# nLab (infinity,1)-Grothendieck construction

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## Idea

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

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

and

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

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

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

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

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

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

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

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

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

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 $(\infty,0)$-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

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

• $C$ an SSet-category;

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

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

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

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

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

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

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

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

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

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

• $C = *$ only has a single point;

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

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

Then

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

and

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

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

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

###### Proposition

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

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

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

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

$St_{\pi \circ \phi} \simeq \pi_! \circ St_\phi \,.$

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

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

be an sSet-enriched functor. Define from this a new sSet-category $C_F$ by setting

• $Obj(C_F) = Obj(C) \coprod \{\nu\}$

• $C_F(c,d) = \left\{ \array{ C(c,d) & for c,d \in Obj(C) \\ F(c) & for c \in Obj(c) and d = \nu \\ \emptyset & for c = \nu and d \in Obj(C) \\ * & for c = d = \nu } \right.$

The composition operation is that induced from the composition in $C$ and the enriched functoriality of $F$.

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

$K(\phi,p) \simeq C_{St_\phi(X)}$

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

To prove the proposition, we need to compute the pushout

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

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

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 $C_F$, the pushout of the canonical inclusion $C\to C_F$ along any $sSet$-functor $\pi : C \to C'$ is $C'_{\pi_! F}$.

This follows by inspection of what a cocone

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

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

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

being the component $F_{c,\nu} : C_F(c,\nu) \to Q(d(c), d(\nu))$ of the $sSet$-functor.

We may write this natural transformation as

$F \to (d|_C)^* Q(-,d(\nu)) = \iota^* d^* \nu Q(-,d(\nu)) \,,$

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

By commutativity of the diagram this is

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

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

$\eta : \pi_! F \to r^* Q(-,d(\nu)) \,.$

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

###### Proposition

This functor has a right adjoint

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

that takes a simplicial presheaf on $C$ to a simplicial set over $S$ – this is the unstraightening functor.

###### Proof

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

###### Theorem

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

The straightening and the unstraightening functor constitute a Quillen adjunction

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

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

If $\phi$ 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: $(\infty,0)$-fibrations over an $\infty$-groupoid

###### Observation

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

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

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

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

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 $\infty Grpd/X$ on the right fibrations is equivalent to all of $\infty Grpd/X$.

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

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

###### Theorem

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

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

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

where

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

### Model category presentation

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

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

###### Definition

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

The straightening functor

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

from marked simplicial sets over $S$ to marked simplicial presheaves on $C^{op}$ 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 $f: d \rightarrow e$ of $X \in sSet/S$ gives rise to an edge $\tilde f \in St_\phi (X)(d) = K(\phi,p)(d,v)$: the join 2-simplex $f \star v$ of $X^{\triangleright}$

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

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

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

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

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

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

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

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

$St_\phi(X)(g) : St_\phi(X)(d) \times \Delta[1] \rightarrow St_\phi(X)(c)$

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

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

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

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

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

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

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

###### Theorem

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

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

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 $\phi$ 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 $C$ happens to be an ordinary category, the $(\infty,1)$-Grothendieck construction can be simplified and given more explicitly.

This is HTT, section 3.2.5.

###### Definition

(relative nerve functor)

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

an $n$-cell of $N_f(C)$ is

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

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

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

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

commutes.

There is a canonical morphism

$N_f(C) \to N(C)$

to the ordinary nerve of $C$, obtained by forgetting the $\tau$s.

This is HTT, def. 3.2.5.2.

###### Remark

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

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

###### Definition

(marked relative nerve functor)

Let $C$ be a small category. Define a functor

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

by

$(C \stackrel{F}{\to} sSet^+) \mapsto (N_f(C), E_F) \,,$

where $f : C^{op} \stackrel{F}{\to} sSet^+ \to sSet$ is $F$ with the marking forgotten, where $N_f(C)$ is the relative nerve of $C$ under $f$, and where the marking $E_F$ is given by …

This is HTT, def. 3.2.5.12.

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

This is HTT, Lemma 3.2.5.17.

###### Proposition

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

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

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 $S$ be a simplicial set.

There is a sequence of Quillen adjunctions

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

Where from left to right we have

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

2. 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 $S$ is a Kan complex.

## Proerties

### As an (op)lax $\infty$-colomit

The $(\infty,1)$-Grothendieck construction on an $\infty$-functor is equivalently its lax (infinity,1)-colimit (Gepner-Haugseng-Nikolaus 15).

## Examples

### Cartesian fibrations over the point

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

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

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

Definition

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

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

where

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

Then: nerve and realization associated to $Q$ yield a Quillen equivalence of $sSet_{Quillen}$ with itself.

### Cartesian fibrations over the interval

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

By the above procedure we can express $F$ as the image of $p$ 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 $K \to \{0 \to 1\}$, we obtain a functor $f : K_1 \to K_0$ between the fibers, by choosing for each object $d \in K_1$ a Cartesian morphism $e_d \to d$. Then the universal property of Cartesian morphism yields for every morphism $d_1 \to d_2$ in $K_1$ the unique left vertical filler in

$\array{ e_{d_1} &\to& d_1 \\ \downarrow && \downarrow \\ e_{d_2} &\to& d_2 } \,.$

And again by universality, this assignment is functorial: $K_1 \to K_0$.

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

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

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

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

such that

• $F|_{1} = Id_D$;

• $F|_{0} = f$;

• and for all $d \in D$, $F(\{d\}\times \{0 \to 1\})$ is a Cartesian morphism in $K$.

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

• $F|_{1} = h_1$;

• $F|_{0} = h_0 \circ f$;

This is HTT, def. 5.2.1.1.

###### Proposition

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

###### Proof

This is HTT, prop 5.2.1.5.

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

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

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

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

Here the left vertical morphism is marked anodyne: it is the smash product of the marked cofibration (monomorphism) $Id : D^\flat \to D^\flat$ with the marked anodyne morphism $\Delta[1]^# \to \Delta[0]$. 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 $d : D^\flat \times \Delta[1]^# \to K^{\sharp}$ against the Cartesian fibration in

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

exists. This exhibits an associated functor $f := s_0$.

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

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

Together this may be arranged to a diagram

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

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

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

$k|_{D \times \{0 \to 1\}} \to K_0$

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

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

###### Proposition

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

###### Proof

This is HTT, prop 5.2.1.3.

The idea is that we obtain $K$ from first forming the cylinder $D \times \Delta[1]$ and the identifying the left boundary of that with $C$, using $f$.

Formally this means that we form the pushout

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

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

$N \to \Delta[1]$

and does have the property that $N_0 = C$, $N_1 = D$ and that $f$ is associated to it in that the restriction of the canonical morphism $D \times \Delta[1] \to K$ to the 0-fiber is $f$. But it may fail to be a Cartesian fibration.

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

$N \to K \to \Delta[1]^# \,,$

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 $\Delta[1]^#$ is marked) a Cartesian fibration.

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

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

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

To see that the $K$ 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

Discussion in terms of lax (infinity,1)-colimits is in

Revised on February 25, 2015 14:56:22 by Urs Schreiber (195.113.30.252)