# nLab (infinity,1)-Grothendieck construction

### Context

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

(∞,1)-category theory

# Contents

## Idea

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

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

and

${\mathrm{Fib}}_{\mathrm{Grpd}}\left(C\right)\simeq 2\mathrm{Func}\left({C}^{\mathrm{op}},\mathrm{Grpd}\right)$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

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

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

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

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

$\mathrm{RFib}\left(C\right)\simeq \mathrm{Func}\left({C}^{\mathrm{op}},\infty \mathrm{Grpd}\right)$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 $\left(\infty ,0\right)$-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, ${\tau }_{\mathrm{hc}}\left(S\right)$ the corresponding SSet-category (under the left adjoint ${\tau }_{\mathrm{hc}}:\mathrm{SSet}\to \mathrm{SSet}\mathrm{Cat}$ of the homotopy coherent nerve, denoted $ℭ$ in HTT);

• $C$ an SSet-category;

• $\varphi :{\tau }_{\mathrm{hc}}\left(S\right)\to C$ a morphism of SSet-categories.

In particular we will be interested in the case that $\varphi$ is the identity, or at least an equivalence, identifying $C$ with ${\tau }_{\mathrm{hc}}\left(S\right)$.

For any object $\left(p:X\to S\right)$ in $\mathrm{sSet}/S$ consider the sSet-category $K\left(\varphi ,p\right)$ obtained as the (ordinary) pushout in SSet Cat

$\begin{array}{ccc}{\tau }_{\mathrm{hc}}\left(X\right)& \stackrel{}{\to }& {\tau }_{\mathrm{hc}}\left({X}^{▹}\right)\\ {}^{\varphi \left(p\right)}↓& & ↓\\ C& \to & K\left(\varphi ,p\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ \tau_{hc}(X) &\stackrel{}{\to}& \tau_{hc}(X^{\triangleright}) \\ {}^{\mathllap{\phi(p)}}\downarrow && \downarrow \\ C &\to& K(\phi,p) } \,,

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

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

${\mathrm{St}}_{\varphi }:\mathrm{sSet}/S\to \left[{C}^{\mathrm{op}},\mathrm{sSet}\right]$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 ${C}^{\mathrm{op}}$ to $\mathrm{sSet}$ by defining it on objects $\left(p:X\to S\right)$ to act as

${\mathrm{St}}_{\varphi }\left(X\right):=K\left(\varphi ,p\right)\left(-,v\right):{C}^{\mathrm{op}}\to \mathrm{SSet}\phantom{\rule{thinmathspace}{0ex}}.$St_\phi(X) := K(\phi,p)(-,v) : C^{op} \to SSet \,.
###### Example

The straightening functor effectively computes the fibers of a Cartesian fibration $\left(p:X\to C\right)$ 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;

• $X=\left\{a\to b\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}c\right\}$ 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

• ${C}^{▹}=\left\{\begin{array}{ccccc}a& \to & b& & c\\ & ↘⇐& ↓& ↙\\ & & v\end{array}\right\}$

and

• ${X}^{▹}{\coprod }_{X}C=\left\{\begin{array}{ccc}& & •\\ & ↙& ↓& ↘\\ ↓& ⇐& ↓\\ & ↘& ↓& ↙\\ & & v\end{array}\right\}$

Therefore the category of morphisms in this pushout from $*$ to $v$ is indeed again the category $\left\{a\to b\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}c\right\}$.

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

###### Proposition

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

${\pi }_{!}:\left[{C}^{\mathrm{op}},\mathrm{sSet}\right]\to \left[{C\prime }^{\mathrm{op}},\mathrm{sSet}\right]$\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 $\pi \circ \varphi$ and the original straightening functor for $\varphi$ followed by left Kan extension along $\pi$:

${\mathrm{St}}_{\pi \circ \varphi }\simeq {\pi }_{!}\circ {\mathrm{St}}_{\varphi }\phantom{\rule{thinmathspace}{0ex}}.$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

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

be an sSet-enriched functor. Define from this a new sSet-category ${C}_{F}$ by setting

• $\mathrm{Obj}\left({C}_{F}\right)=\mathrm{Obj}\left(C\right)\coprod \left\{\nu \right\}$

• ${C}_{F}\left(c,d\right)=\left\{\begin{array}{cc}C\left(c,d\right)& \mathrm{for}c,d\in \mathrm{Obj}\left(C\right)\\ F\left(c\right)& \mathrm{for}c\in \mathrm{Obj}\left(c\right)\mathrm{and}d=\nu \\ \varnothing & \mathrm{for}c=\nu \mathrm{and}d\in \mathrm{Obj}\left(C\right)\\ *& \mathrm{for}c=d=\nu \end{array}$

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

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

$K\left(\varphi ,p\right)\simeq {C}_{{\mathrm{St}}_{\varphi }\left(X\right)}$K(\phi,p) \simeq C_{St_\phi(X)}

(because $K\left(\varphi ,p\right)$ 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 $K\left(\varphi ,p\right)$ is of form ${C}_{F}$ for some $F$; and ${\mathrm{St}}_{\varphi }\left(X\right)$ is that $F$ by definition).

To prove the proposition, we need to compute the pushout

$\begin{array}{ccc}{\tau }_{\mathrm{hc}}\left(X\right)& \to & {\tau }_{\mathrm{hc}}\left({X}^{▹}\right)\\ ↓& & ↓\\ C& \to & K\left(\varphi ,p\right)={C}_{{\mathrm{St}}_{\varphi }\left(X\right)}\\ {}^{\pi }↓& & ↓\\ C\prime & \to & Q\end{array}$\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 $Q\simeq C{\prime }_{{\pi }_{!}{\mathrm{St}}_{\varphi }\left(X\right)}$.

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 $\mathrm{sSet}$-functor $\pi :C\to C\prime$ is $C{\prime }_{{\pi }_{!}F}$.

This follows by inspection of what a cocone

$\begin{array}{ccc}C& \stackrel{\iota }{\to }& {C}_{F}\\ {}^{\pi }↓& & {↓}^{d}\\ C\prime & \underset{r}{\to }& Q\end{array}$\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 $d{\mid }_{C}:C\to Q$, an object $d\left(\nu \right)\in Q$ together with a natural transformation

$F\left(c\right)\to Q\left(d{\mid }_{C}\left(c\right),d\left(\nu \right)\right)$F(c) \to Q(d|_C(c), d(\nu))

being the component ${F}_{c,\nu }:{C}_{F}\left(c,\nu \right)\to Q\left(d\left(c\right),d\left(\nu \right)\right)$ of the $\mathrm{sSet}$-functor.

We may write this natural transformation as

$F\to \left(d{\mid }_{C}{\right)}^{*}Q\left(-,d\left(\nu \right)\right)={\iota }^{*}{d}^{*}\nu Q\left(-,d\left(\nu \right)\right)\phantom{\rule{thinmathspace}{0ex}},$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

$\cdots \simeq {\pi }^{*}{r}^{*}Q\left(-,d\left(\nu \right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\eta :{\pi }_{!}F\to {r}^{*}Q\left(-,d\left(\nu \right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\eta : \pi_! F \to r^* Q(-,d(\nu)) \,.

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

###### Proposition

This functor has a right adjoint

${\mathrm{Un}}_{\varphi }:\left[{C}^{\mathrm{op}},\mathrm{sSet}\right]\to \mathrm{sSet}/S\phantom{\rule{thinmathspace}{0ex}},$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 ${\mathrm{St}}_{\varphi }$ preserves colimits. The claim then follows with the adjoint functor theorem.

###### Theorem

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

The straightening and the unstraightening functor constitute a Quillen adjunction

$\left({\mathrm{St}}_{\varphi }⊣{\mathrm{Un}}_{\varphi }\right):\mathrm{sSet}/S\stackrel{\stackrel{{\mathrm{Un}}_{\varphi }}{←}}{\underset{{\mathrm{St}}_{\varphi }}{\to }}\left[{C}^{\mathrm{op}},\mathrm{sSet}\right]$(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 $\varphi$ 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: $\left(\infty ,0\right)$-fibrations over an $\infty$-groupoid

###### Observation

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

$\infty \mathrm{Grpd}/C\simeq \left[{C}^{\mathrm{op}},\infty \mathrm{Grpd}\right]\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccccc}X& & \stackrel{\simeq }{\to }& & \stackrel{^}{X}\\ & ↘& & {↙}_{\mathrm{fib}}\\ & & C\end{array}\phantom{\rule{thinmathspace}{0ex}},$\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 $\infty \mathrm{Grpd}/X$ on the right fibrations is equivalent to all of $\infty \mathrm{Grpd}/X$.

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

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

###### Theorem

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

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

$\mathrm{Cart}\left(C\right)\simeq \mathrm{Func}\left({C}^{\mathrm{op}},\left(\infty ,1\right)\mathrm{Cat}\right)$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, ${\tau }_{\mathrm{hc}}\left(S\right)$ the corresponding simplicially enriched category (where ${\tau }_{\mathrm{hc}}$ is the left adjoint of the homotopy coherent nerve) and let $\varphi :{\tau }_{\mathrm{hc}}\left(S\right)\to C$ be an sSet-enriched functor.

###### Definition

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

The straightening functor

${\mathrm{St}}_{\varphi }:{\mathrm{sSet}}^{+}/S\to \left[{C}^{\mathrm{op}},{\mathrm{sSet}}^{+}\right]$St_\phi : sSet^+/S \to [C^{op}, sSet^+]

from marked simplicial sets over $S$ to marked simplicial presheaves on ${C}^{\mathrm{op}}$ 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 $f:d\to e$ of $X\in \mathrm{sSet}/S$ gives rise to an edge $\stackrel{˜}{f}\in {\mathrm{St}}_{\varphi }\left(X\right)\left(d\right)=K\left(\varphi ,p\right)\left(d,v\right)$: the join 2-simplex $f\star v$ of ${X}^{▹}$

$\begin{array}{ccccc}d& & \stackrel{f}{\to }& & e\\ & {}_{\stackrel{˜}{d}}↘& \stackrel{\stackrel{˜}{f}}{⇒}& {↙}_{\stackrel{˜}{e}}\\ & & v\end{array}$\array{ d && \stackrel{f}{\to} && e \\ & {}_{\mathllap{\tilde d}}\searrow & \stackrel{\tilde f}{\Rightarrow} & \swarrow_{\mathrlap{\tilde e}} \\ && v }

with image $\stackrel{˜}{f}:\stackrel{˜}{d}\to {f}^{*}\stackrel{˜}{e}$ in the pushout $K\left(\varphi ,p\right)\left(d,v\right)={\mathrm{St}}_{\varphi }X\left(d\right)$.

We define the straightening functor to assign that marking of edges which is the minimal one such that all such morphisms $\stackrel{˜}{f}$ are marked in ${\mathrm{St}}_{\varphi }X\left(d\right)$, for all marked $f:d\to e$ in $X$: this means that this marking is being completed under the constraint that ${\mathrm{St}}_{\varphi }\left(X\right)$ be sSet-enriched functorial.

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

$\mathrm{Map}\left(X,Y{\right)}_{n}={\mathrm{Hom}}_{{\mathrm{sSet}}^{+}}\left(X×\Delta \left[n{\right]}^{#},Y\right)\phantom{\rule{thinmathspace}{0ex}}.$Map(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^#, Y) \,.

So we need to find a marking on the ${\mathrm{St}}_{\varphi }\left(X\right)\left(-\right)$ such that for all $g:\Delta \left[1\right]\to C\left(c,d\right)$ the composite

$\Delta \left[1\right]\stackrel{g}{\to }C\left(c,d\right)\stackrel{{\mathrm{St}}_{\varphi }\left(X\right)\left(c,d\right)}{\to }\mathrm{Map}\left({\mathrm{St}}_{\varphi }\left(X\right)\left(d\right),{\mathrm{St}}_{\varphi }\left(X\right)\left(c\right)\right)$\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

${\mathrm{St}}_{\varphi }\left(X\right)\left(g\right):{\mathrm{St}}_{\varphi }\left(X\right)\left(d\right)×\Delta \left[1\right]\to {\mathrm{St}}_{\varphi }\left(X\right)\left(c\right)$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 $\stackrel{˜}{f}×\left\{0\to 1\right\}$ i.e. 1-cells $\left(\stackrel{˜}{f},\mathrm{Id}\right):\Delta \left[1\right]\to {\mathrm{St}}_{\varphi }\left(X\right)\left(d\right)×\Delta \left[1\right]$ to marked edges

${g}^{*}\stackrel{˜}{f}:\Delta \left[1\right]\stackrel{\left(\stackrel{˜}{f},\mathrm{Id}\right)}{\to }{\mathrm{St}}_{\varphi }\left(X\right)\left(d\right)×\Delta \left[1\right]\stackrel{{\mathrm{St}}_{\varphi }\left(X\right)\left(g\right)}{\to }{\mathrm{St}}_{\varphi }\left(X\right)\left(c\right)$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 ${\mathrm{St}}_{\varphi }\left(X\right)\left(c\right)$. So we need to ensure that the edges of this form are marked:

we define that the straightening functor marks an edge in ${\mathrm{St}}_{\varphi }\left(X\right)\left(c\right)$ iff it is of this form ${g}^{*}\stackrel{˜}{f}$, for $f:d\to e$ a marked edge of $X$ and $g\in C\left(c,d{\right)}_{1}$.

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

${n}_{\varphi }:\left[{C}^{\mathrm{op}},{\mathrm{sSet}}^{+}\right]\to {\mathrm{sSet}}^{+}/S\phantom{\rule{thinmathspace}{0ex}}.$n_\phi : [C^{op}, sSet^+] \to sSet^+/S \,.

This functor ${\mathrm{Un}}_{\varphi }$ exhibits the $\left(\infty ,1\right)$-Grothendieck-construction proper, in that it constructs a Cartesian fibration from a given $\left(\infty ,1\right)$-functor:

###### Theorem

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

This induces a Quillen adjunction

$\left({\mathrm{St}}_{\varphi }⊣{\mathrm{Un}}_{\varphi }\right):{\mathrm{SSet}}^{+}/S\stackrel{\stackrel{{\mathrm{Un}}_{\varphi }}{←}}{\underset{{\mathrm{St}}_{\varphi }}{\to }}\left[{C}^{\mathrm{op}},{\mathrm{SSet}}^{+}\right]$(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 $\varphi$ 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 $\left(\infty ,1\right)$-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 $f:C\to \mathrm{sSet}$ be a functor. The simplicial set ${N}_{f}\left(C\right)$, the relative nerve of $C$ under $f$ is defined as follows:

an $n$-cell of ${N}_{f}\left(C\right)$ is

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

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

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

$\begin{array}{ccc}\Delta \left[j\right]& \stackrel{\tau \left(j\right)}{\to }& f\left(\sigma \left(j\right)\right)\\ ↓& & {↓}^{f\left(\sigma \left(j\to k\right)\right)}\\ \Delta \left[k\right]& \stackrel{\tau \left(k\right)}{\to }& f\left(\sigma \left(k\right)\right)\end{array}$\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

${N}_{f}\left(C\right)\to N\left(C\right)$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 ${N}_{f}\left(C\right)\to N\left(C\right)$ is an isomorphism of simplicial sets, so ${N}_{f}\left(C\right)$ this is the ordinary nerve of $C$.

The fiber of ${N}_{f}\left(C\right)\to N\left(C\right)$ 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 $\tau \left(n\right):\Delta \left[n\right]\to f\left(c\right)$. So the fiber of ${N}_{f}\left(C\right)$ over $c$ is $f\left(c\right)$.

###### Definition

(marked relative nerve functor)

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

${\mathrm{sSet}}^{+}/N\left(C\right)←\left[C,{\mathrm{sSet}}^{+}\right]:{N}^{+}$sSet^+/N(C) \leftarrow [C, sSet^+] : N^+

by

$\left(C\stackrel{F}{\to }{\mathrm{sSet}}^{+}\right)↦\left({N}_{f}\left(C\right),{E}_{F}\right)\phantom{\rule{thinmathspace}{0ex}},$(C \stackrel{F}{\to} sSet^+) \mapsto (N_f(C), E_F) \,,

where $f:{C}^{\mathrm{op}}\stackrel{F}{\to }{\mathrm{sSet}}^{+}\to \mathrm{sSet}$ is $F$ with the marking forgotten, where ${N}_{f}\left(C\right)$ 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 ${ℱ}^{+}$. The value of ${ℱ}^{+}\left(F\right)$ on some $c\in C$ is equivalent to the value of $\mathrm{St}\left(F\right)$.

This is HTT, Lemma 3.2.5.17.

###### Proposition

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

The adjunction

$\left({ℱ}^{+}⊣{N}^{+}\right):{\mathrm{sSet}}_{/N\left(C\right)}^{+}\stackrel{\stackrel{{ℱ}^{+}}{\to }}{\underset{{N}^{+}}{←}}\left[C,{\mathrm{sSet}}^{+}\right]\phantom{\rule{thinmathspace}{0ex}}.$(\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

$\left(\mathrm{sSet}/S{\right)}_{\mathrm{Joyal}}\stackrel{\stackrel{}{\to }}{\stackrel{}{←}}{\mathrm{sSet}}^{+}/S\stackrel{\stackrel{}{\to }}{\stackrel{}{←}}\left({\mathrm{sSet}}^{+}/S{\right)}^{\mathrm{loc}}\stackrel{\stackrel{}{\to }}{\stackrel{}{←}}\left(\mathrm{sSet}/S{\right)}_{\mathrm{rfib}}\stackrel{\stackrel{}{\to }}{\stackrel{}{←}}\left(\mathrm{sSet}/S{\right)}_{\mathrm{Quillen}}\phantom{\rule{thinmathspace}{0ex}}.$(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.

## Examples

### Cartesian fibrations over the point

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

${\mathrm{St}}_{\varphi }:\mathrm{sSet}/*\simeq \mathrm{sSet}\to \left[*,\mathrm{sSet}\right]\simeq \mathrm{sSet}$St_\phi : sSet/* \simeq sSet \to [*,sSet] \simeq sSet

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

Definition

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

$Q\left[n\right]:=\mid {J}^{n}\mid \left(x,v\right)\phantom{\rule{thinmathspace}{0ex}},$Q[n] := |J^n|(x,v) \,,

where

${J}^{n}={C}^{◃}\left(\Delta \left[n\right]\to \left\{x\right\}\right)\phantom{\rule{thinmathspace}{0ex}}.$J^n = C^{\triangleleft}(\Delta[n] \to \{x\}) \,.

Then: nerve and realization associated to $Q$ yield a Quillen equivalence of ${\mathrm{sSet}}_{\mathrm{Quillen}}$ with itself.

### Cartesian fibrations over the interval

A Cartesian fibration $p:K\to \Delta \left[1\right]$ over the 1-simplex corresponds to a morphism $\Delta \left[1{\right]}^{\mathrm{op}}\to$ (∞,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 $K\to \left\{0\to 1\right\}$, 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

$\begin{array}{ccc}{e}_{{d}_{1}}& \to & {d}_{1}\\ ↓& & ↓\\ {e}_{{d}_{2}}& \to & {d}_{2}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccc}{K}_{1}& ↪& K\\ ↓& {↗}_{e}& ↓\\ {K}_{1}×\left\{0\to 1\right\}& \to & \left\{0\to 1\right\}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $p:K\to \Delta \left[1\right]$ with fibers the quasi-categories $C:={K}_{0}$ and $D:={K}_{1}$, an $\left(\infty ,1\right)$-functor associated to the Cartesian fibration $p$ is a functor $f:D\to C$ such that there exists a commuting diagram in sSet

$\begin{array}{ccccc}D×\Delta \left[1\right]& & \stackrel{F}{\to }& & K\\ & ↘& & {↙}_{p}\\ & & \Delta \left[1\right]\end{array}$\array{ D \times \Delta[1] &&\stackrel{F}{\to}&& K \\ & \searrow && \swarrow_{\mathrlap{p}} \\ && \Delta[1] }

such that

• $F{\mid }_{1}={\mathrm{Id}}_{D}$;

• $F{\mid }_{0}=f$;

• and for all $d\in D$, $F\left(\left\{d\right\}×\left\{0\to 1\right\}\right)$ 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

• $F{\mid }_{1}={h}_{1}$;

• $F{\mid }_{0}={h}_{0}\circ f$;

This is HTT, def. 5.2.1.1.

###### Proposition

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

###### 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

$\begin{array}{ccc}{D}^{♭}×\left\{1\right\}& ↪& {K}^{♯}\\ ↓& & {↓}^{p}\\ {D}^{♭}×\Delta \left[1{\right]}^{#}& \to & \Delta \left[1{\right]}^{#}\end{array}$\array{ D^\flat \times \{1\} &\hookrightarrow& K^{\sharp} \\ \downarrow && \downarrow^{\mathrlap{p}} \\ D^{\flat} \times \Delta[1]^{#} &\to& \Delta[1]^# }

in the category ${\mathrm{sSet}}^{+}$ of marked simplicial sets.

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

$\begin{array}{ccc}{D}^{♭}×\left\{1\right\}& ↪& {K}^{♯}\\ ↓& {↗}_{s}& {↓}^{p}\\ {D}^{♭}×\Delta \left[1{\right]}^{#}& \to & \Delta \left[1{\right]}^{#}\end{array}$\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 $f\prime$ is given. It will correspondingly come with its diagram

$\begin{array}{ccc}{D}^{♭}×\left\{1\right\}& ↪& {K}^{♯}\\ ↓& {↗}_{s\prime }& {↓}^{p}\\ {D}^{♭}×\Delta \left[1{\right]}^{#}& \to & \Delta \left[1{\right]}^{#}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccc}{D}^{♭}×\Lambda \left[2{\right]}_{2}& \stackrel{\left(s,s\prime \right)}{\to }& {K}^{♯}\\ ↓& {↗}_{q}& {↓}^{p}\\ {D}^{♭}×\Delta \left[2{\right]}^{#}& \to & \Delta \left[1{\right]}^{#}\end{array}\phantom{\rule{thinmathspace}{0ex}},$\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 $s\prime$, 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 $\left\{d\right\}×\left\{0\to 1\right\}$ to a Cartesian morphism in $K$, and due to the commutativity of the diagram this morphism must be in ${K}_{0}$, sitting over $\left\{0\right\}$. But as discussed there, a Cartesian morphism over a point is an equivalence. This means that the restriction

$k{\mid }_{D×\left\{0\to 1\right\}}\to {K}_{0}$k|_{D \times \{0 \to 1\}} \to K_0

is an invertible natural transformation between $f$ and $f\prime$, 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 $\left(\infty ,1\right)$-functor $f:D\to C$ is associated to some Cartesian fibration $p:K\to \Delta \left[1\right]$, 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 $D×\Delta \left[1\right]$ and the identifying the left boundary of that with $C$, using $f$.

Formally this means that we form the pushout

$N:=\left({D}^{♯}×\Delta \left[1{\right]}^{#}\right)\coprod _{{D}^{♯}×\left\{0{\right\}}^{#}}{C}^{♯}$N := (D^\sharp \times \Delta[1]^#) \coprod_{D^\sharp \times \{0\}^#} C^\sharp

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

$N\to \Delta \left[1\right]$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 $D×\Delta \left[1\right]\to K$ 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 $N\to \Delta \left[1\right]$ as

$N\to K\to \Delta \left[1{\right]}^{#}\phantom{\rule{thinmathspace}{0ex}},$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 $\Delta \left[1{\right]}^{#}$ is marked) a Cartesian fibration.

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

$\begin{array}{ccc}A& \to & {N}_{\alpha }\\ ↓& & ↓& ↘\\ B& \to & {N}_{\alpha +1}& \to & \Delta \left[1\right]\end{array}\phantom{\rule{thinmathspace}{0ex}},$\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 ${N}_{\alpha }{×}_{\Delta \left[1\right]}\left\{0\right\}$ is equivalent to $C$, then so is ${N}_{\alpha +1}{×}_{\Delta \left[1\right]}\left\{0\right\}$ and similarly for $D$. By induction, it follows that $f$ is indeed associated to $K\to \Delta \left[1\right]$.

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 of section 3.2 in

Revised on June 28, 2012 16:54:36 by Stephan Alexander Spahn (79.227.142.182)