###### Proof.

An $\infty$-Categorical Localisation Functor
for Diagrams of Simplicial Sets

Severin Bunk

###### Abstract

Associated to each small category $\mathscr{C}$, there is a category of
$\mathscr{C}$-shaped diagrams of simplicial sets and an $\infty$-category of
$N\mathscr{C}$-shaped homotopy coherent diagrams of spaces. We present a
functor which exhibits the latter as the $\infty$-categorical localisation of
the former at the objectwise weak homotopy equivalences. This builds on a
Quillen equivalence between the projective and covariant model structures
associated to $\mathscr{C}$ due to Heuts-Moerdijk [11], as well as Cisinski’s
theory of $\infty$-categorical localisations [2]. We use the localisation
functor to give simplified proofs that the left (resp. right) homotopy Kan
extension of diagrams of simplicial sets presents the $\infty$-categorical
left (resp. right) Kan extension of coherent diagrams of spaces.

###### Contents

  1. 1 Introduction
  2. 2 The rectification functor
  3. 3 From strict diagrams in ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ to coherent diagrams of spaces
  4. 4 Localisation of simplicial diagrams
  5. 5 Computational properties of the localisation functor
  6. A Background on $\infty$-categorical localisation

##  1 Introduction

The use of homotopy coherent mathematics, and in particular of
$\infty$-categories, is becoming increasingly common. Where available, an
often powerful approach is to combine $\infty$-categorical theory with
stricter classical constructions, such as in the setting of simplicial or
model categories. Presenting $\infty$-categorical constructions by stricter
models can have the upshot of combining the larger scope of coherent
mathematics with the easier and more explicit computability of classical
constructions. To exploit this, it is necessary that one can pass easily
between both frameworks. The main goal of this paper is to provide a short and
explicit transition from strict diagrams of simplicial sets, indexed by a
small category $\mathscr{C}$, to coherent diagrams of spaces, indexed by the
nerve $N\mathscr{C}$.

More concretely, let ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ denote
the category of simplicial sets, and let
${\mathscr{K}\mathrm{an}}\subset{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$
denote the full subcategory on the Kan complexes. We combine the Quillen
equivalences, treated in [11], between the projective and covariant model
structures associated to a small category $\mathscr{C}$ with the model for the
$\infty$-category $\mathscr{S}$ of spaces and the $\infty$-categorical
localisation theory from [2] to write down a functor

$\gamma_{\mathscr{C}}\colon
N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\longrightarrow{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$
(1.1)

from the nerve of the category of $\mathscr{C}$-shaped diagrams of Kan
complexes to the $\infty$-category of $N\mathscr{C}$-shaped diagrams of
spaces. We then prove:

###### Theorem 1.2

The functor $\gamma_{\mathscr{C}}$ exhibits
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ as the
$\infty$-categorical localisation of
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
at the objectwise weak homotopy equivalences. Furthermore, the functor
$\gamma_{\mathscr{C}}$ allows us to compute mapping spaces, internal homs and
Kan extensions in ${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ via the
classical derived constructions in the projective model structure on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.

The mere fact that ${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ is the
$\infty$-categorical localisation of
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
at the objectwise weak equivalences is known: one way to see this is through a
combination of Lurie’s [14, Prop. 4.2.4.4], Hinich’s [9, Prop. 1.2.1] and
classical results by Dwyer and Kan [6, 4, 5]. However, this chain of arguments
is rather indirect. It also does not lend itself easily to being used in
calculations because it passes through various models for $\infty$-categories
(quasicategories, fibrant simplicial categories and marked simplicial sets).
More recently, Cisinski provided an alternative proof of the fact [2, Thm.
7.8.9, Cor. 7.9.9]—entirely in the world of quasicategories—in the course of
developing his theory of $\infty$-categorical localisations. While this proof
is conceptually clearer and includes the provision of a localisation functor
(combining [2, Par. 7.3.14, 7.8.5, 7.8.7]), the latter is still hard to
compute explicitly (for instance, it still involves an $\infty$-categorical
left Kan extension).

The contribution of this paper lies in providing an explicit and computable
localisation functor as well as a much simplified proof of the fact that
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ is the
$\infty$-categorical localisation of
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
One way of approaching the localisation is through Lurie’s relative nerve
functor [14, Def. 3.2.5.2], which is part of a Quillen equivalence between the
projective model structure on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
and the covariant model structure on
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}$; this was
shown in [14, Prop. 3.2.5.18] and a simplified proof was given in [11]. One
could then invoke abstract arguments or further constructions which show that
either of the above model categories present the $\infty$-category
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ (this is proved, for
instance, in the aforementioned [14, Prop. 4.2.4.4] and [2, Thm. 7.8.9, Cor.
7.9.9]).

Here, we circumvent this second step: instead, we enhance the relative nerve
functor to write down the functor $\gamma_{\mathscr{C}}$, which takes its
values directly in the $\infty$-category
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$, and prove that this
already exhibits the $\infty$-categorical localisation (like [11], we avoid
the homotopy coherent nerve [14, 16] in doing so). At the same time, we obtain
a framework which unifies three perspectives on $\mathscr{S}$-valued functors:
they can be presented as strict
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$-valued diagrams, or as left
fibrations over $N\mathscr{C}$, or they can be viewed directly as objects in
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$. Furthermore, we make
manifest the full homotopical meaning of the rectification of simplicial
diagrams $F\colon\mathscr{C}\to{\mathscr{K}\mathrm{an}}$ from [11]: it is
really the left fibration presented by the coherent diagram of spaces
$\gamma_{\mathscr{C}}F\colon N\mathscr{C}\to\mathscr{S}$ described by $F$.

In the case where $\mathscr{C}=*$ is the final category, we obtain an
$\infty$-categorical localisation functor $\gamma\colon
N{\mathscr{K}\mathrm{an}}\to\mathscr{S}$, which exhibits the localisation at
the weak homotopy equivalences. We show an explicit instance of the fact that
localisation commutes with taking functor categories; that is, we show that
there is a commutative diagram of $\infty$-categories and functors (Theorem
3.38)

${N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{C},N{\mathscr{K}\mathrm{an}})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})}$$\scriptstyle{\gamma_{\mathscr{C}}}$$\scriptstyle{\cong}$$\scriptstyle{\gamma_{*}}$
(1.3)

Finally, we remark that shortly before the completion of the present paper,
the preprint [15] appeared on the ArXiv. There, the Quillen equivalences from
[11] are lifted from the projective and covariant model structures based on
the Kan-Quillen model structure to those based on the Joyal model structure on
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$. Employing Sharma’s results it
should be possible to also prove versions of the results in the present paper
for diagrams of $\infty$-categories in place of diagrams of
$\infty$-groupoids.

#### Outline.

In Section 2 we recall the rectification and homotopy colimit functors, as
well as various Quillen equivalences from [11]. We provide alternative
presentations of the rectification functor and analyse its properties. In
Section 3 we enhance the rectification functor to a functor
$\gamma_{\mathscr{C}}$ from strict diagrams of Kan complexes to coherent
diagrams of spaces and investigate its interplay with left fibrations. In
Section 4 we prove our main theorem, namely that $\gamma_{\mathscr{C}}$
implements the $\infty$-categorical localisation of the category of strict
Kan-complex-valued diagrams at the objectwise weak equivalences (Theorem
4.18). We use this result and the techniques developed to prove several nice
computational properties of $\gamma_{\mathscr{C}}$ in Section 5: it allows us
to compute mapping spaces, exponentials and Kan extensions at the level of
strict diagrams. Finally, we recall some of Cisinski’s $\infty$-categorical
localisation theory in Appendix A.

#### Notation

We fix a nested pair ${\mathrm{U}}\in{\mathrm{V}}$ of Grothendieck universes
such that the standard simplex category ${\mathbbe{\Delta}}$ is
${\mathrm{U}}$-small. We will not distinguish notationally between a morphism
$u\colon[k]\to[n]$ in ${\mathbbe{\Delta}}$ and the corresponding morphism
$u\colon\Delta^{k}\to\Delta^{n}$ of simplicial sets under the Yoneda embedding
of ${\mathbbe{\Delta}}$.

$\mathrm{h}\mathscr{D}$ |  homotopy category of an $\infty$-category $\mathscr{D}$
---|---
${\operatorname{Ho}}({\mathscr{M}})$ |  homotopy category of a model category ${\mathscr{M}}$
$\underline{{\mathscr{M}}}(-,-)$ |  simplicially enriched hom space in a simplicial category ${\mathscr{M}}$
$\mathscr{S}$ |  $\infty$-category of ${\mathrm{U}}$-small spaces/$\infty$-groupoids
$\widehat{\mathscr{S}}$ |  $\infty$-category of ${\mathrm{V}}$-small spaces/$\infty$-groupoids
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ |  Kan-Quillen model category of ${\mathrm{U}}$-small simplicial sets
${\mathrm{V}}{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ |  Kan-Quillen model category of ${\mathrm{V}}$-small simplicial sets
${\mathscr{K}\mathrm{an}}$ |  full subcategory of ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ on the ${\mathrm{U}}$-small Kan complexes
${\operatorname{Fun}}(\mathscr{C},\mathscr{D})$ |  1-category of (1-categorical) functors $\mathscr{C}\to\mathscr{D}$, for 1-categories $\mathscr{C}$ and $\mathscr{D}$
${\mathscr{F}\mathrm{un}}({\mathscr{A}},\mathscr{B})$ |  $\infty$-category of functors from ${\mathscr{A}}\in{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ to an $\infty$-category $\mathscr{B}$
$\mathscr{C}[W^{-1}]$ |  1-categorical/Gabriel-Zisman localisation of a 1-category $\mathscr{C}$ at a set $W$ of morphisms
$L^{B}_{S}{\mathscr{M}}$ |  left Bousfield localisation of a model category ${\mathscr{M}}$ at a set $S$ of morphisms
$L_{W}{\mathscr{A}}$ |  $\infty$-categorical localisation of an $\infty$-category ${\mathscr{A}}$ at a set of morphisms $W$

#### Acknowledgements

The author would like to thank Walker Stern for valuable feedback on a
preliminary draft of this paper. This research was funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under project number
468806966.

##  2 The rectification functor

In this section we start by reviewing results from [11]. We develop different
presentations of the rectification functor $r_{\mathscr{C}}^{*}$ from [11] and
analyse its various properties.

###  2.1 Various model structures on simplicial sets and diagrams

We briefly recall the main model structures we will use on categories of
diagrams of simplicial sets.

###### Theorem 2.1

Let $A$ be a simplicial set. There exists a unique model structure on the
slice category ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/A}$ whose
cofibrations are the monomorphisms and whose fibrant objects are the left
fibrations with codomain $A$ [13, Ch. 8]. In this model structure, a morphism
between fibrant objects is a fibration precisely if it is a left fibration,
and it is a weak equivalence if and only if it is a fibrewise weak equivalence
[2, Thm. 4.4.14]. Furthermore, this model structure is simplicial [14, Prop.
2.1.4.8].

###### Definition 2.2

The model structure in Theorem 2.1 is called the _covariant model structure on
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/A}$._

The following results are standard; see, for instance, the textbook [10].

###### Theorem 2.3

Let $\mathscr{C}$ be a small category.

  1. (1)

There exists a simplicial model structure on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
whose weak equivalences (resp. fibrations) are the objectwise weak
equivalences (resp. fibrations).

  2. (2)

There exists a simplicial model structure on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
whose weak equivalences (resp. cofibrations) are the objectwise weak
equivalences (resp. cofibrations).

###### Definition 2.4

The model structures in Theorem 2.3 are called the _projective_ and
_injective_ model structure on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$,
respectively.

###  2.2 The rectification functor and its properties

One of the main tools in this paper is what is called the rectification
functor in [11], or the relative nerve in [14]. Given a ${\mathrm{U}}$-small
(1-)category $\mathscr{C}$, it is a functor
$r_{\mathscr{C}}^{*}\colon{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}$,
which translates diagrams
$\mathscr{C}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ of simplicial
sets on a small category $\mathscr{C}$ into simplicial sets over the nerve
$N\mathscr{C}$. In this section, we recall the rectification functor following
[11] and develop alternative presentations of this functor. These will be
useful in proofs in later sections. We investigate several properties of this
functor, such as its behaviour under changes of the indexing category
$\mathscr{C}$. This is crucial for combining the rectification functor with
Cisinski’s model for the $\infty$-category $\mathscr{S}$ of spaces [2, Sec.
5.2] later on.

Let $\mathscr{C}$ be a ${\mathrm{U}}$-small (1-)category. Let
$\alpha\colon[n]\to\mathscr{C}$ be an $n$-simplex in $N\mathscr{C}$, and let
$\tau_{[n]}\colon{\mathbbe{\Delta}}_{/[n]}\to[n]$ be the last-vertex functor.
Recall that this sends an object $\varphi\colon[k]\to[n]$ to the value
$\varphi(k)\in[n]$ and a morphism

${{[k]}}$${{[k^{\prime}]}}$${{[n]}}$$\scriptstyle{v}$$\scriptstyle{u}$$\scriptstyle{u^{\prime}}$
(2.5)

to the unique morphism $u(k)\to u^{\prime}(k^{\prime})$ in $[n]$. We let
$\mathrm{pr}\colon{\mathbbe{\Delta}}_{/[n]}\to{\mathbbe{\Delta}}$ denote the
canonical projection functor and
$\Delta^{\cdot}\colon{\mathbbe{\Delta}}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$,
$[k]\mapsto\Delta^{k}$ the Yoneda embedding of ${\mathbbe{\Delta}}$.

Given $\alpha\colon[n]\to\mathscr{C}$ and a functor
$F\colon\mathscr{C}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$, define
the set of natural transformations

$(r_{\mathscr{C}}^{*}F)_{n,\alpha}={\operatorname{Fun}}({\mathbbe{\Delta}}_{/[n]},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})(\Delta^{\cdot}\circ\mathrm{pr},\,F\circ\alpha\circ\tau_{[n]})\,.$
(2.6)

This assembles into a functor

$r_{\mathscr{C}}^{*}\colon{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\longrightarrow{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}\,,\qquad(r_{\mathscr{C}}^{*}F)_{n}=\coprod_{\alpha\in
N_{n}\mathscr{C}}(r_{\mathscr{C}}^{*}F)_{n,\alpha}\,.$ (2.7)

Explicitly, an $n$-simplex of $r_{\mathscr{C}}^{*}F$ consists of pairs
$(\alpha,x)$ of an $n$-simplex $\alpha\in N\mathscr{C}$ together with a family
$x=(x_{u})_{u}$ as follows: the family is indexed by all maps
$u\colon[k]\to[n]$, and $x_{u}$ is an element $x_{u}\in F(\alpha_{u(k)})_{k}$,
or equivalently a map $\Delta^{k}\to F(\alpha_{u(k)})$. These data satisfy the
property that, for each factorisation

${{[k]}}$${{[k^{\prime}]}}$${{[n]}}$$\scriptstyle{v}$$\scriptstyle{u}$$\scriptstyle{u^{\prime}}$
(2.8)

we have a commutative diagram

${\Delta^{k}}$${F(\alpha_{u(k)})}$${\Delta^{k^{\prime}}}$${F(\alpha_{u^{\prime}(k^{\prime})})}$$\scriptstyle{x_{u}}$$\scriptstyle{v}$$\scriptstyle{F(f_{v})}$$\scriptstyle{x_{u^{\prime}}}$
(2.9)

where $f_{v}\colon\alpha_{u(k)}\to\alpha_{u^{\prime}(k^{\prime})}$ is the
image under $\alpha\colon[n]\to\mathscr{C}$ of the unique morphism $u(k)\to
u^{\prime}(k^{\prime})$ in $[n]$. The simplicial structure of
$r_{\mathscr{C}}^{*}F$ is induced by composition by maps
$w\colon[n^{\prime}]\to[n]$. Explicitly, $w$ sends an $n$-simplex
$(\alpha,x)\in(r_{\mathscr{C}}^{*}F)_{n}$ to the $n^{\prime}$-simplex
$(w^{*}\alpha,w^{*}x)$, with $w^{*}\alpha=\alpha\circ
w\colon[n^{\prime}]\to\mathscr{C}$ and

$(w^{*}x)_{u}=x_{w\circ u}\in
F\big{(}(w^{*}\alpha)_{u(k)}\big{)}_{k}=F(\alpha_{w\circ u(k)})_{k}$ (2.10)

for $u\colon[k]\to[n^{\prime}]$. The map $r_{\mathscr{C}}^{*}F\to
N\mathscr{C}$ is given by sending a pair $(\alpha,x)$ to the $n$-simplex
$\alpha\in N_{n}\mathscr{C}$. Given a morphism $f\colon F\to G$ in
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$,
the image of $f$ under $r_{\mathscr{C}}^{*}$ acts by sending a pair
$(\alpha,x)$ to the pair $(\alpha,f(x))$, where $f$ is applied to each element
$x_{u}$ in the family $x=(x_{u})_{u}$.

###### Definition 2.11

Let $\mathscr{C}$ be a ${\mathrm{U}}$-small category. The functor
$r_{\mathscr{C}}^{*}\colon{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\longrightarrow{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}$
is called the _rectification functor for $\mathscr{C}$_.

###### Remark 2.12

This agrees with the functor $r^{*}$ from [11, p. 9] (we have chosen to add
the subscript $\mathscr{C}$ to the notation because will need to vary the
indexing category). As Heuts and Moerdijk point out already, this also
coincides with Lurie’s relative nerve functor [14, Def. 3.2.5.2].
$\triangleleft$

In the following we present further equivalent ways of describing the
rectification functor $r_{\mathscr{C}}^{*}$. These require less data and are
therefore more convenient to work with in certain circumstances.

###### Definition 2.13

For $n\in\mathbb{N}_{0}$, let ${\mathbbe{\Sigma}}^{n}$ denote the category
whose objects are pairs $[i,j]$, $i\leq j\in[n]$ (we follow [8] for the
notation). We identify a pair $[i,j]$ with the interval

$[i,j]=\\{l\in[n]\,|\,i\leq l\leq j\\}\subset[n]\,.$ (2.14)

There is a unique morphism $[i,j]\to[i^{\prime},j^{\prime}]$ in
${\mathbbe{\Sigma}}^{n}$ whenever there is an inclusion of intervals
$[i,j]\subset[i^{\prime},j^{\prime}]$, or, equivalently, whenever
$i^{\prime}\leq i\leq j\leq j^{\prime}$.

###### Remark 2.15

The categories ${\mathbbe{\Sigma}}^{n}$, with $n$ ranging over the non-
negative integers, are the foundation for Segal objects of spans in
$\infty$-categories. We refer to [8] for more on this. Below, we rephrase the
simplicial set $r_{\mathscr{C}}^{*}F$ as encoding certain spans of sections of
the functor $F$. $\triangleleft$

Each $n$-simplex $\alpha\in N_{n}\mathscr{C}$ determines a functor

$\Sigma\alpha\colon{\mathbbe{\Sigma}}^{n}\to{\mathbbe{\Delta}}^{\mathrm{op}}\times\mathscr{C}\
$ (2.16)

as follows: $\Sigma\alpha$ sends an object $[i,j]\in{\mathbbe{\Sigma}}^{n}$ to
the pair $([i],\alpha_{j})$. It sends an inclusion
$[i,j]\subset[i^{\prime},j^{\prime}]$ to the pair $(v,\alpha_{j,j^{\prime}})$,
where $v\colon[i^{\prime}]\to[i]$ is defined by $v(l)=l$, for
$l=0,\ldots,i^{\prime}$, and the morphism $\alpha_{j,j^{\prime}}$ in
$\mathscr{C}$ is the image under $\alpha$ of the unique morphism $j\to
j^{\prime}$ in $[n]$. The tensor-hom adjunction in the category of
${\mathrm{U}}$-small categories provides a canonical identification

$(-)^{\dashv}\colon{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\longrightarrow{\operatorname{Fun}}({\mathbbe{\Delta}}^{\mathrm{op}}\times\mathscr{C},{\mathscr{S}\mathrm{et}})\,.$
(2.17)

Given a functor
$F\colon\mathscr{C}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ we have
$F^{\dashv}([i],c)=F(c)_{i}$, for each $i\in{\mathbbe{\Delta}}$ and
$c\in\mathscr{C}$.

###### Lemma 2.18

Let $F\colon\mathscr{C}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ be a
functor. The simplices $(\alpha,x)$ of $r_{\mathscr{C}}^{*}F$ are in canonical
bijection with pairs $(\alpha,y)$, where $y$ is a section of
$(\Sigma\alpha)^{*}F^{\dashv}\colon{\mathbbe{\Sigma}}^{n}\to{\mathscr{S}\mathrm{et}}$,
or, equivalently, with elements of
$\operatorname{lim}((\Sigma\alpha)^{*}F^{\dashv}\colon{\mathbbe{\Sigma}}^{n}\to{\mathscr{S}\mathrm{et}})$.

###### Proof.

The map $(\alpha,x)\mapsto(\alpha,y)$ simply forgets part of the data; it
remains to describe an inverse map to this process. Let $u\colon[k]\to[n]$ be
an object in ${\mathbbe{\Delta}}_{/[n]}$. The morphism $u$ has a unique
factorisation

${{[k]}}$${{[k^{\prime}]}}$${{[k^{\prime\prime}]}}$${{[n]}}$$\scriptstyle{v}$$\scriptstyle{u}$$\scriptstyle{v^{\prime}}$$\scriptstyle{u^{\prime}}$$\scriptstyle{u^{\prime\prime}}$
(2.19)

The pair $(u^{\prime},v)$ forms the image factorisation of $u$; that is, we
may identify $u^{\prime}$ with the inclusion $\operatorname{im}(u)\subset[n]$
(where we view $\operatorname{im}(u)\cong[k^{\prime}]$ as a totally ordered
set with the order induced by this inclusion). The map $v^{\prime}$ can then
be identified with the inclusion
$\operatorname{im}(u)\hookrightarrow\overline{\operatorname{im}(u)}=[u(0),u(k)]\subset[n]$,
where for $i\leq j\in[n]$ we let $[i,j]\subset[n]$ denote the ‘interval from
$i$ to $j$ in $[n]$’, i.e. the totally ordered subset of $[n]$ consisting of
all $l\in[n]$ with $i\leq l\leq j$. Note that
$\overline{\operatorname{im}(u)}\hookrightarrow[n]$ is the smallest interval
in $[n]$ through which $u$ factors. In this way, each order-preserving map
$u\colon[k]\to[n]$ factors uniquely through a minimal interval inclusion
$\overline{\operatorname{im}(u)}\hookrightarrow[n]$.

Given a second object $v\colon[l]\to[n]$ in ${\mathbbe{\Delta}}_{/[n]}$ and a
morphism $w\colon u\to v$ in ${\mathbbe{\Delta}}_{/[n]}$, we obtain an induced
commutative diagram (using that $v\circ w=u$)

${{[k]}}$${\operatorname{im}(u)}$${\overline{\operatorname{im}(u)}}$${{[l]}}$${\operatorname{im}(v)}$${\overline{\operatorname{im}(v)}}$$\scriptstyle{w}$
(2.20)

in ${\mathbbe{\Delta}}_{/[n]}$. The right-hand vertical morphism is an
inclusion of intervals in $[n]$. By these arguments and the compatibility
condition (2.9) on the family $x$, we see that the data $(\alpha,x)$ are
entirely determined by the morphisms given by inclusions of intervals in
$[n]$, i.e. the morphisms in ${\mathbbe{\Sigma}}^{n}$. Explicitly, for any
morphism $u\colon[k]\to[n]$, the compatibility condition (2.9) applied to the
factorisation
$[k]\overset{v}{\to}\overline{\operatorname{im}(u)}\overset{\iota}{\to}[n]$
induces a commutative diagram

${\Delta^{k}}$${F(\alpha_{u(k)})}$${\Delta^{\overline{\operatorname{im}(u)}}}$${F(\alpha_{u(k)})}$$\scriptstyle{x_{u}}$$\scriptstyle{x_{\iota}}$
(2.21)

implying that $x_{u}$ is entirely determined by $x_{\iota}$, i.e. by the value
of the family $x$ on the interval inclusion
$\overline{\operatorname{im}(u)}\hookrightarrow[n]$. ∎

###### Definition 2.22

Let ${\mathbbe{\Lambda}}^{n}\subset{\mathbbe{\Sigma}}^{n}$ denote the full
subcategory on those objects $[i,j]$ where $j-i\in\\{0,1\\}$.

We record the following observation:

###### Lemma 2.23

For each $n\in\mathbb{N}_{0}$, the inclusion
$\iota^{n}\colon{\mathbbe{\Lambda}}^{n}\hookrightarrow{\mathbbe{\Sigma}}^{n}$
is homotopy cofinal; that is, its nerve $N\iota^{n}\colon
N{\mathbbe{\Lambda}}^{n}\hookrightarrow N{\mathbbe{\Sigma}}^{n}$ is a cofinal
morphism in ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$.

###### Proof.

Given an object $[i,j]\in{\mathbbe{\Sigma}}^{n}$, the comma category
$\iota^{n}_{/[i,j]}$ is equivalent to ${\mathbbe{\Lambda}}^{j-i}$, whose nerve
is a weakly contractible simplicial set. ∎

The following is already in [11] (see before Lemma 4.1); we include it here
for completeness.

###### Lemma 2.24

Let $F\colon\mathscr{C}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ be a
functor, and let
$F^{\dashv}\colon{\mathbbe{\Delta}}^{\mathrm{op}}\times\mathscr{C}\to{\mathscr{S}\mathrm{et}}$
denote the associated functor defined by $F^{\dashv}([i],c)=F(c)_{i}$. The
elements $(\alpha,x)$ of $r_{\mathscr{C}}^{*}F$ are in canonical bijection
with pairs $(\alpha,z)$, where $z$ is a section of
$\iota^{n*}(\Sigma\alpha)^{*}F^{\dashv}\colon{\mathbbe{\Lambda}}^{n}\to{\mathscr{S}\mathrm{et}}$,
or, equivalently, with elements of
$\operatorname{lim}(\iota^{n*}(\Sigma\alpha)^{*}F^{\dashv}\colon{\mathbbe{\Lambda}}^{n}\to{\mathscr{S}\mathrm{et}})$.

###### Proof.

This follows from Lemma 2.18 together with Lemma 2.23 and the fact that
homotopy cofinal functors are in particular cofinal. ∎

###### Remark 2.25

The upshot of using ${\mathbbe{\Sigma}}^{n}$ over ${\mathbbe{\Lambda}}^{n}$ is
that the former assemble into a (strict) functor
${\mathbbe{\Sigma}}\colon{\mathbbe{\Delta}}\to{\mathscr{C}\mathrm{at}}$. The
categories ${\mathbbe{\Lambda}}^{n}$ do not produce a functor
${\mathbbe{\Delta}}\to{\mathscr{C}\mathrm{at}}$, but rather we only obtain a
functor ${\mathbbe{\Delta}}_{int}\to{\mathscr{C}\mathrm{at}}$, where
${\mathbbe{\Delta}}_{int}\subset{\mathbbe{\Delta}}$ is the wide subcategory
with only the inert morphisms; see [8, Def. 5.1, Def. 5.2] for further
details. $\triangleleft$

The categories ${\mathbbe{\Sigma}}^{n,\mathrm{op}}$ assemble into a strict
functor

${\mathbbe{\Sigma}}^{\mathrm{op}}\colon{\mathbbe{\Delta}}\longrightarrow{\mathscr{C}\mathrm{at}}\,,\quad[n]\longmapsto{\mathbbe{\Sigma}}^{n,\mathrm{op}}\,.$
(2.26)

It sends a morphism $u\colon[n]\to[m]$ to the functor

${\mathbbe{\Sigma}}^{\mathrm{op}}(u)\colon{\mathbbe{\Sigma}}^{n,\mathrm{op}}\to{\mathbbe{\Sigma}}^{m,\mathrm{op}}\,,\quad[i,j]\longmapsto\big{[}u(i),u(j)\big{]}\,.$
(2.27)

The Grothendieck construction of ${\mathbbe{\Sigma}}^{\mathrm{op}}$ is a
functor
$\textstyle{\int}{\mathbbe{\Sigma}}^{\mathrm{op}}\longrightarrow{\mathbbe{\Delta}}$.
We form the (strict) pullback of categories

${\Big{(}{\mathbbe{\Delta}}_{/N\mathscr{C}}\underset{{\mathbbe{\Delta}}}{\times}\textstyle{\int}{\mathbbe{\Sigma}}^{\mathrm{op}}\Big{)}^{\mathrm{op}}\cong\Big{(}\textstyle{\int}(\mathrm{pr}^{*}{\mathbbe{\Sigma}}^{\mathrm{op}})\Big{)}^{\mathrm{op}}}$${\big{(}\textstyle{\int}{\mathbbe{\Sigma}}^{\mathrm{op}}\big{)}^{\mathrm{op}}}$${\big{(}{\mathbbe{\Delta}}_{/N\mathscr{C}}\big{)}^{\mathrm{op}}}$${{\mathbbe{\Delta}}^{\mathrm{op}}}$$\scriptstyle{\pi}$$\scriptstyle{\mathrm{pr}}$
(2.28)

Consider the functor

$\Sigma^{\mathrm{op}}\colon{\mathbbe{\Delta}}_{/N\mathscr{C}}\underset{{\mathbbe{\Delta}}}{\times}\textstyle{\int}{\mathbbe{\Sigma}}^{\mathrm{op}}\longrightarrow{\mathbbe{\Delta}}\times\mathscr{C}^{\mathrm{op}}$
(2.29)

defined as follows: given an object
$(\alpha\colon[n]\to\mathscr{C},[i,j]\in{\mathbbe{\Sigma}}^{n})$, we set
$\Sigma^{\mathrm{op}}(\alpha,[i,j])=([i],\alpha_{j})$. A morphism
$(\alpha\colon[n]\to\mathscr{C},[i,j])\longrightarrow(\beta\colon[k]\to\mathscr{C},[i^{\prime},j^{\prime}])$
in
${\mathbbe{\Delta}}_{/N\mathscr{C}}\times_{\mathbbe{\Delta}}\textstyle{\int}{\mathbbe{\Sigma}}^{\mathrm{op}}$
is given by a morphism $v\colon[n]\to[k]$ such that $\beta\circ v=\alpha$ and
$[i^{\prime},j^{\prime}]\subset[v(i),v(j)]$, i.e. $v(i)\leq i^{\prime}\leq
j^{\prime}\leq v(j)$. The functor $\Sigma^{\mathrm{op}}$ sends this to the
pair $(v_{|[0,i]},\beta_{j^{\prime},v(j)})$ of the following data:

  1. (1)

$v_{|[0,i]}\colon[i]\to[i^{\prime}]$ is the restriction of the morphism
$v\colon[n]\to[k]$ to the interval $[i]\cong[0,i]\subset[n]$; by the condition
that $[i^{\prime},j^{\prime}]\subset[v(i),v(j)]$ we have that $v(i)\leq
i^{\prime}$ and this restriction factors through
$[i^{\prime}]\cong[0,i^{\prime}]\subset[k]$.

  2. (2)

The morphism $\beta_{j^{\prime},v(j)}$ is the unique morphism
$\beta_{j^{\prime}}\to\beta_{v(j)}=\alpha_{j}$ induced by the functor
$\beta\colon[k]\to\mathscr{C}$; again, this relies on the condition that
$[i^{\prime},j^{\prime}]\subset[v(i),v(j)]$.

###### Lemma 2.30

Let $G\colon\mathscr{D}\to{\mathscr{C}\mathrm{at}}$ be a _strict_ functor, and
let $\pi\colon(\textstyle{\int}G)^{\mathrm{op}}\to\mathscr{D}^{\mathrm{op}}$
be the opposite of the Grothendieck construction of $G$.

  1. (1)

The fibres $\pi^{-1}(d)$ coincide with the values $G(d)^{\mathrm{op}}$; in
particular, they depend strictly functorially on $d\in\mathscr{D}$.

  2. (2)

Let $\mathscr{E}$ be a complete category. For any functor
$S\colon(\textstyle{\int}G)^{\mathrm{op}}\to\mathscr{E}$, the right Kan
extension $\pi_{*}S$ is isomorphic in
${\operatorname{Fun}}(\mathscr{D}^{\mathrm{op}},\mathscr{E})$ to the functor

${\mathrm{R}}_{\pi}S\colon\mathscr{D}^{\mathrm{op}}\to\mathscr{E}\,,\quad
d\longmapsto\operatorname{lim}\big{(}G(d)^{\mathrm{op}}\hookrightarrow(\textstyle{\int}G)^{\mathrm{op}}\overset{S}{\longrightarrow}\mathscr{E}\big{)}\,.$
(2.31)

The fact that ${\mathrm{R}}_{\pi}S$ is functorial makes use of the assumption
that $G\colon\mathscr{D}\to{\mathscr{C}\mathrm{at}}$ is strict.

###### Proof.

Claim (1) is an immediate property of the Grothendieck construction: An object
in $(\textstyle{\int}G)^{\mathrm{op}}$ consists of a pair $(d,x)$, where
$d\in\mathscr{D}$ and $x\in G(d)$. A morphism
$(d,x)\to(d^{\prime},x^{\prime})$ in $(\textstyle{\int}G)^{\mathrm{op}}$ is a
pair $(f,g)$, where $f\in\mathscr{D}(d^{\prime},d)$ and $g\in
G(d)(Gf(x^{\prime}),x)$. The claim then follows by the assumption that $G$ is
strict; in particular, the value of $G$ on the identity morphism of $d$ is the
identity functor on $G(d)$.

The Grothendieck construction produces a functor
$\textstyle{\int}G\to\mathscr{D}$ whose nerve is a proper morphism of
$\infty$-categories [2, Prop. 7.3.2]; the nerve of $\pi$ is therefore a smooth
morphism. In particular, it follows that, for each $d\in\mathscr{D}$, the
nerve of the inclusion $G(d)^{\mathrm{op}}=\pi^{-1}(d)\hookrightarrow d/\pi$
is a cofinal morphism in ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ [2,
Thm. 4.4.36]. Equivalently, the functor
$G(d)^{\mathrm{op}}=\pi^{-1}(d)\hookrightarrow d/\pi$ is homotopy cofinal, and
in particular a cofinal functor of categories. This induces the natural
isomorphism $\pi_{*}S\cong{\mathrm{R}}_{\pi}S$. ∎

Let $\mathscr{D}$ be a small category, and let
$A\colon\mathscr{D}^{\mathrm{op}}\to{\mathscr{S}\mathrm{et}}$ be a presheaf on
$\mathscr{D}$. We recall the standard equivalence (see, for instance, [2, Rmk.
1.1.14])

$\Psi\colon{\operatorname{Fun}}\big{(}(\mathscr{D}_{/A})^{\mathrm{op}},{\mathscr{S}\mathrm{et}}\big{)}\longrightarrow{\operatorname{Fun}}(\mathscr{D}^{\mathrm{op}},{\mathscr{S}\mathrm{et}})_{/A}\,.$
(2.32)

Explicitly, given a presheaf $F$ on $\mathscr{D}_{/A}$ and $d\in\mathscr{D}$,
we have

$\Psi F(d)=\bigg{(}\coprod_{\phi\in
A(d)}F(\phi)\longrightarrow\coprod_{\phi\in A(d)}*\cong A(d)\bigg{)}\,.$
(2.33)

Let
$h\colon\mathscr{D}\to{\operatorname{Fun}}(\mathscr{D}^{\mathrm{op}},{\mathscr{S}\mathrm{et}})$
denote the Yoneda embedding of $\mathscr{D}$. The inverse functor of $\Psi$
sends an object $\psi\colon X\to A$ in
${\operatorname{Fun}}(\mathscr{D}^{\mathrm{op}},{\mathscr{S}\mathrm{et}})_{/A}$
to the presheaf on $\mathscr{D}_{/A}$ whose value on $\phi\colon h_{d}\to A$
is the set of factorisations $\phi=(h_{d}\to X\overset{\psi}{\to}A)$ of $\phi$
through $\psi$.

In the particular case where $\mathscr{D}={\mathbbe{\Delta}}$ and
$A=N\mathscr{C}$, this provides an equivalence

$\Psi\colon{\operatorname{Fun}}\big{(}({\mathbbe{\Delta}}_{/N\mathscr{C}})^{\mathrm{op}},{\mathscr{S}\mathrm{et}}\big{)}\longrightarrow{\operatorname{Fun}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathscr{S}\mathrm{et}})_{/N\mathscr{C}}={\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}\,.$
(2.34)

With $\Sigma$ as in (2.29), we introduce the the composite
$s_{\mathscr{C}}^{*}\coloneqq\Psi\circ\pi_{*}\circ\Sigma^{*}\circ(-)^{\dashv}$,
i.e.

${s_{\mathscr{C}}^{*}\colon{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}$${{\operatorname{Fun}}({\mathbbe{\Delta}}^{\mathrm{op}}\times\mathscr{C},{\mathscr{S}\mathrm{et}})}$${{\operatorname{Fun}}\big{(}({\mathbbe{\Delta}}_{/N\mathscr{C}})^{\mathrm{op}}\times_{{\mathbbe{\Delta}}^{\mathrm{op}}}(\textstyle{\int}{\mathbbe{\Sigma}}^{\mathrm{op}})^{\mathrm{op}},\,{\mathscr{S}\mathrm{et}}\big{)}}$${\cdots}$${\cdots\
{\operatorname{Fun}}\big{(}({\mathbbe{\Delta}}_{/N\mathscr{C}})^{\mathrm{op}},{\mathscr{S}\mathrm{et}}\big{)}}$${{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}\,.}$$\scriptstyle{(-)^{\dashv}}$$\scriptstyle{\Sigma^{*}}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\Psi}$
(2.35)

Analogously, we introduce the composite

$t_{\mathscr{C}}^{*}\coloneqq\Psi\circ{\mathrm{R}}_{\pi}\circ\Sigma^{*}\circ(-)^{\dashv}\colon{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\longrightarrow{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}\,.$
(2.36)

We thus obtain three functors

$r_{\mathscr{C}}^{*}\,,s_{\mathscr{C}}^{*}\,,t_{\mathscr{C}}^{*}\colon{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\longrightarrow{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}\,.$
(2.37)

###### Proposition 2.38

There are canonical isomorphisms

$r_{\mathscr{C}}^{*}(F)\cong s_{\mathscr{C}}^{*}(F)\cong
t_{\mathscr{C}}^{*}(F)\,,$ (2.39)

in ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}$, natural in
$F\in{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.

###### Proof.

This is now a combination of Lemma 2.18, equation (2.33) and Lemma 2.30. Note
that the coproduct in our original definition (2.7) corresponds here to the
coproduct in the construction of the functor $\Psi$ (2.33). ∎

This allows us to show the following properties of $r_{\mathscr{C}}^{*}$,
which are important in applications:

###### Corollary 2.40

The functor
$r_{\mathscr{C}}^{*}\colon{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\longrightarrow{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}$
preserves filtered colimits.

###### Proof.

We use the natural isomorphism
$r_{\mathscr{C}}^{*}F\cong\Psi\circ{\mathrm{R}}_{\pi}\circ\Sigma^{*}(F^{\dashv})$.
Since colimits in functor categories are computed objectwise, $(-)^{\dashv}$
and $\Sigma^{*}$ preserve all colimits. The value of
${\mathrm{R}}_{\pi}\circ\Sigma^{*}(F^{\dashv})$ at an object
$\alpha\colon[n]\to\mathscr{C}$ of ${\mathbbe{\Delta}}_{/[n]}$ is computed as
a finite limit over ${\mathbbe{\Sigma}}^{n}$. Thus, ${\mathrm{R}}_{\pi}$
commutes with filtered colimits. Finally, the equivalence $\Psi$ preserves all
colimits. ∎

###### Lemma 2.41

The functor
$r_{\mathscr{C}}^{*}\colon{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\longrightarrow{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}$
sends injective cofibrations (i.e. objectwise monomorphisms) to cofibrations
(i.e. monomorphisms over $N\mathscr{C}$).

###### Proof.

Let $f\colon F\to G$ be an objectwise monomorphism in
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
By Proposition 2.38, it suffices to prove the same claim for
$s_{\mathscr{C}}^{*}$. Let $(\alpha,x),(\beta,y)\in s_{\mathscr{C}}^{*}F$ and
suppose that
$(s_{\mathscr{C}}^{*}f)(\alpha,x)=(s_{\mathscr{C}}^{*}f)(\beta,y)$. This
readily implies $\alpha=\beta$. The data $x=(x_{i,j})_{i,j=0,\ldots,n}$ and
$y=(y_{i,j})_{i,j=0,\ldots,n}$ are sections of $\Sigma\alpha^{*}F$ (in the
notation of (2.16)). Under $s_{\mathscr{C}}^{*}f$, these are sent to the
sections $f(x)=(f(x_{i,j}))_{i,j=0,\ldots,n}$ and
$f(y)=(f(y)_{i,j})_{i,j=0,\ldots,n}$ of $\Sigma\alpha^{*}G$, which are equal
by our assumption. Since $f$ was an objectwise monomorphism, it follows that
$x=y$. ∎

The following observation is crucial: it will allow us to enhance
$r_{\mathscr{C}}^{*}$ to a functor taking values in the $\infty$-category of
functors with values in Cisinski’s $\infty$-category of spaces (see Definition
3.1 below). We give a short proof for completeness.

###### Lemma 2.42

_[ 14, Rmk. 3.2.5.7]_ Let $\psi\colon\mathscr{D}\to\mathscr{C}$ and
$F\colon\mathscr{C}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ be
functors of (1-)categories.

  1. (1)

There is a pullback square of simplicial sets

${r_{\mathscr{D}}^{*}(\psi^{*}F)}$${r_{\mathscr{C}}^{*}F}$${N\mathscr{D}}$${N\mathscr{C}}$$\scriptstyle{N\psi}$
(2.43)

That is, there is a canonical natural isomorphism

$r_{\mathscr{D}}^{*}\circ\psi^{*}\cong(N\psi)^{*}\circ r_{\mathscr{C}}^{*}\,.$
(2.44)

  2. (2)

For any $\alpha\colon[n]\to\mathscr{D}$, there is a canonical bijection

$(r_{\mathscr{C}}^{*}F)_{\psi_{*}\alpha,n}\cong\big{(}r_{\mathscr{D}}^{*}(\psi^{*}F)\big{)}_{\alpha,n}\,.$
(2.45)

###### Proof.

An $l$-simplex of $r_{\mathscr{D}}^{*}(\psi^{*}F)$ consists of a pair
$(\alpha,\eta)$, where $\alpha\colon[l]\to\mathscr{D}$ is an $l$-simplex of
$N\mathscr{D}$, and where
$\eta\colon\Delta^{\cdot}\circ\mathrm{pr}\longrightarrow
F\circ\psi\circ\alpha\circ\tau_{[l]}$ is a natural transformation of functors
${\mathbbe{\Delta}}_{/[l]}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$
(in the notation from before (2.7)). This is equivalently a triple
$(\alpha,\eta^{\prime},\beta)$ of an $l$-simplex $\alpha$ in $N\mathscr{D}$,
an $l$-simplex $\beta$ of $N\mathscr{C}$ and a natural transformation
$\eta^{\prime}\colon\Delta^{\cdot}\circ\mathrm{pr}\longrightarrow
F\circ\beta\circ\tau_{[l]}$, such that $\beta=\psi_{*}\alpha$. This
identification establishes that diagram (2.43) is a pullback square in
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$. Part (2) can either be seen
explicitly by unravelling the definition of $r_{\mathscr{D}}^{*}F$, or by
applying the pasting law for pullbacks to Part (1). ∎

###  2.3 Homotopy-theoretic properties of the rectification functor

In this section we collect various homotopy-theoretic properties of the
rectification functor $r_{\mathscr{C}}^{*}$ (see Definition 2.11). Given a
${\mathrm{U}}$-small category $\mathscr{C}$, we endow
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
with the projective model structure. The following statement is a combination
of [11, Thm. C, Prop. 5.1, Prop. 5.2]:

###### Theorem 2.46

There is a pair of Quillen equivalences

${{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}$${{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}}$${{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\,.}$$\scriptstyle{h_{\mathscr{C}!}}$$\scriptstyle{\perp}$$\scriptstyle{r_{\mathscr{C}!}}$$\scriptstyle{\perp}$$\scriptstyle{h_{\mathscr{C}}^{*}}$$\scriptstyle{r_{\mathscr{C}}^{*}}$
(2.47)

On the level of total derived functors between homotopy categories, the
functors $\mathbb{L}r_{\mathscr{C}!}$ and $\mathbb{L}h_{\mathscr{C}!}$, and
hence also the functors $\mathbb{R}r_{\mathscr{C}}^{*}$ and
$\mathbb{R}h_{\mathscr{C}}^{*}$, are weak mutual inverses.

The proof that the Quillen adjunctions in Theorem 2.46 are in fact Quillen
equivalences is established by showing that the left derived functors of
$h_{\mathscr{C}!}$ and $r_{\mathscr{C}!}$ are weak mutual inverses. This
relies on the following result:

###### Proposition 2.48

_[ 11, Props. 5.1 and 5.2]_ Let
$X\in{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
and $A\in{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}$.

  1. (1)

There exists a natural transformation $\tau_{X}\colon
r_{\mathscr{C}!}h_{\mathscr{C}!}X\to X$, which is a weak equivalence in
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
whenever $X$ is projectively cofibrant.

  2. (2)

There exists a natural zig-zag of weak equivalences $A\leftarrow
L(A)\rightarrow h_{\mathscr{C}!}r_{\mathscr{C}!}A$ in
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}$.

Heuts-Moerdijk moreover prove the following consequence of Theorem 2.46, which
relates to the Kan-Quillen model structure rather than the covariant model
structure:

###### Theorem 2.49

_[ 11, Cor. D]_ Let $\mathscr{C}$ be a small category, and let
$L^{B}_{\mathscr{C}}{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
denote the left Bousfield localisation of the projective model structure on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
at all morphisms in $\mathscr{C}$ (or, more precisely, at their images under
the coYoneda embedding). The Quillen equivalence $h_{\mathscr{C}!}\dashv
h_{\mathscr{C}}^{*}$ from Theorem 2.46 induces a Quillen equivalence

${L^{B}_{\mathscr{C}}{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}$${\big{(}{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}\big{)}_{KQ}\,,}$$\scriptstyle{h_{\mathscr{C}!}}$$\scriptstyle{\perp}$$\scriptstyle{h_{\mathscr{C}}^{*}}$
(2.50)

where ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}$ is
endowed with the slice model structure induced from the Kan-Quillen model
structure on ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$.

An analogue of this statement also holds true for the right-hand Quillen
equivalence $r_{\mathscr{C}!}\dashv r_{\mathscr{C}}^{*}$ of Theorem 2.46:

###### Proposition 2.51

The Quillen equivalence $r_{\mathscr{C}!}\dashv r_{\mathscr{C}}^{*}$ from
Theorem 2.46 induces a Quillen equivalence

${\big{(}{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}\big{)}_{KQ}}$${L^{B}_{\mathscr{C}}{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\,.}$$\scriptstyle{r_{\mathscr{C}!}}$$\scriptstyle{\perp}$$\scriptstyle{r_{\mathscr{C}}^{*}}$
(2.52)

###### Proof.

Let
$h^{(-)}\colon\mathscr{C}^{\mathrm{op}}\to{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$,
$c\mapsto h^{c}=\mathscr{C}(c,-)$ denote the co-Yoneda embedding of
$\mathscr{C}$. Let $S$ denote the set of morphisms $h^{g}\colon h^{d}\to
h^{c}$ in
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$,
where $g\colon c\to d$ ranges over all morphisms in $\mathscr{C}$. Note that
each
$h^{c}\in{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
is projectively cofibrant. It is shown in [11, Prop. 6.1] and the proof of
Cor. D in [11] that

$\big{(}{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}\big{)}_{KQ}=L^{B}_{h_{\mathscr{C}!}S}\big{(}{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}\big{)}$
(2.53)

as model categories. We first show that the Quillen adjunction
$r_{\mathscr{C}!}:{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}\rightleftarrows{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}):r_{\mathscr{C}}^{*}$
descends to a Quillen adjunction

${r_{\mathscr{C}!}:L^{B}_{h_{\mathscr{C}!}S}\big{(}{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}\big{)}}$${L^{B}_{\mathscr{C}}{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}):r_{\mathscr{C}}^{*}\,.}$$\scriptstyle{\perp}$
(2.54)

By [10, Prop. 3.3.18], it suffices to show that $r_{\mathscr{C}!}$ takes each
morphism $h^{g}\colon h^{d}\to h^{c}$ in $S$ to a weak equivalence in
$L^{B}_{\mathscr{C}}{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
To see this, let $X\in
L^{B}_{\mathscr{C}}{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
be fibrant. By Proposition 2.48 there exists a natural transformation
$\tau_{Y}\colon r_{\mathscr{C}!}h_{\mathscr{C}!}Y\to Y$, which is a projective
weak equivalence whenever
$Y\in{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
is projectively cofibrant. This induces a commutative diagram of simplicial
sets (recall that we denote the simplicial hom sets in a simplicially enriched
category ${\mathscr{M}}$ by $\underline{{\mathscr{M}}}(-,-)$)

${\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(r_{\mathscr{C}!}h_{\mathscr{C}!}h^{c},X)}$${\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(r_{\mathscr{C}!}h_{\mathscr{C}!}h^{d},X)}$${X(c)}$${X(d)}$$\scriptstyle{(r_{\mathscr{C}!}h_{\mathscr{C}!}h^{g})^{*}}$$\scriptstyle{\tau_{(h^{c})}^{*}}$$\scriptstyle{\tau_{(h^{d})}^{*}}$$\scriptstyle{X(g)}$
(2.55)

The vertical morphisms are weak equivalences because $X$ is projectively
fibrant and corepresentables are cofibrant in
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
The bottom morphism is a weak equivalence as well since $X$ is fibrant in
$L^{B}_{\mathscr{C}}{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$;
that is, $X(g)$ is a weak equivalence for each morphism $g$ in $\mathscr{C}$.
It thus follows that the top horizontal morphism is a weak equivalence, and
hence, that the Quillen adjunction $r_{\mathscr{C}!}\dashv h_{\mathscr{C}!}$
descends to the localisations as claimed.

It remains to show that this descended adjunction is even a Quillen
equivalence. However, since the natural weak equivalences in Proposition 2.48
are still weak equivalences in the localisations, the fact that the left
derived functors of $h_{\mathscr{C}!}$ and $r_{\mathscr{C}!}$ are mutual weak
inverses still holds true at the level of the localisations. ∎

###### Remark 2.56

Alternatively, one can employ a different way of presenting
$({\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}})_{KQ}$ as a
Bousfield localisation of the covariant model structure on
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}$: it is shown
in [11, Lemma 6.1] and the proof of [11, Cor. D] that
$({\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}})_{KQ}$ agrees
with the left Bousfield localisation of the covariant model structure at the
morphisms

${\Delta^{\\{1\\}}}$${\Delta^{1}}$${N\mathscr{C}}$$\scriptstyle{y}$$\scriptstyle{f}$
(2.57)

where $f$ ranges over all morphisms $f\colon f_{0}\to f_{1}$ in $\mathscr{C}$.
Note that $r_{\mathscr{C}!}f_{i}=h^{f_{i}}=\mathscr{C}(f_{i},-)$ is the
functor corepresented by $f_{i}$. One thus needs to show that the morphism
$h^{f_{1}}=r_{\mathscr{C}!}(f_{1})\to r_{\mathscr{C}!}f$ is a weak equivalence
in
$L^{B}_{\mathscr{C}}{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
Let $X\in
L^{B}_{\mathscr{C}}{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
be fibrant. We have a triangle

${\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(r_{\mathscr{C}!}f,X)}$${\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(r_{\mathscr{C}!}f_{0},X)\cong
X(f_{0})}$${\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(r_{\mathscr{C}!}f_{1},X)\cong
X(f_{1})}$ (2.58)

and one can show that this commutes up to simplicial homotopy. The left-hand
and bottom morphisms are weak equivalences between Kan complexes, and thus
homotopy equivalences. It then follows that the right-hand morphism is a
homotopy equivalence as well, showing that $r_{\mathscr{C}!}\dashv
r_{\mathscr{C}}^{*}$ descends to a Quillen adjunction between the
localisations. $\triangleleft$

###### Remark 2.59

The fact that $r_{\mathscr{C}!}\dashv r_{\mathscr{C}}^{*}$ descends to a
Quillen adjunction between the localisations has been sketched without proof
in [11] (see the paragraph after Cor. D). We give a proof here for
completeness. $\triangleleft$

Let
${\mathscr{K}\mathrm{an}}\subset{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$
denote the full subcategory on the Kan complexes.

###### Corollary 2.60

Let $\mathscr{C}$ be a small category, and let
$F\colon\mathscr{C}\to{\mathscr{K}\mathrm{an}}$ be a functor which sends each
morphism in $\mathscr{C}$ to a weak equivalence in
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$. Then, the morphism
$r_{\mathscr{C}}^{*}F\to N\mathscr{C}$ is a Kan fibration.

###### Proof.

A functor $F$ with these properties is precisely a fibrant object in
$L^{B}_{\mathscr{C}}{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
∎

##  3 From strict diagrams in ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$
to coherent diagrams of spaces

In this section we start by recalling Cisinski’s $\infty$-category of spaces
$\mathscr{S}$. We describe in detail the data encoding maps from simplicial
sets to $\mathscr{S}$. Combining this with the rectification functor
$r_{\mathscr{C}}^{*}$ from Section 2.2 we build functors of
$\infty$-categories $\gamma\colon N{\mathscr{K}\mathrm{an}}\to\mathscr{S}$ and
$\gamma_{\mathscr{C}}\colon
N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\longrightarrow{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$,
for any ${\mathrm{U}}$-small 1-category $\mathscr{C}$. Given a functor
$F\colon\mathscr{C}\to{\mathscr{K}\mathrm{an}}$, we show that the
$\infty$-functor $\gamma_{\mathscr{C}}F\colon N\mathscr{C}\to\mathscr{S}$
classifies the left fibration $r_{\mathscr{C}}^{*}F\to N\mathscr{C}$.

###### Definition 3.1

_[ 2, Def. 5.2.3], [3]_ We define the following two $\infty$-categories:

  1. (1)

The _$\infty$ -category $\mathscr{S}$ of (${\mathrm{U}}$-small) spaces_ is the
simplicial set given as follows: an $n$-simplex of $\mathscr{S}$ consists of a
left fibration $p\colon X\to\Delta^{n}$ with ${\mathrm{U}}$-small fibres,
together with a choice, for each $f\colon\Delta^{m}\to\Delta^{n}$, of a
pullback square of ${\mathrm{U}}$-small simplicial sets

${f^{*}X}$${X}$${\Delta^{m}}$${\Delta^{n}}$$\scriptstyle{u}$ (3.2)

such that the pullback assigned to the identity on $\Delta^{n}$ is the
identity on $X$. The simplicial structure of $\mathscr{S}$ arises from the
pasting law for pullbacks

  2. (2)

The _$\infty$ -category_ of pointed (${\mathrm{U}}$-small) spaces is the
simplicial set $\mathscr{S}_{*}$ whose $n$-simplices consist of a left
fibration $p\colon X\to\Delta^{n}$ with ${\mathrm{U}}$-small fibres, together
with a choice of a section $s$ of $p$ and, for each
$u\colon\Delta^{m}\to\Delta^{n}$, of a pullback square of ${\mathrm{U}}$-small
simplicial sets as above. The simplicial structure of $\mathscr{S}_{*}$ arises
from the pasting law for pullbacks

We denote the morphism which forgets the sections by
$p_{\mathrm{univ}}\colon\mathscr{S}_{*}\to\mathscr{S}$. We recall the
following two key facts about the $\infty$-categories $\mathscr{S}$ and
$\mathscr{S}_{*}$:

###### Lemma 3.3

_[ 2, Prop. 5.2.4]_ Specifying a cartesian square

${Y}$${\mathscr{S}_{*}}$${X}$${\mathscr{S}}$$\scriptstyle{\tilde{F}}$$\scriptstyle{f}$$\scriptstyle{p_{\mathrm{univ}}}$$\scriptstyle{F}$
(3.4)

where $f$ is a left fibration, is equivalent to choosing a cartesian square

${\varphi^{*}X}$${X}$${\Delta^{n}}$${Y}$$\scriptstyle{\tilde{\varphi}}$$\scriptstyle{\varphi^{*}f}$$\scriptstyle{f}$$\scriptstyle{\varphi}$
(3.5)

for each map $\varphi\colon\Delta^{n}\to Y$, where $\varphi^{*}X$ is
${\mathrm{U}}$-small.

###### Definition 3.6

[2, Def. 5.2.5] In the situation of Lemma 3.3, we say that _the morphism $F$
classifies the left fibration $f$_.

###### Proposition 3.7

_[ 2, Cor. 5.2.8]_ The morphism
$p_{\mathrm{univ}}\colon\mathscr{S}_{*}\to\mathscr{S}$ is a left fibration.
Every left fibration $f\colon Y\to X$ in
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ is classified by some map
$F\colon X\to\mathscr{S}$.

###### Remark 3.8

Let $L$ be a ${\mathrm{U}}$-small simplicial set. Explicitly, each map of
(${\mathrm{V}}$-small) simplicial sets $F\colon L\to\mathscr{S}$ is specified
as follows:

  * •

The map $F_{n}\colon L_{n}\to\mathscr{S}_{n}$: to each $n$-simplex
$a\colon\Delta^{n}\to L$ of $L$, the map $F$ assigns an $n$-simplex $F_{n}(a)$
of $\mathscr{S}$, i.e. a left fibration $A(a)\to\Delta^{n}$ together with
choices of pullback squares

${A(a;v)}$${A(a)}$${\Delta^{r}}$${\Delta^{n}}$$\scriptstyle{v}$ (3.9)

for each morphism $v\colon\Delta^{r}\to\Delta^{n}$ in
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$.

  * •

Compatibility with the simplicial structure maps of $L$ and $\mathscr{S}$: in
order for $F$ to define a morphism of $({\mathrm{V}}$-small) simplicial sets,
these data have to satisfy the following consistency condition: given a
morphism $u\colon\Delta^{k}\to\Delta^{n}$, we obtain the $k$-simplex $u^{*}a$
of $L$ as the composition $a\circ u\colon\Delta^{k}\to L$. We also obtain a
$k$-simplex $u^{*}F(a)$ in $\mathscr{S}$, and we spell out what it means for
these to agree. The assignment $F$ associates to $u^{*}a\in L_{k}$ a left
fibration $A(u^{*}a)\to\Delta^{k}$ and choices of pullback squares

${A(u^{*}a;v)}$${A(u^{*}a)}$${\Delta^{r}}$${\Delta^{k}}$$\scriptstyle{v}$
(3.10)

for maps $v\colon\Delta^{r}\to\Delta^{k}$. The left fibration
$A(u^{*}a)\to\Delta^{k}$ has to coincide with the left vertical morphism in
the chosen pullback

${A(a;u)}$${A(a)}$${\Delta^{k}}$${\Delta^{n}}$$\scriptstyle{v}$ (3.11)

for the $n$-simplex $F(a)$, i.e. $A(u^{*}a)=A(a;u)$. Further, the chosen
pullbacks of $F(a)$ and $u^{*}F(a)$ must agree, which we may describe by the
equation

$A(u^{*}a;v)=A(a,u\circ v)$ (3.12)

for each map $v\colon\Delta^{r}\to\Delta^{k}$. $\triangleleft$

###### Remark 3.13

An equivalent way to describe a morphism $F\colon L\to\mathscr{S}$ is as
follows: Consider the comma category ${\mathbbe{\Delta}}_{/L}$ and the functor
$\Delta^{\cdot}\circ\mathrm{pr}\colon{\mathbbe{\Delta}}_{/L}\to{\mathbbe{\Delta}}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$
(where $\mathrm{pr}$ is the canonical projection functor). The data from
Remark 3.8 are equivalent to that of a functor
$A\colon{\mathbbe{\Delta}}_{/L}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$,
together with a natural transformation $p\colon
A\to\Delta^{\cdot}\circ\mathrm{pr}$ satisfying that

  1. (1)

$p$ is a cartesian natural transformation (i.e. all its naturality squares are
cartesian) and

  2. (2)

$p$ consists of left fibrations. $\triangleleft$

###### Lemma 3.14

Let $A\to L$ be a left fibration of ${\mathrm{U}}$-small simplicial sets.
Suppose we are given choices of pullback squares

${A(a)}$${A}$${\Delta^{n}}$${L}$$\scriptstyle{a}$ (3.15)

for each map $a\colon\Delta^{n}\to L$. These data induce a consistent family
of left fibrations and choices of pullbacks as in Remark 3.8, i.e. the data of
a morphism $L\to\mathscr{S}$.

###### Proof.

We need to provide, for each morphism $v\colon\Delta^{r}\to\Delta^{n}$, a
choice of pullback square

${A(a;v)}$${A(a)}$${\Delta^{r}}$${\Delta^{n}}$$\scriptstyle{v}$ (3.16)

However, we can augment any such diagram to a double square

${A(a;v)}$${A(a)}$${A}$${\Delta^{r}}$${\Delta^{n}}$${L}$$\scriptstyle{v}$$\scriptstyle{a}$
(3.17)

Note that, by assumption, we have already specified a pullback

${A(a\circ v)}$${A}$${\Delta^{r}}$${L}$$\scriptstyle{a\circ v}$ (3.18)

for the outer square. Thus, we see that setting $A(a;v)=A(a\circ v)$ for each
$v\colon\Delta^{r}\to\Delta^{n}$ produces a consistent family as required. ∎

###### Theorem 3.19

The functors

$\big{\\{}r_{[n]}^{*}\colon{\operatorname{Fun}}([n],{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/\Delta^{n}}\,\big{|}\,n\in\mathbb{N}_{0}\big{\\}}$
(3.20)

assemble into a functor of $\infty$-categories

$\gamma\colon N{\mathscr{K}\mathrm{an}}\longrightarrow\mathscr{S}\,.$ (3.21)

###### Proof.

The functor $\gamma$ sends an $n$-simplex
$X\colon[n]\to{\mathscr{K}\mathrm{an}}$ of $N{\mathscr{K}\mathrm{an}}$ to the
following $n$-simplex of $\mathscr{S}$: its underlying left fibration is given
by $r_{[n]}^{*}X\to\Delta^{n}$. Given any functor $u\colon[k]\to[n]$, we
associate to this the square

${r_{[k]}^{*}(u^{*}X)}$${r_{[n]}^{*}X}$${\Delta^{k}}$${\Delta^{n}}$$\scriptstyle{u}$
(3.22)

which is cartesian by Lemma 2.42. (recall that for a functor
$u\colon[k]\to[n]$ we denote the map $\Delta^{k}\to\Delta^{n}$ induced by the
Yoneda embedding of ${\mathbbe{\Delta}}$ again by $u$). In other words, the
left fibrations and choices of pullbacks are generated under Lemma 3.14 from
the left fibration $r_{[k]}^{*}X\to\Delta^{k}$ and the choices of pullbacks as
in the above diagram. Thus, these data constitute an $n$-simplex of
$\mathscr{S}$. ∎

###### Theorem 3.23

Let $\mathscr{C}$ be a ${\mathrm{U}}$-small category, and let
$X\colon\mathscr{C}\to{\mathscr{K}\mathrm{an}}$ be a functor. The diagram

${r_{\mathscr{C}}^{*}X}$${\mathscr{S}_{*}}$${N\mathscr{C}}$${N{\mathscr{K}\mathrm{an}}}$${\mathscr{S}}$$\scriptstyle{p_{\mathrm{univ}}}$$\scriptstyle{NX}$$\scriptstyle{\gamma}$
(3.24)

is cartesian (in ${\mathrm{V}}{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$).
That is, the the functor $NX\circ\gamma\colon N\mathscr{C}\to\mathscr{S}$
classifies the left fibration $r_{\mathscr{C}}^{*}X\to N\mathscr{C}$.

This provides a simplified analogue of [14, 3.2.5.21].

###### Proof.

We have to show that there is an isomorphism of (${\mathrm{V}}$-small)
simplicial sets

$r_{\mathscr{C}}^{*}X\cong
N\mathscr{C}\underset{\mathscr{S}}{\times}\mathscr{S}_{*}\,.$ (3.25)

An $n$-simplex in $N\mathscr{C}\times_{\mathscr{S}}\mathscr{S}_{*}$ consists
of the following data: first, we have an $n$-simplex of $N_{n}\mathscr{C}$, or
equivalently a functor $\alpha\colon[n]\to\mathscr{C}$. The image of $\alpha$
in $\mathscr{S}$ under the functor $\gamma\circ NX$ is, by construction of
$\gamma$, the left fibration $q_{\alpha}\colon
r_{[n]}^{*}(\alpha^{*}X)\longrightarrow\Delta^{n}$, together with the choices
of pullbacks induced by Lemma 2.42. An $n$-simplex in
$N\mathscr{C}\times_{\mathscr{S}}\mathscr{S}_{*}$ then consists of a pair
$(\alpha,s)$, where $s$ is a section of $q_{\alpha}$. By the above and Lemma
2.42 we have a commutative diagram

${\Delta^{n}}$${r_{[n]}^{*}(\alpha^{*}X)}$${r_{\mathscr{C}}^{*}X}$${\Delta^{n}}$${N\mathscr{C}}$$\scriptstyle{1_{\Delta^{n}}}$$\scriptstyle{a}$$\scriptstyle{s}$$\scriptstyle{q_{\alpha}}$$\scriptstyle{q}$$\scriptstyle{N\alpha}$
(3.26)

in ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$. The top morphism now
defines an $n$-simplex in $r_{\mathscr{C}}^{*}X$ from the pair $(\alpha,s)$.
This defines a map on $n$-simplices

$\big{(}(N\mathscr{C})\times_{\mathscr{S}}\mathscr{S}_{*}\big{)}_{n}\longrightarrow\big{(}r_{\mathscr{C}}^{*}X\big{)}_{n}\,.$
(3.27)

In the opposite direction, denote the left fibration $r_{\mathscr{C}}^{*}X\to
N\mathscr{C}$ by $q$. Given an $n$-simplex $a\colon\Delta^{n}\to
r_{\mathscr{C}}^{*}X$, set $\alpha^{\prime}\coloneqq q\circ a$. This is the
nerve of a unique functor $\alpha\colon[n]\to\mathscr{C}$ (since the nerve
functor
$N\colon{\mathscr{C}\mathrm{at}}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$
is fully faithful as a functor of 1-categories). Consider the diagram

${\Delta^{n}}$${r_{[n]}^{*}(\alpha^{*}X)}$${r_{\mathscr{C}}^{*}X}$${\Delta^{n}}$${N\mathscr{C}}$$\scriptstyle{1_{\Delta^{n}}}$$\scriptstyle{a}$$\scriptstyle{s}$$\scriptstyle{q_{\alpha}}$$\scriptstyle{q}$$\scriptstyle{N\alpha}$
(3.28)

By Lemma 2.42 the square in this diagram is cartesian, so that there is a
unique morphism $s$ fitting into this diagram as the dashed morphism. We thus
obtain a map

$\big{(}r_{\mathscr{C}}^{*}X\big{)}_{n}\longrightarrow\big{(}(N\mathscr{C})\times_{\mathscr{S}}\mathscr{S}_{*}\big{)}_{n}\,.$
(3.29)

The universal property of pullbacks ensures that this is the inverse to the
above map on $n$-simplices. Finally, it follows from the pasting law for
pullbacks and Lemma 2.42 that these maps on $n$-simplices form maps of
simplicial sets. ∎

Next, we enhance Theorem 3.19 from providing a functor between
$N{\mathscr{K}\mathrm{an}}$ and the $\infty$-category of spaces $\mathscr{S}$
so that it provides functors between diagrams in both $\infty$-categories.

###### Remark 3.30

Let $L$ be a ${\mathrm{U}}$-small simplicial set. We can describe the
simplicial set ${\mathscr{F}\mathrm{un}}(L,\mathscr{S})$ explicitly as
follows: an $n$-simplex in ${\mathscr{F}\mathrm{un}}(L,\mathscr{S})$ is a
morphism $F\colon\Delta^{n}{\times}L\to\mathscr{S}$. Such a morphism
corresponds to specifying, for each morphism
$a\colon\Delta^{k}\to\Delta^{n}{\times}L$, a left fibration
$A(a)\to\Delta^{k}$ with further choices of pullbacks $A(a;v)$ for each
morphism $v\colon\Delta^{r}\to\Delta^{k}$ (using the notation of Remark 3.8),
satisfying the consistency condition $A(a;v)=A(a\circ v)$ from Remark 3.8. We
denote the data corresponding to the $n$-simplex $F$ by $\\{A\\}$.

Next, we describe the action of the simplicial structure maps of
${\mathscr{F}\mathrm{un}}(L,\mathscr{S})$. Consider a morphism
$u\colon\Delta^{l}\to\Delta^{n}$; this defines a simplicial structure map
$(u{\times}1_{L})^{*}\colon{\mathscr{F}\mathrm{un}}(L,\mathscr{S})_{n}\to{\mathscr{F}\mathrm{un}}(L,\mathscr{S})_{l}$.
It acts by precomposing morphisms $F\colon\Delta^{n}{\times}L\to\mathscr{S}$
with the morphism
$(u{\times}1_{L})\colon\Delta^{l}{\times}L\to\Delta^{n}{\times}L$. Given the
above $n$-simplex $F$, specified by the consistent collection of left
fibrations and chosen pullbacks $\\{A\\}$, we describe its image under
$(u{\times}1_{L})^{*}$ explicitly: for a morphism
$b\colon\Delta^{k}\to\Delta^{l}{\times}L$, the associated left fibration of
the new $l$-simplex is given by $A^{\prime}(b)=A((u{\times}1_{L})\circ
b)\to\Delta^{k}$, with the necessary choices of pullback squares induced in
the same manner. This gives a new consistent family of left fibrations and
pullback squares $\\{A^{\prime}\\}$ which corresponds to the morphism
$(u{\times}1_{L})^{*}F\colon\Delta^{l}{\times}L\to\mathscr{S}$.
$\triangleleft$

We have the following enhancement of Theorem 3.19:

###### Theorem 3.31

Let $\mathscr{C}$ be a small category. The functors

$\big{\\{}r_{[n]{\times}\mathscr{C}}^{*}\colon{\operatorname{Fun}}\big{(}[n]{\times}\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}\big{)}\longrightarrow{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/\Delta^{n}\times
N\mathscr{C}}\,\big{|}\,n\in\mathbb{N}_{0}\big{\\}}$ (3.32)

assemble into a functor between $\infty$-categories

$\gamma_{\mathscr{C}}\colon
N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\longrightarrow{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})\,.$
(3.33)

###### Proof.

There are canonical isomorphisms

$N_{n}{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\cong{\operatorname{Fun}}\big{(}[n]{\times}\mathscr{C},{\mathscr{K}\mathrm{an}}\big{)}\quad\text{and}\quad\big{(}{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})\big{)}_{n}\cong{\mathrm{V}}{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}(\Delta^{n}{\times}N\mathscr{C},\mathscr{S})\,.$
(3.34)

Let $X\colon[n]\to{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$
be an $n$-simplex in
$N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$. Equivalently,
we can write these data as a functor
$X^{\dashv}\colon[n]{\times}\mathscr{C}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$.
We produce from this a consistent family $\\{A\\}$ of left fibrations and
choices of pullbacks as in Remark 3.30 (recall that these families correspond
to $n$-simplices in ${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$).
This will be aided by Lemma 3.14: first, consider the left fibration
$r_{[n]{\times}\mathscr{C}}^{*}X^{\dashv}\longrightarrow\Delta^{n}{\times}N\mathscr{C}$.
Let $\alpha\colon[k]\to[n]{\times}\mathscr{C}$ be any functor. We assign to
this the square

${A(a)=r_{[k]}^{*}(\alpha^{*}X^{\dashv})}$${r_{[n]{\times}\mathscr{C}}^{*}X^{\dashv}=A}$${\Delta^{k}}$${\Delta^{n}{\times}N\mathscr{C}}$$\scriptstyle{N\alpha}$
(3.35)

This is a pullback square by Lemma 2.42. From the left fibration
$r_{[n]{\times}\mathscr{C}}^{*}X^{\dashv}\longrightarrow\Delta^{n}{\times}N\mathscr{C}$
and these choices of pullback squares, Lemma 3.14 produces a consistent family
$\\{A\\}$ as required, i.e. a morphism
$\Delta^{n}{\times}N\mathscr{C}\to\mathscr{S}$ and thus, equivalently, an
$n$-simplex in ${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$.

It remains to show that this assignment $X\longmapsto\\{A\\}$ defines a
morphism
$N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\longrightarrow{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$
of simplicial sets; that is, we need to show that it is compatible with the
simplicial structure maps. Let $u\colon[l]\to[n]$ be a morphism in
${\mathbbe{\Delta}}$. On the domain side, the structure map associated to $u$
sends an $n$-simplex
$X^{\dashv}\colon[n]{\times}\mathscr{C}\to{\mathscr{K}\mathrm{an}}$ to the
$l$-simplex
$(u{\times}1_{\mathscr{C}})^{*}X^{\dashv}\colon[l]{\times}\mathscr{C}\to{\mathscr{K}\mathrm{an}}$.
The coherent family associated to this by the above construction arises (via
Lemma 3.14) from the left fibration

$r_{[l]{\times}\mathscr{C}}^{*}\big{(}(u{\times}1_{\mathscr{C}})^{*}X^{\dashv}\big{)}\longrightarrow\Delta^{l}{\times}N\mathscr{C}\,,$
(3.36)

together with the choices of pullbacks

${r_{[k]}^{*}(\beta^{*}(u{\times}1_{\mathscr{C}})^{*}X^{\dashv})}$${r_{[l]{\times}\mathscr{C}}^{*}\big{(}(u{\times}1_{\mathscr{C}})^{*}X^{\dashv}\big{)}}$${\Delta^{k}}$${\Delta^{l}{\times}N\mathscr{C}}$$\scriptstyle{N\beta}$
(3.37)

for each functor $\beta\colon[k]\to[l]{\times}\mathscr{C}$. Invoking the
explicit formulas for the action of the simplicial structure maps in
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ from Remark 3.30, we see
that this agrees with the result of applying the structure map associated to
$u$ on the codomain side to the consistent family $\\{A\\}$ constructed above.
∎

###### Theorem 3.38

Let $\mathscr{C}$ be a small category, and let
$X\colon\mathscr{C}\to{\mathscr{K}\mathrm{an}}$ be a functor. There is a
strictly commutative diagram of categories and functors

${N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{C},N{\mathscr{K}\mathrm{an}})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})}$$\scriptstyle{\gamma_{\mathscr{C}}}$$\scriptstyle{\cong}$$\scriptstyle{\gamma\circ(-)}$
(3.39)

###### Proof.

Let $X\colon\Delta^{n}\to
N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$ be an $n$-simplex
of $N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$. Under the
tensor-hom adjunction, this corresponds to a unique functor
$X^{\dashv}\colon[n]{\times}\mathscr{C}\to{\mathscr{K}\mathrm{an}}$. The image
of this $n$-simplex under the functor $\gamma_{\mathscr{C}}$ is the
$n$-simplex

$\gamma_{\mathscr{C}}(X)\in{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})_{n}\cong{\mathrm{V}}{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}(\Delta^{n}{\times}N\mathscr{C},\mathscr{S})$
(3.40)

which is induced via Lemma 3.14 from the left fibration
$r_{[n]\times\mathscr{C}}^{*}X^{\dashv}\longrightarrow\Delta^{n}{\times}N\mathscr{C}$
and the choices of pullbacks
$A(N\alpha)=r_{[k]}^{*}(\alpha^{*}X^{\dashv})\to\Delta^{k}$, for functors
$\alpha\colon[k]\to[n]{\times}\mathscr{C}$ (in the notation of Remark 3.8; see
also the proof of Theorem 3.31). More explicitly, $\gamma_{\mathscr{C}}(X)$ is
a morphism of simplicial sets $\Delta^{n}{\times}N\mathscr{C}\to\mathscr{S}$;
it sends a $k$-simplex $\alpha\colon[k]\to[n]{\times}\mathscr{C}$ to the
$k$-simplex of $\mathscr{S}$ which is given by the left fibration
$r_{[k]}^{*}(\alpha^{*}X^{\dashv})\longrightarrow\Delta^{k}$ together with the
choices of pullbacks induced by Lemma 2.42.

We compare this to the $n$-simplex of
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ corresponding to the map

${\Delta^{n}\times N\mathscr{C}\cong
N([n]\times\mathscr{C})}$${N{\mathscr{K}\mathrm{an}}}$${\mathscr{S}\,.}$$\scriptstyle{NX^{\dashv}}$$\scriptstyle{\gamma}$
(3.41)

Using Remark 3.30 and the construction of the functor $\gamma$ in (the proof
of) Theorem 3.19, we can describe the action of this second map on
$k$-simplices explicitly: consider a functor
$\alpha\colon[k]\to[n]{\times}\mathscr{C}$. We obtain a $k$-simplex

$X^{\dashv}\circ\alpha=\alpha^{*}X^{\dashv}\colon[k]\to{\mathscr{K}\mathrm{an}}$
(3.42)

of $N{\mathscr{K}\mathrm{an}}$. By construction, the functor $\gamma$ sends
this $k$-simplex to the left fibration
$r_{[k]}^{*}(\alpha^{*}X^{\dashv})\longrightarrow\Delta^{k}$, together with
the choices of pullbacks induced by Lemma 2.42. Therefore, we see that the
morphisms
$\gamma_{\mathscr{C}}(X)\colon\Delta^{n}{\times}N\mathscr{C}\to\mathscr{S}$
and $\gamma\circ
NX^{\dashv}\colon\Delta^{n}{\times}N\mathscr{C}\to\mathscr{S}$ agree. ∎

###### Corollary 3.43

The left fibration classified by $\gamma_{\mathscr{C}}X\colon
N\mathscr{C}\to\mathscr{S}$ is given by $r_{\mathscr{C}}^{*}X\to
N\mathscr{C}$.

###### Proof.

This now follows immediately from combining Theorems 3.23 and 3.38. ∎

###### Remark 3.44

For a morphism $f\colon K\to L$ of simplicial sets, we can describe the
induced morphism
$f^{*}\colon{\mathscr{F}\mathrm{un}}(L,\mathscr{S})\longrightarrow{\mathscr{F}\mathrm{un}}(K,\mathscr{S})$
as follows: let $\\{A\\}$ be a consistent family of left fibrations
$A(a)\to\Delta^{k}$, for each $k\in\mathbb{N}_{0}$ and
$a\colon\Delta^{k}\to\Delta^{n}{\times}L$, describing an $n$-simplex
$F\colon\Delta^{n}{\times}L\to\mathscr{S}$ in
${\mathscr{F}\mathrm{un}}(L,\mathscr{S})$ as in Remark 3.30. The consistent
family $\\{f^{*}A\\}$, describing the $n$-simplex

${f^{*}F\colon\Delta^{n}\times K}$${\Delta^{n}\times
L}$${\mathscr{S}\,,}$$\scriptstyle{1_{\Delta^{n}}\times f}$$\scriptstyle{F}$
(3.45)

assigns to a map $b\colon\Delta^{k}\to\Delta^{n}{\times}K$ the left fibration

$A\big{(}(1_{\Delta^{n}}\times f)\circ b\big{)}\longrightarrow\Delta^{k}\,,$
(3.46)

which is part of the original consistent family $\\{A\\}$ that describes $F$.
For each morphism $u\colon\Delta^{m}\to\Delta^{n}$ we obtain a commutative
square

${\Delta^{m}\times K}$${\Delta^{m}\times L}$${\Delta^{n}\times
K}$${\Delta^{n}\times L}$$\scriptstyle{1_{\Delta^{m}\times
f}}$$\scriptstyle{u\times 1_{K}}$$\scriptstyle{u\times
1_{L}}$$\scriptstyle{1_{\Delta^{n}\times f}}$ (3.47)

This implies that $f^{*}$ is indeed a morphism of simplicial sets
${\mathscr{F}\mathrm{un}}(L,\mathscr{S})\longrightarrow{\mathscr{F}\mathrm{un}}(K,\mathscr{S})$.
$\triangleleft$

###### Corollary 3.48

Any functor $\psi\colon\mathscr{C}\to\mathscr{D}$ of ${\mathrm{U}}$-small
categories induces a commutative square

${N{\operatorname{Fun}}(\mathscr{D},{\mathscr{K}\mathrm{an}})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{D},\mathscr{S})}$${N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})}$$\scriptstyle{\gamma_{\mathscr{D}}}$$\scriptstyle{\psi^{*}}$$\scriptstyle{N\psi^{*}}$$\scriptstyle{\gamma_{\mathscr{C}}}$
(3.49)

This is compatible with composition of functors.

###### Proof.

Given any $n$-simplex
$X\colon[n]{\times}\mathscr{C}\to{\mathscr{K}\mathrm{an}}$ in
$N{\mathscr{K}\mathrm{an}}$, we readily observe that

$\gamma\circ N\big{(}X^{\dashv}\circ(1_{[n]}\times\psi)\big{)}=\gamma\circ
N(X^{\dashv})\circ(1_{\Delta^{n}}\times N\psi)\,.$ (3.50)

The claim now follows from Theorem 3.38. ∎

##  4 Localisation of simplicial diagrams

In this section we prove our main theorem: the functor $\gamma_{\mathscr{C}}$
exhibits ${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ as the
$\infty$-categorical localisation of
$N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$ at the
objectwise weak homotopy equivalences. To achieve this, we first show that
$\gamma_{\mathscr{C}}$ detects objectwise weak homotopy equivalences and that
it is a left exact functor between $\infty$-categories with weak equivalences
and fibrations (see Appendix A for background).

###### Proposition 4.1

Suppose $f\colon X_{0}\to X_{1}$ is a morphism in
${\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$, corresponding to
a functor $X_{f}\colon[1]{\times}\mathscr{C}\to{\mathscr{K}\mathrm{an}}$.
Then, $f$ is an objectwise weak equivalence in
${\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$ if and only if
the 1-simplex
$\gamma_{\mathscr{C}}(X_{f})\in{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})_{1}$
is an equivalence in the $\infty$-category
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$.

###### Proof.

A morphism
$F\in{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})_{1}\cong{\mathrm{V}}{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}(\Delta^{1}{\times}N\mathscr{C},\mathscr{S})$
is an objectwise equivalence precisely if, for each $c\in\mathscr{C}$, the
restriction $F_{|c}$ of $F$ to $\Delta^{1}{\times}\\{c\\}$ is an equivalence
in $\mathscr{S}$ [2, Cor. 3.5.12].

Suppose first that $f\colon X_{0}\to X_{1}$ is a weak equivalence. In
particular, for each $c\in\mathscr{C}$, the morphism $f_{|c}\colon X_{0}(c)\to
X_{1}(c)$ is a weak equivalence between Kan complexes, and thus a fibrant
object in
$L^{B}_{[1]}{\operatorname{Fun}}([1],{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
in the notation of Proposition 2.51. By Corollary 3.43 the morphism

$\gamma_{\mathscr{C}}(X_{f})\colon\Delta^{1}{\times}N\mathscr{C}\longrightarrow\mathscr{S}$
(4.2)

classifies the left fibration
$r_{[1]{\times}\mathscr{C}}^{*}X_{f}\longrightarrow\Delta^{1}{\times}N\mathscr{C}$.
By Lemma 2.42, there is a cartesian square

${r_{[1]}^{*}(X_{f_{|c}})}$${r_{[1]\times\mathscr{C}}^{*}(X_{f})}$${\Delta^{1}\times\\{c\\}}$${\Delta^{1}\times
N\mathscr{C}}$ (4.3)

That is, the restriction
$(\gamma_{\mathscr{C}}X_{f})_{|c}\colon\Delta^{1}{\times}\\{c\\}\to\mathscr{S}$
classifies the left fibration $r_{[1]}^{*}(X_{f_{|c}})\to\Delta^{1}$.
According to Corollary 2.60, this is even a Kan fibration. Thus, the functor
$(\gamma_{[1]\times\mathscr{C}}X_{f})_{|c}\colon\Delta^{1}\to\mathscr{S}$
factors through the maximal subgroupoid $k(\mathscr{S})\subset\mathscr{S}$, by
[2, Prop. 5.2.13].

Now suppose that $f\colon X_{0}\to X_{1}$ is such that
$\gamma_{\mathscr{C}}(X_{f})$ is an equivalence in
${\operatorname{Fun}}(N\mathscr{C},\mathscr{S})$. By Corollary 3.43, the above
pullback diagram and [2, Prop. 5.2.13], it follows that

$r_{\\{c\\}\times[1]}^{*}(X_{f_{|c}})\cong(r_{\mathscr{C}\times[1]}X_{f})_{|c}\longrightarrow\Delta^{1}$
(4.4)

is a Kan fibration, for each $c\in\mathscr{C}$. We need to show that
$X_{f_{|c}}$ is local in
$L^{B}_{[1]}{\operatorname{Fun}}([1],{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$,
for each $c\in\mathscr{C}$. That will follow if $X_{f_{|c}}$ is isomorphic in
${\operatorname{Ho}}({\operatorname{Fun}}([1],{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}))$
to a diagram $[1]\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$ which sends
the non-identity morphism in $[1]$ to a weak equivalence. Theorem 2.49 shows
that $h_{[1]}^{*}r_{[1]}^{*}X_{f_{|c}}$ is a fibrant object in
$L^{B}_{[1]}{\operatorname{Fun}}([1],{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$,
and Theorem 2.46 implies that there is an isomorphism in
${\operatorname{Ho}}({\operatorname{Fun}}([1],{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}))$
between $X_{f_{|c}}$ and $h_{[1]}^{*}r_{[1]}^{*}X_{f_{|c}}$. ∎

###### Remark 4.5

Alternatively, one can prove Proposition 4.1 by appealing to Corollary 3.43,
Proposition 2.51 and [2, Prop. 5.3.16]. $\triangleleft$

###### Remark 4.6

We turn
$N{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
into an $\infty$-category with weak equivalences and fibrations (see
Definition A.3): we define its class ${\mathsf{F}}$ of fibrations to consist
of the projective fibrations and its class $\mathsf{W}$ of weak equivalences
to consist of the objectwise weak equivalences. That this satisfies the axioms
in Definitions A.1 and A.3 follows directly from the model category axioms for
the projective model structure on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
Its underlying $\infty$-category of fibrant objects (see Definition A.8)
coincides with $N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$
(see also Example A.9). $\triangleleft$

###### Corollary 4.7

The functor $\gamma_{\mathscr{C}}\colon
N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\longrightarrow{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$
factors through the $\infty$-categorical localisation
$L_{\mathsf{W}}N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$
and detects _weak_ equivalences.

###### Remark 4.8

We turn ${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ into an
$\infty$-category with weak equivalences and fibrations by defining its class
of fibrations ${\mathsf{F}}^{\prime}$ to consist of _all_ morphisms and its
class $\mathsf{W}^{\prime}$ to consist of the honest equivalences (i.e. the
objectwise weakly invertible morphisms) in
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$. This satisfies the
axioms in Definitions A.1 and A.3 because $\mathscr{S}$ has finite limits.
$\triangleleft$

The following is a well-known result, but we state it here nevertheless; the
construction of $\gamma_{\mathscr{C}}$ allows us to see this result very
explicitly.

###### Lemma 4.9

Let $\mathscr{I}$ be a ${\mathrm{U}}$-small category, and let
$X\colon\mathscr{I}\to{\mathscr{K}\mathrm{an}}$ be a functor. Then, there is a
canonical equivalence of spaces

$\gamma(\operatorname{holim}_{\mathscr{I}}X)\simeq\operatorname{lim}^{\mathscr{S}}_{\mathscr{I}}(\gamma_{\mathscr{I}}X)\,.$
(4.10)

Here, $\gamma$ is as in Theorem 3.19 and $\gamma_{\mathscr{I}}$ is as in
Theorem 3.31.

###### Proof.

The functor $\gamma_{\mathscr{I}}X$ classifies the left fibration
$r_{\mathscr{I}}^{*}X\to N\mathscr{I}$ (by Corollary 3.43). The
$\infty$-categorical limit
$\operatorname{lim}^{\mathscr{S}}_{\mathscr{I}}(\gamma_{\mathscr{I}}X)$ is
computed by the space of sections of this map by [2, Cor. 5.4.7] under the
equivalence
$\mathrm{h}\mathscr{S}\simeq{\operatorname{Ho}}({\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
(see, for instance, [2, Cor. 5.3.21, Thm. 5.4.5] for $A=*$; note that this
does not rely on the statement that $\mathscr{S}$ is the $\infty$-categorical
localisation of ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$). One checks
that the identity map $1_{N\mathscr{I}}$ coincides with
$r_{\mathscr{I}}^{*}(*)$ as objects of
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{I}}$; here,
$r_{\mathscr{I}}^{*}(*)$ is the functor
$r_{\mathscr{I}}^{*}\colon{\operatorname{Fun}}(\mathscr{I},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\longrightarrow{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{I}}$
applied to the constant diagram with value $\Delta^{0}$. Furthermore, by
Theorem 2.46 and since each object in
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{I}}$ is cofibrant,
the (non-derived) counit $r_{\mathscr{I}!}r_{\mathscr{I}}^{*}(*)\to*$ is a
cofibrant replacement of $*$ in the projective model structure on
${\operatorname{Fun}}(\mathscr{I},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
Consequently, we have the following canonical isomorphisms in
${\operatorname{Ho}}({\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$:

$\displaystyle\underline{{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{I}}}(N\mathscr{I},r_{\mathscr{I}}^{*}X)$
$\displaystyle\cong\underline{{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{I}}}(r_{\mathscr{I}}^{*}(*),X)$
(4.11)
$\displaystyle\cong\underline{{\operatorname{Fun}}(\mathscr{I},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(r_{\mathscr{I}!}r_{\mathscr{I}}^{*}(*),X)$
(4.12) $\displaystyle\simeq\operatorname{holim}_{\mathscr{I}}(X)\,.$ (4.13)

That completes the proof. ∎

###### Proposition 4.14

The functor $\gamma_{\mathscr{C}}\colon
N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\longrightarrow{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$
is left exact as a functor between $\infty$-categories with weak equivalences
and fibrations (see Definition A.10).

###### Proof.

We need to check the axioms of Definition A.10. Axiom (1) is satisfied because
$\gamma_{\mathscr{C}}(*)$ is the functor $N\mathscr{C}\to\mathscr{S}$
classifies the left fibration $1_{N\mathscr{C}}\colon N\mathscr{C}\to
N\mathscr{C}$, which corresponds to the final object in
${\operatorname{Fun}}(N\mathscr{C},\mathscr{S})$. (Alternatively, the final
object of ${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ is the
essentially unique functor whose value on each $c\in\mathscr{C}$ is a final
object $e\in\mathscr{S}$, and we have
$(\gamma_{\mathscr{C}}(*))(c)\simeq\gamma_{[0]}(*(c))\simeq e$.) Axiom (2)
holds true because each morphism in $\mathscr{S}$ is a fibration (see Remark
4.8) and the functor $\gamma_{\mathscr{C}}$ preserves weak equivalences (by
Proposition 4.1). For axiom (3), consider morphisms $v\colon Y^{\prime}\to Y$
and $f\colon X\to Y$ in
${\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$, where $f$ is a
projective fibration. Let $\mathrm{CSp}=\\{0\to 1\leftarrow 2\\}$ denote the
cospan category, and let
$D\colon\mathrm{CSp}\times\mathscr{C}\to{\mathscr{K}\mathrm{an}}$ denote the
diagram corresponding to the cospan $Y^{\prime}\to Y\leftarrow X$ in
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
We have to show that the square

${\gamma_{\mathscr{C}}\big{(}Y^{\prime}\times_{Y}X\big{)}}$${\gamma_{\mathscr{C}}(X)}$${\gamma_{\mathscr{C}}(Y^{\prime})}$${\gamma_{\mathscr{C}}(Y)}$$\scriptstyle{\gamma_{\mathscr{C}}(f)}$$\scriptstyle{\gamma_{\mathscr{C}}(v)}$
(4.15)

is cartesian in ${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$. The
cospan obtained by omitting the top-left corner of this diagram is
equivalently described by a digram $\tilde{D}\colon\Lambda^{2}_{2}\times
N\mathscr{C}\to\mathscr{S}$ (since $N\mathrm{CSp}=\Lambda^{2}_{2}$).
Explicitly, this diagram is given as
$\tilde{D}=\gamma_{\mathrm{CSp}\times\mathscr{C}}D$.

There is a canonical morphism
$\gamma_{\mathscr{C}}(Y^{\prime}\times_{Y}X)\to\operatorname{lim}_{\Lambda^{2}_{2}}D$,
and it suffices to show that this morphism is an equivalence in
${\operatorname{Fun}}(N\mathscr{C},\mathscr{S})$. However, since limits in
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ are computed objectwise,
we may restrict ourselves to the case where $\mathscr{C}=*$ is the final
category. That is, we may replace the $\infty$-category
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ by $\mathscr{S}$. In that
case, we need to show that the canonical morphism

$\gamma(Y^{\prime}\times_{Y}X)\longrightarrow\operatorname{lim}^{\mathscr{S}}_{\Lambda^{2}_{2}}(\gamma_{\mathrm{CSp}}D)$
(4.16)

is an equivalence (with $\gamma$ as in Theorem 3.19). We have shown in Lemma
4.9 that there is a canonical equivalence

$\gamma\big{(}\operatorname{holim}_{\mathrm{CSp}}D\big{)}\simeq\operatorname{lim}^{\mathscr{S}}_{\Lambda^{2}_{2}}(\gamma_{\mathrm{CSp}}D)\,.$
(4.17)

However, by our assumptions on the cospan $Y^{\prime}\to Y\leftarrow X$, the
pullback $Y^{\prime}\times_{Y}X$ already computes the homotopy limit of $D$,
showing that
$Y^{\prime}\times_{Y}X\to\operatorname{lim}^{\mathscr{S}}(\gamma_{\mathrm{CSp}}D)$
is indeed an equivalence. ∎

###### Theorem 4.18

Let $\mathscr{C}$ be a ${\mathrm{U}}$-small category. The functor
$\gamma_{\mathscr{C}}\colon
N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\longrightarrow{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$
exhibits ${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ as the
$\infty$-categorical localisation of
${\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$ at the objectwise
weak equivalences.

###### Proof.

Using the notation of Remarks 4.6 and 4.8, we have a commutative diagram

${N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})}$${L_{\mathsf{W}}N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})}$${L_{\mathsf{W}^{\prime}}{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})}$$\scriptstyle{\gamma_{\mathscr{C}}}$$\scriptstyle{\ell_{\mathsf{W}}}$$\scriptstyle{\mathbb{R}\gamma_{\mathscr{C}}}$
(4.19)

The vertical functors are localisation functors; the equality on the right-
hand side holds true because we have chosen $\mathsf{W}^{\prime}$ to consist
precisely of the honest equivalences in
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$. Our goal is to show that
the right derived functor

$\mathbb{R}\gamma_{\mathscr{C}}\colon
L_{\mathsf{W}}N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\longrightarrow
L_{\mathsf{W}^{\prime}}{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})={\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$
(4.20)

is an equivalence. By Propositions 4.14 and A.14 it suffices to show that the
induced functor

$\mathrm{h}\mathbb{R}\gamma_{\mathscr{C}}\colon\mathrm{h}\big{(}L_{\mathsf{W}}N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\big{)}\longrightarrow\mathrm{h}\big{(}{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})\big{)}$
(4.21)

on homotopy categories is an equivalence (of ordinary categories). By [2, Rmk.
7.1.6] there is a canonical equivalence of categories

$\mathrm{h}\big{(}L_{\mathsf{W}}N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\big{)}\simeq{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})[\mathsf{W}^{-1}]\,,$
(4.22)

where on the right-hand side we have the ordinary 1-categorical localisation.
The latter can be computed has the model-categorical homotopy category
${\operatorname{Ho}}({\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}))$
of the projective model structure.

We claim that there is a commutative diagram

${{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})}$${\mathrm{h}{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})}$${{\operatorname{Ho}}\big{(}{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})\big{)}}$${{\operatorname{Ho}}({\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}})}$$\scriptstyle{\mathrm{h}\gamma_{\mathscr{C}}}$$\scriptstyle{\operatorname{RepFib}}$$\scriptstyle{\mathbb{R}r_{\mathscr{C}}^{*}}$
(4.23)

of (ordinary) categories and functors. The functor $\operatorname{RepFib}$
sends an $\infty$-functor $F\colon N\mathscr{C}\to\mathscr{S}$ to the left
fibration it classifies; we refer to [2, Par. 5.3.14] and the proof of [2,
Thm. 5.4.5] for details. The functor $\mathbb{R}r_{\mathscr{C}}^{*}$ is an
equivalence by Theorem 2.46, as is the functor $\operatorname{RepFib}$ [2,
Thm. 5.4.5] (note that Cisinski writes $\mathit{LFib}(N\mathscr{C})$ for
${\operatorname{Ho}}({\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}})$).
Granted the claim that diagram (4.23) commutes, it follows that the functor
$\mathrm{h}\gamma_{\mathscr{C}}$ exhibits
$\mathrm{h}{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ as the
localisation of
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
at the objectwise weak equivalences, as ordinary categories. Consequently, the
induced functor from the homotopy category of the localisation

${\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})[\mathsf{W}^{-1}]\simeq\mathrm{h}\big{(}L_{\mathsf{W}}N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\big{)}\longrightarrow\mathrm{h}{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$
(4.24)

is an equivalence. By the universal property of 1-categorical localisation
this functor is canonically equivalent to
$\mathrm{h}\mathbb{R}\gamma_{\mathscr{C}}$. We thus infer that
$\mathrm{h}\mathbb{R}\gamma_{\mathscr{C}}$ is an equivalence as well.

It remains to show that diagram (4.23) commutes. The commutativity on objects
follows directly from Corollary 3.43; we can write

$\operatorname{RepFib}\circ\gamma_{\mathscr{C}}(X)=(r_{\mathscr{C}}^{*}X\to
N\mathscr{C})\,.$ (4.25)

To see the commutativity on morphisms, let $f\colon X_{0}\to X_{1}$ be a
morphism in ${\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$.
Equivalently, this is encoded in a functor
$X_{f}\colon[1]\times\mathscr{C}\to{\mathscr{K}\mathrm{an}}$. The
corresponding 1-simplex in
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ classifies the left
fibration

$r_{[1]\times\mathscr{C}}^{*}X_{f}\longrightarrow\Delta^{1}\times
N\mathscr{C}\,.$ (4.26)

The morphism in
${\operatorname{Ho}}({\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}})$
obtained from this by applying the functor $\operatorname{RepFib}$ is
described as follows (see [2, Par. 5.3.14]): first, note that because both
objects are fibrant and cofibrant the set of morphisms
${\operatorname{Ho}}({\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}})(r_{\mathscr{C}}^{*}X_{0},r_{\mathscr{C}}^{*}X_{1})$
coincides with the set
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}(r_{\mathscr{C}}^{*}X_{0},r_{\mathscr{C}}^{*}X_{1})/{\sim_{l}}$
of left homotopy equivalence classes of morphisms in
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}(r_{\mathscr{C}}^{*}X_{0},r_{\mathscr{C}}^{*}X_{1})$
(in the sense of, for instance, [12, Def. 1.2.4]). Consider the following
diagram in ${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$:

${\Delta^{\\{0\\}}\times
r_{\mathscr{C}}^{*}X_{0}}$${r_{[1]\times\mathscr{C}}^{*}X_{f}}$${\Delta^{1}\times
r_{\mathscr{C}}^{*}X_{0}}$${\Delta^{1}\times
N\mathscr{C}}$$\scriptstyle{\varphi}$ (4.27)

It admits a lift $\varphi$ because the left-hand vertical morphism is a
cofinal inclusion and hence left anodyne [2, Cor. 4.1.9], and the right-hand
vertical morphism is a left fibration. Taking the fibre of $\varphi$ over
$\Delta^{\\{1\\}}\hookrightarrow\Delta^{1}$ (and using Lemma 2.42), we obtain
a morphism $\varphi_{1}\colon r_{\mathscr{C}}^{*}X_{0}\longrightarrow
r_{\mathscr{C}}^{*}X_{1}$. Its class in
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}(r_{\mathscr{C}}^{*}X_{0},r_{\mathscr{C}}^{*}X_{1})/{\sim_{l}}$
is the image of $X_{f}$ under
$\operatorname{RepFib}\circ\,\mathrm{h}\gamma_{\mathscr{C}}$ (where $\sim_{l}$
denotes taking left homotopy classes); see [2, Par. 5.3.14] and the proof of
[2, Thm.5.4.5]. This left homotopy class is independent of the choice of lift
$\varphi$ by [2, Prop. 5.3.15].

We have to show that there exists a choice for $\varphi$ such that
$\varphi_{1}\sim_{l}r_{\mathscr{C}}^{*}(f)$. This is greatly simplified by the
description of $r_{\mathscr{C}}^{*}$ from Lemma 2.18: each $n$-simplex of
$\Delta^{1}{\times}N\mathscr{C}$ arises from a unique pair $(u,\alpha)$ of
functors

$u\colon[n]\to[1]\qquad\text{and}\qquad\alpha\colon[n]\to\mathscr{C}$ (4.28)

by applying the Yoneda embedding of ${\mathbbe{\Delta}}$ to $u$ and the nerve
functor to $\alpha$ (recall that we denote the map $\Delta^{n}\to\Delta^{1}$
arising from $u$ again by $u$). We may describe each $n$-simplex in
$r_{[1]\times\mathscr{C}}^{*}X_{f}$ over an $n$-simplex
$(u,N\alpha)\in\Delta^{1}_{n}\times N_{n}\mathscr{C}$ as a family
$z=(z_{ij})_{i,j}$, where $z_{ij}\in X_{f}(u(j),\alpha_{j})_{i}$, for each
$0\leq i\leq j\leq n$, such that $z$ forms a section as in Lemma 2.18 (compare
also (2.16) and surrounding text). That is, $z$ satisfies the identities

$X_{f}(u(j-1,j),\alpha_{j-1,j})(z_{i,j-1})=z_{ij}=d_{i+1}(z_{i+1,j})\,,\quad\forall\
0\leq i<j\leq n\,,$ (4.29)

where $(u(j-1,j),\alpha_{j-1,j})$ is the image under $(u,\alpha)$ of the
unique morphism $(j{-}1)\to j$ in $[n]$.

We can naturally split this family into two parts, $z=(x,y)$, where

$\displaystyle x_{ij}\in X_{0}(\alpha_{j})_{i}\,,$ $\displaystyle u(j)=0\,,$
(4.30) $\displaystyle y_{ij}\in X_{1}(\alpha_{j})_{i}\,,$ $\displaystyle
u(j)=1\,.$ (4.31)

The fact that $z=(x,y)$ forms a section amounts to these data satisfying the
conditions

$\displaystyle X_{0}(\alpha_{j-1,j})_{i}(x_{ij})$
$\displaystyle=x_{ij}=d_{i+1}(x_{i+1,j})\,,$ $\displaystyle j\in[n]\text{ with
}u(j)=0\,,$ (4.32) $\displaystyle(f_{|\alpha_{j}})_{i}\circ
X_{0}(\alpha_{j-1,j})_{i}(x_{i,j-1})$
$\displaystyle=y_{ij}=d_{i+1}(y_{i+1,j})\,,$ $\displaystyle j\in[n]\text{ with
}u(j-1)=0\text{ and }u(j)=1\,,$ (4.33) $\displaystyle
X_{1}(\alpha_{j-1,j})_{i}(y_{ij})$
$\displaystyle=y_{ij}=d_{i+1}(y_{i+1,j})\,,$ $\displaystyle j\in[n]\text{ with
}u(j-1)=1\,.$ (4.34)

With these preparations, we construct a lift $\varphi$ in diagram (4.27) as
follows: consider an $n$-simplex $(u,N\alpha,x=(x_{ij})_{i,j})$ in
$\Delta^{1}\times r_{\mathscr{C}}^{*}X_{0}$ over the $n$-simplex
$(u,N\alpha)\in(\Delta^{1}{\times}N\mathscr{C})_{n}$; that is, $x\in
r_{\mathscr{C}}^{*}X_{0}$ is an $n$-simplex over $N\alpha$. We define
$\varphi(u,N\alpha,x)$ as the pair $z_{ij}=(x^{\prime}_{ij},y_{ij})$, where we
set $x^{\prime}_{ij}=x_{ij}\in X_{0}(\alpha_{j})_{i}$ whenever $u(j)=0$ and
$y_{ij}=f_{|\alpha_{j}}(x_{ij})\in X_{1}(\alpha_{j})_{i}$ whenever $u(j)=1$.
This satisfies the identities (4.32) because $x$ is an $n$-simplex of
$r_{\mathscr{C}}^{*}X_{0}$ and $f\colon X_{0}\to X_{1}$ is a natural
transformation of diagrams of simplicial sets, i.e. it is compatible with the
face and degeneracy maps of $X_{0}$ and $X_{1}$. For the same reasons, it
follows that this defines a morphism

$\varphi\colon\Delta^{1}\times r_{\mathscr{C}}^{*}X_{0}\longrightarrow
r_{[1]\times\mathscr{C}}^{*}X_{f}$ (4.35)

as desired. The resulting map $\varphi_{1}\colon r_{\mathscr{C}}^{*}X_{0}\to
r_{\mathscr{C}}^{*}X_{1}$ coincides with $r_{\mathscr{C}}^{*}(f)$. ∎

###### Corollary 4.36

Let
$R\colon{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}\to{\mathscr{K}\mathrm{an}}\subset{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$
be a fibrant replacement functor with natural weak equivalence $\rho\colon
1_{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}\to R$. Then, the composition

${\gamma^{\prime}_{\mathscr{C}}\colon
N{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}$${N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})}$$\scriptstyle{N(R\circ(-))}$$\scriptstyle{\gamma_{\mathscr{C}}}$
(4.37)

exhibits ${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ as the
$\infty$-categorical localisation of
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
at the objectwise weak equivalences.

###### Proof.

This follows readily from [2, Thm. 7.5.18] and the fact that, viewing
$N{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
as an $\infty$-category with weak equivalences and fibrations, the
$\infty$-subcategory
$N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})\subset
N{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
is precisely the underlying $\infty$-category of fibrant objects (see Remark
4.6). In particular, the functor $N(R\circ(-))$ induces an equivalence on the
level of localisations. ∎

###### Remark 4.38

Recall that the nerve functor
$N\colon{\mathscr{C}\mathrm{at}}\to{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$
is fully faithful as a functor of 1-categories (this holds true within any
universe). We can rephrase Theorem 3.38 by saying that there is a commutative
square of (${\mathrm{V}}$-small) simplicial sets

${N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{C},N{\mathscr{K}\mathrm{an}})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})}$${{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})}$$\scriptstyle{\gamma_{\mathscr{C}}}$$\scriptstyle{\cong}$$\scriptstyle{\gamma_{*}}$
(4.39)

Theorem 4.18 implies that $\gamma_{\mathscr{C}}$ exhibits the localisation of
${\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$ at the objectwise
weak equivalences. Applying this in the case where $\mathscr{C}=*$ and using
Theorem 3.38 we also obtain that $\gamma\colon
N{\mathscr{K}\mathrm{an}}\to\mathscr{S}$ exhibits the $\infty$-categorical
localisation of ${\mathscr{K}\mathrm{an}}$ at the weak equivalences in the
Kan-Quillen model structure. In light of this, we can now interpret diagram
(4.39) as an explicit manifestation of [2, Prop. 7.9.2], i.e. the fact that
localisation commutes with forming functor $\infty$-categories.
$\triangleleft$

##  5 Computational properties of the localisation functor

In this section we exhibit some of the pleasant computational properties of
the localisation functor $\gamma_{\mathscr{C}}$: we show that, for
$F,G\in{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$, the
$\infty$-categorical mapping space between $\gamma_{\mathscr{C}}F$ and
$\gamma_{\mathscr{C}}G$ in
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$ can be computed as the
model-categorical mapping space. The same holds true for the internal hom (or
exponential) in ${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})$.
Moreover, we show that the left and right ($\infty$-categorical) Kan
extensions of $\mathscr{S}$-valued functors can be computed by the left and
right homotopy Kan extensions of
${\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}$-valued functors. These
insights themselves are not new (see, for instance, [14, Thm. 4.2.4.1]), but
here we are able to derive them in a particularly straightforward fashion,
building on [2, Cor. 5.4.7], the results of this paper so far and
manipulations of model-categorical mapping spaces. Throughout this section,
let $\mathscr{C}$ be a ${\mathrm{U}}$-small category. Recall that we denote
the simplicial hom sets in a simplicially enriched category ${\mathscr{M}}$ by
$\underline{{\mathscr{M}}}(-,-)$.

###### Proposition 5.1

Let $F,G\colon\mathscr{C}\to{\mathscr{K}\mathrm{an}}$. There exist canonical
equivalences in $\mathscr{S}$,

${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})(\gamma_{\mathscr{C}}F,\gamma_{\mathscr{C}}G)\simeq\underline{{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}}(r_{\mathscr{C}}^{*}F,r_{\mathscr{C}}^{*}G)\simeq\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(r_{\mathscr{C}!}r_{\mathscr{C}}^{*}F,G)\,,$
(5.2)

and a further equivalence

$\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(r_{\mathscr{C}!}r_{\mathscr{C}}^{*}F,G)\simeq\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(QF,G)$
(5.3)

where $QF$ is any functorial cofibrant replacement of $F$ in the projective
model structure on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
These equivalences are natural in $F$ and $G$.

This can be derived from general arguments about localisations along the lines
of [2, Sec. 7.10], but we give a more direct proof:

###### Proof.

The first equivalence follows directly from Corollary 3.43 and [2, Cor.
5.4.7]. The second equivalence can be obtained, for instance, by using that
$r_{\mathscr{C}!}\dashv r_{\mathscr{C}}^{*}$ is a simplicial Quillen
equivalence: since $F$ is fibrant and $r_{\mathscr{C}}^{*}F$ is cofibrant, the
derived counit of this adjunction provides an equivalence
$r_{\mathscr{C}!}r_{\mathscr{C}}^{*}F\to F$. It thus exhibits
$r_{\mathscr{C}!}r_{\mathscr{C}}^{*}F$ as a cofibrant replacement of $F$ in
the projective model structure on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
For any other functorial cofibrant replacement $QF$, there is a natural zig-
zag of weak equivalences

${r_{\mathscr{C}!}r_{\mathscr{C}}^{*}F}$${Q(r_{\mathscr{C}!}r_{\mathscr{C}}^{*}F)}$${QF}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$
(5.4)

between cofibrant objects, which the functor
$\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(-,G)$
sends to an equivalence in $\mathscr{S}$. ∎

A particular choice of cofibrant replacement functor for the projective model
structure on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
is given by Dugger’s cofibrant replacement functor [7, Sec. 2.6]; we denote
this functor by $Q^{p}$ (note that since here we use simplicial precosheaves
on $\mathscr{C}$ instead of simplicial presheaves, we need to introduce an
additional $(-)^{\mathrm{op}}$ in Dugger’s conventions). Explicitly, we have

$(Q^{p}F)_{n}(c)=\coprod_{c_{0}\to\cdots\to c_{n}\to c}F_{n}(c_{0})\,,$ (5.5)

where the coproduct is taken over chains of morphisms in $\mathscr{C}$. The
functor $Q^{p}$ comes with a natural weak equivalence to the identity functor
on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$,
which we denote by $q^{p}$.

###### Corollary 5.6

For each $F,G\in{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$,
there is an equivalence, natural in $F$ and $G$,

$(\gamma_{\mathscr{C}}G)^{\gamma_{\mathscr{C}}F}\simeq\gamma_{\mathscr{C}}(G^{Q^{p}F})\,.$
(5.7)

###### Proof.

By the Yoneda Lemma, it suffices to show that there is a natural equivalence

${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})\big{(}-,(\gamma_{\mathscr{C}}G)^{\gamma_{\mathscr{C}}F}\big{)}\simeq{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})\big{(}-,\gamma_{\mathscr{C}}(G^{Q^{p}F})\big{)}$
(5.8)

of functors
${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})^{\mathrm{op}}\longrightarrow\widehat{\mathscr{S}}$
(where $\widehat{\mathscr{S}}$ is the $\infty$-category of
${\mathrm{V}}$-small spaces). By Theorem 4.18 $\gamma_{\mathscr{C}}$ is a
localisation functor. It thus induces an equivalence [2, Def. 7.1.2]

$\gamma_{\mathscr{C}}^{*}\colon{\mathscr{F}\mathrm{un}}\big{(}{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})^{\mathrm{op}},\widehat{\mathscr{S}}\big{)}\longrightarrow{\mathscr{F}\mathrm{un}}_{\mathsf{W}}\big{(}N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})^{\mathrm{op}},\widehat{\mathscr{S}}\big{)}\,,$
(5.9)

where on the right-hand side

${\mathscr{F}\mathrm{un}}_{\mathsf{W}}\big{(}N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})^{\mathrm{op}},\widehat{\mathscr{S}}\big{)}\subset{\mathscr{F}\mathrm{un}}\big{(}N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})^{\mathrm{op}},\widehat{\mathscr{S}}\big{)}$
(5.10)

denotes the full $\infty$-subcategory on those functors which send weak
equivalences to equivalences. Therefore, it suffices to show that there is an
equivalence of mapping spaces

${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})\big{(}\gamma_{\mathscr{C}}A,(\gamma_{\mathscr{C}}G)^{\gamma_{\mathscr{C}}F}\big{)}\simeq{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})\big{(}\gamma_{\mathscr{C}}A,\gamma_{\mathscr{C}}(G^{Q^{p}F})\big{)}\,,$
(5.11)

natural in $A\in{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$.

Since $\gamma_{\mathscr{C}}$ is left exact (Proposition 4.14) and both $A$ and
$F$ are fibrant objects in
$N{\operatorname{Fun}}(\mathscr{C},{\mathscr{K}\mathrm{an}})$ (seen as an
$\infty$-category of fibrant objects—see Remark 4.6), we have an equivalence

${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})(\gamma_{\mathscr{C}}A,(\gamma_{\mathscr{C}}G)^{\gamma_{\mathscr{C}}F})\simeq{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})(\gamma_{\mathscr{C}}(A\times
F),\gamma_{\mathscr{C}}G)\,.$ (5.12)

By Proposition 5.1 there is a further equivalence

${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})(\gamma_{\mathscr{C}}(A\times
F),\gamma_{\mathscr{C}}G)\simeq\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}\big{(}Q^{p}(A\times
F),G\big{)}\,.$ (5.13)

For this choice of cofibrant replacement functor there exists a canonical
commutative diagram

${Q^{p}(A\times F)}$${Q^{p}A\times Q^{p}F}$${A\times
F}$$\scriptstyle{\sim}$$\scriptstyle{q^{p}_{A\times
F}}$$\scriptstyle{\sim}$$\scriptstyle{q^{p}_{A}\times q^{p}_{F}}$ (5.14)

whose horizontal arrow stems from the canonical functors of slice categories

$\mathscr{C}_{/A_{n}\times
F_{n}}\longrightarrow\mathscr{C}_{/A_{n}}\times\mathscr{C}_{/F_{n}}$ (5.15)

for each $n\in\mathbb{N}_{0}$. The right-hand arrow is a weak equivalence
since $q^{p}_{A}\colon Q^{p}A\to A$ and $q^{p}_{F}\colon Q^{p}F\to F$ are so
and the injective model structure on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
is cartesian monoidal [1, Prop. 4.51] (it has the same weak equivalences as
the projective model structure). It follows that the top morphism is a weak
equivalence as well. Thus, we have natural equivalences

$\displaystyle{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})(\gamma_{\mathscr{C}}A,(\gamma_{\mathscr{C}}G)^{\gamma_{\mathscr{C}}F})$
$\displaystyle\simeq\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}\big{(}Q^{p}(A\times
F),G\big{)}$ (5.16)
$\displaystyle\simeq\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(Q^{p}A,G^{Q^{p}F})$
(5.17)
$\displaystyle\simeq{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})\big{(}\gamma_{\mathscr{C}}A,\gamma_{\mathscr{C}}(G^{Q^{p}F})\big{)}\,,$
(5.18)

where the last step is again Proposition 5.1. ∎

Let $\mathscr{D}$ be a second ${\mathrm{U}}$-small category. We now compare
$\infty$-categorical and homotopy Kan extensions along functors
$\pi\colon\mathscr{C}\to\mathscr{D}$.

###### Proposition 5.19

Let $\pi\colon\mathscr{C}\to\mathscr{D}$ be a functor, and let
$X\colon\mathscr{C}\to{\mathscr{K}\mathrm{an}}$ be a diagram. The
$\infty$-categorical right Kan extension $(N\pi)_{*}\gamma_{\mathscr{C}}(X)$
of $\gamma_{\mathscr{C}}(X)$ classifies the left fibration
$r_{\mathscr{D}}^{*}\operatorname{hoRan}_{\pi}(X)\longrightarrow
N\mathscr{D}$, where $\operatorname{hoRan}_{\pi}(X)$ denotes the homotopy
right Kan extension of $X$ along $\pi$. Equivalently, there is a canonical
equivalence

$(N\pi)_{*}\gamma_{\mathscr{C}}(X)\simeq\gamma_{\mathscr{D}}(\operatorname{hoRan}_{\pi}X)\,.$
(5.20)

###### Proof.

We show that $\gamma_{\mathscr{D}}(\operatorname{hoRan}_{\pi}X)$ represents
the functor

${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})(N\pi^{*}(-),\gamma_{\mathscr{C}}X)\colon{\mathscr{F}\mathrm{un}}(N\mathscr{D},\mathscr{S})\longrightarrow\widehat{\mathscr{S}}\,.$
(5.21)

To that end, let $F\colon N\mathscr{D}\to\mathscr{S}$ be a functor. By the
same logic as in the proof of Corollary 5.6, we may precompose the left-hand
argument of ${\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})(-,-)$ with the
localisation functor $\gamma_{\mathscr{C}}$. That is, we may assume that $F$
is of the form $\gamma_{\mathscr{D}}Y$ for some functor
$Y\colon\mathscr{D}\to{\mathscr{K}\mathrm{an}}$. Furthermore, we may assume
that $X\colon\mathscr{C}\to{\mathscr{K}\mathrm{an}}$ is fibrant in the
injective model structure on
${\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$;
we can always replace $X$ by an injectively fibrant approximation, up to an
objectwise weak equivalence, which is preserved by both
$\operatorname{hoRan}_{\pi}$ and $\gamma_{\mathscr{D}}$ (see Proposition 4.1).
In particular, we then have that
$\operatorname{hoRan}_{\pi}X\simeq\operatorname{Ran}_{\pi}X$, since

${\pi^{*}:{\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}$${{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}):\operatorname{Ran}_{\pi}}$$\scriptstyle{\perp}$
(5.22)

is a Quillen adjunction with respect to the injective model structures.

There are canonical natural equivalences of spaces

$\displaystyle{\mathscr{F}\mathrm{un}}(N\mathscr{D},\mathscr{S})\big{(}\gamma_{\mathscr{D}}Y,\gamma_{\mathscr{D}}(\operatorname{hoRan}_{\pi}X)\big{)}$
$\displaystyle\overset{(1)}{\simeq}\underline{{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{D}}}\big{(}r_{\mathscr{D}}^{*}Y,r_{\mathscr{D}}^{*}(\operatorname{Ran}_{\pi}X)\big{)}$
(5.23)
$\displaystyle\overset{(2)}{\cong}\underline{{\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(r_{\mathscr{D}!}r_{\mathscr{D}}^{*}Y,\operatorname{Ran}_{\pi}X)$
(5.24)
$\displaystyle\overset{(3)}{\simeq}\underline{{\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(Y,\operatorname{Ran}_{\pi}X)$
(5.25)
$\displaystyle\overset{(4)}{\cong}\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(\pi^{*}Y,X)$
(5.26)
$\displaystyle\overset{(5)}{\simeq}\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(r_{\mathscr{C}!}r_{\mathscr{C}}^{*}\pi^{*}Y,X)$
(5.27)
$\displaystyle\overset{(6)}{\cong}\underline{{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}}(r_{\mathscr{C}}^{*}\pi^{*}Y,r_{\mathscr{C}}^{*}X)$
(5.28)
$\displaystyle\overset{(7)}{\simeq}{\operatorname{Fun}}(N\mathscr{C},\mathscr{S})(\gamma_{\mathscr{C}}\pi^{*}Y,\gamma_{\mathscr{C}}X)$
(5.29)
$\displaystyle\overset{(8)}{\simeq}{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})(N\pi^{*}\gamma_{\mathscr{C}}Y,\gamma_{\mathscr{C}}X)\,.$
(5.30)

Equivalence (1) stems from Proposition 5.1. For step (2), we have used that
the adjunction $r_{\mathscr{C}!}\dashv r_{\mathscr{C}}^{*}$ is simplicial [11,
Rmk. 4.4]. Step (3) uses that $Y$ is projectively fibrant, that
$r_{\mathscr{D}!}\dashv r_{\mathscr{D}}^{*}$ is a Quillen equivalence between
the projective and the covariant model structure [11, Thm. C], and that
injectively fibrant objects are also projectively fibrant in
${\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
Thus, the counit $r_{\mathscr{D}!}r_{\mathscr{D}}^{*}Y\to Y$ is, in
particular, a weak equivalence between cofibrant objects in the injective
model structure, and $\operatorname{Ran}_{\pi}X$ is injectively fibrant. Step
(4) uses the universal property of Kan extensions. Steps (5), (6) and (7) use
the same arguments as steps (3), (2) and (1) (in that order) with
$\mathscr{C}$ in place of $\mathscr{D}$, and the final step (8) is an
application of Corollary 3.48. ∎

###### Proposition 5.31

Let $\pi\colon\mathscr{C}\to\mathscr{D}$ and
$X\colon\mathscr{C}\to{\mathscr{K}\mathrm{an}}$ be functors. Let
$R^{p}_{\mathscr{D}}$ denote a projectively fibrant replacement in
${\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
The $\infty$-categorical left Kan extension
$(N\pi)_{!}\gamma_{\mathscr{C}}(X)$ classifies the left fibration
$r_{\mathscr{D}}^{*}R^{p}_{\mathscr{D}}{\operatorname{hoLan}}_{\pi}(X)\longrightarrow
N\mathscr{D}$. Equivalently, there is a canonical equivalence

$(N\pi)_{!}\gamma_{\mathscr{C}}(X)\simeq\gamma_{\mathscr{D}}(R^{p}_{\mathscr{D}}\circ{\operatorname{hoLan}}_{\pi}X)\,.$
(5.32)

Note that $\gamma_{\mathscr{D}}(R^{p}\circ{\operatorname{hoLan}}_{\pi}X)$ is
the image of ${\operatorname{hoLan}}_{\pi}X$ under the localisation functor in
Corollary 4.36.

###### Proof.

As in the proof of Corollary 5.6 and Proposition 5.19, since
$\gamma_{\mathscr{C}}$ is a localisation functor it suffices to consider
mapping spaces to objects of the form
$F=\gamma_{\mathscr{D}}Y\in{\mathscr{F}\mathrm{un}}(N\mathscr{D},\mathscr{S})$,
for $Y\in{\operatorname{Fun}}(\mathscr{D},{\mathscr{K}\mathrm{an}})$. In fact,
up to weak equivalence, we may assume that $Y$ is even injectively fibrant in
${\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$.
There are canonical natural equivalences (in $\widehat{\mathscr{S}}$)

$\displaystyle{\mathscr{F}\mathrm{un}}(N\mathscr{D},\mathscr{S})(\gamma_{\mathscr{D}}R^{p}_{\mathscr{D}}{\operatorname{hoLan}}_{\pi}X,\gamma_{\mathscr{D}}Y)$
$\displaystyle\overset{(1)}{\simeq}\underline{{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{D}}}(r_{\mathscr{D}}^{*}R^{p}_{\mathscr{D}}{\operatorname{hoLan}}_{\pi}X,r_{\mathscr{D}}^{*}Y)$
(5.33)
$\displaystyle\overset{(2)}{\simeq}\underline{{\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(r_{\mathscr{D}!}r_{\mathscr{D}}^{*}R^{p}_{\mathscr{D}}{\operatorname{hoLan}}_{\pi}X,Y)$
(5.34)
$\displaystyle\overset{(3)}{\simeq}\underline{{\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(R^{p}_{\mathscr{D}}{\operatorname{hoLan}}_{\pi}X,Y)$
(5.35)
$\displaystyle\overset{(4)}{\simeq}\underline{{\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(R^{p}_{\mathscr{D}}{\operatorname{hoLan}}_{\pi}r_{\mathscr{C}!}r_{\mathscr{C}}^{*}X,Y)$
(5.36)

In step (1) we have use Proposition 5.1. Step (2) uses that
$r_{\mathscr{D}!}\dashv r_{\mathscr{D}}^{*}$ is a simplicial adjunction. In
steps (3) and (4) we have used that $Y$ is injectively fibrant as well as that
the injective model structure on
${\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$
is simplicial (as in step (3) of the proof of Proposition 5.19). We further
have equivalences of spaces

$\displaystyle\underline{{\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(R^{p}_{\mathscr{D}}{\operatorname{hoLan}}_{\pi}r_{\mathscr{C}!}r_{\mathscr{C}}^{*}X,Y)$
$\displaystyle\overset{(5)}{\simeq}\underline{{\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}({\operatorname{hoLan}}_{\pi}r_{\mathscr{C}!}r_{\mathscr{C}}^{*}X,Y)$
(5.37)
$\displaystyle\overset{(6)}{\simeq}\underline{{\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(\operatorname{Lan}_{\pi}r_{\mathscr{C}!}r_{\mathscr{C}}^{*}X,Y)$
(5.38)
$\displaystyle\overset{(7)}{\simeq}\underline{{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})}(r_{\mathscr{C}!}r_{\mathscr{C}}^{*}X,\pi^{*}Y)$
(5.39)
$\displaystyle\overset{(8)}{\simeq}\underline{{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}}_{/N\mathscr{C}}}(r_{\mathscr{C}}^{*}X,r_{\mathscr{C}}^{*}\pi^{*}Y)$
(5.40)
$\displaystyle\overset{(9)}{\simeq}{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})\big{(}\gamma_{\mathscr{C}}X,\gamma_{\mathscr{C}}(\pi^{*}Y)\big{)}$
(5.41)
$\displaystyle\overset{(10)}{\simeq}{\mathscr{F}\mathrm{un}}(N\mathscr{C},\mathscr{S})(\gamma_{\mathscr{C}}X,(N\pi)^{*}\gamma_{\mathscr{D}}Y)\,.$
(5.42)

Step (5) again uses the injective model structure on
${\operatorname{Fun}}(\mathscr{D},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$,
the weak equivalence from the identity to $R^{p}$ and that $Y$ is injectively
fibrant. Step (6) uses that $r_{\mathscr{C}!}r_{\mathscr{C}}^{*}X$ is
projectively cofibrant, so that its homotopy left Kan extension is weakly
equivalent to its strict left Kan extension. Step (7) is where the defining
property of the 1-categorical left Kan extension enters (note that the
adjunction $\operatorname{Lan}_{\pi}\dashv\pi^{*}$ is simplicial; this holds
true since, for each
$Z\in{\operatorname{Fun}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{{\mathbbe{\Delta}}}})$,
the functor $Z\times(-)$ commutes with colimits). In step (8) we have used the
simplicial adjunction $r_{\mathscr{C}!}\dashv r_{\mathscr{C}}^{*}$, and step
(9) is Proposition 5.1. Step (9) is an application of Proposition 5.1.
Finally, in step (10) we have used Corollary 3.48. ∎

##  Appendix A Background on $\infty$-categorical localisation

For the reader’s convenience, we recall some definitions and results on
localisation of $\infty$-categories from [2, Ch. 7]

###### Definition A.1

_[ 2, Def. 7.4.6], [3]_ Let ${\mathscr{A}}$ be an $\infty$-category with a
fixed final object $e$. A _class of fibrations in ${\mathscr{A}}$_ is a
subobject ${\mathsf{F}}\subset{\mathscr{A}}$ with the following properties:

  1. (1)

${\mathsf{F}}$ contains all identity morphisms (i.e. all degenerate edges).

  2. (2)

${\mathsf{F}}$ is closed under composition.

  3. (3)

For any morphisms $g\colon y^{\prime}\to y$ and $f\colon x\to y$ in
${\mathscr{A}}$ such that $y^{\prime}\to e$, $y\to e$ and $x\to e$ are in
${\mathsf{F}}$, the pullback $y^{\prime}\times_{y}x$ in ${\mathscr{A}}$
exists, and, for any cartesian square

${x^{\prime}}$${x}$${y^{\prime}}$${y}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g}$
(A.2)

in ${\mathscr{A}}$, the morphism $f^{\prime}$ is in ${\mathsf{F}}$ whenever
$f$ is so.

Morphisms in ${\mathsf{F}}$ are called _fibrations_. Given an
$\infty$-category with a class of fibrations $({\mathscr{A}},{\mathsf{F}})$,
we call an object $y\in{\mathscr{A}}$ _fibrant_ if the morphism $y\to e$ is in
${\mathsf{F}}$.

In particular, since ${\mathsf{F}}$ contains all identity morphisms, the final
object $e\in{\mathscr{A}}$ is necessarily fibrant.

###### Definition A.3

_[ 2, Def. 7.4.12]_ An _$\infty$ -category with weak equivalences and
fibrations_ is a triple $({\mathscr{A}},\mathsf{W},{\mathsf{F}})$, where
$\mathsf{W}\subset{\mathscr{A}}$ is an $\infty$-subcategory (in particular
closed under composition) and ${\mathsf{F}}\subset{\mathscr{A}}$ is a class of
fibrations, such that the following properties are satisfied:

  1. (1)

The class $\mathsf{W}$ satisfies the two-out-of-three property.

  2. (2)

For any cartesian square

${x^{\prime}}$${x}$${y^{\prime}}$${y}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g}$
(A.4)

in ${\mathscr{A}}$, in which $f$ is a fibration between fibrant objects and
$y^{\prime}$ is fibrant, if $f$ is additionally in $\mathsf{W}$, then so is
$f^{\prime}$.

  3. (3)

For each map $f\colon x\to y$ in ${\mathscr{A}}$ with $y$ fibrant, there
exists a commutative diagram (a 2-simplex)

${x^{\prime}}$${x}$${y}$$\scriptstyle{p}$$\scriptstyle{w}$$\scriptstyle{f}$
(A.5)

with $w\in\mathsf{W}$ and $p\in{\mathsf{F}}$.

The morphisms in $\mathsf{W}$ are called _weak equivalences_. The morphisms in
${\mathsf{F}}\cap\mathsf{W}$ are called _trivial fibrations_.

###### Example A.6

Let ${\mathscr{M}}$ be a model category. Let $\mathsf{W},{\mathsf{F}}\subset
N{\mathscr{M}}$ denote the simplicial subsets generated by those edges which
are weak equivalences or fibrations in ${\mathscr{M}}$, respectively. The
triple $(N{\mathscr{M}},\mathsf{W},{\mathsf{F}})$ is an $\infty$-category with
weak equivalences and fibrations. $\triangleleft$

###### Remark A.7

Suppose that ${\mathscr{A}}$ is an $\infty$-category such that defining
${\mathsf{F}}$ to consists of all morphisms in ${\mathscr{A}}$ makes the pair
$({\mathscr{A}},{\mathsf{F}})$ into an $\infty$-category with fibrations. Then
${\mathscr{A}}$ necessarily has finite limits (since it has a final object and
pullbacks [2, Thm. 7.3.27]). $\triangleleft$

###### Definition A.8

_[ 2, Def. 7.5.7]_ An $\infty$-category with weak equivalences and fibrations
$({\mathscr{A}},\mathsf{W},{\mathsf{F}})$ in which each object is fibrant is
called an _$\infty$ -category of fibrant objects_.

###### Example A.9

In the setting of Remark A.6, let ${\mathscr{M}}_{f}\subset{\mathscr{M}}$
denote the full subcategory on the fibrant objects in ${\mathscr{M}}$. Let
$\mathsf{W}_{f}\subset N{\mathscr{M}}_{f}$ (resp. ${\mathsf{F}}_{f}\subset
N{\mathscr{M}}_{f}$) denote the simplicial subset generated by those edges in
$N{\mathscr{M}}$ which are in $\mathsf{W}$ (resp. in ${\mathsf{F}}$). Then,
the triple $(N{\mathscr{M}}_{f},\mathsf{W}_{f},{\mathsf{F}}_{f})$ is an
$\infty$-category of fibrant objects. $\triangleleft$

###### Definition A.10

_[ 2, Def. 7.5.2]_ Let
$({\mathscr{A}},\mathsf{W}_{\mathscr{A}},{\mathsf{F}}_{\mathscr{A}})$ and
$(\mathscr{B},\mathsf{W}_{\mathscr{B}},{\mathsf{F}}_{\mathscr{B}})$ be two
$\infty$-categories with weak equivalences and fibrations. A functor
$F\colon{\mathscr{A}}\to\mathscr{B}$ is called _left exact_ if it satisfies
the following properties:

  1. (1)

$F$ preserves final objects.

  2. (2)

$F$ sends fibrations _between fibrant objects_ to fibrations and trivial
fibrations _between fibrant objects_ to trivial fibrations.

  3. (3)

$F$ maps cartesian squares

${x^{\prime}}$${x}$${y^{\prime}}$${y}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{g}$
(A.11)

in ${\mathscr{A}}$, where $f$ is a fibration and $y$ and $y^{\prime}$ are
fibrant, to cartesian squares in $\mathscr{B}$.

In particular, by properties (1) and (2), $F$ preserves fibrant objects. It
also follows from an adaptation of Ken Brown’s Lemma to the present
$\infty$-categorical context [2, Prop. 7.4.13] that any left exact functor
preserves weak equivalences between fibrant objects [2, Cor. 7.4.14]. As a
consequence, one obtains an $\infty$-categorical notion of derived functors
between $\infty$-categorical localisations: first, given $\infty$-categories
${\mathscr{A}}$ and $\mathscr{B}$ with collections of weak equivalences
$\mathsf{W}_{\mathscr{A}}$ and $\mathsf{W}_{\mathscr{B}}$, respectively, as
well as a functor $F\colon{\mathscr{A}}\to\mathscr{B}$ which preserves weak
equivalences, let $\overline{F}\colon
L_{\mathsf{W}_{\mathscr{A}}}{\mathscr{A}}\longrightarrow
L_{\mathsf{W}_{\mathscr{B}}}\mathscr{B}$ denote the unique functor induced on
localisations.

Now, let $({\mathscr{A}},\mathsf{W}_{\mathscr{A}},{\mathsf{F}}_{\mathscr{A}})$
be an $\infty$-category with weak equivalences and fibrations,
$(\mathscr{B},\mathsf{W}_{\mathscr{B}})$ an $\infty$-category with an
$\infty$-subcategory of weak equivalences and
$F\colon{\mathscr{A}}\to\mathscr{B}$ a functor of $\infty$-categories which
sends weak equivalences between fibrant objects to weak equivalences. Let
${\mathscr{A}}_{f}\subset{\mathscr{A}}$ be the full $\infty$-subcategory on
the fibrant objects; this is an $\infty$-category with weak equivalences and
fibrations when endowed with the restrictions of $\mathsf{W}_{\mathscr{A}}$
and ${\mathsf{F}}_{\mathscr{A}}$. Let $R\colon
L_{\mathsf{W}_{\mathscr{A}}}{\mathscr{A}}\to
L_{\mathsf{W}_{\mathscr{A}}}{\mathscr{A}}_{f}$ be a weak inverse to the
equivalence $\overline{\iota}$ induced by the inclusion
$\iota\colon{\mathscr{A}}_{f}\hookrightarrow{\mathscr{A}}$ (see [2, Thm.
7.5.18]) and let $\ell_{\mathscr{B}}\colon\mathscr{B}\to
L_{\mathsf{W}_{\mathscr{B}}}\mathscr{B}$ denote the localisation functor.

###### Definition A.12

The right derived functor of $F$ is [2, Par. 7.5.25, Def. 7.5.26]

$\mathbb{R}F=R^{*}\overline{\ell_{\mathscr{B}}\circ\iota^{*}F}=\overline{(\ell_{\mathscr{B}}\circ
F\circ\iota)}\circ R\colon
L_{\mathsf{W}_{\mathscr{A}}}{\mathscr{A}}\longrightarrow
L_{\mathsf{W}_{\mathscr{B}}}\mathscr{B}\,.$ (A.13)

The most important result about right derived functors in the context of this
paper is

###### Theorem A.14

_[ 2, Cor. 7.6.11]_ Let
$({\mathscr{A}},{\mathsf{F}}_{\mathscr{A}},\mathsf{W}_{\mathscr{A}})$ and
$(\mathscr{B},{\mathsf{F}}_{\mathscr{B}},\mathsf{W}_{\mathscr{B}})$ be
$\infty$-categories with fibrations and weak equivalences, and let
$F\colon{\mathscr{A}}\to\mathscr{B}$ be a left exact functor. The right
derived functor $\mathbb{R}F\colon
L_{\mathsf{W}_{\mathscr{A}}}{\mathscr{A}}\to
L_{\mathsf{W}_{\mathscr{B}}}\mathscr{B}$ is an equivalence if and only if it
is so on the level of homotopy categories; that is,
$\mathrm{h}\mathbb{R}F\colon\mathrm{h}L_{\mathsf{W}_{\mathscr{A}}}{\mathscr{A}}\to\mathrm{h}L_{\mathsf{W}_{\mathscr{B}}}\mathscr{B}$
is an equivalence of _ordinary_ categories.

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