Into boldhook—> Mapsto |—>

# The formalism of Segal sections

Edouard Balzin

###### Abstract.

Given a family of model categories ${\mathcal{E}}\to{\mathcal{C}}$, we
associate to it a homotopical category of derived, or Segal, sections ${\sf
DSect}({\mathcal{C}},{\mathcal{E}})$ that models the higher-categorical
sections of the localisation $L{\mathcal{E}}\to{\mathcal{C}}$. The derived
sections provide an alternative, strict model for various higher algebra
objects appearing in the work of Lurie. We prove a few results concerning the
properties of the homotopical category ${\sf
DSect}({\mathcal{C}},{\mathcal{E}})$, and as an example, study its behaviour
with respect to the base-change along a select class of functors.

###### Contents

  1. 1 Simplicial Replacements
    1. 1.1 Preliminaries
    2. 1.2 The replacements
    3. 1.3 Families over the simplicial replacement
  2. 2 Derived, or Segal, sections
    1. 2.1 Presections
    2. 2.2 Homotopical category of Segal sections
  3. 3 Higher-categorical aspects
    1. 3.1 Behaviour with respect to the infinity-localisation
    2. 3.2 Higher-categorical Segal sections
  4. 4 Resolutions
    1. 4.1 Relative comma objects
    2. 4.2 Sections over relative comma objects
    3. 4.3 Comparing the Segal sections

## Introduction

### Segal objects

The formalism presented in this paper was developed in the study of homotopy
algebraic structures as described by Segal and generalised by Lurie. We begin
the introduction by describing this context.

Denote by $\Gamma$ the category whose objects are finite sets and morphisms
are given by partially defined set maps. Each such morphism between $S$ and
$T$ can be depicted as $S\supset S_{0}\to T$. A _$\Gamma$ -space_ is simply a
functor $X:\Gamma\to{\rm\bf Top}$ taking values in the category of topological
spaces. Each such functor can be evaluated on the maps in $\Gamma$ that have
the form $S\supset{s}\to 1$, where $s\in S$ and $1$ is the one-element set. As
a result, for each $S\in\Gamma$ we get the map

(i) $X(S)\longrightarrow X(1)^{S}$

and the $\Gamma$-space is called _Segal_ [23] if the map (i) is a homotopy
equivalence for all $S\in\Gamma$.

For each $S$, we also have the map $S\stackrel{{\scriptstyle=}}{{\supset}}S\to
1$, and evaluating $X$ on it, combined with (i), gives the following span in
${\rm\bf Top}$:

(ii)
$X(1)^{S}\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}X(S)\longrightarrow
X(1);$

choosing an inverse equivalence to the left map, we get a non-canonical
operation $X(1)^{S}\to X(1)$. There is no ambiguity in choice of the inverse
in the homotopy category of topological spaces ${\rm Ho}\,{\rm\bf Top}$: the
result is a commutative monoid structure on $X(1)$ in the homotopy category.
The full structure that $X(1)$ carries is that of an ${\mathrm{E}}_{\infty}$,
or a homotopy commutative monoid.

Instead of $\Gamma$ one can work with other categories. For example, take the
category $\Delta$ which consists of finite categories $[n]=0\to 1\to...\to n$
($n\geq 0$) and functors between them. We can then consider functors
$Y:\Delta^{\sf op}\to{\rm\bf Top}$ such that $Y([0])=*$ and for each $n\geq
1$, the map $Y([n])\to Y([1])^{n}$, induced by all the consecutive interval
inclusions $[1]\to[n]$, is a homotopy equivalence. Such Segal $\Delta$-spaces
are known to describe homotopy associative, or ${\mathrm{E}}_{1}$-monoids. The
functor $\Delta^{\sf op}\to\Gamma$ which maps $[n]$ to the set of its $n$
generating arrows, allows us to pull $\Gamma$-spaces back to $\Delta^{\sf
op}$: every commutative monoid is also an associative one. There are further
examples of small categories that parametrise operations up to homotopy. For
instance, [5] introduces the notion of an operator category and gives examples
of categories that model ${\mathrm{E}}_{m}$-algebras. The case $m=2$ can also
be approached geometrically using the exit path category [27] of the $2$-disk.
One can also introduce categories that permit to describe an
${\mathrm{E}}_{m}$-algebra over another algebra. All such categories
${\mathrm{O}}$ come with a functor ${\mathrm{O}}\to\Gamma$, so that we can
pull Segal $\Gamma$-spaces back to ${\mathrm{O}}$, reflecting the “initial”
property of homotopy commutative structures.

Let us attempt to replace ${\rm\bf Top}$ with another homotopical category (by
which we simply mean a category with a distinguished class of weak
equivalences) possessing a monoidal structure. For example, take the category
${\rm\bf DVect}_{k}$ of chain complexes of vector spaces over a field $k$. The
maps like (i) will take the form $X(S)\to\oplus_{S}X(1)$, which is not very
interesting for the purposes of algebra. Indeed, the tensor product
$\otimes_{k}$ is not Cartesian. The observation of [20] addresses this issue
by making the following construction.

### Algebras as sections

Given any symmetric monoidal category $({\mathcal{M}},\otimes,{\mathbb{I}})$,
let us define the category ${\mathcal{M}}^{\otimes}$. Its objects are pairs
$(S,\\{X_{s}\\}_{s\in S})$ where $S\in\Gamma$ and each $X_{s}$ is an object of
${\mathcal{M}}$. A morphism $(S,\\{X_{s}\\}_{s\in S})\to(T,\\{Y_{t}\\}_{t\in
T})$ consists of a partially defined map $f:S\to T$, and for each $t\in T$, of
a morphism $\otimes_{s\in f^{-1}(t)}X_{s}\to Y_{t}$. When $f^{-1}(t)$ is
empty, the monoidal product over it equals ${\mathbb{I}}$. The compositions
can then be defined with the help of the coherence isomorphisms for the
product $\otimes$ and the unit object.

The forgetful functor $p:{\mathcal{M}}^{\otimes}\to\Gamma$ is a _Grothendieck
opfibration_ [15]: the assignment
$S\mapsto{\mathcal{M}}^{\otimes}(S):=p^{-1}(S)={\mathcal{M}}^{S}$ is
functorial, but in a weak way. Each map $f:S\to T$ corresponds to a functor
$f_{!}:{\mathcal{M}}^{\otimes}(S)\to{\mathcal{M}}^{\otimes}(T)$, sending
$(S,\\{X_{s}\\}_{s\in S})$ to $(T,\\{(\otimes_{s\in f^{-1}(t)}X_{s})\\}_{t\in
T})$. We see that for composable maps $f,g$, one has $g_{!}f_{!}\cong(gf)_{!}$
and such isomorphisms, while suitably coherent, are nonetheless not
equalities. It is more convenient to work with Grothendieck (op)fibrations
since this allows to keep all these isomorphisms implicit.

Denoting for simplicity two objects of ${\mathcal{M}}^{\otimes}$ as $X,Y$, any
map $\alpha:X\to Y$ factors as $X\to f_{!}X\to Y$ where $f=p(\alpha)$; the map
$\alpha$ is called _cocartesian_ if the induced map $f_{!}X\to Y$ is an
isomorphism. Consider now a section of $p$, that is a functor
${\mathcal{A}}:\Gamma\to{\mathcal{M}}^{\otimes}$ with $p{\mathcal{A}}=id$.
Demand further that given any _inert_ map in $\Gamma$, that is a map of the
form $S\supset T\stackrel{{\scriptstyle=}}{{\to}}T$, the induced map
${\mathcal{A}}(S)\to{\mathcal{A}}(T)$ is cocartesian. It is elementary to
verify that each ${\mathcal{A}}(S)$ is isomorphic to $S$ copies of
${\mathcal{A}}(1)\in{\mathcal{M}}^{\otimes}(1)={\mathcal{M}}$. The remaining
information provided by the section ${\mathcal{A}}$ equips ${\mathcal{A}}(1)$
with the structure of a commutative monoid in ${\mathcal{M}}$: the value of
${\mathcal{A}}$ on the map $S\stackrel{{\scriptstyle=}}{{\supset}}S\to 1$
gives multiplication ${\mathcal{A}}(1)^{S}\to{\mathcal{A}}(1)$, the
composition property insures that everything is determined by this map for
$|S|=2$ (which is forced to be associative) and the action of automorphisms in
$\Gamma$ forces the multiplication to be commutative.

Setting ${\mathcal{M}}={\rm\bf DVect}_{k}$, it is well known that the homotopy
theory of commutative $dg$-algebras is an inadequate model for
${\mathrm{E}}_{\infty}$-algebras in ${\rm\bf DVect}_{k}$ if $chark\neq 0$.
From the fibrational perspective, the reason for this is that the transition
functors of the opfibration ${\mathcal{M}}^{\otimes}\to\Gamma$ lack adjoints
(they are multifold tensor products). One cannot thus apply the model
structure results of [3] or pull back to a suitable Reedy category and use
[1]. Another problem arises if we try different parametrising categories
${\mathrm{O}}$ described above. We can consider functors
${\mathrm{O}}\to{\mathcal{M}}^{\otimes}$ that commute with the projection to
$\Gamma$, subject to some cocartesian conditions. This way one can indeed
obtain associative algebra in the case of $\Delta^{\sf op}\to\Gamma$, but the
operator categories parametrising ${\mathrm{E}}_{m}$-structures will give
commutative algebras for $m\geq 2$, due to the Eckmann-Hilton argument.

In order to deal with the problems described above, [20] passes to the world
of symmetric monoidal infinity-categories, and performs all the constructions
on that level. More precisely the machinery of [20] is developed for the
infinity-operads, and is not immediately applicable to arbitrary operation-
indexing categories ${\mathrm{O}}\to\Gamma$ some of which [2] have no operadic
analogues (technically speaking, one can attempt taking a fibrant replacement
of ${\mathrm{O}}$ in the category of infinity-operads, but this will result in
the loss of control over the combinatorics of ${\mathrm{O}}$). This approach
is also, arguably, quite technical: the chapters of [20] devoted to colimits
of algebras and adjoint functors contain heavy combinatorics necessary even
for colimits of algebras in the presentable setting (in the sense of [19,
5.5]). These issues motivated us to look for an alternative approach.

### Segal sections

Let us pass from ${\mathcal{M}}^{\otimes}\to\Gamma$ to more general
Grothendieck opfibrations ${\mathcal{E}}\to{\mathcal{C}}$ over small
categories ${\mathcal{C}}$. For each $c\in{\mathcal{C}}$ we thus have the
fibre category ${\mathcal{E}}(c)$, and each map $f:c\to c^{\prime}$ produces a
transition functor $f_{!}:{\mathcal{E}}(c)\to{\mathcal{E}}(c^{\prime})$ (the
compositions are respected up to an isomorphism). Define ${\sf
Sect}({\mathcal{C}},{\mathcal{E}})$ to be the category of sections of
${\mathcal{E}}\to{\mathcal{C}}$; it is defined as the fibre of the post-
composition functor
$\operatorname{\mathsf{Fun}}({\mathcal{C}},{\mathcal{E}})\to\operatorname{\mathsf{Fun}}({\mathcal{C}},{\mathcal{C}})$
(with $\operatorname{\mathsf{Fun}}$ denoting the functor categories) over the
identity functor. Any section $X:{\mathcal{C}}\to{\mathcal{E}}$ associates to
a map $f:c\to c^{\prime}$ the map $f_{!}X(c)\to X(c^{\prime})$ in
${\mathcal{E}}(c^{\prime})$; if the latter map is an isomorphism, the section
$X$ is cocartesian along $f$. We can impose the cocartesian condition along a
subset ${\mathcal{S}}$ of maps of ${\mathcal{C}}$, and denote the resulting
subcategory as ${\sf Sect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})$. This
category is a generalisation of the algebra objects category in the case of
${\mathcal{M}}^{\otimes}\to\Gamma$.

Assume now that each ${\mathcal{E}}(c)$ has weak equivalences
${\mathcal{W}}(c)$. For simplicity we ask here that the transition functors
$f_{!}$ preserve weak equivalences. We can then take the higher-categorical
localisation of ${\mathcal{E}}$ with respect to all ${\mathcal{W}}(c)$, and as
observed in [16, 21], the resulting infinity-functor
$L{\mathcal{E}}\to{\mathcal{C}}$ is a cocartesian fibration in infinity-
categories as defined in [19]. One can use the language of [19] to define the
higher categories ${\sf Sect}({\mathcal{C}},L{\mathcal{E}})$ and ${\sf
Sect}_{\mathcal{S}}({\mathcal{C}},L{\mathcal{E}})$. We would like to
understand the properties of these higher categories as a function of the
properties of ${\mathcal{E}}$, and ideally have convenient strict models for
them.

Our answer to this problem is the notion of a _derived_ , or _Segal_ , section
(we will use both names interchangeably in this text). Informally, such an
object $X$ associates to $c\in{\mathcal{C}}$ an object
$X(c)\in{\mathcal{E}}(c)$, and to each map $f:c\to c^{\prime}$, a diagram

(iii)
$f_{!}X(c)\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}X_{f}\longrightarrow
X(c^{\prime})$

with $f_{!}:{\mathcal{E}}(c)\to{\mathcal{E}}(c^{\prime})$ being the transition
functor and the left map in (iii) belonging to ${\mathcal{W}}(c)$. If we were
to invert it, we would get a map $f_{!}X(c)\to X(c^{\prime})$: in this sense,
a Segal section is a derived version of the concept of a section. There are
also additional data that ensure the coherence with respect to the
compositions in ${\mathcal{C}}$. Finally, if $f\in{\mathcal{S}}$, then both
maps in the diagram (iii) are required to be weak equivalences.

Below we explain more formally our definition of Segal sections; those readers
that are only interested in the properties of the homotopical category of
derived sections are invited to skip further.

The start is to replace the base category ${\mathcal{C}}$. The well-known
construction associates to it the category $\Delta/{\mathcal{C}}$, whose
objects are functors $[n]\to{\mathcal{C}}$ (in other words, strings of arrows
$c_{0}\to...\to c_{n}$), and the morphisms are functors $[n]\to[m]$ over
${\mathcal{C}}$. The _simplicial replacement_ ${\mathbb{C}}$ of
${\mathcal{C}}$ is defining by simply putting
${\mathbb{C}}:=(\Delta/{\mathcal{C}})^{\sf op}$.

Among the maps in $\Delta$, we can consider those interval inclusions
$[m]\hookrightarrow[n]$ that send $0$ to $0$. The maps in ${\mathbb{C}}$ with
such underlying $\Delta$-map look like

$(c_{0}\to...\to c_{n})\longrightarrow(c_{0}\to...\to c_{m})$

and are called Segal maps in this paper. The reason for this terminology is
the known fact that the functor $h:{\mathbb{C}}\to{\mathcal{C}}$ sending
$c_{0}\to...\to c_{n}$ to $c_{0}$, is the (higher-categorical) localisation of
${\mathbb{C}}$ along Segal maps (Proposition 1.15). As a result, given a
category ${\mathcal{M}}$ with weak equivalences ${\mathcal{W}}$, we can
represent the functors from ${\mathcal{C}}$ to the localisation of
${\mathcal{M}}$ with respect to ${\mathcal{W}}$ (in ordinary or higher sense)
as functors from ${\mathbb{C}}$ to ${\mathcal{M}}$ sending the Segal maps to
${\mathcal{W}}$. Given such functor $X$, its evaluation on the span
$c_{0}\leftarrow(c_{0}\stackrel{{\scriptstyle f}}{{\to}}c_{1})\to c_{1}$ gives
a diagram

(iv)
$X(c_{0})\stackrel{{\scriptstyle\in{\mathcal{W}}}}{{\longleftarrow}}X(c_{0}\stackrel{{\scriptstyle
f}}{{\to}}c_{1})\longrightarrow X(c_{1})$

that can be compared with (ii) and is a version of (iii) when no non-trivial
opfibration is involved. In this way, $X:{\mathbb{C}}\to{\mathcal{M}}$ can be
viewed as a “Segal functor” from ${\mathcal{C}}$ to ${\mathcal{M}}$.

The next step of the construction concerns the opfibration
${\mathcal{E}}\to{\mathcal{C}}$. Its extension to ${\mathbb{C}}$ can be done
using the functor $h:{\mathbb{C}}\to{\mathcal{C}}$. While this technique is
useful when dealing with Quillen presheaves [1], the result is not
illuminating in the case of ${\mathcal{M}}^{\otimes}\to\Gamma$ due to the
already mentioned lack of adjoints. Instead, one has to turn to the
_transpose_ fibration ${\mathcal{E}}^{\top}\to{\mathcal{C}}^{\sf op}$
(Definition 1.17): it has the same fibres ${\mathcal{E}}(c)$, and has the same
transition functors inherited from ${\mathcal{E}}\to{\mathcal{C}}$, but read
in a contravariant direction over ${\mathcal{C}}^{\sf op}$. Now, any object
$c_{0}\to...\to c_{n}\in{\mathbb{C}}$, viewed as a functor
$\mathbf{c}:[n]\to{\mathcal{C}}$, allows us to write

$\mathbf{E}(\mathbf{c}):={\sf Sect}([n],\mathbf{c}^{{\sf
op},*}{\mathcal{E}}^{\top}),$

in other words, we consider the sections of the transpose fibration over the
string $c_{0}\to...\to c_{n}$. Moreover the assignment
$\mathbf{c}\mapsto\mathbf{E}(\mathbf{c})$ defines a functor: for each map
$\alpha:\mathbf{c}\to\mathbf{c}^{\prime}$ in ${\mathbb{C}}$, we have the
restriction
$\alpha_{!}:\mathbf{E}(\mathbf{c})\to\mathbf{E}(\mathbf{c}^{\prime})$. Using
the Grothendieck construction [26] we can take the associated opfibration
denoted $\mathbf{E}\to{\mathbb{C}}$.

Given an object $\mathbf{c}=c_{0}\to...\to c_{n}$, the functor
$\mathbf{E}(\mathbf{c})\to\mathbf{E}(c_{0})={\mathcal{E}}(c_{0})$ of
evaluation at $c_{0}$ admits a right adjoint that induces an equivalence
$\mathbf{E}(c_{0})\cong{\sf Cart}([n],\mathbf{c}^{{\sf
op},*}{\mathcal{E}}^{\top})\subset\mathbf{E}(\mathbf{c})$ of
$\mathbf{E}(c_{0})$ with the full subcategory of ${\sf
Sect}([n],\mathbf{c}^{{\sf op},*}{\mathcal{E}}^{\top})$ consisting of sections
taking all maps of $[n]$ to cartesian maps. A similar situation happens along
any Segal map $\alpha:\mathbf{c}\to\mathbf{c}^{\prime}$, and the existing
adjunctions
$\alpha_{!}:\mathbf{E}(\mathbf{c})\rightleftarrows\mathbf{E}(\mathbf{c}^{\prime}):\alpha^{*}$
mean that $\mathbf{E}\to{\mathbb{C}}$ is also a fibration along Segal maps.

The following is Proposition 2.2 in the main text.

######  Proposition 1.

The category ${\sf Sect}({\mathcal{C}},{\mathcal{E}})$ is equivalent to the
full subcategory of ${\sf Sect}({\mathbb{C}},\mathbf{E})$ consisting of those
$X$ such that for each Segal map $\alpha:\mathbf{c}\to\mathbf{c}^{\prime}$,
the image $X(\alpha)$ is cartesian, meaning
$X(\mathbf{c})\stackrel{{\scriptstyle\sim}}{{\to}}\alpha^{*}X(\mathbf{c}^{\prime})$.

If ${\mathcal{E}}\to{\mathcal{C}}$ has the weak equivalences as before, each
category $\mathbf{E}(\mathbf{c})$ also comes with weak equivalences, defined
value-by-value. The category of _derived sections_ ${\sf
DSect}({\mathcal{C}},{\mathcal{E}})$ is then defined as the full subcategory
of ${\sf Sect}({\mathbb{C}},\mathbf{E})$ consisting of all $X$ such that each
Segal $\alpha:\mathbf{c}\to\mathbf{c}^{\prime}$ induces a weak equivalence
$X(\mathbf{c})\stackrel{{\scriptstyle{\mathcal{W}}}}{{\longrightarrow}}\alpha^{*}X(\mathbf{c}^{\prime})$
(more generally, in the precedent expression we will replace $\alpha^{*}$ by
its right derived functor ${\mathbb{R}}\alpha^{*}$). The weak equivalences of
derived sections are defined value-by-value, giving a homotopical structure on
${\sf DSect}({\mathcal{C}},{\mathcal{E}})$. Without further explanation, we
also mention that one can take the subset ${\mathcal{S}}$ of maps of
${\mathcal{C}}$ into account (Definition 2.16), resulting in a full
homotopical subcategory ${\sf
DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})\subset{\sf
DSect}({\mathcal{C}},{\mathcal{E}})$ of ${\mathcal{S}}$_-locally constant_
derived sections.

### Summary of properties

The summary of the discussion above is that given an opfibration
${\mathcal{E}}\to{\mathcal{C}}$, there exists another opfibration
$\mathbf{E}\to{\mathbb{C}}$ that inherits weak equivalences from
${\mathcal{E}}$, and the homotopical category of derived sections ${\sf
DSect}({\mathcal{C}},{\mathcal{E}})$ is defined as a full subcategory of ${\sf
Sect}({\mathbb{C}},\mathbf{E})$ using a weakened version of cartesian section
condition. Let us now describe some of its properties.

We first study the case when the fibres of ${\mathcal{E}}\to{\mathcal{C}}$ are
model categories, and the transition functors preserve fibrations and trivial
fibrations (but not necessarily have adjoints): such opfibrations are called
_model_ in this text. One important example is the case of ${\rm\bf
DVect}_{k}^{\otimes}\to\Gamma$ with $k$ any field, whose infinity-sections
model ${\mathrm{E}}_{\infty}$-algebras over $k$ (one can use operation-
indexing categories to include ${\mathrm{E}}_{n}$-algebras over $k$ and other
structures). Another example is ${\rm\bf DVect}_{k}^{{\sf
op},\otimes}\to\Gamma$, corresponding to the monoidal structure on the
opposite category of chain complexes over $k$: studying infinity-sections in
this case corresponds to higher coalgebra objects in $L{\rm\bf DVect}_{k}$.

The following proposition is the summary of subsection 2.2:

######  Proposition 2.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration. Then the functor
$\mathbf{E}\to{\mathbb{C}}$ is a bifibration in model categories and Quillen
pairs, its sections ${\sf Sect}({\mathbb{C}},\mathbf{E})$ is a model category
for the Reedy structure of [1], and the full subcategory ${\sf
DSect}({\mathcal{C}},{\mathcal{E}})\subset{\sf Sect}({\mathbb{C}},\mathbf{E})$
is closed under homotopy limits.

Moreover when the model categories ${\mathcal{E}}(c)$ are combinatorial and
the transition functors $f_{!}$ are accessible, there is a Bousfield
localisation of the Reedy model structure on ${\sf
Sect}({\mathbb{C}},\mathbf{E})$ whose fibrant objects are satisfy the derived
section condition. In particular, there exist homotopy colimits of derived
sections.

If the transition functors along the maps from the subset ${\mathcal{S}}$
preserve limits, then the inclusion ${\sf
DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})\subset{\sf
Sect}({\mathbb{C}},\mathbf{E})$ is also stable under limits, yet the existence
of a localised model structure is not guaranteed (the situation is similar to
that encountered in [4]). Nonetheless Proposition 2 allows to establish the
following important result. Given a model opfibration
${\mathcal{E}}\to{\mathcal{C}}$, denote by
${\mathcal{E}}_{f}\subset{\mathcal{E}}$ the subopfibration spanned by the
fibrewise-fibrant objects. We can apply the infinity-localisation of [16] to
get a cocartesian fibration in quasicategories
$L{\mathcal{E}}_{f}\to{\mathcal{C}}$. The following result is a consequence of
the comparison theorem given in [1] and is a combination of Theorem 3.3 and
Corollary 3.5 in the main text:

######  Theorem 3.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration and ${\mathcal{S}}$
a (possibly empty) subset of maps of ${\mathcal{C}}$. The infinity-category
${\sf Sect}_{\mathcal{S}}({\mathcal{C}},L{\mathcal{E}}_{f})$ consisting of
sections ${\mathcal{C}}\to L{\mathcal{E}}_{f}$ that send the maps of
${\mathcal{S}}$ to cocartesian maps of $L{\mathcal{E}}_{f}$ is equivalent to
the infinity-localisation $L{\sf
DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})$, in a manner compatible
with the base-change.

Assume that the fibres of ${\mathcal{E}}\to{\mathcal{C}}$ are combinatorial
model categories, the transition functors are accessible, and the transition
functors along the maps in ${\mathcal{S}}$ preserve limits. Then both $L{\sf
DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})$ and ${\sf
Sect}_{\mathcal{S}}({\mathcal{C}},L{\mathcal{E}}_{f})$ are presentable
infinity-categories.

We remark that even in the non-homotopical context, it is not obvious why the
sections of ${\mathcal{E}}\to{\mathcal{C}}$ have colimits, as they are indeed
not given fibrewise. One can view the idea of a Segal section to be a smarter
way to represent the information encoded by a higher-categorical section
${\mathcal{C}}\to L{\mathcal{E}}_{f}$. Returning to the case of ${\rm\bf
DVect}_{k}^{\otimes}\to\Gamma$ with inert maps denoted as $In$, the
homotopical category ${\sf DSect}_{In}(\Gamma,{\rm\bf DVect}_{k}^{\otimes})$
provides a strictification of the infinity-category of
${\mathrm{E}}_{\infty}$-algebras in ${\rm\bf DVect}_{k}$ and brings with it an
alternative to [20]’s proof of the existence of (homotopy) colimits. The same
is true for more general operation-indexing categories
${\mathrm{O}}\to\Gamma$, whether or not they have corresponding operadic
analogues.

The procedure of associating $\mathbf{E}\to{\mathbb{C}}$ to the opfibration
${\mathcal{E}}\to{\mathcal{C}}$ can be also carried out in the situation when
${\mathcal{E}}$ is an infinity-category (and the functor to ${\mathcal{C}}$ is
a cocartesian fibration), and that is explained in subsection 3.2. One then
has that the infinity-category ${\sf Sect}({\mathcal{C}},{\mathcal{E}})$ is
identified with a full subcategory of ${\sf Sect}({\mathbb{C}},\mathbf{E})$ in
a way similar to that explained in Proposition 1. This allows to establish the
presentability part of Theorem 3 more generally, as stated in Theorem 3.13 in
the main text.

######  Theorem 4.

Let ${\mathcal{E}}\to{\mathcal{B}}$ be a cocartesian fibration over a small
infinity category ${\mathcal{B}}$, such that each fibre ${\mathcal{E}}(b)$ is
presentable and each transition functor is accessible. Let ${\mathcal{S}}$ be
a subset of maps of ${\mathcal{B}}$ such that each transition functor $f_{!}$
along $f\in{\mathcal{S}}$ preserves limits (and thus satisfies the right
adjoint functor theorem).

Then the infinity-category of ${\mathcal{S}}$-cocartesian sections ${\sf
Sect}_{\mathcal{S}}({\mathcal{B}},{\mathcal{E}})$ is presentable, with limits
and (sufficiently large) filtered colimits calculated fibrewise.

This result does not appear in [19, 5.5], and is relatively immediate if one
adapts the Segal section perspective. It gives, again, an alternative to
[20]’s proof of presentability of the categories of algebras over infinity-
operads in presentably symmetric monoidal categories.

### Resolutions

The final section can be viewed as an exercise in the calculus of simplicial
replacements, and involves proving a certain descent statement. Given a
functor $F:{\mathcal{D}}\to{\mathcal{C}}$, one can readily induce the pullback
$F^{*}:{\sf DSect}({\mathcal{C}},{\mathcal{E}})\to{\sf
DSect}({\mathcal{D}},F^{*}{\mathcal{E}})$. If $F$ presents ${\mathcal{C}}$ as
a higher-categorical localisation of ${\mathcal{D}}$ with respect to
$F$-isomorphisms, denoted $F-iso$, then Theorem 3 implies that $F^{*}$ is
homotopically fully faithful, and its image consists of ${\sf
DSect}_{F-iso}({\mathcal{D}},F^{*}{\mathcal{E}})$. When $F$ belongs to a
special class of localisations (called resolutions in this paper, a condition
due to [16, Key Lemma]), the same statement can be proven without passing by
the comparison with the higher-categorical sections, which reveals interesting
simplicial combinatorics on the way (and that we shall use in our subsequent
work).

### Organisation of the paper

The first section introduces the combinatorics relevant to the procedure of
replacing ${\mathcal{E}}\to{\mathcal{C}}$ with its “simplicial extension”
$\mathbf{E}\to{\mathbb{C}}$, and we proceed to define and study the first
properties of derived/Segal sections, proving Propositions 1 and 2 in the
second section. The third section compares the Segal sections with the higher-
categorical sections, which leads to Theorem 3, and then gives a version of
the purely higher-categorical Segal section construction, resulting in Theorem
4. The resolutions example is treated in the last, fourth section.

### Acknowledgements

The author is grateful to Yonatan Harpaz, Dmitry Kaledin and Carlos Simpson
for the discussions around the content of this paper. I thank both Fondation
Jacques Hadamard (FMJH) and Centre de Mathématiques Laurent Schwartz for
financial, and my friends for personal support. This paper uses Paul Taylor’s
diagrams package.

##  1\. Simplicial Replacements

###  1.1. Preliminaries

For the dictionary of Grothendieck fibrations, and a definition of a
factorisation category, in particular of a Reedy category, we invite the
reader to consult the first section of [1]. Due to the existing ambiguity we
shall often interchange _op_ and _co_ when speaking of opfibrations: they
shall also be called cocartesian fibrations, and the maps will be either
cocartesian or opcartesian. The terms fibration, cartesian fibration will mean
the classical, contravariant Grothendieck fibrations.

One way to produce new factorisation categories out of existing ones consists
of considering presheaves, interpreted as discrete opfibrations.

######  Definition 1.1.

Let ${\mathcal{B}}$ be any small category. A _${\mathcal{B}}$ -indexed
category_ is a (small) discrete opfibration
${\mathcal{X}}\to{\mathcal{B}}^{\sf op}$. A morphism of
${\mathcal{B}}$-indexed categories ${\mathcal{X}}\to{\mathcal{Y}}$ is given by
a cocartesian morphism of discrete opfibrations over ${\mathcal{B}}^{\sf op}$.

We denote by ${\rm\bf Cat}({\mathcal{B}})$ the category of small
${\mathcal{B}}$-indexed categories. When ${\mathcal{B}}=[0]$ the one-object
category, ${\rm\bf Cat}([0])$ is simply denoted as ${\rm\bf Cat}$.

######  Remark 1.2.

Conventionally (as for instance in topos theory [18]) an indexed category is
yet another name for a contravariant pseudofunctor from ${\mathcal{B}}$ to
categories. We adopt a more rigid notion, which is equivalent to a presheaf of
sets over ${\mathcal{B}}$.

Let now ${\mathcal{B}}$ be a factorisation category, with the factorisation
structure given by $(\mathscr{G},\mathscr{D})$.

######  Lemma 1.3.

For any ${\mathcal{B}}$-indexed category
$\pi:{\mathcal{X}}\to{\mathcal{B}}^{\sf op}$ There exists a unique
factorisation system $(\mathscr{L}_{\mathcal{X}},\mathscr{R}_{\mathcal{X}})$
on ${\mathcal{X}}$ such that $\pi$ becomes a factorisation functor
$({\mathcal{X}},\mathscr{L}_{\mathcal{X}},\mathscr{R}_{\mathcal{X}})\to({\mathcal{B}}^{\sf
op},\mathscr{D}^{\sf op},\mathscr{G}^{\sf op})$. Moreover, each morphism of
${\mathcal{B}}$-indexed categories becomes a factorisation functor, as well.

###### Proof.

Set $\mathscr{L}_{\mathcal{X}}:=\pi^{-1}(\mathscr{D}^{\sf op})$ and
$\mathscr{R}_{\mathcal{X}}:=\pi^{-1}(\mathscr{G}^{\sf op})$. ∎

######  Definition 1.4.

We shall call the pair $(\mathscr{L}_{\mathcal{X}},\mathscr{R}_{\mathcal{X}})$
the factorisation system canonically induced from
$({\mathcal{B}},\mathscr{G},\mathscr{D})$.

######  Notation 1.5.

If the factorisation category structure on ${\mathcal{B}}$ has a name (as, for
example, the Reedy factorisation system on $\Delta$), then we shall also adopt
the same name for the factorisation system on the ${\mathcal{B}}$-indexed
categories ${\mathcal{X}}\to{\mathcal{B}}^{\sf op}$.

######  Definition 1.6.

Let $F:{\mathcal{B}}^{\prime}\to{\mathcal{B}}$ be a functor. A _$F$
-reindexing_ of a ${\mathcal{B}}$-indexed category
$\pi:{\mathcal{X}}_{\mathcal{B}}\to{\mathcal{B}}^{\sf op}$ is the pull-back of
$\pi$ along $F^{\sf op}$. In other words, it is the left vertical arrow in the
pullback square

######  Lemma 1.7.

Let
$F:({\mathcal{B}}^{\prime},\mathscr{G}^{\prime},\mathscr{D}^{\prime})\to({\mathcal{B}},\mathscr{G},\mathscr{D})$
be a factorisation functor and ${\mathcal{X}}\to{\mathcal{B}}^{\sf op}$ a
${\mathcal{B}}$-indexed category. Then the functor
$F_{\mathcal{X}}:{\mathcal{X}}_{{\mathcal{B}}^{\prime}}\to{\mathcal{X}}_{{\mathcal{B}}}$
induced by the reindexing operation is a factorisation functor
$({\mathcal{X}}_{{\mathcal{B}}^{\prime}},\mathscr{L}_{{\mathcal{X}}_{{\mathcal{B}}^{\prime}}},\mathscr{R}_{{\mathcal{X}}_{{\mathcal{B}}^{\prime}}})\to({\mathcal{X}}_{{\mathcal{B}}},\mathscr{L}_{{\mathcal{X}}_{{\mathcal{B}}}},\mathscr{R}_{{\mathcal{X}}_{{\mathcal{B}}}})$
between the canonically induced factorisation systems.

###### Proof.

Immediate. ∎

As we see, ${\mathcal{B}}$-indexed categories naturally inherit the
factorisation structure from ${\mathcal{B}}$, and the interaction with
factorisation functors is equally natural.

######  Proposition 1.8 (Inheritance for indexed categories).

Let
$F:({\mathcal{B}}^{\prime},\mathscr{G}^{\prime},\mathscr{D}^{\prime})\to({\mathcal{B}},\mathscr{G},\mathscr{D})$
be a factorisation functor. Then, for any ${\mathcal{B}}$-indexed category
${\mathcal{X}}_{\mathcal{B}}$, we have the following:

  1. (1)

Given any square  with
${\mathcal{Y}}_{{\mathcal{B}}^{\prime}}\to{\mathcal{B}}^{\prime{\sf op}}$ a
${\mathcal{B}}^{\prime}$-indexed category, the functor $G$ is naturally a
factorisation functor
$G:({\mathcal{Y}}_{{\mathcal{B}}^{\prime}},\mathscr{L}_{{\mathcal{Y}}_{{\mathcal{B}}^{\prime}}},\mathscr{R}_{{\mathcal{Y}}_{{\mathcal{B}}^{\prime}}})\to({\mathcal{X}}_{{\mathcal{B}}},\mathscr{L}_{{\mathcal{X}}_{{\mathcal{B}}}},\mathscr{R}_{{\mathcal{X}}_{{\mathcal{B}}}})$
between the canonically induced factorisation systems.

  2. (2)

If $\mathscr{D}^{\sf op}$ is a locally Noetherian category (see [1]), then so
is the induced category $\mathscr{L}_{{\mathcal{X}}_{{\mathcal{B}}}}$. There
is also a dual result for the right class.

  3. (3)

If $F$ is such that the induced functor $\mathscr{D}^{\prime{\sf
op}}\to\mathscr{D}^{\sf op}$ is a closed immersion of Noether categories (see
[1]), then the induced functor
$\mathscr{L}_{{\mathcal{X}}_{{\mathcal{B}}^{\prime}}}\to\mathscr{L}_{{\mathcal{X}}_{{\mathcal{B}}}}$
has the same property, as well.

  4. (4)

If $F^{\sf op}:{\mathcal{B}}^{\prime{\sf op}}\to{\mathcal{B}}^{\sf op}$ is
right-closed (see [1]), then so is
$F_{\mathcal{X}}:{\mathcal{X}}_{{\mathcal{B}}^{\prime}}\to{\mathcal{X}}_{{\mathcal{B}}}$.
Dually for left-closed.

  5. (5)

If $({\mathcal{B}},\mathscr{G},\mathscr{D})$ is a Reedy category, then so is
$({\mathcal{X}}_{{\mathcal{B}}},\mathscr{L}_{{\mathcal{X}}_{{\mathcal{B}}}},\mathscr{R}_{{\mathcal{X}}_{{\mathcal{B}}}})$.

###### Proof.

Most proofs are elementary, using the fact that an indexed category is
discretely opfibred over the base. This for example leads to isomorphisms of
comma categories
${\mathcal{X}}_{\mathcal{B}}/x\cong(\pi(x)\backslash{\mathcal{B}})^{\sf op}$,
that allows to verify both the Noether property and the closed immersion
condition. ∎

###  1.2. The replacements

######  Definition 1.9.

Given a small category ${\mathcal{C}}$, its _simplicial replacement_ is the
unique $\Delta$-indexed category ${\mathbb{C}}\to\Delta^{\sf op}$ such that
the fibre ${\mathbb{C}}([n])$ is the set ${\rm
Ob}\,\operatorname{\mathsf{Fun}}([n],{\mathcal{C}})$ of functors from $[n]$ to
${\mathcal{C}}$, with morphisms over $[n]\leftarrow[m]$ given by
precomposition
$\operatorname{\mathsf{Fun}}([n],{\mathcal{C}})\to\operatorname{\mathsf{Fun}}([m],{\mathcal{C}})$.

######  Lemma 1.10.

For ${\mathcal{C}}\in{\rm\bf Cat}$, the simplicial replacement
${\mathbb{C}}\to\Delta^{\sf op}$ can be obtained as the opfibrational
Grothendieck construction $\int N{\mathcal{C}}$ of the nerve
$N{\mathcal{C}}:\Delta^{\sf op}\to{\rm\bf Set}\subset{\rm\bf Cat}$. The
assignment ${\mathcal{C}}\mapsto{\mathbb{C}}$ defines a functor from ${\rm\bf
Cat}$ to the category ${\rm\bf Cat}(\Delta)$ of $\Delta$-indexed categories.

###### Proof.

Clear. ∎

######  Remark 1.11.

The assignment ${\mathcal{C}}\mapsto{\mathbb{C}}$ does not commute with the
operation of taking opposite categories. For any ${\mathcal{C}}$, one can
construct an isomorphism of categories $\int N{\mathcal{C}}\cong\int
N({\mathcal{C}}^{\sf op})$ that commutes with the $\Delta$-indexing up to an
involution reversing the order. Taking the simplicial replacements does not
send equivalences of categories to equivalences over $\Delta^{\sf op}$.

######  Notation 1.12.

An object of ${\mathbb{C}}$ is given by a sequence $c_{0}\to...\to c_{n}$ of
composable morphisms in ${\mathcal{C}}$. It will often be denoted as
$\mathbf{c}_{[n]}$ or simply as $\mathbf{c}$ when the $\Delta$-index is not
important. We shall also write $\mathbf{c}_{[n]}:[n]\to{\mathcal{C}}$ for the
associated functor.

For a functor $F:{\mathcal{D}}\to{\mathcal{C}}$ the induced functor will often
be denoted $\mathbb{F}:{\mathbb{D}}\to{\mathbb{C}}$. One has
$\mathbb{F}(d_{0}\to...\to d_{n})=Fd_{0}\to...\to Fd_{n}$.

The following result already appears in [25, 1] and will be revisited in the
section devoted to resolutions.

######  Lemma 1.13.

The assignments $\mathbf{c}_{[n]}\mapsto c_{0}$ and $\mathbf{c}_{[n]}\mapsto
c_{n}$ determine functors $h_{\mathcal{C}}:{\mathbb{C}}\to{\mathcal{C}}$ and
$t_{\mathcal{C}}:{\mathbb{C}}\to{\mathcal{C}}^{\sf op}$ that are higher-
categorical localisations of ${\mathbb{C}}$ along the maps that are sent by
these functors to identities.

###### Proof.

The opposite of the functor $t_{\mathcal{C}}$ is seen to be an opfibration
with contractible fibres, one then applies [1, Lemma 3.53]. The functor
$h_{\mathcal{C}}$ is seen to be isomorphic, in ${\rm\bf Cat}/{\mathcal{C}}$,
to the functor $t_{{\mathcal{C}}^{\sf op}}$, via the isomorphism relating the
simplicial replacements of ${\mathcal{C}}$ and ${\mathcal{C}}^{\sf op}$. ∎

Recall (for example, from Appendix of [1]) that $\Delta$ possesses the Reedy
factorisation system given by surjective and injective maps. By Proposition
1.8, we get the induced Reedy factorisation system
$({\mathbb{C}}_{-},{\mathbb{C}}_{+})$ on ${\mathbb{C}}$. We shall use the term
degeneracies to call the maps belonging to ${\mathbb{C}}_{+}$: those are the
maps that lie over surjections in $\Delta$.

There is another factorisation system on $\Delta$, that is given by initial
element preserving maps and left interval inclusions. Let us spell out the
induced structure on ${\mathbb{C}}$ in detail.

######  Definition 1.14.

A map $\zeta:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$ is _Segal_ iff its
projection in $\Delta$, $\pi(\zeta):[m]\to[n]$, is an interval inclusion of
$[m]$ as first $m+1$ elements of $[n]$, i.e. $\pi(\zeta)(i)=i$ for $0\leq
i\leq m$. In particular, $m$ should be less or equal to $n$.

A map $\zeta:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$ is _anchor_ , or
_endpoint-preserving_ , iff the underlying map in $\Delta^{\sf op}$ preserves
the endpoints: $\pi(\zeta)(m)=n$.

We denote by $\mathscr{S}_{\mathbb{C}}$ and $\mathscr{A}_{\mathbb{C}}$ the
sets of all Segal and anchor maps respectively. Note that $h_{\mathcal{C}}$
sends ${\mathbb{C}}_{+}$ and $\mathscr{S}_{\mathbb{C}}$ to the identity maps
of ${\mathcal{C}}$, and $t_{\mathcal{C}}$ sends ${\mathbb{C}}_{+}$ and
$\mathscr{A}_{\mathbb{C}}$ to the identity maps in ${\mathcal{C}}$. The
importance of Segal maps is summarised by the following:

######  Proposition 1.15.

For a small category ${\mathcal{C}}$, the functor
$h_{\mathcal{C}}:{\mathbb{C}}\to{\mathcal{C}}$ is a higher-categorical
localisation of ${\mathbb{C}}$ along $\mathscr{S}_{\mathbb{C}}$.

###### Proof.

Lemma 1.13 establishes that $h_{\mathcal{C}}$ is a localisation along the maps
of ${\mathbb{C}}$ that project to the identities of ${\mathcal{C}}$. It will
suffice to show that given an infinity-category ${\mathcal{X}}$ and any
functor $F:{\mathbb{C}}\to{\mathcal{X}}$ inverting $\mathscr{S}_{\mathbb{C}}$,
one has that $F$ also inverts all $h_{\mathcal{C}}$-identities.

Let $\alpha:\mathbf{c}\to\mathbf{c}^{\prime}$ be a map in ${\mathbb{C}}$ such
that $h_{\mathcal{C}}(\alpha)=id_{c}$. This implies that $\alpha$ takes the
form

$\alpha:(c\to c^{\prime}\to...\to c\to c_{0}\to...)\longrightarrow(c\to
c^{\prime}_{0}\to...)$

with $c\to c^{\prime}\to...\to c$ composing into
$c\stackrel{{\scriptstyle=}}{{\to}}c$. Since Segal maps are inverted by $F$,
it is clear that the first element preserving maps are $F$-invertible. It will
thus suffice to show that the following map in ${\mathbb{C}}$ is
$F$-invertible:

$\alpha^{\prime}:(c\stackrel{{\scriptstyle=}}{{\to}}c\to
c_{0}\to...)\longrightarrow(c\to c^{\prime}_{0}\to...),$

as there is a pair of endpoint-preserving maps connecting $\alpha$ and
$\alpha^{\prime}$ by the means of a commutative diagram. In turn it will
suffice to show that the following map (connected to $\alpha^{\prime}$ by
Segal maps) in ${\mathbb{C}}$ is $F$-invertible:

$\beta:(c\stackrel{{\scriptstyle=}}{{\to}}c)\longrightarrow(c).$

The map $\beta$ is a one-sided inverse of the degeneracy
$(c)\to(c\stackrel{{\scriptstyle=}}{{\to}}c)$. The latter is a one-sided
inverse of the Segal map $(c\stackrel{{\scriptstyle=}}{{\to}}c)\to(c)$. All
these maps are hence $F$-invertible. ∎

######  Remark 1.16.

Neither the set of Segal maps nor the set of $h_{\mathcal{C}}$-identities is
saturated in the sense one applies when one speaks of localisation [12]: the
saturated set of maps that gives the localisation
$h_{\mathcal{C}}:{\mathbb{C}}\to{\mathcal{C}}$ is given by those
$\alpha:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$ that project to
isomorphisms of ${\mathcal{C}}$.

Proposition 1.15 justifies the idea that, given a homotopical category
$({\mathcal{M}},{\mathcal{W}})$, a functor $F:{\mathbb{C}}\to{\mathcal{M}}$
sending $\mathscr{S}_{\mathbb{C}}$ to ${\mathcal{W}}$ is a suitable weakening
of the concept of a functor from ${\mathcal{C}}$ to ${\mathcal{M}}$. The
action of $F$ on spans in ${\mathbb{C}}$ like

$c\longleftarrow(c\stackrel{{\scriptstyle f}}{{\to}}c^{\prime})\longrightarrow
c^{\prime},$

where the left arrow is Segal, gives a span
$F(c)\stackrel{{\scriptstyle{\mathcal{W}}}}{{\leftarrow}}F(c\to
c^{\prime})\rightarrow F(c^{\prime})$, where the left map is a weak
equivalence. On the level of ${\rm Ho}\,{\mathcal{M}}$, this span gives a map
$F(c)\to F(c^{\prime})$, which one can denote $F(f)$. Applying $F$ to higher-
length objects then ensures higher coherences for the ‘weak functor’ $F$. This
is summarised by saying that given a relative [8] functor
$F:({\mathbb{C}},\mathscr{S}_{\mathbb{C}})\to({\mathcal{M}},{\mathcal{W}})$
(meaning that $F$ takes $\mathscr{S}_{\mathbb{C}}$ to ${\mathcal{W}}$), after
passing to infinity-localisations one gets an induced infinity-functor
$LF:{\mathcal{C}}\cong L_{\mathscr{S}_{\mathbb{C}}}{\mathbb{C}}\to
L_{\mathcal{W}}{\mathcal{M}}$.

###  1.3. Families over the simplicial replacement

We now turn to the question of extending an opfibration
${\mathcal{E}}\to{\mathcal{C}}$ to ${\mathbb{C}}$. This can be done in a few
ways. Since we have a functor $h_{\mathcal{C}}:{\mathbb{C}}\to{\mathcal{C}}$,
there is a pull-back opfibration
$h_{\mathcal{C}}^{*}{\mathcal{E}}\to{\mathbb{C}}$. However, unless
${\mathcal{E}}\to{\mathcal{C}}$ is a Quillen presheaf, it is not apparent why
the category of sections of $h_{\mathcal{C}}^{*}{\mathcal{E}}\to{\mathbb{C}}$
carries a model structure. In particular, we cannot use this approach to study
higher algebra in the language of model categories.

Any opfibration $p:{\mathcal{E}}\to{\mathcal{C}}$ can be described, up to an
equivalence, as a Grothendieck construction of a covariant functor from
${\mathcal{C}}$ to categories. The latter is the same thing as a contravariant
functor from ${\mathcal{C}}^{\sf op}$ to categories. The way to capture this
duality without passing to category-valued functors, is the following.

######  Definition 1.17.

Fix an opfibration $p:{\mathcal{E}}\to{\mathcal{C}}$. Define a category
denoted as ${\mathcal{E}}^{\top}$ as follows:

  1. (1)

$Ob({\mathcal{E}}^{\top})=Ob({\mathcal{E}})$

  2. (2)

A morphism from $x\to z$ in ${\mathcal{E}}^{\top}$ is an isomorphism class of
cospans in ${\mathcal{E}}$

$x\longrightarrow y\longleftarrow z$

such that the left arrow is fibrewise, $p(x\to y)=id_{p(x)}$, and the right
arrow is opcartesian.

There is an evident functor
$p^{\top}:{\mathcal{E}}^{\top}\to{\mathcal{C}}^{\sf op}$ which sends a map
$x\longrightarrow y\longleftarrow z$ to $p(y\longleftarrow z)$. A morphism of
${\mathcal{E}}^{\top}$ is $p^{\top}$-cartesian iff it can be represented by a
span of the form
$y\stackrel{{\scriptstyle=}}{{\longrightarrow}}y\longleftarrow z$. The functor
$p^{\top}$ is a fibration, which we call the _transpose fibration_ of $p$.

If ${\mathcal{E}}\to{\mathcal{C}}$ equals $\int E\to{\mathcal{C}}$ for a
functor $E:{\mathcal{C}}\to{\rm\bf Cat}$, then
${\mathcal{E}}^{\top}\to{\mathcal{C}}^{\sf op}$ is equivalent to the
(fibrational) Grothendieck construction applied to $E:({\mathcal{C}}^{\sf
op})^{\sf op}\to{\rm\bf Cat}$ viewed as a contravariant functor on
${\mathcal{C}}^{\sf op}$. In particular, if
${\mathcal{E}}\cong{\mathcal{E}}_{0}\times{\mathcal{C}}$ and the opfibration
structure is given by the ${\mathcal{C}}$-projection, then
${\mathcal{E}}^{\top}$ is simply ${\mathcal{E}}_{0}\times{\mathcal{C}}^{\sf
op}$. The construction of a transpose fibration that we outlined can be also
done higher-categorically, see [6].

After taking the transpose fibration ${\mathcal{E}}^{\top}\to{\mathcal{C}}$ of
the opfibration ${\mathcal{E}}\to{\mathcal{C}}$, we can also consider the
pull-back fibration $t_{\mathcal{C}}^{*}{\mathcal{E}}^{\top}\to{\mathbb{C}}$.
The following proposition is a manifestation of the duality between
${\mathcal{E}}$ and ${\mathcal{E}}^{\top}$:

######  Proposition 1.18.

Given an opfibration $p:{\mathcal{E}}\to{\mathcal{C}}$, there is a morphism
$T:h^{*}_{\mathcal{C}}{\mathcal{E}}\to
t^{*}_{\mathcal{C}}{\mathcal{E}}^{\top}$ commuting with functors to
${\mathbb{C}}$ which sends opcartesian maps of
$h^{*}_{\mathcal{C}}{\mathcal{E}}$ to cartesian maps of
$t^{*}_{\mathcal{C}}{\mathcal{E}}$.

###### Proof.

Consider the category ${\mathcal{X}}$ defined as follows.

  * •

An object of ${\mathcal{X}}$ is a pair $(\mathbf{c}_{[n]},\alpha)$ where
$\mathbf{c}_{[n]}=c_{0}\to...\to c_{n}$ is an object of ${\mathbb{C}}$ and
$\alpha:x\to y$ is an opcartesian map in ${\mathcal{E}}$ which covers the
composition $c_{0}\to c_{n}$ in ${\mathcal{C}}$ (i.e. $p(\alpha)=c_{0}\to
c_{n}$),

  * •

A morphism $(\mathbf{c}_{[n]},\alpha:x\to
y)\to(\mathbf{c}^{\prime}_{[m]},\beta:x^{\prime}\to y^{\prime})$ consists of a
map $\mathbf{c}\to\mathbf{c}^{\prime}$ in ${\mathbb{C}}$ and a map
$\gamma:x\to x^{\prime}$ which covers the induced map $c_{0}\to
c^{\prime}_{0}$.

One can check that the natural functor ${\mathcal{X}}\to{\mathbb{C}}$ is an
opfibration, and that the assignment $(\mathbf{c},\alpha:x\to
y)\mapsto(\mathbf{c},x)$ defines an equivalence over ${\mathbb{C}}$ of
opfibrations
${\mathcal{X}}\stackrel{{\scriptstyle\sim}}{{\to}}h_{\mathcal{C}}^{*}{\mathcal{E}}$.

On the other hand, consider the assignment $(\mathbf{c},\alpha:x\to
y)\mapsto(\mathbf{c},y)$. We claim that it defines a functor
$\bar{T}:{\mathcal{X}}\to t^{*}_{\mathcal{C}}{\mathcal{E}}$ commuting with
projections to ${\mathbb{C}}$. Let $(f,t):(\mathbf{c},\alpha:x\to
y)\to(\mathbf{c}^{\prime},\beta:x^{\prime}\to y^{\prime})$ be a map. In
particular, we have the following diagram in ${\mathcal{E}}$:

(1.1) $\begin{diagram}$

Suppose first that the map $t$ is fibrewise. Then by opcartesian property
there exists a map $t^{\prime}:y\to y^{\prime}$ rendering the diagram (1.1)
commutative. Remembering the description of arrows in Definition 1.17, we
define $\bar{T}(f,t)=(f,y\stackrel{{\scriptstyle
t^{\prime}}}{{\to}}y^{\prime}\stackrel{{\scriptstyle
id}}{{\leftarrow}}y^{\prime})$; in other words, we view $t^{\prime}$ as a
fibrewise map of ${\mathcal{E}}^{\top}$.

Next, if $t$ is opcartesian, find an opcartesian map $k:y^{\prime}\to z$ in
${\mathcal{E}}$ covering $c^{\prime}_{m}\to c_{n}$ (which is induced from
$f:\mathbf{c}\to\mathbf{c}^{\prime}$). The composition $k\beta t$ and $\alpha$
both project along ${\mathcal{E}}\to{\mathcal{C}}$ to the map $c_{0}\to
c_{n}=c_{0}\to c^{\prime}_{0}\to c^{\prime}_{m}\to c_{n}$, hence there is a
(fibrewise) isomorphism $z\cong y$. This implies that the diagram (1.1) can be
completed as with all arrows opcartesian in ${\mathcal{E}}$. We put, again,
$\bar{T}(f,t)=(f,y\stackrel{{\scriptstyle id}}{{\to}}y\stackrel{{\scriptstyle
t^{\prime}}}{{\leftarrow}}y^{\prime})$, thus viewing $t^{\prime}$ as a
cartesian map of ${\mathcal{E}}^{\top}$. Any other case of $(f,t)$ can be
treated by reducing to these two cases.

Inverting the equivalence
${\mathcal{X}}\stackrel{{\scriptstyle\sim}}{{\to}}h_{\mathcal{C}}^{*}{\mathcal{E}}$
and composing with $\bar{T}$, we obtain the desired functor
$T:h_{\mathcal{C}}^{*}{\mathcal{E}}\to t^{*}_{\mathcal{C}}{\mathcal{E}}$. ∎

As we can see from the construction, the functor
$T:h_{\mathcal{C}}^{*}{\mathcal{E}}\to t^{*}_{\mathcal{C}}{\mathcal{E}}$ acts
by pushing an object $x$ over $c_{0}$ along the composition of the string

$c_{0}\stackrel{{\scriptstyle f_{1}}}{{\to}}...\stackrel{{\scriptstyle
f_{n}}}{{\to}}c_{n},$

which gives $(f_{n}...f_{1})_{!}x\in{\mathcal{E}}(c_{n})$. There exists a
refinement of this procedure that remembers much more information.

######  Definition 1.19.

Given $\mathbf{c}_{[n]}\in{\mathbb{C}}$, consider it as a functor
$\mathbf{c}_{[n]}^{\sf op}:[n]^{\sf op}\to{\mathcal{C}}^{\sf op}$, and define
$\mathbf{E}(\mathbf{c}_{[n]})$ to be ${\sf Sect}([n]^{\sf
op},\mathbf{c}_{[n]}^{{\sf op},*}{\mathcal{E}}^{\top})$, the category of
sections of the transpose fibration pulled back along $\mathbf{c}_{[n]}^{\sf
op}$.

######  Definition 1.20.

For an opfibration ${\mathcal{E}}\to{\mathcal{C}}$, its _simplicial extension_
is an opfibration $\mathbf{E}\to{\mathbb{C}}$ which is the (covariant)
Grothendieck construction of the functor
$\mathbf{c}\mapsto\mathbf{E}(\mathbf{c})$. It is characterised by the fibres
$\mathbf{E}(\mathbf{c}_{[n]})\cong{\sf Sect}([n]^{\sf
op},\mathbf{c}_{[n]}^{{\sf op},*}{\mathcal{E}}^{\top})$ and by the transition
adjunctions
$\alpha_{!}:\mathbf{E}(\mathbf{c}_{[n]})\rightarrow\mathbf{E}(\mathbf{c}^{\prime}_{[m]})$
given by pullbacks along maps
$\alpha:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$.

######  Lemma 1.21.

The assignment $\mathbf{c}_{[n]}\mapsto\mathbf{E}(\mathbf{c}_{[n]})={\sf
Sect}([n]^{\sf op},\mathbf{c}_{[n]}^{{\sf op},*}{\mathcal{E}}^{\top})$ has the
following properties: for each Segal map
$\alpha:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$ there is an induced
adjunction

$\alpha_{!}:\mathbf{E}(\mathbf{c}_{[n]})\rightleftarrows\mathbf{E}(\mathbf{c}^{\prime}_{[m]}):\alpha^{*}.$

When the fibres of ${\mathcal{E}}\to{\mathcal{C}}$ are complete, the right
adjoint $\alpha^{*}$ exists for all maps $\alpha$ of ${\mathbb{C}}$.

###### Proof.

Given $\alpha:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$, the functor
$\alpha_{!}:\mathbf{E}(\mathbf{c}_{[n]})\rightarrow\mathbf{E}(\mathbf{c}^{\prime}_{[m]})$
is induced as a pullback along the underlying map $[m]\to[n]$. It will thus
suffice to prove that each $\alpha_{!}$ has an adjoint. To do this, consider
four different cases:

  1. (1)

The map $\alpha$ is a degeneracy, covering a surjection
$a:[m]\twoheadrightarrow[n]$. Given a section
$A\in\mathbf{E}(\mathbf{c}^{\prime}_{[m]})$, which can be represented by a
diagram $A_{m}\to...\to A_{0}$ (we use this order to underline the opposite
character), the section $\alpha^{*}A$ is then represented by the diagram

$A_{a^{-1}(m)}\to...\to A_{a^{-1}(0)}$

where $a^{-1}(i)$ denotes the first element of the inverse image of $i\in[n]$
under $a$.

  2. (2)

The map $\alpha$ covers an injection $a:[m]\hookrightarrow[n]$ which preserves
both endpoints. We use $a$ to identify $[m]$ with its image in $[n]$.
Representing again $A\in\mathbf{E}(\mathbf{c}^{\prime}_{[m]})$ as
$A_{m}\to...\to A_{0}$, we set where $A(j)\to B_{i}\to A(k)$ is the
factorisation of the arrow $A(j)\to A(k)$, with $j$ being the maximal closest
to $i$ (respectively $k$ being the minimal closest to $i$) value belonging to
$[m]$. Intuitively, the procedure consists of “filling the holes”, the
elements which are not in the image of $a$, in the natural and minimal way.

  3. (3)

For the remaining two injections, the one omitting the first or the last
element respectively, the situation is as follows. The Segal maps

$(c_{0}\to...\to c_{m}\stackrel{{\scriptstyle
f}}{{\to}}c_{m+1})\,\,\,\longrightarrow(c_{1}\to...\to c_{m})$

omitting the last element are treated similarly to the previous case: from

$A_{0}\longleftarrow...\longleftarrow A_{m}$

we get a section

$A_{0}\longleftarrow...\longleftarrow A_{m}\longleftarrow f_{!}A_{m}.$

Suppose we have a map in ${\mathbb{C}}$ that looks like

$(c_{0}\to...\to c_{n})\,\,\,\longrightarrow(c_{1}\to...\to c_{n});$

given $A_{1}\leftarrow...\leftarrow A_{n}$, we define
$B_{0}\leftarrow...\leftarrow B_{n}$ by setting $B_{0}=*$ to be the terminal
object of ${\mathcal{E}}(c_{0})$. Next, $B_{1}:=A_{1}\times f_{1,!}*$, to
correct for the lack of map to $f_{1,!}*$. The procedure then continues, with

$B_{2}:=A_{2}\prod_{f_{2,!}A_{1}}f_{2,!}B_{1}\text{ and
}B_{k}:=A_{k}\prod_{f_{k,!}A_{k-1}}f_{k,!}B_{k-1},$

which is indeed the application of inductive principle to construct sections,
providing us with the right adjoint just like in Proposition [1, Proposition
1.31]. Intuitively, the fact that the terminal object is not preserved by
transition functors forces us to “correct” the rest of the section. Due to the
fact that this rarely happens in practice, the procedure is in fact trivial,
and consists of putting the final object in $c_{0}$-th position. ∎

The functor $\mathbf{E}\to{\mathbb{C}}$ will be our chosen way to extend
${\mathcal{E}}\to{\mathcal{C}}$ to ${\mathbb{C}}$. It is an opfibration and
always a fibration over the Segal maps (and a bifibration if
${\mathcal{E}}\to{\mathcal{C}}$ is fibrewise-complete).

######  Remark 1.22.

Note that $\mathbf{E}$ is not a simplicial replacement of ${\mathcal{E}}$ or
${\mathcal{E}}^{\top}$. In particular, the fibre of
$\mathbf{E}\to{\mathbb{C}}$ over an object $\mathbf{c}_{[n]}$ is equivalent to
${\sf Sect}([n]^{\sf op},\mathbf{c}_{[n]}^{*}{\mathcal{E}}^{\top})$, with
$\mathbf{c}_{[n]}$ regarded as a functor $[n]^{\sf op}\to{\mathcal{C}}^{\sf
op}$. One could view such sections over an $n$-simplex as a relativisation and
a thickening of the nerve construction. For example, when
${\mathcal{E}}\cong{\mathcal{E}}_{0}\times{\mathcal{C}}$ is a constant
opfibration, then $\mathbf{E}(\mathbf{c}_{[n]})$ is simply the category
$\operatorname{\mathsf{Fun}}([n]^{\sf op},{\mathcal{E}}_{0})$.

Given any $\mathbf{c}_{[n]}\in{\mathbb{C}}$, the map $\mathbf{c}_{[n]}\to
c_{0}$ is Segal. In the induced adjunction,

(1.2)
$\mathbf{E}(\mathbf{c}_{[n]})\rightleftarrows\mathbf{E}(c_{0})={\mathcal{E}}(c_{0}),$

observe that the right adjoint takes values in the full subcategory
$\mathbf{E}^{cart}(\mathbf{c}_{[n]})={\sf Cart}([n]^{\sf
op},\mathbf{c}_{[n]}^{*}{\mathcal{E}}^{\top})$ spanned by cartesian sections
of ${\sf Sect}([n]^{\sf op},\mathbf{c}_{[n]}^{*}{\mathcal{E}}^{\top})$. Note
that for any map $\alpha:\mathbf{c}\to\mathbf{c}^{\prime}$, the left adjoint
preserves cartesian sections, giving
$\alpha_{!}:\mathbf{E}^{cart}(\mathbf{c})\to\mathbf{E}^{cart}(\mathbf{c}^{\prime})$.
The induced opfibration $\mathbf{E}^{cart}\to{\mathbb{C}}$ is in fact
equivalent to $h_{\mathcal{C}}^{*}{\mathcal{E}}\to{\mathbb{C}}$.

######  Lemma 1.23.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be an opfibration. Then there is a diagram
$h_{\mathcal{C}}^{*}{\mathcal{E}}\stackrel{{\scriptstyle
S}}{{\to}}\mathbf{E}\to t_{\mathcal{C}}^{*}{\mathcal{E}}^{\top}$ of functors
over ${\mathbb{C}}$ factoring the functor $T$ of Proposition 1.18. The functor
$S$ preserves opcartesian arrows, and for each $\mathbf{c}$, the induced
functor $S_{\mathbf{c}}:{\mathcal{E}}(c_{0})\cong
h_{\mathcal{C}}^{*}{\mathcal{E}}(\mathbf{c})\to\mathbf{E}(\mathbf{c})$ is
fully faithful. The image of $S$ consists of the _subopfibration_
$\mathbf{E}^{cart}\subset\mathbf{E}$ given by the subcategories of cartesian
sections in $\mathbf{E}(\mathbf{c}_{[n]})$.

###### Proof.

The functors of the second assertion are defined as follows. Given
$\mathbf{c}_{[n]}:[n]^{\sf op}\to{\mathcal{C}}^{\sf op}$, the second functor
$\mathbf{E}\to t_{\mathcal{C}}^{*}{\mathcal{E}}^{\top}$ is induced by
evaluating a section $X\in{\sf Sect}([n]^{\sf
op},\mathbf{c}_{[n]}^{*}{\mathcal{E}}^{\top})$ at $n$, which corresponds to
$c_{n}$. The first functor, $S:h_{\mathcal{C}}^{*}{\mathcal{E}}\to\mathbf{E}$,
is induced by the inclusion $\mathbf{E}^{cart}\subset\mathbf{E}$ and the
equivalence $\mathbf{E}^{cart}\cong h_{\mathcal{C}}^{*}{\mathcal{E}}$. It can
be constructed explicitly via the procedure that sends
$X\in{\mathcal{E}}(c_{0})$ to the following diagram in ${\mathcal{E}}^{\top}$:

$X\longleftarrow(f_{1})_{!}X\longleftarrow...\longleftarrow(f_{n}...f_{1})_{!}X;$

here $\mathbf{c}_{[n]}=c_{0}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}...\stackrel{{\scriptstyle
f_{n}}}{{\longrightarrow}}c_{n}$ and $(f_{i})_{!}$ and the like denote the
transition functors of ${\mathcal{E}}\to{\mathcal{C}}$. The verification of
the rest is elementary. ∎

######  Remark 1.24.

Given an opfibration $p:{\mathcal{E}}\to{\mathcal{C}}$ and a string of
composable arrows $\mathbf{c}_{[n]}:[n]\to{\mathcal{C}}$, one can consider the
category of sections ${\sf Sect}([n],\mathbf{c}_{[n]}^{*}{\mathcal{E}})$
directly induced from the opfibration $p$. Without going into much detail, we
note that the bifibration over $\int N({\mathcal{C}}^{\sf op})$ resulting from
the assignment $\mathbf{c}_{n}\mapsto{\sf
Sect}([n],\mathbf{c}_{[n]}^{*}{\mathcal{E}})$ is related to studying (weak)
sections of the transpose fibration ${\mathcal{E}}^{\top}\to{\mathcal{C}}^{\sf
op}$.

##  2\. Derived, or Segal, sections

###  2.1. Presections

######  Definition 2.1.

Given an opfibration ${\mathcal{E}}\to{\mathcal{C}}$, its category of
_presections_ is the category

${\sf PSect}({\mathcal{C}},{\mathcal{E}}):={\sf
Sect}({\mathbb{C}},\mathbf{E}).$

of sections of the simplicial extension $\mathbf{E}\to{\mathbb{C}}$.

To relate ${\sf Sect}({\mathcal{C}},{\mathcal{E}})$ to ${\sf
PSect}({\mathcal{C}},{\mathcal{E}})$, recall the functor
$S:h_{\mathcal{C}}^{*}{\mathcal{E}}\to\mathbf{E}$ of Lemma 1.23. The functor
$h_{\mathcal{C}}$ also induces the pull-back functor $h_{\mathcal{C}}^{*}:{\sf
Sect}({\mathcal{C}},{\mathcal{E}})\to{\sf
Sect}({\mathbb{C}},h_{\mathcal{C}}^{*}{\mathcal{E}})$.

######  Proposition 2.2.

The assignment $X\mapsto S\circ(h_{\mathcal{C}}^{*}X)$ defines a functor
$i:{\sf Sect}({\mathcal{C}},{\mathcal{E}})\to{\sf
PSect}({\mathcal{C}},{\mathcal{E}})$. Its essential image consists of the
presections sending the Segal maps $\mathscr{S}_{\mathbb{C}}$ to _cartesian_
morphisms in $\mathbf{E}$.

###### Proof.

The functor $h_{\mathcal{C}}:{\mathbb{C}}\to{\mathcal{C}}$ is a localisation
along Segal maps. This readily implies that the natural pull-back functor
${\sf Sect}({\mathcal{C}},{\mathcal{E}})\to{\sf
Sect}({\mathbb{C}},h_{\mathcal{C}}^{*}{\mathcal{E}})$ is fully faithful, and
its image consists of sections $X:{\mathbb{C}}\to
h_{\mathcal{C}}^{*}{\mathcal{E}}$ that send the Segal maps of ${\mathbb{C}}$
to the opcartesian maps of $h_{\mathcal{C}}^{*}{\mathcal{E}}$. Lemma 1.23 then
implies that the category ${\sf Sect}({\mathcal{C}},{\mathcal{E}})$ is
identified with a full subcategory of ${\sf Sect}({\mathbb{C}},\mathbf{E})$
consisting of such sections $X$ that

  1. (1)

the section $X$ factors through $\mathbf{E}^{cart}\to{\mathbb{C}}$: for each
$\mathbf{c}\in{\mathbb{C}}$, the value $X(\mathbf{c})$ is a cartesian section
in $\mathbf{E}(\mathbf{c})$,

  2. (2)

for each Segal map $\mathbf{c}\to\mathbf{c}^{\prime}$, the induced map
$X(\mathbf{c})\to X(\mathbf{c}^{\prime})$ is opcartesian.

In the presence of the first condition, the second condition is equivalent to
requiring that for each Segal map $\mathbf{c}\to\mathbf{c}^{\prime}$, the
induced map $X(\mathbf{c})\to X(\mathbf{c}^{\prime})$ is _cartesian_. This
happens because the adjunction
$\mathbf{E}(\mathbf{c})\rightleftarrows\mathbf{E}(\mathbf{c}^{\prime})$
restricts to an adjoint equivalence of categories
$\mathbf{E}(\mathbf{c})^{cart}\cong\mathbf{E}(\mathbf{c}^{\prime})^{cart}$
whenever the map $\mathbf{c}\to\mathbf{c}^{\prime}$ is Segal. Thus for Segal
$\alpha:\mathbf{c}\to\mathbf{c}^{\prime}$, the map $\alpha_{!}X(\mathbf{c})\to
X(\mathbf{c}^{\prime})$ is an isomorphism iff
$X(\mathbf{c})\to\alpha^{*}X(\mathbf{c}^{\prime})$ is such.

However, any section $X:{\mathbb{C}}\to\mathbf{E}$ sending the Segal maps of
${\mathbb{C}}$ to cartesian maps of $\mathbf{E}$ automatically takes values in
$\mathbf{E}^{cart}$, as verified by computing the value of $X$ on
$\mathbf{c}\to c_{0}$. ∎

######  Remark 2.3.

Consider an object $\mathbf{c}_{[n]}=c_{0}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}c_{1}\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}...\stackrel{{\scriptstyle
f_{n}}}{{\longrightarrow}}c_{n}$ of ${\mathbb{C}}$. Then $S\in
Sect({\mathcal{C}},{\mathcal{E}})$ is sent by the functor above to $i(S)$ such
that $i(S)(\mathbf{c}_{[n]})$ is represented by the diagram

$S(c_{0})\longleftarrow(f_{1})_{!}S(c_{0})\longleftarrow...\longleftarrow(f_{n}...f_{1})_{!}S(c_{0})$

where $(f_{k}...f_{1})_{!}:{\mathcal{E}}(c_{0})\to{\mathcal{E}}(c_{k})$ is a
transition functor along the composition of $f_{i}$.

We now put a homotopical structure on ${\mathcal{E}}\to{\mathcal{C}}$. The
notion that leads to the model structure on presections is the following one:

######  Definition 2.4.

A _model opfibration_ ${\mathcal{E}}\to{\mathcal{C}}$ is an opfibration such
that each fibre ${\mathcal{E}}(c)$ is a model category and the transition
functors preserve fibrations and trivial fibrations of the model structure.
Equivalently, given a diagram with horizontal maps opcartesian and vertical
maps in fibres, if $X\to Z$ is a (trivial) fibration then so is $Y\to T$.

######  Definition 2.5.

Say that a map $f:\mathbf{c}_{n}\to\mathbf{c}_{m}$ is

  1. (1)

anti-Segal if the underlying $\Delta$-map is an interval inclusion of $[m]$ as
last $m+1$ elements of $[n]$,

  2. (2)

convex if the underlying $\Delta$-map preserves initial and final elements.

######  Lemma 2.6.

Any map $\alpha:\mathbf{c}\to\mathbf{c}^{\prime}$ can be factored as
$\mathbf{c}\to\mathbf{c}^{\prime\prime}\to\mathbf{c}^{\prime}$, where the
first map is anti-Segal and the second map preserves the initial elements.

###### Proof.

Clear. ∎

######  Lemma 2.7.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration. Then

  1. (1)

The bifibration $\mathbf{E}\to{\mathbb{C}}$ is a Quillen presheaf [17]: the
fibres $\mathbf{E}(\mathbf{c}_{[n]})={\sf Sect}([n]^{\sf
op},\mathbf{c}_{[n]}^{{\sf op},*}{\mathcal{E}}^{\top})$ are model categories
with cofibrations and weak equivalences given valuewise, and for each
$f:\mathbf{c}\to\mathbf{c}^{\prime}$, the adjunction
$f_{!}:\mathbf{E}(\mathbf{c})\rightleftarrows\mathbf{E}(\mathbf{c}^{\prime}):f^{*}$
is a Quillen pair.

  2. (2)

The following base change condition holds. For a commutative square in
${\mathbb{C}}$,  with vertical arrows Segal, and horizontal arrows projecting
to initial element preserving maps in $\Delta$, the induced derived natural
transformation
${\mathbb{L}}f_{!}{\mathbb{R}}\alpha^{*}\to{\mathbb{R}}\beta^{*}{\mathbb{L}}g_{!}$
is an isomorphism.

  3. (3)

The following base change condition holds. For a commutative square in
${\mathbb{C}}$,  with vertical arrows anti-Segal, and horizontal arrows
convex, the induced derived natural transformation
${\mathbb{L}}\alpha_{!}{\mathbb{R}}f^{*}\to{\mathbb{R}}g^{*}{\mathbb{L}}\beta_{!}$
is an isomorphism.

###### Proof.

The first assertion is an application of [1, Theorem 1] to the fibres of
$\mathbf{E}$. Observe that the fibres $\mathbf{E}(\mathbf{c}_{[n]})$ are the
categories of sections of the fibration
$\mathbf{c}^{*}_{[n]}{\mathcal{E}}^{\top}\to[n]^{\sf op}$, which is also an
admissible model semifibration in the terminology of [1]. The admissibility is
true because the category $[n]^{\sf op}$ and the associated matching
categories have initial objects. We thus can apply the aforementioned theorem
to establish the existence of a model structure on $\mathbf{E}(\mathbf{c})$ in
which cofibrations and weak equivalences are defined objectwise. This readily
implies that the functors

$f_{!}:\mathbf{E}(\mathbf{c})\longrightarrow\mathbf{E}(\mathbf{c}^{\prime})$

are left Quillen.

The second and the third assertion follow from the explicit description of
functors in Lemma 1.21. ∎

######  Corollary 2.8.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration, then the
presection category ${\sf PSect}({\mathcal{C}},{\mathcal{E}})={\sf
Sect}({\mathbb{C}},\mathbf{E})$ has the Reedy model structure of [1, Theorem
1].

######  Remark 2.9.

The fact that in Definition 2.4, the transition functors preserve the
fibrational part, and not the cofibrations, reflects the dual, “coalgebraic”
character of the notions of a presection and of a Segal section (introduced
below).

If we consider opfibrations ${\mathcal{M}}^{\otimes}\to\Gamma$ associated to
monoidal categories as explained in the introduction, then to get a model
opfibration, one can take ${\mathcal{M}}$ to be a Cartesian monoidal model
category like ${\rm\bf SSet}$, but also the categories ${\rm\bf DVect}_{k}$ of
chain complexes of vector spaces, or simplicial vector spaces, since over a
field $k$ the tensor product is exact (note that there is no restriction
imposed on the characteristic). Another example involves taking
${\mathcal{M}}={\mathcal{N}}^{\sf op}$, where ${\mathcal{N}}$ is a monoidal
model category with all objects cofibrant. For example, we can take
${\mathcal{M}}={\rm\bf DVect}_{k}^{\sf op}$.

###  2.2. Homotopical category of Segal sections

######  Definition 2.10.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration. A presection
$S:{\mathbb{C}}\to\mathbf{E}$ is _derived_ , or _Segal_ , if for any Segal map
$\alpha:\mathbf{c}\to\mathbf{c}^{\prime}$ the morphism $S(\mathbf{c})\to
S(\mathbf{c}^{\prime})$ is weakly cartesian, meaning that the induced morphism
$S(\mathbf{c})\to{\mathbb{R}}\alpha^{*}S(\mathbf{c}^{\prime})$ is an
isomorphism in ${\rm Ho}\,\mathbf{E}(\mathbf{c})$.

We denote by ${\sf DSect}({\mathcal{C}},{\mathcal{E}})$ the full subcategory
of ${\sf PSect}({\mathcal{C}},{\mathcal{E}})$ consisting of Segal sections,
with the weak equivalences being those induced from presections.

######  Lemma 2.11.

Let $S\to S^{\prime}$ be a weak equivalence in ${\sf
PSect}({\mathcal{C}},{\mathcal{E}})={\sf Sect}({\mathbb{C}},\mathbf{E})$.
Then, if one of $S,S^{\prime}$ is Segal, so is the other.

######  Lemma 2.12.

Let $X$ be a derived section and $\alpha:\mathbf{c}\to\mathbf{c}^{\prime}$ be
a map in ${\mathbb{C}}$ covering an initial element preserving map. Then both
$X(\mathbf{c})\to{\mathbb{R}}\alpha^{*}X(\mathbf{c}^{\prime})$ and
${\mathbb{L}}\alpha_{!}X(\mathbf{c})\to X(\mathbf{c}^{\prime})$ are
isomorphisms in corresponding homotopy categories.

###### Proof.

Since $\alpha$ preserves initial elements, we have the following commutative
diagram using this diagram, the base change of Lemma 2.7 and the fact that $X$
is Segal, we can base change from the two maps of this lemma to studying what
happens at $c_{0}=c^{\prime}_{0}$. And there, everything trivially coincides.
∎

######  Remark 2.13.

Using a similar argument, it is easy to see that a presection $X$ is Segal iff
it satisfies the condition of Definition 2.10 for Segal maps of the form
$\mathbf{c}_{[n]}\to c_{0}$.

Consider the following span in ${\mathbb{C}}$: A derived section $X$ would
provide us with the span so that on homotopy level, one has a morphism
${\mathbb{L}}\beta_{!}{\mathbb{R}}\alpha^{*}X(c)\to X(c^{\prime})$. However,
in this case, ${\mathbb{L}}\beta_{!}{\mathbb{R}}\alpha^{*}X(c)$ is isomorphic
to ${\mathbb{R}}f_{!}X(c)$, where ${\mathbb{R}}f_{!}:{\rm
Ho}\,{\mathcal{E}}(c)\to{\rm Ho}\,{\mathcal{E}}(c^{\prime})$ is the derived
transition functor along $f$. That is, the data of a derived section give us a
section of the opfibration ${\rm Ho}\,{\mathcal{E}}\to{\mathcal{C}}$, with
${\rm Ho}\,$ applied fibrewise:

######  Lemma 2.14.

Let $X\in{\sf DSect}({\mathcal{C}},{\mathcal{E}})$ be fibrant as a presection.
Then the assignment $c\mapsto X(c)$ naturally prolongs to a section $hX\in{\sf
Sect}({\mathcal{C}},{\rm Ho}\,({\mathcal{E}}))$, where the opfibration ${\rm
Ho}\,{\mathcal{E}}\to{\mathcal{C}}$ has localised categories as fibres and
right derived functors of transition functors of ${\mathcal{E}}$, as
transition functors.

###### Proof.

One can check the compositions using Lemmas 2.7 and 2.12. ∎

######  Lemma 2.15.

Let $X$ be a derived section. Then for any anti-Segal map
$\alpha:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$ such that the maps
$c_{i-1}\to c_{i}$, $1\leq i\leq n-m$, are isomorphisms, the induced map

${\mathbb{L}}\alpha_{!}X(\mathbf{c}_{[n]})\to X(\mathbf{c}^{\prime}_{[m]})$

is an isomorphism in ${\rm Ho}\,\mathbf{E}(\mathbf{c}^{\prime}_{[m]})$.

###### Proof.

Lemmas 2.7 and 2.12 allow to reduce everything to the case of a single
isomorphism, ${\mathbb{L}}\alpha_{!}X(c_{0}\cong c_{1})\to X(c_{1})$, and the
latter is invertible in the homotopy category due to Lemma 2.14. ∎

Lemma 2.15 motivates to consider the following definition. Denote by
${\mathcal{S}}$ a subset of maps of ${\mathcal{C}}$.

######  Definition 2.16.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration. A derived section
$X\in{\sf DSect}({\mathcal{C}},{\mathcal{E}})$ is _${\mathcal{S}}$ -locally
constant_ if for any anti-Segal map
$\alpha:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$ such that the maps
$c_{i-1}\to c_{i}$, $1\leq i\leq n-m$ belong to ${\mathcal{S}}$, the induced
map

${\mathbb{L}}\alpha_{!}X(\mathbf{c}_{[n]})\to X(\mathbf{c}^{\prime}_{[m]})$

is an isomorphism in ${\rm Ho}\,\mathbf{E}(\mathbf{c}^{\prime}_{[m]})$.

The anti-Segal maps appearing in this definition will sometimes be called
${\mathcal{S}}$-decolouring. We denote by ${\sf
DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})$ the full subcategory of
${\mathcal{S}}$-locally constant derived sections.

######  Lemma 2.17.

A derived section $X\in{\sf DSect}({\mathcal{C}},{\mathcal{E}})$ is
${\mathcal{S}}$-locally constant iff for any map
$\alpha:\mathbf{c}\to\mathbf{c}^{\prime}$ projecting to an element of
${\mathcal{S}}$, the induced map ${\mathbb{L}}\alpha_{!}X(\mathbf{c})\to
X(\mathbf{c}^{\prime})$ is an isomorphism in ${\rm
Ho}\,\mathbf{E}(\mathbf{c}^{\prime})$.

###### Proof.

Use of base-change similar to the proof of previous lemmas coupled with the
fact that any map of ${\mathbb{C}}$ factors into the composition of an anti-
Segal map and an initial element preserving map. ∎

######  Lemma 2.18.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration and ${\mathcal{S}}$
a subset of maps of ${\mathcal{C}}$ such that for each $f\in{\mathcal{S}}$,
the induced transition functor $f_{!}$ preserves limits. Then for each anti-
Segal map $\alpha:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$ such that the
maps $c_{i-1}\to c_{i}$, $1\leq i\leq n-m$ belong to ${\mathcal{S}}$, the
induced functor
$\alpha_{!}:\mathbf{E}(\mathbf{c}_{[n]})\to\mathbf{E}(\mathbf{c}^{\prime}_{[m]})$
preserves fibrations, trivial fibrations and limits, and hence homotopy
limits.

###### Proof.

It follows from [1, Proposition 1.27] that the limits in
$\mathbf{E}(\mathbf{c}_{[n]})$ are calculated objectwise in $c_{0},...c_{n-m}$
in this particular case, and after that by the means of the same inductive
procedure as in $\mathbf{E}(\mathbf{c}^{\prime}_{[m]})$. The observation for
the (trivial) fibrations is similar, given their explicit description [1,
Definition 2.8]. The conclusion about the homotopy limits follows from the
general results of [12] that reduce the question to the computation of limits
of Reedy-fibrant diagrams that are preserved by the functors like
$\alpha_{!}$. ∎

######  Proposition 2.19.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration, ${\mathcal{S}}$ a
subset of maps of ${\mathcal{C}}$, and $\mathbf{E}\to{\mathbb{C}}$ the
associated bifibration. Then

  1. (1)

if $X\in{\sf DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})$, then any
fibrant and cofibrant replacement of $X$ is also a ${\mathcal{S}}$-locally
constant derived section,

  2. (2)

if $X\in{\sf DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})$ is fibrant as
a section in ${\sf PSect}({\mathcal{C}},{\mathcal{E}})={\sf
Sect}({\mathbb{C}},\mathbf{E})$ and $f:x\to y$ is a Segal map in
${\mathbb{C}}$, then the induced morphism $X(x)\to f^{*}X(y)$ is a trivial
fibration of fibrant objects,

  3. (3)

if $X_{\bullet}:I\to{\sf DSect}({\mathcal{C}},{\mathcal{E}})$ is a diagram of
derived sections, then its homotopy limit in ${\sf
PSect}({\mathcal{C}},{\mathcal{E}})$ is a derived section. If the transition
functors along ${\mathcal{S}}$ preserve limits, then the same result is true
for the diagrams valued in ${\sf
DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})$.

###### Proof.

The first assertion is a direct consequence of Lemma 2.11.

For the second assertion, we know that $X$ fibrant implies $X(x)$ fibrant for
each $x\in{\mathcal{X}}$ of degree $0$. In general, we know that
$X(x)\to\mathscr{Mat}_{x}X$ is a fibration. Similarly as for simplicial
objects in a model category, one can prove that for any $\alpha:x\to y$
covering an injection $[n]\hookleftarrow[m]$ in $\Delta$, the map
$\mathscr{Mat}_{x}X\to\alpha^{*}X(y)$ is a fibration. This implies that $X(x)$
is fibrant and any Segal map $x\to y$ goes to $X(x)\to X(y)$, a fibration and
a weak equivalence.

The final assertion is easily checked due to the fact that the (homotopy)
limits are calculated fibrewise [17] in ${\sf Sect}({\mathbb{C}},\mathbf{E})$
and the commutativity between ${\mathbb{R}}f^{*}$ for a Segal map $f:x\to y$
and homotopy colimits:

$\mathbb{R}\operatorname{\varprojlim}X_{\bullet}(x)\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{R}\operatorname{\varprojlim}\mathbb{R}f^{*}X_{\bullet}(y)\cong\mathbb{R}f^{*}\mathbb{R}\operatorname{\varprojlim}X_{\bullet}(y).$

When the transition functors along ${\mathcal{S}}$ preserve limits, Lemma 2.18
and a dual calculation permit to check the same for the local constancy
condition. ∎

The colimits of derived sections, if exist, are usually not calculated in a
fibrewise way. However, some abstract existence results can be established if
the original model opfibration ${\mathcal{E}}\to{\mathcal{C}}$ has
combinatorial fibres and accessible transition functors. For example, one can
state the following:

######  Proposition 2.20.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration such that each
${\mathcal{E}}(c)$ is left proper combinatorial, and each transition functor
$f_{!}:{\mathcal{E}}(c)\to{\mathcal{E}}(c^{\prime})$ is accessible. Then there
exists a left proper combinatorial model structure on ${\sf
PSect}({\mathcal{C}},{\mathcal{E}})$ with Reedy cofibrations and fibrant
objects given by Reedy-fibrant objects of ${\sf
DSect}({\mathcal{C}},{\mathcal{E}})$.

###### Proof.

According to [1, Proposition 2.30], each model category
$\mathbf{E}(\mathbf{c})$, being the category of sections over a Reedy
category, is combinatorial. It is moreover left proper, as is easily checked
using the fact that the colimits are computed fibrewise. Thus the Quillen
presheaf $\mathbf{E}\to{\mathbb{C}}$ has left proper combinatorial fibres.
This again implies that the category ${\sf
PSect}({\mathcal{C}},{\mathcal{E}})={\sf Sect}({\mathbb{C}},\mathbf{E})$ is
left proper combinatorial for the Reedy model structure.

The rest is done similarly to [3, Theorem 2.42]. As explained there or at [1,
Lemma 2.31], for each object $X\in\mathbf{E}(\mathbf{c})$ there exists a
section $i(X)$ such that ${\sf
PSect}({\mathcal{C}},{\mathcal{E}})(i(X),Y)=\mathbf{E}(\mathbf{c})(X,Y(\mathbf{c}))$:
indeed,
$i(X)(\mathbf{c}^{\prime})\cong\coprod_{\alpha:\mathbf{c}\to\mathbf{c}^{\prime}}\alpha_{!}X$.
For each map $f:\mathbf{c}\to\mathbf{c}^{\prime}$, one has a canonical map
$f_{!}X\to i(X)(\mathbf{c}^{\prime})$, and hence $i(f_{!}X)\to i(X)$. The
required model structure is then obtained by left Bousfield-localising the
Reedy structure on ${\sf PSect}({\mathcal{C}},{\mathcal{E}})$ along the set

$\\{i(f_{!}X)\to
i(X)\,\,|\,\,f:\mathbf{c}\to\mathbf{c}^{\prime}\in\mathscr{S}_{\mathbb{C}},\,X\in
G(\mathbf{c})\\}$

with $G(\mathbf{c})$ denoting, as in [3], the set of cofibrant homotopy
generators of $\mathbf{E}(\mathbf{c})$. ∎

Proposition 2.20 implies in particular the existence of homotopy colimits of
derived sections, even if the latter are calculated inexplicitly. To see the
same for ${\mathcal{S}}$-locally constant derived sections, one can attempt to
use the formalism of right model categories of Barwick [4]. However, the
question of homotopy colimits can be approached differently, using the
language of higher categories.

##  3\. Higher-categorical aspects

###  3.1. Behaviour with respect to the infinity-localisation

Our higher-categorical conventions are the same as those adopted in the third
section of [1], and we will make use of the results of [1] in what follows
below.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration. The subcategory
${\mathcal{E}}_{f}$ of fibrewise fibrant objects forms a sub-opfibration over
${\mathcal{C}}$ with weak equivalence preserving transition functors.
Localising ${\mathcal{E}}_{f}$ along the totality of fibrewise weak
equivalences yields [1, 16, 21] a cocartesian fibration in quasicategories
$L{\mathcal{E}}_{f}\to{\mathcal{C}}$. The goal of this subsection is to show
that ${\sf Sect}({\mathcal{C}},L{\mathcal{E}}_{f})$ is equivalent to $L{\sf
DSect}({\mathcal{C}},{\mathcal{E}})$.

Let us first consider the following diagram:

(3.1) $\begin{diagram}$

The functor $S$ was considered in Lemma 1.23, and is the restriction to
fibrant objects of the right adjoint in the adjunction

$S:{\mathcal{E}}(c_{0})={\sf
Sect}([0],c_{0}^{*}{\mathcal{E}}^{\top})\leftrightarrows{\sf Sect}([n]^{\sf
op},\mathbf{c}_{[n]}^{*}{\mathcal{E}}^{\top}):ev_{0}.$

Note that the resulting infinity-adjunction (existing by [1, Corollary 3.31]
[21, Theorem 5.1.1]),

$LS:L{\mathcal{E}}(c_{0})\leftrightarrows
L\mathbf{E}(\mathbf{c}_{[n]}):Lev_{0}$

has the property that its right adjoint $LS$ is full and faithful, and its
essential image, when restricted ${\mathcal{E}}(c_{0})_{f}$, coincides with
the full subcategory $L(\mathbf{E}(\mathbf{c}_{[n]})^{wcart}_{f})$ consisting
of fibrant weakly cartesian sections in ${\sf Sect}([n]^{\sf
op},\mathbf{c}_{[n]}^{*}{\mathcal{E}}^{\top})$, that is, of those $X:[n]^{\sf
op}\to{\mathcal{E}}^{\top}$ such that each $X(k)$ is fibrant and each
$X(i)\to\alpha^{*}X(i-1)$ is a weak equivalence. Indeed, the functors

$S:{\mathcal{E}}(c_{0})_{f}\leftrightarrows\mathbf{E}(\mathbf{c}_{[n]})^{wcart}_{f}:ev_{0}$

are inverse weak equivalences [8] of relative categories.

Given a map $\mathbf{c}\to\mathbf{c}^{\prime}$, the covariant transition
functor $\mathbf{E}(\mathbf{c})\to\mathbf{E}(\mathbf{c}^{\prime})$, being
merely a pullback, induces
$\mathbf{E}(\mathbf{c})^{wcart}_{f}\to\mathbf{E}(\mathbf{c}^{\prime})^{wcart}_{f}$
and thus an opfibration $\mathbf{E}^{wcart}_{f}\to{\mathbb{C}}$. Combining
that with the precedent remarks, using the functoriality of the infinity-
localisation (it can be realised [16] as a functorial fibrant replacement in
marked simplicial sets) we localise the diagram (3.1) to obtain a diagram of
cocartesian fibrations of quasicategories

(3.2) $\begin{diagram}$

with the arrow $LS$ being an equivalence by [19, Proposition 3.1.3.5].

The naturality of the Grothendieck construction for quasicategories [21,
Remark 3.1.13] together with [21, Proposition 5.2.3] imply that the
localisation of the opfibration ${\mathcal{E}}_{f}\to{\mathcal{C}}$ is
universal, meaning that the map $Lh^{*}{\mathcal{E}}_{f}\to
h^{*}L{\mathcal{E}}_{f}$ is a cocartesian equivalence over ${\mathbb{C}}$. On
the other hand, since the functor $h:{\mathbb{C}}\to{\mathcal{C}}$ is an
infinity-localisation along the Segal maps (Proposition 1.15), we can identify
certain sections over ${\mathbb{C}}$ with the sections over ${\mathcal{C}}$.
Indeed, given any categorical fibration ${\mathcal{F}}\to{\mathcal{C}}$, there
is an induced homotopy pull-back diagram

(3.3) $\begin{diagram}$

with $\operatorname{\mathsf{Fun}}_{\mathscr{S}}({\mathbb{C}},{\mathcal{F}})$
(and similarly for ${\mathcal{C}}$) being the infinity-category of functors
${\mathbb{C}}\to{\mathcal{F}}$ that send Segal maps of ${\mathbb{C}}$ to
equivalences of ${\mathcal{F}}$. Both vertical maps in (3.3) are categorical
fibrations, so taking the pull-back induces an equivalence ${\sf
Sect}({\mathcal{C}},{\mathcal{F}})\cong{\sf
Sect}_{\mathscr{S}}({\mathbb{C}},{\mathcal{F}})$, with the latter denoting the
sub-category of functors ${\mathbb{C}}\to{\mathcal{F}}$ over ${\mathcal{C}}$
sending Segal maps to equivalences of ${\mathcal{F}}$.

Consequently, the infinity-category ${\sf
Sect}({\mathcal{C}},L{\mathcal{E}}_{f})$ is identified with the full
subcategory of ${\sf Sect}({\mathbb{C}},h^{*}L{\mathcal{E}}_{f})$ consisting
of those $X$ such that for each Segal map $\alpha$ in ${\mathbb{C}}$, the map
$X(\alpha)$ is cocartesian. For the last statement, one uses the fact that a
morphism of $h^{*}L{\mathcal{E}}_{f}$ is cocartesian iff its image in
$L{\mathcal{E}}_{f}$ is such, a direct consequence of [19, Remark 2.4.1.4]. We
conclude the precedent discussion by stating the following.

######  Lemma 3.1.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration. Then there is a
canonically induced equivalence between the infinity-category ${\sf
Sect}({\mathcal{C}},L{\mathcal{E}}_{f})$ and the full subcategory of ${\sf
Sect}({\mathbb{C}},L\mathbf{E})$ consisting of those sections
$X:{\mathbb{C}}\to L\mathbf{E}$ such that

  1. (1)

For each $\mathbf{c}\in{\mathbb{C}}$, the object $X(\mathbf{c})$ belongs to
$L\mathbf{E}^{wcart}(\mathbf{c})$.

  2. (2)

For each Segal map $\alpha$ in ${\mathbb{C}}$, the induced map $X(\alpha)$ is
cocartesian.

Furthermore, these two conditions are equivalent to requiring that $X$ sends
Segal maps to _cartesian_ maps of $L\mathbf{E}$.

###### Proof.

Only the last sentence requires proof. For this, note that just as in the
1-categorical case, the infinity-adjunction
$L\mathbf{E}(\mathbf{c})\rightleftarrows L\mathbf{E}(\mathbf{c}^{\prime})$
restricts to an equivalence of quasicategories
$L\mathbf{E}(\mathbf{c})^{wcart}\cong
L\mathbf{E}(\mathbf{c}^{\prime})^{wcart}$ whenever the map
$\mathbf{c}\to\mathbf{c}^{\prime}$ is Segal. Thus for Segal
$\alpha:\mathbf{c}\to\mathbf{c}^{\prime}$ and $X$ taking value in
$L\mathbf{E}^{wcart}$, the map ${\mathbb{L}}\alpha_{!}X(\mathbf{c})\to
X(\mathbf{c}^{\prime})$ is an equivalence iff
$X(\mathbf{c})\to{\mathbb{R}}\alpha^{*}X(\mathbf{c}^{\prime})$ is such. ∎

######  Proposition 3.2.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration. Then the natural
comparison functor $L{\sf Sect}({\mathbb{C}},\mathbf{E})\to{\sf
Sect}({\mathbb{C}},L\mathbf{E})$ induces an equivalence

$L{\sf DSect}({\mathcal{C}},{\mathcal{E}})\cong{\sf
Sect}({\mathcal{C}},L{\mathcal{E}}_{f}).$

This equivalence is compatible with the base change.

###### Proof.

Since $\mathbf{E}\to{\mathbb{C}}$ is a Quillen presheaf, Theorem 2 (or equally
Proposition 3) of [1] imply that $L{\sf
Sect}({\mathbb{C}},\mathbf{E})\cong{\sf Sect}({\mathbb{C}},L\mathbf{E})$.
Everything then follows from Lemma 3.1 and [1, Lemma 3.44]. ∎

A similar result can be proven for the locally constant derived sections.

######  Theorem 3.3.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration and ${\mathcal{S}}$
a subset of maps of ${\mathcal{C}}$. The infinity-category ${\sf
Sect}_{\mathcal{S}}({\mathcal{C}},L{\mathcal{E}}_{f})$ consisting of sections
${\mathcal{C}}\to L{\mathcal{E}}_{f}$ that send the maps of ${\mathcal{S}}$ to
cocartesian maps of $L{\mathcal{E}}_{f}$ is equivalent to the full subcategory
of ${\sf Sect}({\mathbb{C}},\mathbf{E})$ consisting of those $X$ such that

  1. (1)

$X$ sends the Segal maps of ${\mathbb{C}}$ to cartesian maps of $L\mathbf{E}$,

  2. (2)

$X$ sends the maps of ${\mathbb{C}}$ $h_{\mathcal{C}}$-projecting to
${\mathcal{S}}$, to cocartesian maps of $L\mathbf{E}$.

In particular, one has the equivalence $L{\sf
DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})\cong{\sf
Sect}_{\mathcal{S}}({\mathcal{C}},L{\mathcal{E}}_{f})$ which is compatible
with the base-change.

###### Proof.

Combine Lemmas 3.1, 2.17 and [1, Lemma 3.44]. ∎

As an application of this result, let us see what happens in the presentable
setting. The following is a refinement of [19, Proposition 5.5.3.17].

######  Lemma 3.4.

Let $p:{\mathcal{X}}\to{\mathcal{Y}}$ be a presentable bicartesian fibration
[19, Definition 5.5.3.2] of quasicategories with ${\mathcal{Y}}$ small,
${\mathcal{L}},{\mathcal{R}}$ two sets of maps such that the covariant
transition functors of $p$ along ${\mathcal{L}}$ preserve limits.

Let ${\sf Sect}_{{\mathcal{L}},{\mathcal{R}}}({\mathcal{Y}},{\mathcal{X}})$
denote the full subcategory of ${\sf Sect}({\mathcal{Y}},{\mathcal{X}})$
consisting of those sections that send ${\mathcal{L}}$ to cocartesian and
${\mathcal{R}}$ to cartesian maps in ${\mathcal{X}}$. Then ${\sf
Sect}_{{\mathcal{L}},{\mathcal{R}}}({\mathcal{Y}},{\mathcal{X}})$ is an
accessible localisation of the presentable infinity-category ${\sf
Sect}({\mathcal{Y}},{\mathcal{X}})$, and hence is presentable itself.

###### Proof.

The presentability of ${\sf Sect}({\mathcal{Y}},{\mathcal{X}})$ is [19,
Proposition 5.5.3.17], and the proof of the same proposition shows that it
suffices to consider ${\mathcal{Y}}=[1]$ and prove the accessible localisation
condition in the following two cases:

_Case 1:_ the inclusion ${\sf Sect}_{(\emptyset,\\{0\to
1\\})}([1],{\mathcal{X}})\subset{\sf Sect}([1],{\mathcal{X}})$ of cartesian
sections into all sections of a presentable fibration ${\mathcal{X}}\to[1]$.
This is [19, Lemma 5.5.3.16] and the fact that the evaluation at $1\in[1]$
induces an equivalence ${\sf Sect}_{(\emptyset,\\{0\to
1\\})}([1],{\mathcal{X}})\stackrel{{\scriptstyle\sim}}{{\to}}{\mathcal{X}}(1)$.

_Case 2:_ the inclusion ${\sf Sect}_{(\\{0\to
1\\},\emptyset)}([1],{\mathcal{X}})\subset{\sf Sect}([1],{\mathcal{X}})$ of
cocartesian sections into all sections of a presentable fibration
${\mathcal{X}}\to[1]$ whose associated covariant transition functor
$f_{!}:{\mathcal{X}}(0)\to{\mathcal{X}}(1)$ preserves limits. This can be
solved by applying [20, Proposition 5.4.7.11] or by the following explicit
argument. Note that the evaluation at zero induces a categorical equivalence
${\sf Sect}_{(\\{0\to 1\\},\emptyset)}([1],{\mathcal{X}})\to{\mathcal{X}}(0)$
(this functor is obtained by applying cocartesian sections functor to a left
marked anodyne map). The association of a functor $f_{!}$ to
${\mathcal{X}}\to[1]$ provides, by [19, Definition 5.2.1.1], an inverse
equivalence $s:{\mathcal{X}}(0)\to{\sf Sect}_{(\\{0\to
1\\},\emptyset)}([1],{\mathcal{X}})$ such that for $X\in{\mathcal{X}}(0)$,
$sX(0)=X$ and $sX(1)=f_{!}X$. Thus given any diagram $\mathbf{Y}:I\to{\sf
Sect}_{(\\{0\to 1\\},\emptyset)}([1],{\mathcal{X}})$ we can assume, that up to
an equivalence, $\mathbf{Y}=s\mathbf{X}$, for some
$\mathbf{X}:I\to{\mathcal{X}}(0)$. Taking limits induces the canonical map

$f_{!}\operatorname{\varprojlim}_{I}\mathbf{Y}(0)=f_{!}\operatorname{\varprojlim}_{I}\mathbf{X}\longrightarrow\operatorname{\varprojlim}_{I}f_{!}\mathbf{X}=\operatorname{\varprojlim}_{I}\mathbf{Y}(1);$

and by assumption the only non-invertible arrow here is an equivalence. This
map can be viewed as a section of ${\mathcal{X}}\to[1]$, and by the dual of
(2) of [19, Proposition 5.1.2.2] it serves as a limit of $\mathbf{Y}$. The
category of cocartesian sections of ${\mathcal{X}}\to[1]$ is thus closed under
limits in ${\sf Sect}([1],{\mathcal{X}})$. By a similar argument, using the
fact that $f_{!}$ is a left adjoint, the cocartesian section condition is also
stable under arbitrary colimits. Since ${\sf Sect}_{(\\{0\to
1\\},\emptyset)}([1],{\mathcal{X}})\to{\mathcal{X}}(0)$ is an equivalence, the
category of cocartesian sections is presentable. Thus we have a (co)limit-
preserving inclusion of a presentable category into a presentable category,
and such a functor admits a left adjoint.

The general case follows from

${\sf
Sect}_{{\mathcal{L}},{\mathcal{R}}}({\mathcal{Y}},{\mathcal{X}})=\cap_{f\in{\mathcal{L}}}{\sf
Sect}_{(\\{f\\},\emptyset)}({\mathcal{Y}},{\mathcal{X}})\bigcap\cap_{g\in{\mathcal{R}}}{\sf
Sect}_{(\emptyset,\\{g\\})}({\mathcal{Y}},{\mathcal{X}})$

and the same reduction argument as in [19, Proposition 5.5.3.17]. ∎

######  Corollary 3.5.

Let ${\mathcal{E}}\to{\mathcal{C}}$ be a model opfibration and ${\mathcal{S}}$
a subset of maps of ${\mathcal{C}}$. Assume that the fibres are combinatorial
model categories, the transition functors are accessible, and the transition
functors along the maps in ${\mathcal{S}}$ preserve limits. Then both $L{\sf
DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})$ and ${\sf
Sect}_{\mathcal{S}}({\mathcal{C}},L{\mathcal{E}}_{f})$ are presentable
infinity-categories, with limits and sufficiently filtered colimits calculated
fibrewise.

###### Proof.

Proposition 2.30 of [1] implies that the fibres of $\mathbf{E}\to{\mathbb{C}}$
are combinatorial model categories, and Lemma 2.18 shows that the transition
functors along the anti-Segal maps that forget the ${\mathcal{S}}$-part
preserve homotopy limits. Consequently $L\mathbf{E}\to{\mathbb{C}}$ is a
presentable fibration satisfying the condition of Lemma 3.4 for the sets
$\mathscr{A}\mathscr{S}_{{\mathcal{S}}}$ (of anti-Segal maps of ${\mathbb{C}}$
that forget the ${\mathcal{S}}$-labelled part) and $\mathscr{S}_{\mathbb{C}}$
(of Segal maps). It follows from Lemma 3.4 and [1, Lemma 3.44] that the
infinity-category $L{\sf
DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})\cong{\sf
Sect}_{(\mathscr{A}\mathscr{S}_{\mathcal{S}},\mathscr{S}_{\mathbb{C}})}({\mathbb{C}},L\mathbf{E})$
is presentable, and hence the same is true for ${\sf
Sect}_{\mathcal{S}}({\mathcal{C}},L{\mathcal{E}}_{f})$. ∎

######  Remark 3.6.

The are other ways to use Theorem 3.3 to conclude presentability of the
infinity-category $L{\sf DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})$ or
of its opposite. Return to the example of Remark 2.9 given by the opfibration
${\mathcal{M}}^{\otimes}\to\Gamma$, where ${\mathcal{M}}={\mathcal{N}}^{\sf
op}$ for a combinatorial monoidal model category ${\mathcal{N}}$ with all
objects cofibrant. Denoting inert maps of $\Gamma$ by $In$, it is easy to show
that the infinity-category ${\sf
Sect}_{In}(\Gamma,L{\mathcal{M}}_{f}^{\otimes})^{\sf op}$ that describes
homotopy coalgebra objects in ${\mathcal{N}}$ is presentable. To do so,
observe that

${\sf Sect}_{In}(\Gamma,L{\mathcal{M}}_{f}^{\otimes})^{\sf op}\cong{\sf
Sect}_{In}(\Gamma^{\sf op},L{\mathcal{N}}_{c}^{\otimes,\top})$

which follows from the relation $L{\mathcal{M}}_{f}^{\otimes,{\sf op}}\cong
L{\mathcal{N}}_{c}^{\otimes,\top}$ (it follows from [21, Proposition 5.2.3]
that both these fibrations classify the same infinity-functor) and the
interaction of the functor category with the ${\sf op}$-involution. Then one
can further note that due to [19, Proposition 5.4.7.11 and Remark 5.4.7.14]
the categories ${\sf Sect}_{In}(\Gamma^{\sf
op},L{\mathcal{N}}_{c}^{\otimes,\top})$ and ${\sf Sect}(\Gamma^{\sf
op},L{\mathcal{N}}_{c}^{\otimes,\top})$ are accessible, and [19, Proposition
5.1.2.2] together with the preservation of colimits along inert maps imply
that both ${\sf Sect}_{In}(\Gamma^{\sf op},L{\mathcal{N}}_{c}^{\otimes,\top})$
and ${\sf Sect}(\Gamma^{\sf op},L{\mathcal{N}}_{c}^{\otimes,\top})$ are
cocomplete. Theorem 3.3 then implies that the corresponding derived sections
$L{\sf DSect}_{In}(\Gamma,{\mathcal{M}}^{\otimes})^{\sf op}$ that serve as a
rigid model for infinity-coalgebra objects in ${\mathcal{N}}$ form a
presentable infinity-category.

The same proof works if we replace ${\mathcal{M}}^{\otimes}\to\Gamma$ by any
other model opfibration ${\mathcal{E}}\to{\mathcal{C}}$ whose fibres are
opposite-combinatorial, whose transition functors are opposite-accessible, and
that the transition functors along the subset ${\mathcal{S}}$ preserve limits.
We get that $L{\sf DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})^{\sf op}$
is presentable.

###  3.2. Higher-categorical Segal sections

The combinatorial trick that leads to derived sections can be reproduced
purely in the higher-categorical setting, and that permits to prove statements
like Corollary 3.5 for more general higher-categorical fibrations. Let
${\mathcal{E}}\to{\mathcal{C}}$ be a cocartesian fibration, with
${\mathcal{E}}$ an infinity-category and ${\mathcal{C}}$ an ordinary category.
In this subsection we will explicitly construct a cocartesian fibration
$\mathbf{E}\to{\mathbb{C}}$ which is also cartesian over Segal maps, and
identify ${\sf Sect}({\mathcal{C}},{\mathcal{E}})$ with Segal sections inside
${\sf Sect}({\mathbb{C}},\mathbf{E})$.

######  Remark 3.7.

In the higher-categorical context, we will usually only use the word Segal
when referring to sections, as everything is “derived” already.

The explicit version of the construction that we present uses the relative
nerve functor of Lurie, recalled below. We can also assume that there is a
functor ${\mathrm{E}}:{\mathcal{C}}\to{\rm\bf SSet}_{+}$ taking value in
fibrant objects (that is, $X^{\natural}$ for $X$ a quasicategory), such that
${\mathcal{E}}$ is equivalent over ${\mathcal{C}}$ to the covariant relative
nerve of ${\mathrm{E}}$.

######  Lemma 3.8.

Let ${\mathcal{D}}:I^{\sf op}\to{\rm\bf Cat}$ be a diagram of small categories
such that each ${\mathcal{D}}(i)$ has an initial object $d_{i}$. Let
${\mathrm{F}}_{i}:{\mathcal{D}}(i)\to{\rm\bf SSet}_{+}$ be a family of
fibrant-valued functors, compatible with ${\mathcal{D}}$ in the sense that for
each $f:i\to i^{\prime}$, the induced diagram commutes up to a canonical
isomorphism, with $f^{*}={\mathcal{D}}(f)$. Then one has a map of projectively
fibrant functors in $\operatorname{\mathsf{Fun}}(I,{\rm\bf SSet}_{+})$ whose
value at $i$ is equal to

${\mathrm{F}}_{i}(d_{i})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\sf
Cart}\left({\mathcal{D}}(i)^{\sf op},\int{\mathrm{F}}_{i}\right),$

an inverse of the equivalence between the cartesian sections of the
_contravariant_ relative nerve $\int{\mathrm{F}}_{i}\to{\mathcal{D}}(i)^{\sf
op}$, and ${\mathrm{F}}_{i}(d_{i})$.

Note that the assignment $i\mapsto F_{i}(d_{i})$ defines a covariant functor,
since a map $f:i\to i^{\prime}$ induces a unique map $d_{i}\to
f^{*}d_{i^{\prime}}$ in ${\mathcal{D}}(i)$. The assignment $i\mapsto{\sf
Cart}\left({\mathcal{D}}(i)^{\sf op},\int{\mathrm{F}}_{i}\right)$ is also
covariant, since in view of the marked version of [20, Remark 3.2.5.7], there
is a pullback diagram that induces the corresponding map of cartesian
sections.

###### Proof.

Let us recall the construction of
$\int{\mathrm{F}}_{i}\to{\mathcal{D}}(i)^{\sf op}$. Dualising [20, Definition
3.2.5.2], a map $\Delta^{n}\to\int{\mathrm{F}}_{i}$ is given by

  1. (1)

a map $\sigma:[n]\to{\mathcal{D}}(i)^{\sf op}$,

  2. (2)

for each (nonempty) subset $S\subset[n]$ with the minimal element $s$, a map
$\Delta^{S}\to{\mathrm{F}}_{i}(\sigma(s))$,

  3. (3)

for each such pair $S^{\prime}\subset S\subset[n]$, the commutativity property
of the diagram

An object of $\int{\mathrm{F}}_{i}$ is thus uniquely determined by a pair
consisting of an object $d\in{\mathcal{D}}(i)^{\sf op}$ and
$x\in{\mathrm{F}}_{i}(d)$. A morphism $(d,x)\to(d^{\prime},x^{\prime})$ is the
data of a map $\alpha:d^{\prime}\to d$ in ${\mathcal{D}}(i)$ and a map
$x\to{\mathrm{F}}_{i}(\alpha)x^{\prime}$ in $F_{i}(d)$, just like one expects
from the Grothendieck construction. The forgetful infinity-functor
$\int{\mathrm{F}}_{i}\to{\mathcal{D}}(i)^{\sf op}$ is a cartesian fibration,
with $(d,x)\to(d^{\prime},x^{\prime})$ being cartesian iff
$x\to{\mathrm{F}}_{i}(\alpha)x^{\prime}$ is an equivalence.

To construct the map from the proposition, let us instead construct the
adjoint map ${\mathcal{D}}(i)^{\sf
op}\times{\mathrm{F}}_{i}(d_{i})\to\int{\mathrm{F}}_{i}$ over
${\mathcal{D}}(i)^{\sf op}$. This is done as follows. A map
$\Delta^{n}\to{\mathcal{D}}(i)^{\sf op}\times{\mathrm{F}}_{i}(d_{i})$ is just
a pair of $\sigma:[n]\to{\mathcal{D}}(i)^{\sf op}$ and
$\tau:\Delta^{n}\to{\mathrm{F}}_{i}(d_{i})$. We send it to
$\int{\mathrm{F}}_{i}$ by taking

  1. (1)

the same map $\sigma:[n]\to{\mathcal{D}}(i)^{\sf op}$,

  2. (2)

for each (nonempty) subset $S\subset[n]$ with the minimal element $s$, take
the composition

$\Delta^{S}\subset\Delta^{n}\stackrel{{\scriptstyle\tau}}{{\to}}{\mathrm{F}}_{i}(d_{i})\to{\mathrm{F}}_{i}(\sigma(s))$

to get a map $\Delta^{S}\to{\mathrm{F}}_{i}(\sigma(s))$ (here we use that
$d_{i}$ is initial in ${\mathcal{D}}(i)$).

  3. (3)

for each such pair $S^{\prime}\subset S\subset[n]$, all the squares are
commutative in the diagram below,

(3.4) $\begin{diagram}$

and this implies the commutativity of the outer rectangle.

A map $\Delta^{1}\to{\mathcal{D}}(i)^{\sf op}\times{\mathrm{F}}_{i}(d_{i})$ is
cartesian iff the image of $\tau:\Delta^{1}\to{\mathrm{F}}_{i}(d_{i})$ is an
equivalence in ${\mathrm{F}}_{i}(d_{i})$. It is evidently sent to a cartesian
map in $\int{\mathrm{F}}_{i}$. As a result, the adjoint map
${\mathrm{F}}_{i}(d_{i})\to{\sf Sect}\left({\mathcal{D}}(i)^{\sf
op},\int{\mathrm{F}}_{i}\right)$ factors through the cartesian sections.
Moreover composing with the evaluation map ${\sf
Sect}\left({\mathcal{D}}(i)^{\sf
op},\int{\mathrm{F}}_{i}\right)\to{\mathrm{F}}_{i}(d_{i})$ gives identity. In
light of [20, Corollary 3.3.3.2] we have thus constructed an equivalence

${\mathrm{F}}_{i}(d_{i})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\sf
Cart}\left({\mathcal{D}}(i)^{\sf op},\int{\mathrm{F}}_{i}\right),$

and it remains to check its $I$-naturality.

Given a map $f:i\to i^{\prime}$, observe that we have the following diagram
coming from base-change with top row projecting to ${\mathcal{D}}(i)^{\sf op}$
and the bottom square to ${\mathcal{D}}(i^{\prime})^{\sf op}$. The middle
horizontal functor is induced by the pullback property of the relative nerve;
we can construct it exactly in such a way that the bottom square commutes.
Indeed, for each object $d^{\prime}\in{\mathcal{D}}(i^{\prime}))$, we have
maps $F_{i}(d_{i})\to F_{i}(f^{*}d^{\prime})=F_{i^{\prime}}(d^{\prime})$ that
are natural in the $d^{\prime}$-argument. We can use them to construct
${\mathcal{D}}(i^{\prime})^{\sf
op}\times{\mathrm{F}}_{i}(d_{i})\to\int{\mathrm{F}}_{i^{\prime}}$ similarly to
we did above at (3.4) and verify everything by hand. Taking the cartesian
sections then gives the diagram One has a canonical isomorphism of infinity-
categories

${\sf Cart}({\mathcal{D}}(i)^{\sf op},{\mathcal{D}}(i)^{\sf
op}\times{\mathrm{F}}_{i}(d_{i}))\cong\operatorname{\mathsf{Fun}}_{lc}({\mathcal{D}}(i)^{\sf
op},{\mathrm{F}}_{i}(d_{i})),$

with the right hand side denoting functors taking all arrows to equivalences.
Under this identification the maps ${\mathrm{F}}_{i}(d_{i})\to{\sf
Cart}({\mathcal{D}}(i)^{\sf op},{\mathcal{D}}(i)^{\sf
op}\times{\mathrm{F}}_{i}(d_{i}))$ correspond to constant diagram functors,
and this proves the commutativity of the diagram above. Taking its outer
rectangle proves the sought-after naturality. ∎

The next lemma will provide adjoints along Segal maps.

######  Lemma 3.9.

Let ${\mathcal{F}}\to[n]$ be a cartesian fibration. Let $\alpha:[m]\subset[n]$
be an interval inclusion such that $\alpha(m)=n$ and for each $i\in[n]$ not in
the image of $\alpha$, the fibre ${\mathcal{F}}_{i}$ has an initial object.
Then the pull-back functor $\alpha^{*}:{\sf Sect}([n],{\mathcal{F}})\to{\sf
Sect}([m],{\mathcal{F}})$ has both right and left adjoints $\alpha_{*}$ and
$\alpha_{!}$. The right adjoint furthermore factors through the full
subcategory of ${\sf Sect}([n],{\mathcal{F}})$ consisting of those sections
that are cartesian over the complement of the image of $\alpha$.

###### Proof.

Begin first with the case of $i:\\{1\\}\subset[1]$. Given a cartesian
fibration ${\mathcal{F}}\to[1]$ and associating to it a functor as in [20,
Definition 5.2.1.1] provides, among other things, us with a functor
$i_{*}:{\mathcal{F}}_{1}\to{\sf Cart}([1],{\mathcal{F}})\subset{\sf
Sect}([1],{\mathcal{F}})$ such that $i^{*}i_{*}=id$. The diagram of [20, Lemma
5.5.3.15] (or the relative Kan extension formalism) then tells us that $i_{*}$
is right adjoint to $i^{*}$.

For the left adjoint, note that the initial object
$\emptyset\in{\mathcal{F}}_{0}$ is also initial in the whole ${\mathcal{F}}$
thanks to [20, Lemma 5.2.3.4]. Thus for any $X\in{\mathcal{F}}_{1}$ there
exists a map $\emptyset\to X$ in ${\mathcal{F}}$ that projects to $0\to 1$; we
can view this map as a section. Using [20, Corollary 4.3.2.16 and Proposition
4.3.2.17] we see that there thus exists a functor
$i_{!}:{\mathcal{F}}_{1}\to{\sf Sect}([1],{\mathcal{F}})$ left adjoint to
$i^{*}$ such that $i^{*}i_{!}=id$.

For the general case, it will be enough to show the existence of adjoints with
required properties for the inclusions $\beta:[n-1]\subset[n]$. This inclusion
induces the pushout-product map $\bar{\beta}:[1]\cup_{[0]}[n-1]\to[n]$ where
$[0]$ includes as $0$ in $[n-1]$ and as $1$ in $[1]$. Using [20, Lemma
2.1.2.3] for $\emptyset\to[0]$ and the left anodyne map $[0]\subset[n-1]$, we
conclude that the map $\bar{\beta}$ is inner anodyne and thus

${\sf Sect}([n],{\mathcal{F}})\to{\sf
Sect}([1],{\mathcal{F}})\times_{{\mathcal{F}}_{1}}{\sf
Sect}([n-1],{\mathcal{F}})$

is an equivalence of infinity-categories. It moreover respects the partial
cartesian property: a section $[n]\to{\mathcal{F}}$ is cartesian over $0\to 1$
iff its image in the fibred product has a cartesian ${\sf
Sect}([1],{\mathcal{F}})$-component. It will thus suffice to construct the
adjoints for the functor

${\sf Sect}([1],{\mathcal{F}})\times_{{\mathcal{F}}_{1}}{\sf
Sect}([n-1],{\mathcal{F}})\to{\mathcal{F}}_{1}\times_{{\mathcal{F}}_{1}}{\sf
Sect}([n-1],{\mathcal{F}})\cong{\sf Sect}([n-1],{\mathcal{F}}),$

but we can do this simply by taking fibred products of the adjoint triple
$i_{!}\dashv i^{*}\dashv i_{*}$ constructed before with ${\sf
Sect}([n-1],{\mathcal{F}})\to{\mathcal{F}}_{1}$ and remembering that mappings
spaces of fibred products are fibred products of mapping spaces. ∎

######  Remark 3.10.

Implicit in the proof of Lemma 3.9 is the fact that given a cartesian (or
cocartesian) fibration ${\mathcal{E}}\to{\mathcal{B}}$, the functor ${\sf
Sect}(-,{\mathcal{E}})$ is left Quillen with respect to the Joyal model model
structure over ${\mathcal{B}}$, as follows from, for example, [20, Remark
3.1.4.5 and Proposition 3.1.5.3]. As a consequence all the fibred products
appearing in the proof are also homotopy fibred products.

We are now ready to formulate our higher-categorical variation of derived
sections. Note that given a cocartesian fibration
${\mathcal{E}}\to{\mathcal{C}}$, we will denote by
${\mathcal{E}}^{\top}\to{\mathcal{C}}^{\sf op}$ its transpose or dual, as per
usual. It can be obtained through the Grothendieck construction procedure, or
with the help of [6]. The following proposition summarises the Segal section
construction:

######  Proposition 3.11.

Let ${\mathcal{C}}$ be a small category and ${\mathcal{E}}\to{\mathcal{C}}$ a
cocartesian fibration. Denoting $h_{\mathcal{C}}:{\mathbb{C}}\to{\mathcal{C}}$
the initial element functor as before, there exists a sequence

$h^{*}_{\mathcal{C}}{\mathcal{E}}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathbf{E}^{cart}\longrightarrow\mathbf{E}$

of cocartesian fibrations and functors over ${\mathbb{C}}$, with the first
functor being a cocartesian equivalence and the second also preserving
cocartesian arrows. The fibre of this diagram at
$\mathbf{c}:[n]\to{\mathcal{C}}$ is

${\mathcal{E}}(c_{0})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\sf
Cart}([n]^{\sf op},\mathbf{c}^{*}{\mathcal{E}}^{\top})\longrightarrow{\sf
Sect}([n]^{\sf op},\mathbf{c}^{*}{\mathcal{E}}^{\top})$

with the first functor being an inverse of the evaluation equivalence. The
transition functors between the diagrams for different fibres are induced by
the transition functors of ${\mathcal{E}}$ and the evident restrictions of
(cartesian) sections.

Finally, the cocartesian fibrations $\mathbf{E}\to{\mathbb{C}}$ and
$\mathbf{E}^{cart}\to{\mathbb{C}}$ are also cartesian along Segal maps. The
composition $h^{*}_{\mathcal{C}}{\mathcal{E}}\to\mathbf{E}$ induces an
equivalence between ${\sf Sect}({\mathcal{C}},{\mathcal{E}})$ and the
subcategory ${\sf Sect}_{\mathscr{S}}({\mathbb{C}},\mathbf{E})$ consisting of
those sections $X:{\mathbb{C}}\to\mathbf{E}$ that send Segal maps to cartesian
maps. This equivalence is compatible with the base change along 1-functors.

###### Proof.

We can replace ${\mathcal{E}}\to{\mathcal{C}}$ by
$\int_{\mathcal{C}}{\mathrm{E}}\to{\mathcal{C}}$, the covariant relative nerve
of a (projectively fibrant) functor ${\mathrm{E}}:{\mathcal{C}}\to{\rm\bf
SSet}_{+}$. The transpose fibration ${\mathcal{E}}^{\top}\to{\mathcal{C}}^{\sf
op}$ can be then replaced by $\int_{{\mathcal{C}}^{\sf
op}}{\mathrm{E}}\to{\mathcal{C}}^{\sf op}$, the contravariant relative nerve
of ${\mathrm{E}}$. Given $\mathbf{c}_{[n]}:[n]\to{\mathcal{C}}$, an object of
${\mathbb{C}}$, let us denote
${\mathrm{E}}_{\mathbf{c}}:={\mathrm{E}}\circ\mathbf{c}_{[n]}$. We can then
apply Lemma 3.8 for the diagram ${\mathbb{C}}^{\sf op}\to{\rm\bf
Cat},\mathbf{c}_{[n]}\mapsto[n]$ and the system of functors
${\mathrm{E}}_{\mathbf{c}_{[n]}}$. The result is a diagram of functors from
${\mathbb{C}}$ to ${\rm\bf SSet}_{+}$ whose value at
$\mathbf{c}_{[n]}:[n]\to{\mathcal{C}}$ is

${\mathrm{E}}(c_{0})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\sf
Cart}\left([n]^{\sf
op},\int{\mathrm{E}}_{\mathbf{c}_{[n]}}\right)\longrightarrow{\sf
Sect}\left([n]^{\sf op},\int{\mathrm{E}}_{\mathbf{c}_{[n]}}\right)$

with the contravariant relative nerve underneath. All what remains is to take
the covariant relative nerve and apply Lemma 3.9 for the extra adjoints.

The section identification then proceeds as usual. Given a bicartesian
fibration ${\mathcal{M}}\to[1]$ representing an equivalence
${\mathcal{M}}_{0}\cong{\mathcal{M}}_{1}$, a section $X:[1]\to{\mathcal{M}}$
is cartesian iff it is cocartesian. Thus any section
$X:{\mathbb{C}}\to\mathbf{E}^{cart}$ sends Segal maps to cartesian maps iff it
sends them to cocartesian maps. Since
$h_{\mathcal{C}}:{\mathbb{C}}\to{\mathcal{C}}$ is an infinity-localisation
along the Segal maps, we have a diagram

${\sf
Sect}({\mathcal{C}},{\mathcal{E}})\stackrel{{\scriptstyle\cong}}{{\longleftarrow}}{\sf
Sect}_{\mathscr{S}}({\mathbb{C}},\mathbf{E}^{cart})\longrightarrow{\sf
Sect}_{\mathscr{S}}({\mathbb{C}},\mathbf{E})$

with first arrow an equivalence, and the $\mathscr{S}$-index denoting the
Segal section condition. However, the second arrow is also an equivalence, as
it follows from the description of right adjoints along $(c_{0}\to...\to
c_{n})\to c_{0}$, as given in Lemma 3.9, that any Segal section
${\mathbb{C}}\to\mathbf{E}$ factors through $\mathbf{E}^{cart}$. ∎

######  Corollary 3.12.

Let ${\mathcal{I}}$ be a subset of morphisms of ${\mathcal{C}}$. Then the
infinity-category ${\sf Sect}_{\mathcal{I}}({\mathcal{C}},{\mathcal{E}})$ of
${\mathcal{I}}$-cocartesian sections is naturally identified with the
infinity-category ${\sf
Sect}_{{\mathcal{I}}-loc,\mathscr{S}}({\mathbb{C}},\mathbf{E})$ consisting of
Segal sections that send the ${\mathcal{I}}$-coloured anti-Segal maps,

$(c_{0}\stackrel{{\scriptstyle{\mathcal{I}}}}{{\to}}...\stackrel{{\scriptstyle{\mathcal{I}}}}{{\to}}c_{k}\to...\to
c_{n})\longrightarrow(c_{k}\to...\to c_{n})$

(see Definition 2.16), to cocartesian maps.

###### Proof.

A map of ${\mathbb{C}}$ that projects to ${\mathcal{I}}$ can be factored as a
composition of an ${\mathcal{I}}$-coloured anti-Segal map and of an initial
element preserving map. Any Segal section is automatically cocartesian along
the latter type of maps. ∎

Let us now use the higher-categorical Segal sections to see the presentability
of section categories.

######  Theorem 3.13.

Let ${\mathcal{E}}\to{\mathcal{B}}$ be a cocartesian fibration over a small
infinity category ${\mathcal{B}}$, such that each fibre ${\mathcal{E}}(b)$ is
presentable and each transition functor is accessible. Let
${\mathcal{S}}\subset{\mathcal{B}}_{1}$ be a subset of maps such that each
transition functor $f_{!}$ along $f\in{\mathcal{S}}$ preserves limits (thus
satisfies the right adjoint functor theorem).

Then the infinity-category of ${\mathcal{S}}$-cocartesian sections ${\sf
Sect}_{\mathcal{S}}({\mathcal{B}},{\mathcal{E}})$ is presentable, with limits
and (sufficiently large) filtered colimits calculated fibrewise.

This result is not found in [20, 5.5], and can be used to abstractly conclude
the existence of colimits and adjoints of algebras over infinity-operads and
operator categories.

###### Proof.

We already know that ${\sf Sect}_{\mathcal{S}}({\mathcal{B}},{\mathcal{E}})$
is accessible, with limits and filtered colimits of sections calculated
fibrewise, as follows from [20, Proposition 5.1.2.2 and Corollary 5.4.7.17].
Choose an infinity-localisation $F:{\mathcal{C}}\to{\mathcal{B}}$ surjective
on objects and morphisms, where ${\mathcal{C}}$ is an ordinary category. This
can be done by taking for example the simplex category of ${\mathcal{B}}$, see
[24]. Denote by $F^{*}{\mathcal{S}}$ the subset of maps of ${\mathcal{C}}$
that $F$-project to ${\mathcal{S}}$ or to ${\mathcal{B}}$-equivalences. The
induced functor ${\sf Sect}_{\mathcal{S}}({\mathcal{B}},{\mathcal{E}})\to{\sf
Sect}_{F^{*}{\mathcal{S}}}({\mathcal{C}},F^{*}{\mathcal{E}})$ is then seen to
be an equivalence of infinity-categories: if $X\cong F^{*}Y$ and $X$ is
$F^{*}{\mathcal{S}}$-cocartesian, then $Y$ is ${\mathcal{S}}$-cocartesian
since any map in ${\mathcal{S}}$ has a preimage in $F^{*}{\mathcal{S}}$.

Having a cocartesian fibration ${\mathcal{E}}\to{\mathcal{C}}$ over an
ordinary category and a subset of maps ${\mathcal{I}}$, we can identify ${\sf
Sect}_{\mathcal{I}}({\mathcal{C}},{\mathcal{E}})$ with ${\sf
Sect}_{{\mathcal{I}}-loc,\mathscr{S}}({\mathbb{C}},\mathbf{E})$ as per
Corollary 3.12. Furthermore, the fibres of $\mathbf{E}\to{\mathbb{C}}$ are
sections of pullbacks of ${\mathcal{E}}^{\top}\to{\mathcal{C}}^{\sf op}$. The
latter is a cartesian fibration in presentable categories and accessible
transition functors. By [20, Proposition 5.1.2.2 and Corollary 5.4.7.17] in
this case we have that the fibres $\mathbf{E}(\mathbf{c})$ are presentable,
with colimits calculated objectwise, which in turn implies that the transition
functors of $\mathbf{E}\to{\mathbb{C}}$ preserve colimits. This in turn
implies that $\mathbf{E}\to{\mathbb{C}}$ is a presentable fibration. All that
remains is to check that the ${\mathcal{I}}$-coloured anti-Segal maps lead to
limit-preserving transition functors.

The following will suffice. Let ${\mathcal{F}}\to[n]$ be a cartesian fibration
in presentable infinity-categories and accessible functors. Let
$\omega:[m]\subset[n]$ be an interval inclusion such that $\omega(0)=0$.
Assume that for each $i\geq m$, the transition functors
${\mathcal{F}}_{i+1}\to{\mathcal{F}}_{i}$ preserve limits. Then the pullback
functor $\omega^{*}:{\sf Sect}([n],{\mathcal{F}})\to{\sf
Sect}([m],{\mathcal{F}})$ also preserves limits.

To prove this statement, fix an inclusion $[n-m]\subset[m]$ that is complement
to $\omega$. With the help of [20, Lemma 2.1.2.3], it will be enough to show
that the functor ${\sf Sect}([m],{\mathcal{F}})\times_{{\mathcal{F}}_{m}}{\sf
Sect}([n-m],{\mathcal{F}})\to{\sf Sect}([m],{\mathcal{F}})$ preserves limits.
Given a fibration in complete infinity-categories and limit-preserving
functors, the category of sections is also complete, with limits calculated
fibrewise: this is [20, Proposition 5.4.7.11] together with [20, Remark
5.4.7.13], but can also be proven by hand. Thus the category ${\sf
Sect}([n-m],{\mathcal{F}})$ has fibrewise limits. The category ${\sf
Sect}([m],{\mathcal{F}})$ is presentable, and the functor of evaluation at $m$
preserves all limits and colimits by Lemma 3.9. Thus the (homotopy) fibred
product ${\sf Sect}([m],{\mathcal{F}})\times_{{\mathcal{F}}_{m}}{\sf
Sect}([n-m],{\mathcal{F}})$ is the limit of a diagram of complete categories
and limit-preserving functors. By [20, Proposition 5.4.7.11 and Remark
5.4.7.13] we once again have that this fibre product is complete with limits
calculated in each category separately, and thus preserved by any projection.

Returning to the proof of the theorem, since the transition functors along
${\mathcal{I}}$ preserve limits, we can apply the precedent discussion and
then conclude everything using Lemma 3.4. ∎

##  4\. Resolutions

Given a functor $F:{\mathcal{D}}\to{\mathcal{C}}$ and a model opfibration
${\mathcal{E}}\to{\mathcal{C}}$, we get an induced pullback functor
$\mathbb{F}^{*}:{\sf PSect}({\mathcal{C}},{\mathcal{E}})={\sf
Sect}({\mathbb{C}},\mathbf{E})\to{\sf Sect}({\mathbb{D}},\mathbf{E})={\sf
PSect}({\mathcal{D}},{\mathcal{E}})$ on the categories of presections. This
functor trivially preserves weak equivalences, and also the condition of being
a derived section. We thus get the functor

$\mathbb{F}^{*}:{\rm Ho}\,{\sf DSect}({\mathcal{C}},{\mathcal{E}})\to{\rm
Ho}\,{\sf DSect}({\mathcal{D}},{\mathcal{E}})$

on the level of localisations. In some cases, the functor $\mathbb{F}^{*}$ is
full and faithful, and its essential image is easy to characterise.

######  Definition 4.1.

For a functor $F:{\mathcal{D}}\to{\mathcal{C}}$ and
$\mathbf{c}_{[n]}=c_{0}\to...\to c_{n}$ of ${\mathbb{C}}$, denote by
${\mathcal{D}}(\mathbf{c}_{[n]})$ the category

  * •

with objects being pairs of $\mathbf{d}_{[n]}=d_{0}\to...\to d_{n}$ and of a
commutative diagram so that the vertical maps are isomorphisms,

  * •

with morphisms given by commutative diagrams such that for each $0\leq i\leq
n$, the diagram commutes.

The categories ${\mathcal{D}}(c_{0}\to...\to c_{n})$ are extensions of the
notion of an essential fibre of a functor.

######  Definition 4.2.

  1. (1)

A functor $F:{\mathcal{D}}\to{\mathcal{C}}$ is a _resolution_ if for each
$\mathbf{c}_{[n]}\in{\mathbb{C}}$, the category
${\mathcal{D}}(\mathbf{c}_{[n]})$ is contractible (that is, has a contractible
nerve).

  2. (2)

A functor $F:{\mathcal{D}}\to{\mathcal{C}}$ is a _right resolution_ if

     * •

for each $c\in{\mathbb{C}}$ over $[0]\in\Delta$, the category
${\mathcal{D}}(c)$ is contractible, and

     * •

for each $f:c^{\prime}\to c$ in ${\mathbb{C}}$ over $[1]\in\Delta$ and
$d\in{\mathcal{D}}(c)$, the subcategory
$F(f,d)\subset{\mathcal{D}}(c^{\prime}\stackrel{{\scriptstyle f}}{{\to}}c)$
given by the (strict) fibre of
${\mathcal{D}}(c^{\prime}\stackrel{{\scriptstyle
f}}{{\to}}c)\to{\mathcal{D}}(c)$ over $d$, is contractible.

  3. (3)

A functor $F:{\mathcal{D}}\to{\mathcal{C}}$ is a _left resolution_ if

     * •

for each $c\in{\mathbb{C}}$ over $[0]\in\Delta$, the category
${\mathcal{D}}(c)$ is contractible, and

     * •

for each $f:c^{\prime}\to c$ in ${\mathbb{C}}$ over $[1]\in\Delta$ and
$d\in{\mathcal{D}}(c)$, the subcategory
$F(d^{\prime},f)\subset{\mathcal{D}}(c^{\prime}\stackrel{{\scriptstyle
f}}{{\to}}c)$ given by the (strict) fibre of
${\mathcal{D}}(c^{\prime}\stackrel{{\scriptstyle
f}}{{\to}}c)\to{\mathcal{D}}(c^{\prime})$ over $d^{\prime}$, is contractible.

######  Lemma 4.3.

If $F:{\mathcal{D}}\to{\mathcal{C}}$ is a right or left resolution, then $F$
is also a resolution.

###### Proof.

We prove the right part, the left part being dual, Inductively, assume we have
proven the resolution property for each $\mathbf{c}^{\prime}_{[k]}$ with
$0\leq k<n$. Then for an object $\mathbf{c}_{[n]}=c_{0}\stackrel{{\scriptstyle
f}}{{\to}}c_{1}\to...\to c_{n}$ we have the associated functor
${\mathcal{D}}(c_{0}\stackrel{{\scriptstyle f}}{{\to}}c_{1}\to...\to
c_{n})\to{\mathcal{D}}(c_{1}\to...\to c_{n})$. This is an opfibration over a
contractible category, with fibres equivalent to $F(f,d)$ for some
$d\in{\mathcal{D}}(c_{1})$. Quillen’s Theorem A implies then the
contractibility of ${\mathcal{D}}(\mathbf{c}_{[n]})$. ∎

######  Lemma 4.4.

If $F:{\mathcal{D}}\to{\mathcal{C}}$ is a prefibration (and, by convention, an
isofibration) with contractible fibres, then it is a right resolution.

Dually, if $F:{\mathcal{D}}\to{\mathcal{C}}$ is a preopfibration (and, by
convention, an isofibration) with contractible fibres, then it is a left
resolution.

###### Proof.

Since $F$ is an isofibration, the categories ${\mathcal{D}}(c)$ and
${\mathcal{D}}(c^{\prime}\to c)$ are equivalent to their strict analogues: the
strict fibre $F^{-1}(c)$ and the category of arrows $d^{\prime}\to d$ with
$F(d^{\prime}\to d)$ equal to $c^{\prime}\to c$. It is then easy to see that
the fibres of ${\mathcal{D}}(c^{\prime}\to c)\to{\mathcal{D}}(c)$ have
terminal objects for a prefibration, and the fibres of
${\mathcal{D}}(c^{\prime}\to c)\to{\mathcal{D}}(c^{\prime})$ have initial
objects for a preopfibration, and hence are contractible. Quillen’s Theorem A,
again, implies the result. ∎

This example motivates the intuition behind left (respectively right)
resolutions as covariant (respectively contravariant) families of contractible
homotopy types, represented in categories, indexed by the base category
${\mathcal{C}}$.

######  Lemma 4.5.

If $p:{\mathcal{D}}\rightleftarrows{\mathcal{C}}:i$ is an adjunction and $i$
is full and faithful, then $p$ is a resolution. An equivalence of categories
is a resolution.

###### Proof.

Every fibre ${\mathcal{D}}(c_{0}\to...\to c_{n})$ has a terminal object given
by $ic_{0}\to...\to ic_{n}$ (note that $pi(c_{0})\to...\to pi(c_{n})$ is
isomorphic to $c_{0}\to...\to c_{n}$). ∎

In [16, Key Lemma], it is shown that a resolution
$F:{\mathcal{D}}\to{\mathcal{C}}$ exhibits the category ${\mathcal{C}}$ as a
higher-categorical localisation of ${\mathcal{D}}$ with respect to
$F$-isomorphisms. Let us reproduce the argument here.

######  Proposition 4.6 (Hinich’s Lemma).

Let $F:{\mathcal{D}}\to{\mathcal{C}}$ be a resolution. Then $F$ induces an
equivalence $L{\mathcal{D}}\cong{\mathcal{C}}$, where we infinity-localise
${\mathcal{D}}$ along the set $F^{*}Iso({\mathcal{C}})$ of all maps of
${\mathcal{D}}$ that are sent by $F$ to isomorphisms in ${\mathcal{C}}$.

###### Proof.

It is enough to show that

$({\mathcal{D}},F^{*}Iso({\mathcal{C}}))\longrightarrow({\mathcal{C}},Iso({\mathcal{C}}))$

is a weak equivalence of relative categories. Following [8], one has to show
that for each $[n]$, the induced functor

$\operatorname{\mathsf{Fun}}_{F^{*}Iso({\mathcal{C}})}([n],{\mathcal{D}})\to\operatorname{\mathsf{Fun}}_{Iso({\mathcal{C}})}([n],{\mathcal{C}})$

is a homotopy equivalence; here the sub-index of $\operatorname{\mathsf{Fun}}$
means that we take only those natural transformations that belong to
($F$-)isomorphisms. The comma-fibres of the functor above are given by
${\mathcal{D}}(c_{0}\to...\to c_{n})$. ∎

The comparison between ordinary sections and derived sections carried out in
3.1 and the Key Lemma of Hinich readily implies that derived sections on
${\mathcal{C}}$ are identified with the derived sections on ${\mathcal{D}}$
which are locally constant along the $F$-isomorphisms. It turns out however
that proving the same statement internally to the language of derived
sections, without passing by the comparison of [1, Theorem 2], leads to some
interesting simplicial combinatorics.

Denote as usual by ${\sf DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})$
the category of ${\mathcal{S}}$-locally constant derived sections. We prove
the following result independently of the comparison carried out in 3.1:

######  Proposition 4.7.

Let $F:{\mathcal{D}}\to{\mathcal{C}}$ be a resolution,
${\mathcal{S}}\subset{\mathcal{C}}$ a subset, and
${\mathcal{E}}\to{\mathcal{C}}$ a model opfibration. Then the functor
$\mathbb{F}^{*}:{\sf DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})\to{\sf
DSect}_{F^{*}{\mathcal{S}}}({\mathcal{D}},{\mathcal{E}})$ is a weak
equivalence of relative categories, meaning the equivalence of induced
simplicial (or higher-categorical) localisations.

######  Remark 4.8.

For many uses it is already sufficient to know the weaker statement, that the
functor $\mathbb{F}^{*}:{\rm Ho}\,{\sf
DSect}_{\mathcal{S}}({\mathcal{C}},{\mathcal{E}})\to{\rm Ho}\,{\sf
DSect}_{F^{*}{\mathcal{S}}}({\mathcal{D}},{\mathcal{E}})$ is an equivalence.
However proving such a weaker statement often gives more for free. To verify
that a given relative functor is a weak equivalence, one can often use the
following criterion. Let $F:{\mathcal{M}}\rightleftarrows{\mathcal{N}}:G$ be a
Quillen adjunction with $G$ preserving all weak equivalences, and assume that
there is a full subcategory ${\mathcal{M}}_{0}\subset{\mathcal{M}}$ closed
under the weak equivalences of ${\mathcal{M}}$. If $G$ factors through
${\mathcal{M}}_{0}$ and the homotopy category adjunction ${\mathbb{L}}F:{\rm
Ho}\,{\mathcal{M}}\rightleftarrows{\rm Ho}\,{\mathcal{N}}:{\mathbb{R}}G$
restricts to an equivalence between ${\rm Ho}\,{\mathcal{M}}_{0}$ and ${\rm
Ho}\,{\mathcal{N}}$, then the same is true for the induced higher-categorical
adjunction existing thanks to [11, 21], and so
$G:{\mathcal{M}}\to{\mathcal{M}}_{0}$ gives an equivalence after applying the
higher-categorical localisation functor. Thus in practice one is left working
with Quillen adjunctions and their restrictions, as we shall indeed do later
in this section.

###  4.1. Relative comma objects

######  Notation 4.9.

For a $\Delta$-indexed category $\pi:{\mathcal{X}}\to\Delta^{\sf op}$, the
notation $x\twoheadrightarrow y$ means that the map is a degeneracy, that is
the underlying map $\pi(y)\twoheadrightarrow\pi(x)$ is a surjection in
$\Delta$.

######  Definition 4.10.

Let $F:{\mathcal{X}}\rightarrow{\mathcal{Z}}\leftarrow{\mathcal{Y}}:G$ be a
diagram of $\Delta$-indexed categories. The associated _$\Delta$ -relative
comma object_ $F\sslash G$ is the full subcategory of the ordinary comma-
category $F/G$ consisting of all triples $(x,y,F(x)\twoheadrightarrow G(y))$
where the map $F(x)\twoheadrightarrow G(y)$ is a degeneracy.

By definition, $F\sslash G$ comes with projections to both ${\mathcal{X}}$ and
${\mathcal{Y}}$. We will also use the notation $F\bbslash G$ to denote
$G\sslash F$. If one of the functors is an identity, we will write, as is
customary for comma categories, $F\sslash{\mathcal{Y}}$ instead of $F\sslash
id_{\mathcal{Y}}$ and the like. To know more about the relative comma objects,
the following may be of use.

######  Lemma 4.11.

Let $[1]$ be the usual arrow category. Then the full subcategory
$Arr_{s}(\Delta)$ of the arrow category
$\operatorname{\mathsf{Fun}}([1],\Delta)$ consisting of surjective arrows, is
naturally equipped with a Reedy category structure. The natural source and
target projections from $Arr_{s}(\Delta)$ to $\Delta$ are compatible with the
Reedy structure.

###### Proof.

Any map between two surjective arrows in $\Delta$ is represented by a
commutative square. Using the injection-surjection factorisation system, we
get the diagram in which it is easy to check that the middle vertical arrow is
also a surjection. To see the Reedy category structure, set
$deg([n]\twoheadrightarrow[m])=n+m$. ∎

######  Corollary 4.12.

Let $F:{\mathcal{X}}\rightarrow{\mathcal{Z}}\leftarrow{\mathcal{Y}}:G$ be a
diagram of $\Delta$-indexed categories. Then the category $F\bbslash G$ is a
$Arr_{s}(\Delta)$-indexed category, hence a Reedy category, and both functors
$F\bbslash G\to{\mathcal{X}}$ and $F\bbslash G\to{\mathcal{Y}}$ are compatible
with the Reedy structure.

###### Proof.

Immediate from Proposition 1.8. ∎

######  Lemma 4.13.

For a $\Delta$-indexed functor $F:{\mathcal{Y}}\to{\mathcal{X}}$, the
projection $F\bbslash{\mathcal{X}}\to{\mathcal{X}}$ is a Reedy functor which
is moreover an opfibration over face maps of ${\mathcal{X}}$.

###### Proof.

Corollary 4.12 implies that the projection is Reedy. Now, fix a face map
$f:x\to x^{\prime}$ and $o\in F\bbslash{\mathcal{X}}$ with $p(o)=x$.
Projecting these data to $\Delta$, we find ourselves with an injection
$\varphi:[n]\hookrightarrow[m]$ and a surjection
$\omega:[k]\twoheadrightarrow[m]$ representing the object $o$. Form the
pullback square which exists for this particular configuration of arrows, and
use the fact that $F\bbslash{\mathcal{X}}$ is discretely fibred over
$Arr_{s}(\Delta)$ to uniquely reconstruct the map $o\to f_{!}o$ in
$F\bbslash{\mathcal{X}}$. ∎

In the proof above, note that if $f$ is a degree-lowering map, then so is
$o\to f_{!}o$.

######  Definition 4.14.

A functor $G:{\mathcal{R}}\to{\mathcal{R}}^{\prime}$ between Reedy categories
is called _right-compatible_ , if it preserves the Reedy structure and in
addition for each $x\in{\mathcal{R}}$, the induced map $Mat(x)\to Mat(Gx)$ is
initial relative to ${\mathcal{R}}^{\prime}$, by which we mean that for any
functor $X$ from ${\mathcal{R}}^{\prime}$ to a complete category, there is a
(naturally induced) isomorphism

$\operatorname{\varprojlim}_{Mat(Gx)}X|_{Mat(Gx)}\cong\operatorname{\varprojlim}_{Mat(x)}G^{*}X|_{Mat(x)}.$

######  Corollary 4.15.

The functor
$p_{\mathcal{X}}:{\mathcal{O}}:=F\bbslash{\mathcal{X}}\to{\mathcal{X}}$ is a
right-compatible functor of Reedy categories.

###### Proof.

The opfibration property over face maps yields us an adjunction between the
matching categories

$L:Mat(p_{\mathcal{X}}o)\rightleftarrows Mat(o):R$

where $R$ is the canonical functor induced by $p_{\mathcal{X}}$. The functor
$L$, being a (fully faithful) left adjoint, is initial. The composition of $L$
with the natural projection $p:Mat(o)\to{\mathcal{O}}\to{\mathcal{X}}$
coincides with the projection $q:Mat(p_{\mathcal{X}}o)\to{\mathcal{X}}$. We
thus can observe that for any functor $X:{\mathcal{X}}\to{\mathcal{M}}$ valued
in a complete category, there is the following sequence of isomorphisms

$\operatorname{\varprojlim}_{Mat(o)}p^{*}X\cong\operatorname{\varprojlim}_{Mat(p_{\mathcal{X}}o)}L^{*}p^{*}X\cong\operatorname{\varprojlim}_{Mat(p_{\mathcal{X}}o)}q^{*}X$

which implies right-compatibility. ∎

The main result of this subsection is the interpretation of the higher
categorical localisation property of resolutions in the context of simplicial
replacements:

######  Proposition 4.16.

Let $F:{\mathcal{D}}\to{\mathcal{C}}$ be a functor. Then for the functor
$\mathbb{F}:{\mathbb{D}}\to{\mathbb{C}}$, the projection
$p_{\mathbb{C}}:\mathbb{F}\bbslash{\mathbb{C}}\to{\mathbb{C}}$ is a right-
compatible Reedy functor. Moreover, if $F$ is an isofibration and a
resolution, then the functor $p_{\mathbb{C}}$ is a left resolution.

A proof is given below. We can thus produce the following technical
definition.

######  Definition 4.17.

Let $F:{\mathcal{Y}}\to{\mathcal{X}}$ be a $\Delta$-indexed functor. $F$ is
called a _resolution_ if the associated functor
$F\bbslash{\mathcal{X}}\to{\mathcal{X}}$ is a left resolution.

######  Remark 4.18.

Due to ${\mathcal{X}}$ being a Reedy category any functor
${\mathcal{Y}}\to{\mathcal{X}}$ is automatically an isofibration.

In the course of the proof, we shall need an auxiliary lemma about cofinal
maps.

######  Lemma 4.19.

Consider the diagram and assume that $p,q$ are opfibrations, and both $g$ and
all $f_{x}:{\mathcal{E}}(x)\to{\mathcal{F}}(g(x))$ (induced by taking fibres
over $x\in{\mathcal{D}}$) are homotopy cofinal. Then the map $f$ is homotopy
cofinal.

###### Proof.

For a diagram $X:{\mathcal{F}}\to{\rm\bf Top}$, we have that the map
$\operatorname{hocolim}_{\mathcal{E}}f^{*}X\to\operatorname{hocolim}_{\mathcal{F}}X$
decomposes in ${\rm Ho}\,{\rm\bf Top}$ as follows:

$\operatorname{hocolim}_{\mathcal{E}}f^{*}X\cong\operatorname{hocolim}_{{\mathcal{D}}}{\mathbb{L}}p_{!}f^{*}X\cong\operatorname{hocolim}_{\mathcal{D}}g^{*}{\mathbb{L}}q_{!}X$
$\cong\operatorname{hocolim}_{\mathcal{C}}{\mathbb{L}}q_{!}X\cong\operatorname{hocolim}_{\mathcal{F}}X.$

Here the first and the last isomorphisms are due to the properties of homotopy
left Kan extensions as homotopy adjoints, the third one uses homotopy
cofinality of $g$ and the second one follows from the fact that the base
change ${\mathbb{L}}p_{!}f^{*}\to g^{*}{\mathbb{L}}q_{!}$ being an
isomorphism, which follows from the requirement of the homotopy cofinality
condition on the fibres and the fact that one calculates
$({\mathbb{L}}p_{!}Y)(d)$ as
$\operatorname{hocolim}_{{\mathcal{E}}(d)}Y|_{{\mathcal{E}}(d)}$, and
similarly for ${\mathbb{L}}q_{!}$. ∎

###### Proof of Proposition 4.16..

An object of $\mathbb{F}\bbslash{\mathbb{C}}$ is represented by a surjection
$\sigma:[k_{0}+...+k_{n}]\twoheadrightarrow[n]$, an object

$\mathbf{d}_{[k_{0}+...+k_{n}]}=d_{0}^{0}\to...\to d^{k_{0}}_{0}\to
d^{0}_{1}\to...\to d^{k_{n}}_{n}$

of maps in ${\mathcal{D}}$ such that
$p_{\mathbb{C}}(\mathbf{d}_{[k_{0}+...+k_{n}]})$ is equal to

$\sigma^{*}\mathbf{c}_{[n]}=c_{0}\to...\to c_{0}\to c_{1}...\to c_{n}$

with each $c_{i}$ appearing $k_{i}+1$ times in a row. Sending
$\mathbf{d}_{[k_{0}+...+k_{n}]}$ to $d^{0}_{0}\to d^{0}_{1}\to...\to
d^{0}_{n}$ is seen to produce a functor