(infinity,1)-Grothendieck construction in nLab

Context

(∞,1)(\infty,1)-Category theory

• Idea
• For fibrations in ∞\infty-groupoids
• Model category presentation
• For general fibered (∞,1)(\infty,1)-categories
• Model category presentation
• Over an ordinary category
• Relation beween the model structures
• Properties
• As an (op)lax ∞\infty-colimit
• Examples
• Cartesian fibrations over the point
• Cartesian fibrations over the interval
• Cartesian fibrations over simplices
• The universal Cartesian fibration
• Related concepts
• References

Proposition

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

Let 𝒞\mathcal{C} be a small (∞,1)-category. The operation of homotopy pullback of the universal fibration of ∞ \infty -groupoids constitutes an equivalence of (∞,1)-categories

Func(𝒞 op,∞Grpd)≃RFib(C) Func\big(\mathcal{C}^{op}, \infty Grpd\big) \;\simeq\; RFib(C)

• from the (∞,1)-category of (∞,1)-functors from the opposite category 𝒞 op\mathcal{C}^{op} to ∞Grpd, i.e. the (∞,1)-category of (∞,1)-presheaves on CC;

• to the (∞,1)-category of right fibrations RFib(C)RFib(C) – incarnated as the full SSet-subcategory of the overcategory SSet/CSSet/C on right fibrations;

Proposition

((∞,0)(\infty,0)-fibrations over an ∞\infty-groupoid)
If CC itself be an ∞ \infty -groupoid, then RFib(C)≃∞Grpd/CRFib(C) \simeq \infty Grpd/C is equivalently the slice ∞ \infty -category of ∞Grpd over CC, and hence the above Prop. reduces to

∞Grpd/C≃Func(C op,∞Grpd). \infty Grpd/C \;\simeq\; Func\big(C^{op}, \infty Grpd\big) \,.

Proof

By the fact that there is the classical model structure on simplicial sets we have that every morphism of ∞\infty-groupoids X→CX \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.

Definition

(extracting a simplicial presheaf from a fibration)

Let

• SS be a simplicial set, τ hc(S)\tau_hc(S) the corresponding SSet-category (under the left adjoint τ hc:SSet→SSetCat\tau_{hc} : SSet \to SSet Cat of the homotopy coherent nerve, denoted ℭ\mathfrak{C} in HTT);

• CC an SSet-category;

• ϕ:τ hc(S)→C\phi : \tau_{hc}(S) \to C a morphism of SSet-categories.

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:X→S)(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:X→S)(p : X \to S) to act as

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

Example

The straightening functor effectively computes the fibers of a Cartesian fibration (p:X→C)(p : X \to C) over every point x∈Cx \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={a→bc}X = \left\{ a \to b \;\;\; c\right\} is a category with three objects, two of them connected by a morphism

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

Then

• X ▹={a → b c ↘⇐ ↓ ↙ v}X^{\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 {a→bc}\{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:C→C′\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 π∘ϕ\pi \circ \phi and the original straightening functor for ϕ\phi followed by left Kan extension along π\pi:

St π∘ϕ≃π !∘St ϕ. St_{\pi \circ \phi} \simeq \pi_! \circ St_\phi \,.

Proof

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

let

F:C op→sSet 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,d∈Obj(C) F(c) forc∈Obj(c)andd=ν ∅ forc=νandd∈Obj(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 Q≃C′ 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 C→C FC\to C_F along any sSetsSet-functor π:C→C′\pi \colon 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:C→Qd|_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 !F→r *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:C′→Qr : C' \to Q the canonical inclusion and C F→C′ π !FC_{F} \to C'_{\pi_! F} given by π\pi on the restriction to CC and by the unit F→π *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.

Theorem

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

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

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

• on the left we have the (∞,1)(\infty,1)-category of Cartesian fibrations over CC with small fibers and cartesian functors between them – incarnated as a sSet-subcategory of the overcategory sSet/CsSet/C on Cartesian fibrations;

• and on the right the (∞,1)-category of (∞,1)-functors from C opC^{op} to the (∞,1)-category of (∞,1)-categories.

Furthermore, this is equivalence is natural for CC, where Cart(−)Cart(-) acts by taking pullbacks and Func(− op,(∞,1)Cat)Func(-^{op}, (\infty,1) Cat) acts by composition.

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:d→ef: d \rightarrow e of X∈sSet/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 f⋆vf \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:d→ef : 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˜×{0→1}\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:d→ef : d \to e a marked edge of XX and g∈C(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 \,.

Definition

(relative nerve functor)

Let CC be a small category and let f:C→sSetf : 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(j→k)) 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.

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

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

where f:C op→FsSet +→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 …

Definition

Given a Cartesian fibration p:K→Δ[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:D→Cf : 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 d∈Dd \in D, F({d}×{0→1})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:C→≃K 0h_0 : C \stackrel{\simeq}{\to} K_{0} and h 1:D→≃K 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 0∘fF|_{0} = h_0 \circ f;

Proposition

For p:K→Δ[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 f′f' 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 s′s', 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 d∈Dd \in D the edge {d}×{0→1}\{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×{0→1}→K 0 k|_{D \times \{0 \to 1\}} \to K_0

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

Proposition

Every (∞,1)(\infty,1)-functor f:D→Cf : D \to C is associated to some Cartesian fibration p:K→Δ[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^\natural \times \Delta[1]^#) \coprod_{D^\natural \times \{0\}^#} C^\natural

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

N→Δ[1] ♯ N \to \Delta[1]^{\sharp}

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 N→Δ[1]N \to \Delta[1] as

N→K→Δ[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 K→Δ[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 K→Δ[1]K \to \Delta[1].

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