow& K^{\sharp} \\ \downarrow &\nearrow_{s'}& \downarrow^{\mathrlap{p}} \\ D^{\flat} \times \Delta[1]^{#} &\to& \Delta[1]^# } \,.

Together this may be arranged to a diagram

D ×Λ[2] 2 (s,s) K q p D ×Δ[2] # Δ[1] #, \array{ D^\flat \times \Lambda[2]_2 &\stackrel{(s,s')}{\to}& K^{\sharp} \\ \downarrow &\nearrow_{q}& \downarrow^{\mathrlap{p}} \\ D^{\flat} \times \Delta[2]^{#} &\to& \Delta[1]^# } \,,

where the top horizontal morphism picks the 2-horn in KK whose two edges are labeled by ss and ss', respectively.

Now, the left vertical morphism is still marked anodyne, and hence the lift kk exists, as indicated. Being a morphism of marked simplicial sets, it must map for each dDd \in D the edge {d}×{01}\{d\}\times \{0\to 1\} to a Cartesian morphism in KK, and due to the commutativity of the diagram this morphism must be in K 0K_0, sitting over {0}\{0\}. But as discussed there, a Cartesian morphism over a point is an equivalence. This means that the restriction

k| D×{01}K 0 k|_{D \times \{0 \to 1\}} \to K_0

is an invertible natural transformation between ff and ff', hence these are equivalent in the functor category.

Conversely, every functor f:DCf : D \to C gives rise to a Cartesian fibration that it is associated to, in the above sense.

Proposition

Every (,1)(\infty,1)-functor f:DCf : D \to C is associated to some Cartesian fibration p:KΔ[1]p : K \to \Delta[1], and this is unique up to equivalence.

Proof

This is HTT, prop 5.2.1.3.

The idea is that we obtain KK from first forming the cylinder D×Δ[1]D \times \Delta[1] and the identifying the left boundary of that with CC, using ff.

Formally this means that we form the pushout

N:=(D ×Δ[1] #) D ×{0} #C N := (D^\sharp \times \Delta[1]^#) \coprod_{D^\sharp \times \{0\}^#} C^\sharp

in sSet +sSet^+, where C C^\sharp and D D^\sharp are CC and DD with precisely the equivalences marked. This comes canonically with a morphism

NΔ[1] N \to \Delta[1]

and does have the property that N 0=CN_0 = C, N 1=DN_1 = D and that ff is associated to it in that the restriction of the canonical morphism D×Δ[1]KD \times \Delta[1] \to K to the 0-fiber is ff. But it may fail to be a Cartesian fibration.

To fix this, use the small object argument to factor NΔ[1]N \to \Delta[1] as

NKΔ[1] #, N \to K \to \Delta[1]^# \,,

where the first morphism is marked anodyne and the second has the right lifting property with respect to all marked anodyne morphisms and is hence (since every morphism in Δ[1] #\Delta[1]^# is marked) a Cartesian fibration.

It then remains to check that ff is still associated to this KΔ[1] #K \to \Delta[1]^#. This is done by observing that in the small object argument KK is built succesively from pushouts of the form

A N α B N α+1 Δ[1], \array{ A &\to& N_\alpha \\ \downarrow && \downarrow & \searrow \\ B &\to& N_{\alpha+1} &\to& \Delta[1] } \,,

where the morphisms on the left are the generators of marked anodyne morphisms (see here). from this one checks that if the fiber N α× Δ[1]{0}N_\alpha \times_{\Delta[1]} \{0\} is equivalent to CC, then so is N α+1× Δ[1]{0}N_{\alpha +1} \times_{\Delta[1]} \{0\} and similarly for DD. By induction, it follows that ff is indeed associated to KΔ[1]K \to \Delta[1].

To see that the KK obtained this way is unique up to equivalence, consider…

Cartesian fibrations over simplices

… for the moment see HTT, section 3.2.2

The universal Cartesian fibration

for the moment see

References

The construction for \infty-groupoid fibrations i.e. left/right fibrations is the content of section 2.2.1, that of quasi-category fibrations i.e. Cartesian fibrations that

More on model-category theoretic construction of the \infty-Grothendieck construction with values in \infty-groupoids is in

Revised on March 4, 2014 20:03:27 by Urs Schreiber (89.204.153.222)