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# Derived sections of Grothendieck fibrations and the problems of homotopical
algebra

Edouard Balzin

###### Contents

  1. 1 Generalities on geometric realisation
    1. 1.1 Homotopy colimits
    2. 1.2 Tensors and $\Delta$-categories
    3. 1.3 Homotopical $\Delta$-categories
  2. 2 Fibrations, opfibrations, sections
    1. 2.1 Basic notions
    2. 2.2 Homotopical $\Delta$-opfibrations
  3. 3 Derived sections
    1. 3.1 Simplicial replacements
    2. 3.2 Homotopical category of derived sections
  4. 4 The pushforward functor
    1. 4.1 Main construction
    2. 4.2 Unit and counit correspondences
  5. 5 Case of a resolution
    1. 5.1 Fullness and faithfullness
    2. 5.2 Essential surjectivity

## Introduction

The formalism of topological operads [21] appeared as a way to describe the
algebraic structure of $n$-fold loop spaces. A topological operad ${\cal O}$
is a symmetric sequence of spaces $\\{{\cal O}(l)\\}_{l\in\mathbb{N}}$, where
each ${\cal O}(l)$ should be thought as the space of operations with $n$
inputs and one output, supplied with composition maps ${\cal O}(l)\times{\cal
O}(m)\to{\cal O}(l+m-1)$ satifsying symmetry, associativity and unitality
conditions. Of particular interest are little $n$-disk operads ${\cal D}_{n}$
(defined as little cube operads in [21]), for which ${\cal D}_{n}(l)$ is
(homotopy equivalent to) the configuration space of $l$ points in
$\mathbb{R}^{n}$. Any $n$-fold loop space $X$ has a structure of an algebra
over ${\cal D}_{n}$, which means there are maps ${\cal D}_{n}(l)\times
X^{l}\to X$ satisfying certain conditions.

Denote by ${\rm\bf DVect}_{k}$ the category of chain complexes over a field
$k$. Taking singular chains of ${\cal D}_{n}(l)$ produces the ${\rm\bf
DVect}_{k}$-operad $\mathbb{E}_{n}$ (and indeed, ${\cal D}_{n}$ are often
called $\mathbb{E}_{n}$-operads in topological spaces). Algebras over such
operads $\mathbb{E}_{n}$, that is, chain complexes $M$ together with action
maps $\mathbb{E}_{n}(l)\otimes M^{\otimes l}\to M$, have been studied with
great interest especially in the recent years [18]. An example of an
$\mathbb{E}_{2}$-algebra is the cohomological Hochschild complex $CH(A)$ of a
DG-algebra $A$, which appears in many settings, for example in two-dimensional
topological conformal field theories [7]. It is important, however, to remark
that $CH(A)$ is an $\mathbb{E}_{2}$ algebra _up to quasiisomorphism_ : the
Deligne Conjecture [3, 19, 25] only implies that there exists an operad ${\cal
O}$ in ${\rm\bf DVect}_{k}$, quasiisomorphic to $\mathbb{E}_{2}$, which acts
on $CH(A)$. The proofs of this result involve, subsequently, a lot of
combinatorial work to construct ${\cal O}$, its action on $CH(A)$, and the
(chain of) quasiisomorphisms ${\cal O}\cong\mathbb{E}_{2}$.

The bulkiness of the formalism of operads comes from the fact that two quasi-
isomorphic or homotopy equivalent operads can be of very different size and
complexity yet describe equivalent structures. There is a different approach
to $\mathbb{E}_{n}$-algebras, and more generally to structures related to
configuration spaces, which relies on the machinery of factorisation algebras
of [2]. A factorisation algebra ${\cal F}$ over a space $X$ consists of,
roughly speaking, a presheaf ${\cal F}_{n}$ of complexes of vector spaces on
$X^{n}$ for each power $n$, together with additional structure. First, there
is a map

$\Delta^{n}_{*}{\cal F}_{n}\longrightarrow{\cal F}_{1}$ (i)

between the restriction $\Delta_{n}^{*}{\cal F}_{n}$ of ${\cal F}_{n}$ along
the smallest diagonal $\Delta_{n}:X\to X^{n}$, and ${\cal F}_{1}$. Second, if
we denote by $i_{n}:U_{n}\subset X_{n}$ the complement $\\{(x_{i})|x_{k}\neq
x_{l}\\}$ to all diagonals, then there are factorisation maps

$i_{n}^{*}{\cal F}_{n}\longrightarrow{\cal F}_{1}\boxtimes...\boxtimes{\cal
F}_{1}$ (ii)

between the restriction of ${\cal F}_{n}$ to $U_{n}$ and the external product
of ${\cal F}_{1}$ [2], which are required to be quasiisomorphisms. When $X$ is
a $k$-disk, one can prove that $\mathbb{E}_{n}$-algebras correspond to those
factorisation algebras on $X$ which are moreover constructible (which means
that ${\cal F}_{n}$ is locally constant on the strata for the standard
stratification of $X^{n}$). The notion of factorisation algebra has proved its
use and is arguably more natural and canonical than algebras over topological
operads. It leads one to ask if there is a general ‘homotopic-algebraic’
formalism which does not suffer from the noncanonicity issues of topological
operads and naturally reproduces factorisation algebra approach to a variety
of algebraic structures.

For the context of loop spaces, an approach alternative to operads does exist
and is very useful in practical applications. In [24], Graeme Segal introduced
the notion of a $\Gamma$-space. Denote by ${\rm\bf Fin}_{*}$ the category of
finite sets and partially defined maps: a map $S\to T$ in ${\rm\bf Fin}_{*}$
is a map of sets $U\to T$ defined on a subset $U\subset S$. A $\Gamma$-space
$A$ is then defined as a functor to the category of topological spaces
${\rm\bf Top}$, satisfying Segal conditions. To formulate them, fix a one-
element set $1$. For a set $S$ and an element $x\in S$, we have the
corresponding partial map $\rho_{x}:S\to 1$ defined on the subset $\\{x\\}$.
The Segal conditions say that for each $S\in{\rm\bf Fin}_{*}$, the induced map

$\begin{diagram}$ (iii)

is a homotopy equivalence of topological spaces.

For each $S\in{\rm\bf Fin}_{*}$ there is one map $\pi_{S}:S\to 1$ defined on
the whole of $S$. We can consider the following span

$\begin{diagram}$ (iv)

The Segal conditions imply that this span represents a morphism in the
homotopy category ${\rm Ho}\,{\rm\bf Top}$. A choice of a homotopy inverse for
the left map gives, noncanonically, a multiplication operation
$m_{S}:A(1)^{S}\to A(1)$. After projecting to ${\rm Ho}\,{\rm\bf Top}$ one can
check that $A(1)$ becomes a commutative monoid.

However, a $\Gamma$-space $A$ carries more information than the homotopy
monoid $A(1)$. Segal, just like May with topological operads, used
$\Gamma$-spaces to describe infinite loop spaces and his delooping machinery.
From the modern perspective, a $\Gamma$-space is a proper description of a
homotopy coherent commutative monoid in topological spaces. In particular,
$\Gamma$-spaces describe the same structure as $\mathbb{E}_{\infty}$ [21]
algebras in ${\rm\bf Top}$.

Instead of ${\rm\bf Fin}_{*}$ we can consider the opposite of the usual
category of simplices $\Delta$. A suitable modification of definitions then
permits to model homotopy coherent monoids with no commutativity as certain
simplicial spaces $\Delta^{\sf op}\to{\rm\bf Top}$. Explicit examples include
ordinary loop spaces. Moreover, the work of [1] implies that there are
categories $C$ such that $n$-fold loop spaces — examples of ${\cal
D}_{n}$-algebras — can be also modeled as Segal-type objects $C\to{\rm\bf
Top}$ for a suitable choice of the category $C$. In place of ${\rm\bf Top}$,
one can consider any homotopical category, that is a category ${\cal M}$ with
a subcategory of weak equivalences ${\cal W}$, such that ${\cal M}$ has
(homotopy) products, and define Segal objects as functors to ${\cal M}$ with
maps (iii) being weak equivalences.

The Segal space approach contrasts with operadic approach in that
multiplicative operations $m_{S}:A(1)^{S}\to(1)$ for a $\Gamma$-space $A$ are
not defined canonically and instead are constructed using the properties of
$A$, while specifying a model ${\cal O}$ for $\mathbb{E}_{\infty}$-operad in
${\rm\bf Top}$ and an algebra over it means supplementing a lot of structure.
In particular, for a $|S|$-element set $S$, $A(S)$ need not to be equal to
${\cal O}(|S|)\times A(1)^{S}$. The information about multiplication
properties in Segal formalism is thus entirely contained in the category
${\rm\bf Fin}_{*}$. There is much less arbitrary choice left available, and
one might hope it would be easier to construct and work with Segal structures
rather than with operadic structures. Moreover, there is a great similarity
between Segal $\Gamma$-spaces and factorisation algebras: for a factorisation
algebra ${\cal F}$, the maps (i) and (ii) provide, after passing to stalks,
spans just like (iv).

However, if we attempt to extend the formalism of Segal objects to chain
complexes, we immediately run into difficulties. To produce maps like (iii) in
the $\Gamma$-space picutre we used the universal property of Cartesian product
$\times$ which is not satisfied by the tensor product $\otimes_{k}$ of
${\rm\bf DVect}_{k}$.

To deal with this issue, one observes [24, 27] that any symmetric monoidal
category ${\cal M}$ is a weakly commutative monoid object in ${\rm\bf Cat}$,
the category of categories of suitable size. It is then, up to an equivalence,
described by a ‘$\Gamma$-category’ $M:{\rm\bf Fin}_{*}\to{\rm\bf Cat}$, with
maps (iii) being equivalences of categories. In order not to choose an
equivalence between ${\cal M}$ and $M(1)$, one has to work either with
pseudofunctors from ${\rm\bf Fin}_{*}$ to ${\rm\bf Cat}$, or equivalently,
with Grothendieck opfibrations [11, 28] over ${\rm\bf Fin}_{*}$: either notion
encodes a weak ${\rm\bf Fin}_{*}$-indexed family of categories.

The way [18] to directly produce a Grothendieck opfibration out of a symmetric
monoidal category ${\cal M}$ with monoidal product $\otimes$ is as follows.
Define ${\cal M}^{\otimes}$ to be the category with objects
$(S,\\{M_{x}\\}_{x\in S})$ where $S\in{\rm\bf Fin}_{*}$ and each $M_{x}$ is an
object of ${\cal M}$. A morphism $(S,\\{M_{x}\\}_{x\in
S})\to(T,\\{N_{y}\\}_{y\in T})$ consists of a partially defined map $f:S\to
T$, and for each $y\in T$, of a morphism $\otimes_{x\in f^{-1}(y)}M_{x}\to
N_{y}$; when $f^{-1}(y)$ is empty, the monoidal product over it is the unit
object. The compositions can then be defined with the help of the coherence
isomorphisms for the product $\otimes:{\cal M}\times{\cal M}\to{\cal M}$ and
the unit object. The natural functor $p:{\cal M}^{\otimes}\to{\rm\bf Fin}_{*}$
is a Grothendieck opfibration, which means that the assignment $S\mapsto
p^{-1}(S)={\cal M}^{S}$ is functorial in a weak but coherent way.

Given a monoid object $A\in{\cal M}$, we may define a section ${\rm\bf
Fin}_{*}\to{\cal M}^{\otimes}$ of $p:{\cal M}^{\otimes}\to{\rm\bf Fin}_{*}$ as
$S\mapsto(S,\\{P_{x}\\})$ with each $P_{x}=A$. Sections of this type can be
characterised by putting suitable normalisation conditions on sections of $p$.
However, there is still no evident way to write diagrams for Segal conditions
using the language of sections. If we take a map $f:S\to T$ in ${\rm\bf
Fin}_{*}$, the value of a section $A$ on $f$ is determined by a map
$f_{!}A(S)\to A(T)$ in ${\cal M}^{T}$, where $f_{!}:{\cal M}^{S}\to{\cal
M}^{T}$ is the functor

$f_{!}:(S,\\{P_{x}\\}_{x\in S})\mapsto(T,\\{Q_{y}\\}_{y\in
T}),\,\,\,Q_{y}=\otimes_{x\in f^{-1}y}P_{x}.$ (v)

In particular, for the map $\pi_{S}:S\to 1$ with $\pi_{S}^{-1}(1)=S$, we
directly get multiplication operations $A(1)^{\otimes S}=(\pi_{S})_{!}A(S)\to
A(1)$. To have a Segal description, we would instead like to have an object
which produces diagrams like the following:

$\begin{diagram}$ (vi)

The left map may then be required to be a weak equivalence if ${\cal M}$ has
such.

More generally, in place of ${\cal M}^{\otimes}\to{\rm\bf Fin}_{*}$ we can
consider general Grothendieck opfibrations ${\cal E}\to{\cal C}$ ans ask if
there is a way to define objects which produce, out of a map $f:c\to
c^{\prime}$ in ${\cal C}$, spans of the form $f_{!}A(c)\longleftarrow
A_{f}\longrightarrow A(c^{\prime})$ (here $f_{!}:{\cal E}(c)\to{\cal
E}(c^{\prime})$ is the transition functor induced by the opfibration property
of ${\cal E}\to{\cal C}$).

In this paper we define _derived_ sections of opfibrations with weak
equivalences, which in particular produce diagrams like (vi) above. Let us
briefly describe the construction. Recall first that for a category ${\cal
C}$, its simplicial replacement [5] ${\mathbb{C}}$ is defined as the category
whose objects are composable sequences $c_{0}\to...\to c_{n}$ of arrows of
${\cal C}$ of arbitrary finite length $n\geq 0$. A morphism between
$c_{0}\to...\to c_{n}$ and $c^{\prime}_{0}\to...\to c^{\prime}_{m}$ consists
of an order-preserving map of ordinals $a:[m]\to[n]$ (here $[i]$ denotes a
totally ordered set of $i+1$ elements $0,1,...,i$) such that
$c_{a(k)}=c^{\prime}_{k}$ for $0\leq k\leq m$. If, as before, one denotes by
$\Delta$ the standard category of simplices, then ${\mathbb{C}}$ is the
opposite of the simplex category of the nerve $N{\cal C}:\Delta^{\sf
op}\to{\rm\bf Set}$. The assignments $(c_{0}\to...\to c_{n})\mapsto c_{0}$ or
$(c_{0}\to...\to c_{n})\mapsto c_{n}$ determine two ‘head’ and ‘tail’ functors
$h:{\mathbb{C}}\to{\cal C}$ and $t:{\mathbb{C}}\to{\cal C}^{\sf op}$.

Consider a functor $F:{\mathbb{C}}\to{\cal M}$ where ${\cal M}$ is an
arbitrary category with weak equivalences ${\cal W}$. If we take a morphism
$f:c\to c^{\prime}$ of ${\cal C}$, we then can consider the following span in
${\mathbb{C}}$:

$\begin{diagram}$ (vii)

Evaluating $F$ on this diagram gives the corresponding span in ${\cal M}$:

$\begin{diagram}$ (viii)

If one requires that $F(c)\longleftarrow F(c\stackrel{{\scriptstyle
f}}{{\to}}c^{\prime})$, is an isomorphism, then the span (viii) defines a map
from $F(c)$ to $F(c^{\prime})$, which we denote as $F(f)$. It then makes sense
to ask if $F(gf)=F(g)F(f)$ for a composable pair of arrows
$c\stackrel{{\scriptstyle f}}{{\to}}c^{\prime}\stackrel{{\scriptstyle
g}}{{\to}}c^{\prime\prime}$, or whether $F(id_{c})=id_{F(c)}$. Both those
conditions will be satisfied if $F$ sends to isomorphisms those maps of
${\mathbb{C}}$ which have the form

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

for $0\leq k\leq n$ (that is, those maps which are determined by the inclusion
of $[k]$ as first $k+1$ elements of $[m]$). Such maps of ${\mathbb{C}}$ are
called _anchor maps_ in this article. We observe that such a functor $F$
factors uniquely as $\bar{F}\circ h$, where $\bar{F}:{\cal C}\to{\cal M}$ is a
functor from the original category ${\cal C}$.

If $F$ sends the anchor maps of ${\mathbb{C}}$ to weak equivalences of ${\cal
M}$, spans like (viii) define morphisms in ${\rm Ho}\,{\cal M}$, the
localisation of ${\cal M}$ with respect to its weak equivalences. We may view
such a functor $F:{\mathbb{C}}\to{\cal M}$ as a weakening of the notion of a
functor from ${\cal C}$ to ${\cal M}$, with spans obtained from objects
$c_{0}\to...\to c_{n}$ of greater length ensuring the coherence of
compositions.

Assume now given an opfibration $p:{\cal E}\to{\cal C}$. Moreover, we assume
that each fibre ${\cal E}(c):=p^{-1}(c)$ has weak equivalences, and for each
map $f:c\to c^{\prime}$, the functor $f_{!}:{\cal E}(c)\to{\cal
E}(c^{\prime})$ induced by the opfibration property, preserves those weak
equivalences. Then there exists a functor
$p_{\mathbb{C}}:\mathbf{E}\to{\mathbb{C}}$, such that
$\mathbf{E}(\mathbf{c}_{[n]}):=p_{\mathbb{C}}^{-1}(\mathbf{c}_{[n]})\cong{\cal
E}(c_{n})$, and that for each
$\alpha:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$, there is a naturally
induced functor111Unlike $p$, the functor $p_{\mathbb{C}}$ is a Grothendieck
_fibration_ and describes a contravariant family over ${\mathbb{C}}$.
$\mathbf{E}(\mathbf{c}^{\prime}_{[m]})\to\mathbf{E}(\mathbf{c}_{[n]})$
isomorphic to $t(\alpha)_{!}:{\cal E}(c^{\prime}_{m})\to{\cal E}(c_{n})$.

We define a presection of $p:{\cal E}\to{\cal C}$ to be a section
$B:{\mathbb{C}}\to\mathbf{E}$ of the functor $p_{\mathbb{C}}$. Presections
form a category ${\sf PSect}({\cal C},{\cal E})$ which can be equipped with
weak equivalences if $p$ has such. A presection $B$ acting on spans like (vii)
produces this diagram in ${\cal E}(c^{\prime})$: If the left map of this span
and all other produced by applying $B$ to the anchor maps of ${\mathbb{C}}$,
are isomorphisms, then one can prove that $B$ defines an ordinary section
${\cal C}\to{\cal E}$ of the original opfibration $p:{\cal E}\to{\cal C}$. If
$B$ takes anchor maps to weak equivalences, we call such $B$ a derived
section. They form a category222The notation $\mathbb{R}$ comes from the
analogy with $\mathbb{R}Hom$ in homological and homotopical algebra.
${\sf\mathbb{R}Sect}({\cal C},{\cal E})\subset{\sf PSect}({\cal C},{\cal E})$
with induced weak equivalences. This is our category of derived sections.
Denote by ${\rm Ho}\,{\sf\mathbb{R}Sect}({\cal C},{\cal E})$ its localisation
along the weak equivalences.

The standard way to work with localisations like ${\rm
Ho}\,{\sf\mathbb{R}Sect}({\cal C},{\cal E})$ is to use model categories.
However, in examples such as ${\rm\bf DVect}_{k}^{\otimes}\to{\rm\bf
Fin}_{*}$, while the fibres ${\rm\bf DVect}_{k}^{\otimes}(S)={\rm\bf
DVect}_{k}^{S}$ are model categories, the transition functors (v)
$f_{!}:{\rm\bf DVect}_{k}^{S}\to{\rm\bf DVect}_{k}^{T}$ do not usually
preserve limits or colimits: in the basic case when $f:\\{x,y\\}\to 1$ is the
map sending $x$ and $y$ to the element of $1$, $f_{!}:{\rm\bf
DVect}_{k}^{\\{x,y\\}}={\rm\bf DVect}_{k}\times{\rm\bf DVect}_{k}\to{\rm\bf
DVect}_{k}$ is the tensor product $\otimes$, which preserves neither products
nor direct sums. This makes impossible applying the existing techniques [12]
for opfibrations and putting a model structure on ${\sf PSect}({\cal C},{\cal
E})$, let alone derived sections. As a consequence, for example, it is
difficult to understand the behaviour of ${\sf\mathbb{R}Sect}({\cal C},{\cal
E})$ under the base change. Namely, given a functor $F:{\cal D}\to{\cal C}$,
denote by $F^{*}p:F^{*}{\cal E}\to{\cal D}$ the pullback of the opfibration
$p:{\cal E}\to{\cal C}$ ($F^{*}p$ is again an opfibration). There is an
induced functor $\mathbb{F}^{*}:{\sf PSect}({\cal C},{\cal E})\to{\sf
PSect}({\cal D},F^{*}{\cal E})$, it preserves weak equivalences and respects
derived sections. For a given $F$, we may ask if $\mathbb{F}^{*}$ admits a
homotopy adjoint, or is homotopically full and faithful when restricted to
derived sections. Unfortunately, answering these questions without any
additional structure on $p$ seems impossible.

To deal with this problem, we propose the approach of homotopical
$\Delta$-categories, which axiomatise the notion of geometric realisation of
simplicial objects. Admittedly, this is a fairly restrictive technical tool
specifically adapted to our setting, but nonetheless, examples of
$\Delta$-categories include ${\rm\bf DVect}_{k}$, simplicial vector spaces
$\Delta^{\sf op}\mathbf{Vect}_{k}$ and some other categories which admit a
sufficiently reasonable action of simplicial sets. For ${\rm\bf DVect}_{k}$
and $\Delta^{\sf op}\mathbf{Vect}_{k}$, the $\Delta$-category structure
interacts nicely with the tensor product, giving rise to what we call
homotopical $\Delta$-(op)fibrations. With the use of this technology, we are
able to construct, for each $F:{\cal D}\to{\cal C}$ and a homotopical
$\Delta$-opfibration ${\cal E}\to{\cal C}$, a weak equivalence preserving
functor $\mathbb{F}_{!}:{\sf PSect}({\cal D},F^{*}{\cal E})\to{\sf
PSect}({\cal C},{\cal E})$. This functor is not adjoint to $\mathbb{F}^{*}$
and does not, in general, preserve derived sections. However, $\mathbb{F}_{!}$
comes together with additional structure, for example a natural transformation
$\epsilon:\mathbb{F}_{!}\mathbb{F}^{*}\to id$ on the level of
localisation333Precisely, $\epsilon$ is a natural transformation of functors
$\mathbb{F}_{!}\mathbb{F}^{*},id:{\rm Ho}\,{\sf\mathbb{R}Sect}({\cal C},{\cal
E})\to{\rm Ho}\,{\sf PSect}({\cal C},{\cal E})$. ${\rm
Ho}\,{\sf\mathbb{R}Sect}({\cal C},{\cal E})$ which behaves like the counit of
an adjunction data. In this regard, our approach is ideologically similar to
the result of Costello [7], who constructs a derived equivalence by providing
explicitly two functors together with natural maps which become isomorphisms
on the level of localisation. However, Costello’s construction is rather ad-
hoc; we attempt to be more systematic.

We then restrict our attention to one specific class of examples which is
motivated by algebraic geometry [16, 17] and called resolutions in this paper.
A functor $F:{\cal D}\to{\cal C}$ is a _resolution_ if it is an opfibration
and for each $c\in{\cal C}$ the fibre ${\cal D}(c):=F^{-1}(c)$ has
contractible nerve. For an example of a resolution, consider a finite CW-
complex $Y$ of homotopy type $K(G,1)$ and denote by ${\textrm{B}}G$ the
fundamental groupoid of $Y$. Take $I$ to be the partially ordered set
associated to a chosen regular cellular decomposition of $Y$. Choosing a point
(say, the centre) of each cell of $I$ and connecting these points by paths
when one cell is included in the other defines a functor

$F:I\to{\textrm{B}}G$ (ix)

which is (equivalent to) a resolution. The functor $F$ induces the pullback
functor $F^{*}:{\cal D}({\textrm{B}}G,k)\to{\cal D}(I,k)$, where ${\cal
D}({\textrm{B}}G,k)$ and ${\cal D}(I,k)$ are the derived categories of
functors from ${\textrm{B}}G$ and $I$ to ${\rm\bf DVect}_{k}$ correspondingly
(note that ${\cal D}({\textrm{B}}G,k)$ is the same thing as $Loc(Y,k)$, the
derived category of complexes of locally constant sheaves on $Y$). One can
prove that $F^{*}$ is full and faithful, with its image consisting of those
functors $I\to{\rm\bf DVect}_{k}$ which are ’locally constant’, in the sense
that they send all morphisms of $I$ to quasiisomorphisms. We also see that
${\cal D}(I,k)$ is a good object: it is the category of modules over the
finite-dimensional algebra generated by $I$. In particular, it is simple to
construct objects of ${\cal D}(I,k)$, which, if locally constant, can provide
examples of $G$-representations.

An example of a functor (ix) arises for the configuration spaces ${\cal
D}_{2}^{n}$ of $n$ points on a $2$-disk, which are the classifying spaces for
$n$-braid groups $Br_{n}$, and which admit interesting cellular decompositions
using the combinatorics of planar trees [15]. This example is of particular
importance for Deligne conjecture and factorisation algebras.

For resolutions, the result we prove in this paper is the following. Given a
homotopical $\Delta$-opfibration ${\cal E}\to{\cal C}$ and a resolution
$F:{\cal D}\to{\cal C}$, the induced pullback functor for derived sections,
$\mathbb{F}^{*}:{\sf\mathbb{R}Sect}({\cal C},{\cal
E})\to{\sf\mathbb{R}Sect}({\cal D},F^{*}{\cal E})$, is homotopically full and
faithful. In some specific cases we may also characterise the essential
homotopical image of $\mathbb{F}^{*}$. It consists of those derived sections
which are locally constant (in the weak sense) when restricted to the
simplicial replacements ${\mathbb{D}}(c)$ of fibres ${\cal D}(c)$. In the case
when our opfibration ${\cal E}\to{\cal C}$ is the constant opfibration
${\rm\bf DVect}_{k}\times{\cal C}\to{\cal C}$, we reproduce the result for
derived categories discussed above. In general, resolutions serve as a testing
case, indicating that ${\sf\mathbb{R}Sect}$ is a reasonable thing to consider.
For the example of ${\rm\bf DVect}_{k}^{\otimes}\to{\rm\bf Fin}_{*}$, it seems
that resolutions can be used to provide the proof of Deligne conjecture
without operadic considerations; the details are being investigated by the
author.

Organisation of the paper. In the first section, we introduce the formalism of
homotopical $\Delta$-categories which we need in order to treat the issues
arising from the breakdown of model-categorical formalism. The content of this
section is not genuinely new and is somehow present in the folklore. For
instance, the geometric realisation functor for homotopical categories, in the
setting which goes beyond simplicial model categories, has been considered in
[4, Appendix].

In the second section, we recall some of the basic notions and constructions
related to the theory of Grothendieck (op)fibrations, including the less known
constructions of transpose and power (op)fibrations. Since Grothendieck
opfibrations are natural tools for encoding the notion of families of
categories, we introduce a class of suitably structured opfibrations, called
homotopical $\Delta$-opfibrations, which formalise the notion of a covariant
family of $\Delta$-categories.

In the third section, we introduce simplicial replacements and then use them
to define derived sections. The fourth section deals with the construction of
the pushforward functor $\mathbb{F}_{!}$ and the map
$\epsilon:\mathbb{F}_{!}\mathbb{F}^{*}\to id$, the data which one can use to
verify if the ‘right adjoint’ $\mathbb{F}^{*}$ is full and faithful.

Finally, the fifth section consists of the analysis of the case of a
resolution, stating the Theorems 5.4 and 5.5 and outlining their proof. It is
proven that in this case, the inverse image on the derived sections is full
and faithful on the homotopy level. In addition, under mild assumptions we can
characterise the essential image of the inverse image functor
$\mathbb{F}^{*}$.

Acknowledgements. The author is enormously grateful to his co-advisors, Dmitry
Kaledin and Carlos Simpson, for their immense support and patience. In
particular, Dmitry’s insights on the problem were and are of great importance
for the project. In the course of research, the author benefitted a lot from
conversations with Michael Batanin, Clemens Berger, Emily Riehl, Bruno
Vallette and Gabriele Vezzosi. The unknown referee has provided the comments
of enormous value which, hopefully, led to the improvement of this text. This
work has been carried out in and with the help of University of Nice and
Higher School of Economics, and the author enjoyed the friendly atmosphere of
quite a few conferences that took place in Paris, Nice, Copenhagen, Lausanne
and Yaroslavl. The author is grateful for the aid of TOFIGROU and HOGT ANR
grants, which helped to cover some of his travel expenses. This paper uses
Paul Taylor’s diagrams package.

##  1 Generalities on geometric realisation

Apart from fixing notation, this section sets up the formalism of homotopical
$\Delta$-categories. A $\Delta$-category ${\cal M}$ is a category together
with a comultiplicative action of the simplex category $\Delta$, and for
homotopical $\Delta$-categories this action respect weak equivalences both in
${\cal M}$ and simplicial sets, and provides us with a convenient version of
geometric realisation functor.

Our primary motivation for introducing such formalism is the structure on the
category of chain complexes of vector spaces ${\rm\bf DVect}_{k}$. It is not a
simplicial monoidal model category, but nonetheless comes with a
$\Delta$-action given by tensor products with the chain complex of
$n$-simplexes and an associated version of geometric realisation functor (see
[4] for an alternative discussion of this example).

###  1.1 Homotopy colimits

###### Notation 1.1

For any category ${\cal C}$, $x\in{\cal C}$ means that $x$ is an object of
${\cal C}$. We also write $f\in{\cal C}$ for morphisms $f:x\to y$ of ${\cal
C}$ if there is no confusion. The set of morphisms between two objects $x,y$
of ${\cal C}$ is denoted ${\cal C}(x,y)$. The category of functors
$Fun(I,{\cal M})$ between two categories $I$ and ${\cal M}$ is often denoted
as ${\cal M}^{I}$. Sometimes, given an object $x\in{\cal C}$, we denote again
by $x$ the functor from the terminal category to ${\cal C}$ which picks out
$x$.

From now on, $\Delta$ denotes the usual category of simplexes, i.e. the full
subcategory of the category of small categories ${\rm\bf Cat}$ spanned, for
$n\geq 0$, by categories $[n]$ with $n+1$ objects $0,...,n$ and exactly one
morphism from $i$ to $j$ whenever $i\leq j$.

By ${\rm\bf SSet}=Fun(\Delta^{\sf op},{\rm\bf Set})$ we denote the category of
simplicial sets. We often identify $\Delta$ with its image in ${\rm\bf SSet}$
by the Yoneda embedding

$\Delta^{\bullet}:\Delta\to{\rm\bf SSet}=Fun(\Delta^{\sf op},{\rm\bf
Set}),\,\,\,[n]\mapsto\Delta^{n}:=\Delta^{\bullet}([n])=\Delta(-,[n]).$ (1.1)

For a simplicial object $X:\Delta^{\sf op}\to{\cal M}$ in a category ${\cal
M}$, denote $X_{n}:=X([n])$ for any $[n]\in\Delta$, and similarly, for
bisimplicial objects $Y:\Delta^{\sf op}\times\Delta^{\sf op}\to{\cal M}$, we
write $Y_{nm}=Y(([n],[m]))$. We also write $\Delta^{\sf op}{\cal
M}:=Fun(\Delta^{\sf op},{\cal M})$, $(\Delta\times\Delta^{\sf op}){\cal
M}:=Fun(\Delta\times\Delta^{\sf op},{\cal M})$ and so on.

###### Definition 1.2

A _homotopical category_ is a pair $({\cal M},{\cal W})$ where ${\cal M}$ is a
category and ${\cal W}$ is a subcategory of ${\cal M}$ which contains all
objects and isomorphisms. We moreover require that for a composable pair of
morphisms $f,g$ of ${\cal M}$, if any two elements of $\\{f,g,gf\\}$ are in
${\cal W}$, then the third one is in ${\cal W}$ as well.

We call ${\cal W}$ the category of _weak equivalences_. A morphism $f:x\to y$
of ${\cal M}$ is a _weak equivalence_ if it belongs to ${\cal W}$.

###### Definition 1.3

For a homotopical category $({\cal M},{\cal W})$ its _localisation_ [8, 14]
${\cal W}^{-1}{\cal M}$ is the category together with a functor $p:{\cal
M}\to{\cal W}^{-1}{\cal M}$ such that any functor $F:{\cal M}\to{\cal N}$
which sends maps of ${\cal W}$ to isomorphisms of ${\cal N}$, factors through
$p$ up to a canonical isomorphism.

We also denote ${\cal W}^{-1}{\cal M}$ by ${\rm Ho}\,{\cal M}$.

The existence of localisation for homotopical categories $({\cal M},{\cal W})$
where ${\cal M}$ is not small is a known set-theoretical issue, and one can
check that for the examples of interest, in this paper it will not arise.

###### Definition 1.4

Given two homotopical categories $({\cal M},{\cal W}_{\cal M})$, $({\cal
N},{\cal W}_{\cal N})$ a functor $F:{\cal M}\to{\cal N}$ is _homotopical_ iff
$F({\cal W}_{M})\subset{\cal W}_{N}$. Equivalently, $F$ takes weak
equivalences of ${\cal M}$ to weak equivalences of ${\cal N}$.

Any homotopical functor $F:{\cal M}\to{\cal N}$ produces to a functor
$\overline{F}:{\rm Ho}\,{\cal M}\to{\rm Ho}\,{\cal N}$.

###### Example 1.5

Some well known examples of homotopical categories are

  * •

the category ${\rm\bf SSet}$ of simplicial sets which can be equipped with a
homotopical structure by defining ${\cal W}$ to be the subcategory of weak
homotopy equivalences [9] of simplicial sets,

  * •

the category ${\rm\bf DVect}_{k}$ of unbounded chain complexes over a field
$k$, with ${\cal W}$ being the subcategory of quasiisomorphisms [14],

  * •

any model category ${\cal M}$ with its subcategory of weak equivalences ${\cal
W}$.

###### Definition 1.6

A subcategory ${\cal W}\subset{\cal M}$ satisfies the _two-out-of-six_
property, if given three maps in ${\cal M}$ denoted $f,g,h$, so that they are
composable with compositions $gf,hg,hgf$, if $gf,hg$ are in ${\cal W}$, then
all other maps $f,g,h$ and $hgf$ are in ${\cal W}$.

The subcategory of isomorphisms in any category satisfies two-out-of-six. The
subcategory of weak equivalences in any model category satisfies two-out-of-
six as well [8].

###### Definition 1.7

For $I\in{\rm\bf Cat}$ and a homotopical category $({\cal M},{\cal W})$, the
standard homotopical structure $({\cal M}^{I},{\cal W}_{I})$ on the category
of functors $A:I\to{\cal M}$ consists of those natural transformations
$\alpha:A\to B$ which are valued in the maps of ${\cal W}$. That is, for each
$i\in I$, the map $\alpha(i):A(i)\to B(i)$ is a weak equivalence.

###  1.2 Tensors and $\Delta$-categories

Denote by $\delta:\Delta\to\Delta\times\Delta$ the diagonal functor for
$\Delta$.

###### Definition 1.8

A $\Delta$-structure on a category ${\cal M}$ consists of

  1. 1.

a functor

$\otimes:\Delta\times{\cal M}\to{\cal M},\,\,\,([n],x)\mapsto\Delta^{n}\otimes
x,$

  2. 2.

a natural transformation $diag$ depicted as a 2-square

  3. 3.

a natural isomorphism of ${\cal M}$-endofunctors:
$\Delta^{0}\otimes-\stackrel{{\scriptstyle\sim}}{{\to}}id_{\cal M}$.

These data should satisfy the obvious coassociativity and counitality
identities. A category ${\cal M}$ with a $\Delta$-structure is called a
_$\Delta$ -category_ if ${\cal M}$ is cocomplete and the functor $\otimes$
preserves colimits in the second argument.

###### Remark 1.9

It is immediate that a $\Delta$-category ${\cal M}$ has a ${\rm\bf
SSet}$-enrichment given by the mapping spaces

$\operatorname{Map}_{\cal M}(x,y)_{n}:={\cal M}(\Delta^{n}\otimes x,y).$

###### Example 1.10

The terminal category $[0]$ can be equipped with a (trivial)
$\Delta$-structure.

###### Example 1.11

The category ${\rm\bf DVect}_{k}$ is a $\Delta$-category for
$\Delta^{n}\otimes M:=C_{\bullet}(\Delta^{n})\otimes_{k}M$, where
$C_{\bullet}$ is the chain complex functor. The natural transformation $diag$
comes from the Alexander-Whitney map as follows:

$diag:C_{\bullet}(\Delta^{n})\stackrel{{\scriptstyle
C_{\bullet}(\delta)}}{{\to}}C_{\bullet}(\Delta^{n}\times\Delta^{n})\to
C_{\bullet}(\Delta^{n})\otimes_{k}C_{\bullet}(\Delta^{n}).$

###### Example 1.12

Any simplicial model category ${\cal M}$ is a $\Delta$-category in the obvious
way.

###### Proposition 1.13

If ${\cal M}$ is a $\Delta$-category then $\otimes:\Delta\times{\cal
M}\to{\cal M}$ can be extended uniquely to a functor $\otimes:{\rm\bf
SSet}\times{\cal M}\to{\cal M}$ such that

  1. 1.

$\otimes$ preserves colimits in each argument,

  2. 2.

there is a family of maps

$a(S,T,x):(S\times T)\otimes x\to S\otimes(T\otimes x)$ (1.2)

natural in $S,T\in{\rm\bf SSet}$ and $x\in{\cal M}$, associative in a suitable
sense and so that for each $[n]$, the composition

$\Delta^{n}\otimes x\to(\Delta^{n}\times\Delta^{n})\otimes
x\to\Delta^{n}\otimes(\Delta^{n}\otimes x)$

equals $diag(n,x)$ of Definition 1.8. Moreover, $a(S,T,x)$ is an isomorphism
whenever $S$ or $T$ is discrete.

We sometimes call the natural map $a(S,T,x)$ the _action map_.

###### Proof.

Recall that to each simplicial set $S$ we can associate its category of
simplexes $\Delta/S$. Its objects are all simplexes of $S$, represented as
maps $\Delta^{n}\to S$, and a morphism between two such objects is given by a
map $[n]\to[m]$ in $\Delta$ compatible with morphisms to $S$. Let
$s:\Delta/S\to\Delta$ denote the functor $(\Delta^{n}\to S)\mapsto[n]$, and
define $S\otimes x:=\operatorname{\varinjlim}_{\Delta/S}s\otimes x$.

For two $S,T\in{\rm\bf SSet}$, we have a canonical map $\Delta/(S\times
T)\to\Delta/S\times\Delta/T$ induced by the two projections
$d_{s}:\Delta/(S\times T)\to\Delta/S$ and $d_{t}:\Delta/(S\times
T)\to\Delta/T$. Denote again by $s:\Delta/S\to\Delta$, $t:\Delta/T\to\Delta$
and also by $st:\Delta/(S\times T)\to\Delta$ the corresponding forgetful
functors. Then we have a sequence of maps

$(S\times T)\otimes x\cong\operatorname{\varinjlim}_{\Delta/(S\times
T)}st\otimes x\to\operatorname{\varinjlim}_{\Delta/S\times
T}st\otimes(st\otimes x)\cong\operatorname{\varinjlim}_{\Delta/(S\times
T)}(s\circ d_{s})\otimes((t\circ d_{t})\otimes x)\to$
$\to\operatorname{\varinjlim}_{\Delta_{S}}\operatorname{\varinjlim}_{\Delta_{T}}s\otimes(t\otimes
x)\cong S\otimes(T\otimes x).$

Given the construction of this morphism, one can witness the naturality and
check its associativity; due to the third condition of Definition 1.8, the
action map becomes an isomorphism,

$(S\times T)\otimes x\stackrel{{\scriptstyle\sim}}{{\to}}S\otimes(T\otimes
x),$

when any of the two $S,T\in{\rm\bf SSet}$ is discrete. This proves the last
assertion. $\square$

###### Example 1.14

For any cocomplete category ${\cal M}$, there is canonical $\Delta$-structure
on $\Delta^{\sf op}{\cal M}=Fun(\Delta^{\sf op},{\cal M})$, which produces a
strict associative action of simplicial sets. Given a simplicial set $K$ and a
simplicial object $X\in\Delta^{\sf op}{\cal M}$, we define

$(K\otimes X)_{n}=K_{n}\otimes X_{n}=\coprod_{K_{n}}X_{n}.$

###### Definition 1.15

Given two categories ${\cal M},{\cal N}$ with $\Delta$-structures, a _$\Delta$
-functor _$F:{\cal M}\to{\cal N}$ is a functor between underlying categories
together with a family of morphisms

$m_{F}([n],x):\Delta^{n}\otimes F(x)\to F(\Delta^{n}\otimes x)$

natural in both $[n]$ and $x$. It is required to be compatible with the
diagonal maps and unit isomorphisms.

###### Remark 1.16

Equivalently, a $\Delta$-functor $F:{\cal M}\to{\cal N}$ is a simplical
functor for the simplicial enrichment mentioned in Remark 1.9. In particular,
it is evident that the composition of $\Delta$-functors is naturally a
$\Delta$-functor.

###### Example 1.17

The tensor product of chain complexes

$\otimes_{k}:{\rm\bf DVect}_{k}\times{\rm\bf DVect}_{k}\to{\rm\bf DVect}_{k},$

or more generally, for any finite444$S=*$ corresponds to the identity functor,
$S=\emptyset$ corresponds to the inclusion of $k$ in ${\rm\bf DVect}_{k}$. set
$S$, the $S$-fold tensor product

$\otimes_{k}:{\rm\bf DVect}^{S}_{k}\to{\rm\bf DVect}_{k}$

can be naturally equipped with the structure of a $\Delta$-functor.

###### Proposition 1.18

A $\Delta$-functor on $F:{\cal M}\to{\cal N}$ between $\Delta$-categories
determines a family of maps $m_{F}(S,x):S\otimes F(x)\to F(S\otimes x)$
natural in $S\in{\rm\bf SSet}$ and $x\in{\cal M}$, which restricts to
$m_{F}([n],x)$ for $S=\Delta^{n}$ and respects the action maps of Proposition
1.13.

###### Proof.

Define $m_{F}(S,x)$ as

$S\otimes F(x)\cong\operatorname{\varinjlim}_{\Delta/S}s\otimes
F(x)\stackrel{{\scriptstyle
m_{F}}}{{\longrightarrow}}\operatorname{\varinjlim}_{\Delta/S}F(s\otimes x)\to
F(\operatorname{\varinjlim}_{\Delta/S}s\otimes x)\cong F(S\otimes x).$

Then the result follows. $\square$

Recall that for a functor $F:I^{\sf op}\times I\to{\cal M}$, its _coend_ [20,
IX.6] is defined as the universal object $\int^{I}F$ in ${\cal M}$ together
with maps $F(i,i)\to\int^{I}F$ for each $i\in I$, such that for any morphism
$i\to i^{\prime}$, the induced diagram commutes: Coends exist in ${\cal M}$
when ${\cal M}$ is cocomplete.

###### Definition 1.19

Let ${\cal M}$ be a $\Delta$-category, and $X:\Delta^{\sf op}\to{\cal M}$ a
simplicial object in ${\cal M}$. Its _geometric realisation_ is defined as

$|X|:=\int^{\Delta^{\sf op}}\Delta^{\bullet}\otimes X$

Where $\Delta^{\bullet}$ is the Yoneda functor (1.1). Varying $X$, we get a
functor $|-|:\Delta^{\sf op}{\cal M}\to{\cal M}$.

For $S\in{\rm\bf SSet}$ and $A\in{\cal M}$, it is evident that the realisation
of the simplicial object $[n]\mapsto S_{n}\otimes A$ is canonically isomorphic
to $S\otimes A$.

###### Proposition 1.20

For a $\Delta$-functor $f:{\cal M}\to{\cal N}$ we have a canonical natural
transformation

$s_{f}:|f(-)|_{{\cal N}}\to f|-|_{{\cal M}}$

between the corresponding geometric realisations, where $f:\Delta^{\sf
op}{\cal M}\to\Delta^{\sf op}{\cal N}$ is the induced functor. It is
compatible with the composition in the following sense: the pasting of with
vertical functors given by realisations, is equal to $s_{gf}$.

###### Proof.

A tedious but straightforward check. $\square$

###  1.3 Homotopical $\Delta$-categories

For any bisimplicial object $X\in(\Delta^{{\sf op}}\times\Delta^{\sf op}){\cal
M}$ denote by $\delta^{*}X\in\Delta^{\sf op}{\cal M}$ the diagonal simplicial
object, that is, the pullback of $X$ along the diagonal map
$\delta:\Delta^{\sf op}\to\Delta^{\sf op}\times\Delta^{\sf op}$.

###### Definition 1.21

A _homotopical $\Delta$-structure_ on a category ${\cal M}$ consists of

  * •

a homotopical structure given by a subcategory ${\cal W}\subset{\cal M}$,

  * •

a $\Delta$-structure with the functor $\otimes:\Delta\times{\cal M}\to{\cal
M}$,

so that the following conditions are satisfied:

  1. 1.

the subcategory ${\cal W}$ satisfies two-out-of-six (Definition 1.6),

  2. 2.

${\cal M}$ is a $\Delta$-category and ${\cal W}$ is preserved by small
coproducts,

  3. 3.

the induced functor $\otimes:{\rm\bf SSet}\times{\cal M}\to{\cal M}$ respects
weak equivalences in each variable,

  4. 4.

the induced action map (1.2) $a(S,T,x):(S\times T)\otimes x\to
S\otimes(T\otimes x)$ is a weak equivalence for each $x\in{\cal M}$ and
$S,T\in{\rm\bf SSet}$,

  5. 5.

the geometric realisation functor $|-|:\Delta^{{\sf op}}{\cal M}\to{\cal M}$
preserves pointwise weak equivalences and for each bisimplicial object
$X\in(\Delta^{{\sf op}}\times\Delta^{\sf op}){\cal M}$, the natural composite
map

$\int^{\Delta^{\sf op}}\Delta^{\bullet}\otimes\delta^{*}X\to\int^{\Delta^{\sf
op}}\Delta^{\bullet}\otimes(\Delta^{\bullet}\otimes\delta^{*}X)\to\int^{\Delta^{\sf
op}\times\Delta^{\sf op}}\Delta^{\bullet}\otimes(\Delta^{\bullet}\otimes X)$
(1.3)

is a weak equivalence.

A category together with a homotopical $\Delta$-structure is called a
_homotopical $\Delta$-category_.

###### Remark 1.22

The identity (1.3) implies that for any bisimplicial object $B:\Delta^{\sf
op}\times\Delta^{\sf op}\to{\cal M}$, one may take geometric realisations in
any order. See Proposition 1.26 for details.

###### Example 1.23

Some simplicial model categories ${\cal M}$, for instance simplicial
presheaves with injective model structure or simplicial vector spaces, produce
examples of homotopical $\Delta$-categories. The nontrivial point here is that
the realisation functor $\Delta^{\sf op}{\cal M}\to{\cal M}$ only preserves
weak equivalences between Reedy cofibrant objects [9, VII.3.6], but for the
model categories just mentioned, all objects of $\Delta^{\sf op}{\cal M}$ are
automatically cofibrant.

###### Example 1.24

The category ${\rm\bf DVect}_{k}$ is a homotopical $\Delta$-category for the
$\Delta$-structure of Example 1.11 and ${\cal W}$ being the class of
quasiisomorphisms. In this case, all simplicial objects are Reedy-cofibrant,
and the functor of geometric realisation is known to be left Quillen for the
Reedy model structure on simplicial objects [4, Lemma 9.8].

We assemble together some of the properties of geometric realisation. Define
the category $\Delta_{\infty}$ as a subcategory of $\Delta$ consisting of all
objects and maps $f:[m]\to[n]$ such that $f(m)=n$. One has the adjunction

$j:\Delta\rightleftharpoons\Delta_{\infty}:i$

where $j([n])=[n+1]$ and the adjunction map $id\to i\circ j$ evaluated on
$[n]$ is the inclusion $[n]\hookrightarrow[n+1]$ of $[n]$ as first $n+1$
elements of $[n+1]$. Intuitively, $j$ attaches one more, maximal, element to
each $[n]$.

###### Definition 1.25

A _split-augmented_ simplicial object is a functor $\bar{X}:\Delta^{{\sf
op}}_{\infty}\to{\cal M}$. A simplicial object $X:\Delta^{\sf op}\to{\cal M}$
admits a (split) augmentation iff $X\cong j^{*}\bar{X}$ for some
$\bar{X}:\Delta_{\infty}^{\sf op}\to{\cal M}$.

For a _bisimplicial_ object $X:\Delta^{\sf op}\times\Delta^{\sf op}\to{\cal
M}$, we denote by $||X|_{2}|_{1}$ its repeated realisation, that is the coend
of the functor

$([i],[j],[k],[l])\mapsto\Delta^{i}\otimes(\Delta^{j}\otimes X_{kl})$

and by $||X|_{1}|_{2}|$ its transpose realisation, which is just a repeated
realisation of a transposed bisimplicial object $([n],[m])\mapsto X_{mn}$.

###### Proposition 1.26

For a homotopical $\Delta$-category ${\cal M}$, the following is true:

  1. 1.

For any simplicial object $X$ admitting an augmentation $\bar{X}$, its
realisation is weakly equivalent to $X_{-1}:=\bar{X}_{0}$. Precisely, there
are weak equivalences

$X_{-1}\to|X|\to X_{-1}$

with composition identity that come from the extra maps $X_{-1}\to X_{n}$ and
$X_{n}\to X_{-1}$.

  2. 2.

Given a morphism $X\to Y$ of bisimplicial objects, we have

$\left(||X|_{2}|_{1}\to||Y|_{2}|_{1}\right)\in{\cal
W}\Leftrightarrow\left(||X|_{1}|_{2}\to||Y|_{1}|_{2}\right)\in{\cal W}$

To distinguish simplicial and bisimplicial objects, we write $X_{\bullet}$
($[n]\mapsto X_{n}$) and $X_{\bullet\bullet}$ ($([n],[m])\mapsto X_{nm})$ for
a simplicial and a bisimplicial object, correspondingly.

###### Proof.

The first statement is proven in a few steps. Denote the tensoring of Example
1.14 by $\circledast:{\rm\bf SSet}\times\Delta^{\sf op}{\cal M}\to\Delta^{\sf
op}{\cal M}$. We now prove that the structure of a homotopical
$\Delta$-category on ${\cal M}$ gives rise to a family of weak equivalences
$|K\circledast X_{\bullet}|\to K\otimes|X_{\bullet}|$ natural in $K\in{\rm\bf
SSet}$ and $X_{\bullet}\in\Delta^{\sf op}{\cal M}$. The maps are constructed
as the following sequence:

$K\otimes|X_{\bullet}|=\left(\int^{\Delta^{\sf op}}\Delta^{\bullet}\otimes
K_{\bullet}\right)\otimes\left(\int^{\Delta^{\sf op}}\Delta^{\bullet}\otimes
X_{\bullet}\right)\cong\int^{\Delta^{\sf op}\times\Delta^{\sf
op}}\Delta^{\bullet}\otimes(\Delta^{\bullet}\otimes\coprod_{K_{\bullet}}X_{\bullet})$
$\leftarrow\int^{\Delta^{\sf
op}}\Delta^{\bullet}\otimes\delta^{*}(\coprod_{K_{\bullet}}X_{\bullet})\cong\int^{\Delta^{\sf
op}}\Delta^{\bullet}\otimes(K\circledast X_{\bullet})=|K\circledast
X_{\bullet}|.$

The only non-invertible map in the chain above,

$\int^{\Delta^{\sf op}\times\Delta^{\sf
op}}\Delta^{\bullet}\otimes(\Delta^{\bullet}\otimes\coprod_{K_{\bullet}}X_{\bullet})\leftarrow\int^{\Delta^{\sf
op}}\Delta^{\bullet}\otimes\delta^{*}(\coprod_{K_{\bullet}}X_{\bullet})$

is a weak equivalence by Definition 1.21.

By definition, a simplicial homotopy equivalence in $\Delta^{\sf op}{\cal M}$
consists of two maps $f:X_{\bullet}\to Y_{\bullet}$ and $g:Y_{\bullet}\to
X_{\bullet}$, and two diagrams where the vertical maps are induced from the
two inclusions $[0]\rightrightarrows[1]$ in $\Delta$. The natural weak
equivalence $|K\circledast X_{\bullet}|\to K\otimes|X_{\bullet}|$ then implies
that, after the realisation, the compositions $|g||f|$ and $|f||g|$ are weak
equivalences. By two-out-of-six we get that $|g|$ and $|f|$ are weak
equivalences as well.

It is known [23, Lemma 4.5.1] that $X_{\bullet}$ admitting an augmentation
$\bar{X}_{\bullet}$ in the sense of Definition 1.25 leads to a diagram in
$\Delta^{\sf op}{\cal M}$

$\bar{X}_{0}\to X_{\bullet}\to\bar{X}_{0}$

naturally appearing from the extra morphisms in $\bar{X}_{\bullet}$. The
composition of these maps is the identity $id_{X_{0}}$, and both maps can be
shown to be simplicial homotopy equivalences in $\Delta^{\sf op}{\cal M}$;
they thus become weak equivalences after applying geometric realisation.

For the last statement of the proposition, observe that both maps in question
are weakly equivalent (that is, weakly equivalent as objects of the category
${\cal M}^{[1]}$ of maps in ${\cal M}$) to the map

$\int^{\Delta^{\sf
op}}\Delta^{\bullet}\otimes\delta^{*}(X_{\bullet\bullet})\to\int^{\Delta^{\sf
op}}\Delta^{\bullet}\otimes\delta^{*}(Y_{\bullet\bullet})$

which finishes the proof. $\square$

##  2 Fibrations, opfibrations, sections

The main aim of this section is to provide background material for the
language of Grothendieck fibrations (contravariant families of categories) and
opfibrations (covariant families). Along with the basic notions, we also
explain some operations such as taking powers and transpose fibrations. Some
parts of this material are largely folklore with no well-known reference
available, so we supply proofs. We finish by considering families of
homotopical $\Delta$-categories as described by homotopical
$\Delta$-opfibrations.

###  2.1 Basic notions

###### Definition 2.1

Let $p:{\cal E}\to{\cal C}$ be a functor. A morphism $\alpha:x\to y$ in ${\cal
E}$ is _$p$ -Cartesian_ [11], or simply Cartesian, if, for every morphism
$\beta:z\to y$ of ${\cal E}$ such that $p(\beta)=p(\alpha)$, there exists a
unique morphism $\gamma:z\to x$ such that $\beta=\alpha\gamma$ and
$p(\gamma)=id_{p(z)}$.

A morphism $\alpha:x\to y$ in ${\cal E}$ is _$p$ -opCartesian_ if it is
Cartesian for $p^{{\sf op}}:{\cal E}^{{\sf op}}\to{\cal C}^{{\sf op}}$.

###### Definition 2.2

A functor $p:{\cal E}\to{\cal C}$ is called a _Grothendieck fibration_ [11,
28] (or simply a fibration) of categories iff the following two conditions are
satisfied:

  * •

For every morphism $f:a\to b$ of ${\cal C}$ and $y\in{\cal E}$ such that
$p(y)=b$ there exists a Cartesian morphism $\alpha:x\to y$ in ${\cal E}$
covering $\alpha$, that is, $p(\alpha)=f$.

  * •

The composition of Cartesian morphisms is a Cartesian morphism.

Dually, $p$ is called an _opfibration_ of categories iff $p^{{\sf op}}:{\cal
E}^{{\sf op}}\to{\cal C}^{{\sf op}}$ is a fibration of categories.

An (op)fibration $q:{\cal O}\to{\cal C}$ is _small_ if both ${\cal C}$ and
${\cal O}$ are small.

###### Construction 2.3

Given a functor $E$ from ${\cal C}$ to categories, we produce an opfibration,
which we denote $\int E\to{\cal C}$ and call the _Grothendieck construction_
[28] of $E$. An object of $\int E$ is a pair $(c,x)$ of $c\in{\cal C}$ and
$x\in E(c)$, and a morphism $(c,x)\to(c^{\prime},x^{\prime})$ consists of
$f:c\to c^{\prime}$ together with a map $\alpha:E(f)(x)\to x^{\prime}$ in
$E(x^{\prime})$.

In this paper, we already used the symbol $\int$ for coends; in fact, $\int E$
can be reproduced as a certain coend.

Similar considerations apply in the case of fibrations. For a contravariant
category-valued functor $F$ defined on ${\cal C}$, the Grothendieck
construction is a fibration $\int F\to{\cal C}$ with same pairs $(c,x)$
serving as objects, but with maps given by $f:c\to c^{\prime}$ and $\beta:x\to
F(f)x^{\prime}$.

###### Example 2.4

The category $\Delta/S$ for a simplicial set $S$ is exactly the domain of the
fibration $\int S\to\Delta$, where we view $S$ as a functor $\Delta^{\sf
op}\to{\rm\bf Set}\subset{\rm\bf Cat}$ contravariant on $\Delta$.

Denote by ${\rm\bf Fin}_{*}$ the category with objects finite sets, and
morphisms $f:S\to T$ given by partially defined maps. That is, $f$ is a pair
$(D,\tilde{f})$ of a subset $D\subset S$ and a map of sets $\tilde{f}:D\to T$.

###### Example 2.5

Consider a strict symmetric monoidal category ${\cal M}$, that is, a category
together with an associative commutative unital product $\otimes$. From this
data, we can form a functor $M$ to categories defined on ${\rm\bf Fin}_{*}$ by
the assignment $S\mapsto M(S):={\cal M}^{S}.$ For each partially defined map
$f:S\to T$, there is a functor $f_{!}:{\cal M}^{S}\to{\cal M}^{T}$, which
sends an $\\{X_{s}\\}_{s\in S}\in{\cal M}^{S}$ to $\\{Y_{t}\\}_{t\in
T}\in{\cal M}^{T}$ with $Y_{t}=\otimes_{s\in f^{-1}t}\,X_{s}.$ When the
inverse image is empty, the product is equal to the unit object.

Now we take the Grothendieck construction of $M$, obtaining $\int M\to{\rm\bf
Fin}_{*}$. The category ${\cal M}^{\otimes}:=\int M$ is otherwise described as
follows. Its objects are $(S,\\{X_{s}\\}_{s\in S})$ where $S\in{\rm\bf
Fin}_{*}$ and each $X_{s}$ is an object of ${\cal M}$. A morphism
$(S,\\{X_{s}\\}_{s\in S})\to(T,\\{Y_{t}\\}_{t\in T})$ is a partially defined
map $f:S\to T$, and a morphism $\otimes_{s\in f^{-1}(t)}X_{s}\to Y_{t}$ for
each $t\in T$.

A general (op)fibration is not equal to $\int E\to{\cal C}$ for some functor
$E$ to categories. Nonetheless, consider an opfibration $p:{\cal E}\to{\cal
C}$. For $c\in{\cal C}$, denote by ${\cal E}(c)=p^{-1}(c)$ the fibre of $p$
over $c$. Let $f:c\to c^{\prime}$ be a morphism in ${\cal C}$ and $x\in{\cal
E}(c)$. Then we can choose an opCartesian morphism $\alpha:x\to y$ such that
$p(\alpha)=f$. This specifies an object $y\in{\cal E}(c^{\prime})$. By the
universal property of opCartesian maps, the assignment $x\mapsto y$ defines a
functor $f_{!}:{\cal E}(c)\to{\cal E}(c^{\prime})$, which is called a
transition functor along $f$. One can check [11, 28] that for each composable
pair $f,g$, there exists a coherence isomorphism $g_{!}\circ f_{!}\cong(g\circ
f)_{!}$ such that for any composable triple of arrows $f,g,h$, any choice of
coherence isomorphisms renders commutative the following diagram: In the
literature, such choice of an assignment $f\mapsto f_{!}$ together with
coherence isomporphisms is called a cleavage.

###### Example 2.6

Take an arbitrary symmetric monoidal category ${\cal M}$ Define the category
${\cal M}^{\otimes}$ with the same objects and morphisms as in Example 2.5,
but now with compositions defined with the help of coherence isomorphisms for
$\otimes$. The forgetful functor ${\cal M}^{\otimes}\to{\rm\bf Fin}_{*}$ is
then an opfibration of categories. It is possible to characterise exactly the
opfibrations arising from symmetric monoidal categories using Segal conditions
[18, 27].

###### Definition 2.7

Let $p:{\cal E}\to{\cal C}$ and $q:{\cal E}^{\prime}\to{\cal C}$ be two
(op)fibrations. A _lax morphism_ between $p$ and $q$ is a functor $F:{\cal
E}\to{\cal E}^{\prime}$ such that $q\circ F=p$. Such $F$ is called a
_Cartesian morphism_ if it takes (op)Cartesian morphisms of ${\cal E}$ to
(op)Cartesian morphisms of ${\cal E}^{\prime}$. A _section_ of an
(op)fibration $p$ is a lax morphism from the (op)fibration $id_{\cal C}:{\cal
C}\to{\cal C}$ to $p$.

A morphism between two lax or Cartesian morphisms is a natural transformation
$\alpha:F\to F^{\prime}$ such that for each $x$ in the domain ${\cal E}$,
$\alpha_{x}$ projects to $id_{p(x)}$.

We denote by ${\sf Cart}({\cal E},{\cal E}^{\prime})$ the category of
Cartesian morphisms and by ${\sf Sect}({\cal C},{\cal E})$ the category of
sections.

###### Construction 2.8

Take an opfibration $p:{\cal E}\to{\cal C}$, and for each $c\in{\cal C}$,
denote by $c\backslash{\cal C}$ the category of objects under $c$ [20]. The
forgetful functor $c\backslash{\cal C}\to{\cal C}$ is an opfibration. Then the
assignment $c\mapsto{\sf Cart}(c\backslash{\cal C},{\cal E})$ defines a
covariant category-valued functor on ${\cal C}$. When ${\cal C}$ is small,
this construction is inverse up to an equivalence [28] to (Grothendieck)
Construction 2.3.

This implies that any opfibration (and, similarly, a fibration) $p:{\cal
E}\to{\cal C}$ can be, up to an equivalence, replaced by an opfibration
$\tilde{p}:\tilde{\cal E}\to{\cal C}$, for which the assignment $c\mapsto{\cal
E}(c)$ can be made into a strict functor by a choice of transition functors
along maps in ${\cal C}$. We call the opfibrations (similarly, fibrations)
with later property _strictly cleavable_.

###### Definition 2.9

In ${\rm\bf Fin}_{*}$, a map $\rho:S\to T$ is _inert_ [18] if it is defined on
a subset $T^{\prime}$ of $S$ isomorphic to $T$, and the restriction
$\tilde{\rho}:T^{\prime}\to T$ is a bijection.

###### Example 2.10

Any algebra object $A$ in a symmetric monoidal category ${\cal M}$ gives a
section $A:{\rm\bf Fin}_{*}\to{\cal M}^{\otimes}$ by the rule
$S\mapsto(A,...,A)\in{\cal M}^{\otimes}(S)$. Conversely, consider a section
$B:{\rm\bf Fin}_{*}\to{\cal M}^{\otimes}$ which sends the inert maps to
opCartesian maps. Then the value $B(1)\in{\cal M}$ of $B$ on a one-element set
$1$ becomes a commutative monoid. To show this, note that the ‘inert-to-
opCartesian’ condition implies that $B(S)\cong(S,{B(1)}_{s\in S})$ by the
means of all opCartesian maps out of $B(S)$ lying over all inert maps $S\to
1$. Next, take the map $S\to 1$ defined on each element of $S$, and consider
its image, $B(S)\to B(1)$, in ${\cal M}^{\otimes}$. It factors as

$B(S)\to B(1)^{\otimes S}\to B(1)$

and gives multiplication maps $B(1)^{\otimes S}\to B(1)$. Examining the
composition of maps in ${\cal M}^{\otimes}$, it can be checked that everything
is determined when $S$ is a two-element set, and associated operation
$B(1)\otimes B(1)\to B(1)$ must be associative, with a unit $I_{\cal M}\to
B(1)$ given by the value of $B$ on the map $\emptyset\to 1$ in ${\rm\bf
Fin}_{*}$, and commutative. with commutativity following from the action of
$B$ on the nontrivial automorphism of $S$.

###### Example 2.11

To explain the term ‘lax morphism’, consider a lax symmetric monoidal functor
$F:{\cal M}\to{\cal N}$ between two symmetric monoidal categories. It means
that there is a natural family of maps $FX\otimes FY\to F(X\otimes Y)$ and a
map $I_{\cal N}\to FI_{\cal M}$ with $I_{\cal N},I_{\cal M}$ unit objects,
which satisfy suitable coherence conditions. For opfibrations of Example 2.6
the assignment $(S,\\{X_{s}\\}_{s\in S})\mapsto(S,\\{FX_{s}\\}_{s\in S})$ then
induces a functor $F^{\otimes}:{\cal M}^{\otimes}\to{\cal N}^{\otimes}$ over
${\rm\bf Fin}_{*}$ which does not necessarily preserve Cartesian maps. For
example, consider the map $(X,Y)\to X\otimes Y$, which is opCartesian in
${\cal M}^{\otimes}$. Its image in ${\cal N}^{\otimes}$, $(FX,FY)\to
F(X\otimes Y)$, factors through the ${\cal N}^{\otimes}$-opCartesian map
$(FX,FY)\to FX\otimes FY$ exactly by the lax functor structure of $F$.

###### Example 2.12

Let $L:\int{\cal E}\to\int{\cal E}^{\prime}$ be a lax morphism between two
Grothendieck constructions of covariant functors ${\cal E},{\cal
E}^{\prime}:{\cal C}\to{\rm\bf Cat}$. For each $c\in{\cal C}$, $L$ specifies a
functor $L_{c}:{\cal E}(c)\to{\cal E}^{\prime}(c)$. For each morphism $f:c\to
c^{\prime}$, we get a $2$-square The natural transformation appears because
the image under $L$ of an opCartesian map $X\to{\cal E}(f)X$ ($X\in{\cal
E}(c)$) may not be opCartesian. Factoring $LX\to L{\cal E}(f)X$,

$LX\to{\cal E}^{\prime}(f)LX\to L{\cal E}(f)X,$

gives ${\cal E}^{\prime}(f)LX\to L{\cal E}(f)X$; for each $X\in{\cal E}(c)$,
all such maps assemble into $L_{f}$. For two composable arrows $f:c\to
c^{\prime}$, $g:c^{\prime}\to c^{\prime\prime}$, there is a pasting property
relating $L_{f},L_{g}$ and $L_{gf}$ similar to the one of Proposition 1.20.

For fibrations, there is a difference on the level of 2-diagrams. Consider
${\cal F},{\cal F}^{\prime}:{\cal C}^{\sf op}\to{\rm\bf Cat}$ and take a lax
morphism $M:\int{\cal F}\to\int{\cal F}^{\prime}$ of fibrations over ${\cal
C}$. For $f:c\to c^{\prime}$, we obtain a diagram with $M_{f}$ given by arrows
of the form $M{\cal F}(f)X\to{\cal F}^{\prime}(f)MX$.

###### Definition 2.13

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

  1. 1.

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

  2. 2.

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

$x\longrightarrow y\longleftarrow z$

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

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

If ${\cal E}\to{\cal C}$ equals $\int E\to{\cal C}$ for a functor $E:{\cal
C}\to{\rm\bf Cat}$, then ${\cal E}^{\top}\to{\cal C}^{\sf op}$ is equivalent
to the (fibrational) Grothendieck construction applied to $E:({\cal C}^{\sf
op})^{\sf op}\to{\rm\bf Cat}$ viewed as a contravariant functor on ${\cal
C}^{\sf op}$.

###### Example 2.14

The transpose fibration $({\cal M}^{\otimes})^{\top}\to{\rm\bf Fin}_{*}^{\sf
op}$ of ${\cal M}^{\otimes}\to{\rm\bf Fin}_{*}$ from Example 2.6 can be
constructed by hand just like the original opfibration. Now, to define a lax
functor between two such fibrations, $({\cal M}^{\otimes})^{\top}\to({\cal
N}^{\otimes})^{\top}$, we would have to consider _oplax_ symmetric monoidal
functors between symmetric monoidal categories. Such functors preserve
(cocommutative) coalgebra objects, as opposed to lax symmetric monoidal
functors, which preserve commutative algebra objects. In particular, any
coalgebra object $C$ of ${\cal M}$ produces a section of $({\cal
M}^{\otimes})^{\top}\to{\rm\bf Fin}_{*}^{\sf op}$ by the assignment
$S\mapsto(S,\\{C\\}_{S})$.

Given a functor $F:{\cal D}\to{\cal C}$, we can pull back (op)fibrations over
${\cal C}$ to ${\cal D}$, with the result again being (op)fibrations.
Similarly, given a section $A:{\cal C}\to{\cal E}$ of an (op)fibration ${\cal
E}\to{\cal C}$ we obtain from it the section $F^{*}A:{\cal D}\to F^{*}{\cal
E}$ of the pullback (op)fibration $F^{*}{\cal E}\to{\cal D}$. This operation
defines the pullback functor $F^{*}:{\sf Sect}({\cal C},{\cal E})\to{\sf
Sect}({\cal D},F^{*}{\cal E})$.

###### Proposition 2.15

Assume given a _fibration_ ${\cal F}\to{\cal C}$ and a natural transformation
$\alpha:F\to G$ of functors $F,G:{\cal D}\to{\cal C}$. Then

  * •

there is a natural Cartesian map of fibrations $R_{\alpha}:G^{*}{\cal F}\to
F^{*}{\cal F}$, which we call the _restriction_ map,

  * •

given a section $A:{\cal C}\to{\cal F}$, there is a natural morphism of
sections

$F^{*}A\to R_{\alpha}G^{*}A.$

The fact that ${\cal F}\to{\cal C}$ is a fibration, and not an opfibration, is
important for the direction of the arrows in the proposition.

###### Proof.

Up to an equivalence we can assume ${\cal F}\to{\cal C}$ to be strictly
cleavable. Take $d\in{\cal D}$. For each object $X$ of ${\cal
E}(G(d))=G^{*}{\cal F}(d)$, we have a Cartesian arrow $Y\to X$ in ${\cal F}$
over $\alpha_{d}:F(d)\to G(d)$. The value $R_{\alpha}X$ is then defined to be
equal to $Y$; its action on morphisms can be defined similarly.

Given a section $A$, its value on $\alpha_{d}:F(d)\to G(d)$ can be naturally
factored as

$F^{*}A(d)=A(F(d))\to R_{\alpha}A(G(d))\to A(G(d))=G^{*}A(d).$

Varying $d$, the arrows $F^{*}A(d)\to R_{\alpha}A(G(d))=(R\alpha G^{*}A)(d)$
define the natural transformation in question. $\square$

###### Definition 2.16

Let $p:{\cal E}\to{\cal C}$ be an (op)fibration and $I\in{\rm\bf Cat}$ a
category.

  * •

A product of $I$ and $p:{\cal E}\to{\cal C}$ is the functor $I\times
p:I\times{\cal E}\to{\cal C},\,\,\,\,\,(i,x)\mapsto p(x).$

  * •

A powering of $p$ with $I$ is the functor $p^{I}:{\cal E}^{I}\to{\cal C}$
where ${\cal E}^{I}$ is the subcategory of $Fun(I,{\cal E})$ consisting of all
functors $F:I\to{\cal E}$ such that $p\circ F$ is a constant functor
$I\to{\cal C}$.

Both these functors are (op)fibrations.

Unfortunately, the choice of notation such as ${\cal E}^{I}$ may lead to
confusion as before we used it to denote the whole category of functors
$I\to{\cal E}$. We thus adopt a convention that for (op)fibrations, the
powering notation works in the sense of definition above and not
otherwise555If we think of ordinary categories as Grothendieck fibrations over
a point, then there is no notational ambiguity..

###### Definition 2.17

Given a fibration $p:{\cal F}\to{\cal C}$ and an opfibration $q:{\cal
O}\to{\cal C}$ with small fibres, a _power fibration_ $p^{q}:{\cal F}^{\cal
O}\to{\cal C}$ is defined as follows. An object of ${\cal F}^{\cal O}$ is a
pair of $c\in{\cal C}$ and a functor $X:{\cal O}(c)\to{\cal F}$ such that $pX$
is constant of value $c$. A morphism $(c,X)\to(c^{\prime},Y)$ consists of
$f:c\to c^{\prime}$ and a natural transformation $X\to Y\circ f_{!}$ of
functors ${\cal O}(c)\to{\cal F}$ for some choice of transition functor
$f_{!}:{\cal O}(c)\to{\cal O}(c^{\prime})$. The functor ${\cal F}^{\cal
O}\to{\cal C}$ is the natural projection.

One can verify that ${\cal F}^{\cal O}\to{\cal C}$ is again a fibration, with
fibres equivalent to $Fun({\cal O}(c),{\cal F}(c))$. A fibrational transition
functor ${\cal F}^{\cal O}(c^{\prime})\to{\cal F}^{\cal O}(c)$ is given by
precomposing an object $F:{\cal O}(c)\to{\cal F}(c)$ with $f_{!}:{\cal
O}(c)\to{\cal O}(c^{\prime})$ and postcomposing with $f^{*}:{\cal
F}(c^{\prime})\to{\cal F}(c)$ for some choice of transition functors $f_{!}$
and $f^{*}$ in ${\cal O}$ and ${\cal F}$ respectively.

###### Lemma 2.18

For a functor $F:{\cal D}\to{\cal C}$, and $p:{\cal F}\to{\cal C},q:{\cal
O}\to{\cal C}$ as above,

  1. 1.

There is an equivalence of categories

${\sf Sect}({\cal O},q^{*}{\cal F})\cong{\sf Sect}({\cal C},{\cal F}^{\cal
O}).$

  2. 2.

There is a Cartesian map

$(F^{*}{\cal F})^{F^{*}{\cal O}}\to F^{*}({\cal F}^{\cal O})$

which is moreover an equivalence over ${\cal C}$.

###### Proof.

Clear. $\square$

###### Proposition 2.19

Let $p:{\cal F}\to{\cal C}$ be a fibration with cocomplete fibres.

  1. 1.

For any functor $X:I^{\sf op}\times I\to{\cal F}$ such that $pX$ is constant
of value $c\in{\cal C}$, its coend $\int^{I}X$ exists in ${\cal F}$ and can be
calculated in ${\cal F}(c)$, defining a lax morphism $\int^{I}:{\cal
F}^{I^{\sf op}\times I}\to{\cal F}$ of fibrations over ${\cal C}$.

  2. 2.

Let  be an opCartesian morphism of small strictly cleavable opfibrations. Then
the obvious functor

$P^{*}:{\sf Sect}({\cal C},{\cal F}^{I})\to{\sf Sect}({\cal C},{\cal F}^{\cal
O})$

admits a left adjoint $P_{!}$.

The fact that ${\cal F}\to{\cal C}$ is a fibration is important in this
proposition.

###### Proof.

Let $X:I^{\sf op}\times I\to{\cal F}$ be such that $pX$ is constant at
$c\in{\cal C}$. Take its coend in ${\cal F}(c)$, and denote it
$\int^{I}_{c}X$. We need to check that it satisfies the universal property of
a coend in ${\cal F}$. Let $Z\in{\cal F}$ be an object together with maps
$X(i,i)\to Z$ such that for any morphism $i\to i^{\prime}$, the diagram below
commutes: Applying $p:{\cal F}\to{\cal C}$ to diagrams like above, we find
that for each $i\in I$, the map $X(i,i)\to Z$ lies over a fixed morphism of
${\cal C}$ with domain $c$; we denote it as $f:c\to c^{\prime}$. The fact that
$p$ is a fibration then implies that there exists an object $f^{*}Z$ over $c$
and a Cartesian map $\alpha:f^{*}Z\to Z$ covering $f$, so that each map
$X(i,i)\to Z$ factors through $\alpha$ and the resulting squares are all
commutative. This implies the existence of a map $\int_{c}^{I}X\to f^{*}Z$.
One can then see that the composition $\int_{c}^{I}X\to f^{*}Z\to Z$ is unique
and independent of the choice of $f^{*}Z\to Z$ made above. One can see then
that this map is the one which ensures the universality of $\int_{c}^{I}X$ in
the whole of ${\cal F}$.

We now turn to the second statement. We define a functor on the level of
fibrations, $P_{!}:{\cal F}^{\cal O}\to{\cal F}^{I}$, by setting $P_{!}X$ to
be the left Kan extension [20] of $X:{\cal O}(c)\to{\cal F}(c)$ along the map
$P_{c}:{\cal O}(c)\to I$. Following the line of argumentation as for the
coends before, we can see that this Kan extension exists and moreover can be
calculated in the fibre ${\cal F}(c)$. Given a section $S:{\cal C}\to{\cal
F}^{\cal O}$, we apply $P_{!}$ to its values, inducing the sought-after
functor $P_{!}$, left adjoint to the pullback $P^{*}:{\sf Sect}({\cal C},{\cal
F}^{I})\to{\sf Sect}({\cal C},{\cal F}^{\cal O})$. $\square$

###  2.2 Homotopical $\Delta$-opfibrations

###### Definition 2.20

A _homotopical structure_ on an opfibration ${\cal E}\to{\cal C}$ consists of
a homotopical structure on ${\cal E}$, given by a subcategory ${\cal
W}\subset{\cal E}$ of weak equivalences, compatible with the opfibration in
the following sense:

  1. 1.

the image of ${\cal W}$ in ${\cal C}$ consists of identity morphisms,

  2. 2.

in a commutative square  if we have $f\in{\cal W}$ and
$\alpha,\alpha^{\prime}$ are opCartesian, then $f^{\prime}\in{\cal W}$.

The definition of a homotopical fibration is dual, with implication in the
diagram above going into opposite direction. One can see that for an
(op)fibration with homotopical structure, ${\cal W}=\coprod_{c\in{\cal
C}}{\cal W}(c)$, with $({\cal E}(c),{\cal W}(c))$ being a homotopical category
for each $c\in{\cal C}$. The transition functors of the (op)fibration send
${\cal W}(c)$ to ${\cal W}(c^{\prime})$.

###### Definition 2.21

Given an opfibration ${\cal E}\to{\cal C}$ with a homotopical structure, a
morphism $\alpha:x\to y$ of ${\cal E}$ is _weakly opCartesian_ if it can be
factored as an opCartesian morphism $x\to\alpha_{!}x$ followed by a weak
equivalence $\alpha_{!}x\to y$. Dually, one has the notion of a weakly
Cartesian morphism for homotopical fibrations, where the order of
factorisation is reversed.

Before proceeding with the $\Delta$-structure, we need some extra notation.
Take an opfibration ${\cal E}\to{\cal C}$ and consider its transpose fibration
(Definition 2.13) ${\cal E}^{\top}\to{\cal C}^{\sf op}$. Now, take the product
fibration $\Delta\times{\cal E}^{\top}\to{\cal C}^{\sf op}$. If we have any
functor $\otimes:\Delta\times{\cal E}^{\top}\to{\cal E}^{\top}$ over ${\cal
C}^{\sf op}$, then, for a map $f:c\to c^{\prime}$ of the original base
category ${\cal C}$, we get, after relabelling ${\cal E}^{\top}(c)={\cal
E}(c)$ and ${\cal E}^{\top}(f)=f_{!}:{\cal E}(c)\to{\cal E}(c^{\prime})$, the
following diagram so that each $m_{f}:-\otimes_{c^{\prime}}f_{!}-\to
f_{!}(-\otimes_{c}-)$ appears in the same way as in Example 2.12. Moreover,
$f\mapsto m_{f}$ is suitably functorial in $f$.

###### Definition 2.22

A $\Delta$-structure on an _fibration_ ${\cal F}\to{\cal C}$ is a
$\Delta$-structure $\otimes:\Delta\times{\cal F}\to{\cal F}$ such that

  1. 1.

$\otimes$ is a lax morphism of fibrations over ${\cal C}$,

  2. 2.

the natural transformation $diag$ and unitality isomorphism (see Definition
1.8) are fiberwise.

  3. 3.

For each $f:c\to c^{\prime}$ of ${\cal C}$ and a choice of $f^{*}:{\cal
F}(c^{\prime})\to{\cal F}(c)$ via the fibration property, the induced diagram
makes $f^{*}$ into a $\Delta$-functor.

A $\Delta$-structure on an opfibration ${\cal E}\to{\cal C}$ consists of a
$\Delta$-structure on its transpose fibration. In particular, the notion is
not dual.

###### Example 2.23

The prototype example for how one ought to think about such
$\Delta$-structures is the following. Consider a covariant functor $F$ on
${\cal C}$ such that each $F(c)$ is a $\Delta$-category and each $F(c\to
c^{\prime})$ is a $\Delta$-functor. Forgetting the $\Delta$-structure, we can
apply the Grothendieck construction, obtaining an opfibration $\int F\to{\cal
C}$. This opfibration is, then, equipped with a $\Delta$-structure inherited
from the values of $F$ on the objects and morphisms of ${\cal C}$.

###### Definition 2.24

A _homotopical_ $\Delta$-(op)fibration is an (op)fibration ${\cal E}\to{\cal
C}$ together with a homotopical structure and a $\Delta$-structure, such that
for each $c\in{\cal C}$, the induced structure on the fiber ${\cal E}(c)$ is
that of a homotopical $\Delta$-category.

###### Example 2.25

Chain complexes give us homotopical $\Delta$-opfibration ${\rm\bf
DVect}^{\otimes}\to{\rm\bf Fin}_{*}$. The $\Delta$-structure on the
opfibration is essentially explained in Examples 1.11 and 1.17, and the weak
equivalences are simply induced from the quasiisomorphisms of ${\rm\bf
DVect}_{k}$. In the same way, those simplicial model categories which give us
a homotopical $\Delta$-structure (Example 1.23) can as well give us
homotopical $\Delta$-opfibrations. If such a category ${\cal M}$ in addition
possesses a compatible monoidal structure (this is true, for example, both for
simplicial presheaves and for simplicial vector spaces), then the associated
opfibration ${\cal M}^{\otimes}\to{\rm\bf Fin}_{*}$ is a homotopical
$\Delta$-opfibration.

###### Proposition 2.26

Let ${\cal F}\to{\cal C}$ be a homotopical $\Delta$-fibration. Then

  * •

For any functor $F:{\cal D}\to{\cal C}$, the pullback $F^{*}{\cal F}\to{\cal
D}$ is again a homotopical $\Delta$-fibration.

  * •

There is a lax _realisation_ morphism of fibrations  such that on each fiber,
the functor $\Delta^{\sf op}{\cal F}(c)\to{\cal F}(c)$ is the geometric
realisation for the $\Delta$-category ${\cal F}(c)$.

###### Proof.

Recall that ${\cal F}^{\Delta^{\sf op}}$ is the subcategory of
$Fun(\Delta^{\sf op},{\cal F})$ consisting of $X:\Delta^{\sf op}\to{\cal F}$
which become constant after composing with ${\cal F}\to{\cal C}$. Since ${\cal
F}$ has a $\Delta$-structure, we can consider the assignment
$X\mapsto\Delta^{\bullet}\otimes X$, which defines a lax morphism ${\cal
F}^{\Delta^{\sf op}}\to{\cal F}^{\Delta\times\Delta^{\sf op}}$ of fibrations
over ${\cal C}$ (it can be seen to not preserve Cartesian arrows). We then use
the first part of Proposition 2.19 and compose the just-obtained functor with
the coend ${\cal F}^{\Delta\times\Delta^{\sf op}}\to{\cal F}$ to obtain the
realisation functor of this proposition. $\square$

###### Remark 2.27

From the perspective of $\Delta$-structures, we see that it is preferable to
consider fibrations and not opfibrations. The motivation for opfibrations as
basic ingredients of the play comes from Example 2.10, where algebra objects
in a monoidal category ${\cal M}$ are presented as sections of ${\cal
M}^{\otimes}\to{\rm\bf Fin}_{*}$. Suitably normalised sections of a transpose
fibration $({\cal M}^{\otimes})^{\top}\to{\rm\bf Fin}_{*}^{\sf op}$
correspond, on the other hand, to coalgebra objects in ${\cal M}$, as can be
seen from Example 2.14. The fact that formalism of the subsequent chapter
presents derived sections of an opfibration as certain sections of a related
fibration may be perceived as an example of certain ’Koszul’ duality relating
algebraic and coalgebraic objects.

##  3 Derived sections

This section is devoted to our definition of derived sections. Starting from
the introduction of simplicial replacements of categories, we assemble
together a few homotopy-theoretic facts to be used later. We also outline how
one prolongs an opfibration ${\cal E}\to{\cal C}$ to the simplicial
replacement ${\mathbb{C}}$ of ${\cal C}$. We finish by defining presections
and derived sections.

As shown in [5] and later in [23], simplicial replacements of categories allow
calculating homotopy colimits and may be viewed as certain cofibrant
replacements in homotopy-algebraic sense. We will make this perspective on
simplicial replacements concrete in our setting through the constructions of
the next section.

###  3.1 Simplicial replacements

######  Definition 3.1 ([5])

Given a small category ${\cal C}$, its _simplicial replacement_ , denoted
${\mathbb{C}}$, is the _opposite_ of $\Delta/N{\cal C}=\int N{\cal C}$, that
is the opposite of the category of simplexes of the simplicial set $N{\cal C}$
(cf. Example 2.4).

An object of ${\mathbb{C}}$ is a sequence $c_{0}\to...\to c_{n}$ of composable
morphisms in ${\cal C}$. Any functor $F:{\cal D}\to{\cal C}$ induces a functor
$\mathbb{F}:{\mathbb{D}}\to{\mathbb{C}}:$ by the rule
$\mathbb{F}(d_{0}\to...\to d_{n})=Fd_{0}\to...\to Fd_{n}$. Observe that
$\mathbb{F}$ commutes with the projections from ${\mathbb{D}}$ and
${\mathbb{C}}$ to $\Delta^{\sf op}$.

The assignment ${\cal C}\mapsto{\mathbb{C}}$ defines a functor from ${\rm\bf
Cat}$ to the full subcategory of ${\rm\bf Cat}/(\Delta^{\sf op})$, consisting
of opfibrations over $\Delta^{\sf op}$ with discrete fibers.

###### Notation 3.2

We often denote by $\pi:{\mathbb{C}}\to\Delta^{\sf op}$ the natural
projection. An object $c_{0}\to...\to c_{n}$ of ${\mathbb{C}}$ will be denoted
as $\mathbf{c}_{[n]}$ (so that $\pi(\mathbf{c}_{[n]})=[n]$) or simply as
$\mathbf{c}$ when its underlying $\Delta$-object is not important. Given two
objects $\mathbf{c}_{[n]}$, $\mathbf{c}^{\prime}_{[m]}$, and a map
$\alpha:c_{n}\to c^{\prime}_{0}$, we denote by
$\mathbf{c}_{[n]}*^{\alpha}\mathbf{c}^{\prime}_{[m]}$ the ’concatenated’
object

$c_{0}\to...\to
c_{n}\stackrel{{\scriptstyle\alpha}}{{\to}}c_{0}^{\prime}\to...\to
c_{n}^{\prime}.$

###### Lemma 3.3

There are functors $h_{\cal C}:{\mathbb{C}}\to{\cal C}$ and $t_{\cal
C}:{\mathbb{C}}\to{\cal C}^{\sf op}$ given by $\mathbf{c}_{[n]}\mapsto c_{0}$
or $\mathbf{c}_{[n]}\mapsto c_{n}$ respectively. $\square$

###### Definition 3.4

A map $\zeta:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$ is _anchor_ 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 than $n$.

A map $\zeta:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$ is _structural_ iff
its image under $t_{\cal C}$ is an identity and the underlying map in
$\Delta^{\sf op}$ preserves the endpoints: $\pi(\zeta)(m)=n$.

We denote by $A_{\mathbb{C}}$ and $S_{\mathbb{C}}$ the sets of all anchor and
structure maps respectively.

######  Proposition 3.5 (Factorisation and localisation properties)

For a small category ${\cal C}$,

  1. 1.

Every map $\mathbf{c}\to\mathbf{c}^{\prime}$ can be uniquely factored as an
anchor map $\mathbf{c}\to\mathbf{c}^{\prime\prime}$ followed by a structural
map $\mathbf{c}^{\prime\prime}\to\mathbf{c}$.

  2. 2.

Any functor $F:{\mathbb{C}}\to{\cal N}$ which sends anchor maps of
${\mathbb{C}}$ to isomorphisms factors essentially uniquely as
$F=\tilde{F}\circ h_{\cal C}$ for $\tilde{F}:{\cal C}\to{\cal N}$. In other
words, ${\cal C}$ is a localisation of ${\mathbb{C}}$ with respect to anchor
maps.

###### Proof.

The factorisation property is clear and is inherited from $\Delta$. For the
second statement, we first note that the functor $h_{\cal C}^{*}:{\cal
N}^{\cal C}\to{\cal N}^{\mathbb{C}}$ is full and faithful (see e.g. [23,
Section 4.4]). This implies that ${\cal C}$ is (equvalent to) a localisation
of ${\mathbb{C}}$ with respect to those maps which become isomorphic after
applying $h_{\cal C}$.

Let $F:{\mathbb{C}}\to{\cal N}$ be a functor which sends $A_{\mathbb{C}}$ to
isomorphisms. Define a new functor $\bar{F}:{\cal C}\to{\cal N}$. On objects,
$\bar{F}(c)=F(c)$ where $c$ is viewed as an object of ${\mathbb{C}}$ of zero
length. Take a span

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

the action of $F$ on it gives a span $F(c)\leftarrow F(c\to c^{\prime})\to
F(c^{\prime})$. Inverting the left arrow we get a map
$\bar{F}(f):\bar{F}(c)\to\bar{F}(c^{\prime})$. The action of $F$ on objects of
higher length, $c\to c^{\prime}\to c^{\prime\prime}$, and on degenerate
objects, $c\stackrel{{\scriptstyle id}}{{\to}}c$, then ensures that $\bar{F}$
is indeed a functor and $F=\bar{F}h_{\cal C}$. $\square$

###### Remark 3.6

To stress, the class of anchor maps is not saturated in the sense one applies
when one speaks of localisation [8]. Indeed, not every map which becomes an
isomorphism under $h_{\cal C}$ is an anchor map.

The proposition permits to justify the idea that, given a homotopical category
${\cal M}$ with weak equivalences ${\cal W}$, a functor
$F:{\mathbb{C}}\to{\cal M}$ sending $A_{\mathbb{C}}$ to ${\cal W}$ is a
suitable weakening of the concept of a functor from ${\cal C}$ to ${\cal M}$.
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 an anchor map, gives a span
$F(c)\stackrel{{\scriptstyle{\cal W}}}{{\leftarrow}}F(c\to
c^{\prime})\rightarrow F(c^{\prime})$, where the left map is a weak
equivalence. On the level of ${\rm Ho}\,{\cal M}$, 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’ $F$.

The spans of the form $X\stackrel{{\scriptstyle{\cal
W}}}{{\leftarrow}}Y\rightarrow Z$ have appeared before many times in the
context of localisation (for example, they are known under the name ’cocycles’
in [13]). For an arbitrary homotopical category ${\cal M}$, such spans may not
constitute a good presentation of morphisms in ${\rm Ho}\,{\cal M}$. In
practice one may need to make additional assumptions about ${\cal M}$. For our
purposes, homotopical $\Delta$-categories provide a sufficient answer.

Given two functors ${\cal D}\stackrel{{\scriptstyle F}}{{\to}}{\cal
C}\stackrel{{\scriptstyle G}}{{\leftarrow}}{\cal B}$, there is a useful
categorical notion called the _comma category_ $F/G$ [20]. Its objects are
triples $(d,b,\alpha:F(d)\to G(b))$ for $d\in{\cal D}$ and $b\in{\cal B}$. We
need the following adaptation of this notion:

###### Definition 3.7

Given two a diagram ${\cal D}\stackrel{{\scriptstyle F}}{{\to}}{\cal
C}\stackrel{{\scriptstyle G}}{{\leftarrow}}{\cal B}$, the associated
_simplicial comma object_ $\mathbb{F}/\\!\\!/{\mathbb{G}}$ is defined as the
_opposite_ of the category $\int F/\\!\\!/G$, where $F/\\!\\!/G:\Delta^{\sf
op}\times\Delta^{\sf op}\to{\rm\bf Set}$ is the bisimplicial set

$F/\\!\\!/G([n],[m])=\\{\mathbf{d}_{[n]},\mathbf{b}_{[m]},\alpha:F(d_{n})\to
G(b_{0})\\}$

viewed as a contravariant functor to ${\rm\bf Cat}$. In other words, it is the
category with objects given by triples
$(\mathbf{d}_{[n]},\mathbf{b}_{[m]},\alpha:F(d_{n})\to G(b_{0}))$ and a map

$(\mathbf{d}_{[n]},\mathbf{b}_{[m]},\alpha:F(d_{n})\to
G(b_{0}))\to(\mathbf{d}^{\prime}_{[k]},\mathbf{b}^{\prime}_{[l]},\beta:F(d^{\prime}_{k})\to
G(b^{\prime}_{0}))$

consists of two maps $\mathbf{d}_{[n]}\to\mathbf{d}^{\prime}_{[k]}$,
$\mathbf{b}_{[m]}\to\mathbf{b}^{\prime}_{[l]}$ such that the induced square
commutes:

We often write ${\mathbb{D}}/\\!\\!/{\mathbb{G}}$ or
$\mathbb{F}/\\!\\!/{\mathbb{C}}$ instead of $\mathbb{F}/\\!\\!/{\mathbb{G}}$
if $F$ or $G$ is the identity functor. Given an object $c\in{\cal C}$, we also
consider $\mathbb{F}/\\!\\!/c$ where we treat $c$ as a functor $[0]\to{\cal
C}$ and denote its simplicial replacement by the same letter. The canonical
functor $\mathbb{F}/\\!\\!/{\mathbb{G}}\to\Delta^{\sf op}\times\Delta^{\sf
op}$ is an opfibration with discrete fibers
$\mathbb{F}/\\!\\!/{\mathbb{G}}([n],[m])=F/\\!\\!/G([n],[m])$.

There is a _concatenation_ functor $con:\Delta\times\Delta\to\Delta$,
$([n],[m])\mapsto[n]*[m]=[n+m+1]$, and we think that $[n]$ is included as
first $n+1$ elements of $[n+m+1]$ and $[m]$ as last $[m+1]$ elements. The
action of $con$ on morphisms is then evident.

Then we observe the following. There is a diagram in ${\rm\bf Cat}$

$\begin{diagram}$ (3.1)

with the middle map denoted $pr_{\mathbb{F}/\\!\\!/{\mathbb{G}}}$, covering
the diagram

$\begin{diagram}$ (3.2)

with the middle map acting as $([n],[m])\mapsto[n]*[m]$. Moreover,

  * •

the left natural transformation
$pr_{\mathbb{F}/\\!\\!/{\mathbb{G}}}\to\mathbb{F}\circ pr_{\mathbb{D}}$ is
valued in anchor maps $A_{\mathbb{C}}$,

  * •

the right natural transformation
$pr_{\mathbb{F}/\\!\\!/{\mathbb{G}}}\to{\mathbb{G}}\circ pr_{\mathbb{B}}$ is
valued in structural maps $S_{\mathbb{C}}$,

  * •

$pr_{{\mathbb{B}}}$ is an opfibration whose classifying functor
${\mathbb{B}}\to{\rm\bf Cat}$ sends anchor maps to equivalences of categories.

All this is evident from Definition 3.7: $pr_{\mathbb{D}}$ maps
$(\mathbf{d}_{[n]},\mathbf{b}_{[m]},\alpha:F(d_{n})\to G(b_{0}))$ to
$\mathbf{d}_{[n]}$, $pr_{\mathbb{B}}$ maps it to $\mathbf{b}_{[m]}$, and
$pr_{\mathbb{F}/\\!\\!/{\mathbb{G}}}$ maps it to
$\mathbb{F}(\mathbf{d}_{[n]})*^{\alpha}{\mathbb{G}}(\mathbf{b}_{[m]})$.

The fact that $pr_{{\mathbb{B}}}$ sends anchor maps to equivalences and
Proposition 3.5 suggests that that $pr_{{\mathbb{B}}}$ can be obtained as a
pullback of an opfibration over ${\cal B}$ along the first element map
$h_{\cal B}:{\mathbb{B}}\to{\cal B}$. This opfibration $X\to{\cal B}$ consists
of the category $X$ whose objects are triples
$(\mathbf{d}_{[n]},b,\alpha:F(d_{n})\to G(b))$ and morphisms are given by
$\mathbf{d}_{[n]}\to\mathbf{d}^{\prime}_{[m]}$ in ${\mathbb{D}}$, $b\to
b^{\prime}$ in ${\cal B}$, such that the square commutes in ${\cal B}$, and
the functor $X\to{\cal B}$ is the projection
$(\mathbf{d}_{[n]},b,\alpha)\mapsto b$.

Let $I$ be a small category and denote by $\mathbb{I}$ its simplicial
replacement.

###### Definition 3.8

Let ${\cal M}$ be a homotopical $\Delta$-category. For $X:\mathbb{I}\to{\cal
M}$, its _realisation_ is defined as $|\Pi X|$, where $|-|$ is the geometric
realisation for ${\cal M}$ and $\Pi:Fun({\mathbb{I}},{\cal M})\to\Delta^{\sf
op}{\cal M}$ is the left Kan extension along the canonical projection
$\pi:\mathbb{I}\to\Delta^{\sf op}$.

For any object $i\in I$, there is naturally a map $X(i)\to|\Pi X|$.

###### Lemma 3.9

Let $I$ be a category with a terminal object $1$, and ${\cal M}$ be a
homotopical $\Delta$-category. Then any $X:\mathbb{I}\to{\cal M}$ sending the
anchor maps $A_{\mathbb{I}}$ to maps in ${\cal W}$, the natural map
$X(1)\to|\Pi X|$ is an equivalence.

###### Proof.

Consider an ’augmented’ functor $X^{aug}:\mathbf{i}\mapsto
X(\mathbf{i}*^{x}1)$ (here $x$ corresponds to the canonical map to the
terminal object $t_{I}(\mathbf{i})\to 1$). It is then easy to see that there
is a canonical equivalence $X^{aug}\to X$ coming from the maps
$X(\mathbf{i}*^{x}1)\to X(\mathbf{i})$. It then becomes an equivalence of
realisations. The object $\Pi X^{aug}$, however, can be completed to a split-
augmented simplicial object $\tilde{X}^{aug}:\Delta_{\infty}^{\sf op}\to{\cal
M}$ defined by the formula

$\tilde{X}^{aug}_{n}=\Pi X^{aug}_{n-1},\,\,\,n>0,$ $\tilde{X}^{aug}_{0}=X(1).$

in particular, one augmentation map
$X(1)\to\tilde{X}^{aug}_{1}=\coprod_{i}X(i\to 1)$ comes from the image
$X(1)\to X(1\to 1)$ of the degeneracy map $1\to(1\to 1)$ and the other map

$\tilde{X}^{aug}_{1}=\coprod_{i}X(i\to 1)\to X(1)$

is just the coproduct of the natural maps $X(i\to 1)\to X(1)$. By Proposition
1.26 we have the equivalences

$X(1)\to|\Pi X^{aug}|\to X(1)$

and we can see that the composite map $X(1)\to|\Pi X^{aug}|\to|\Pi X|$, which
is an equivalence, is equal to the map in question. $\square$

###### Lemma 3.10

Let $I$ be a category with contractible nerve and ${\cal M}$ be a homotopical
$\Delta$-category. If a functor $X:\mathbb{I}\to{\cal M}$ takes all morphisms
of $\mathbb{I}$ to isomorphisms, then the natural map $X(i)\to|\Pi X|$ is an
equivalence for any $i\in I$.

###### Proof.

Fix $i\in I$. Proposition 3.5 implies that the functor $X$ can be factored as
$\overline{X}\circ h_{I}$ with $\overline{X}:I\to{\cal M}$. $X$ moreover
factors through the fundamental groupoid of $I$, which is contractible. One
can then see that

$\Pi
X_{n}=\coprod_{\mathbf{i}_{[n]}}X(\mathbf{i}_{[n]})\cong\coprod_{\mathbf{i}_{[n]}}(\overline{X}\circ
h_{I})(\mathbf{i}_{[n]})\cong\coprod_{\mathbf{i}_{[n]}}X(i_{0})\cong\coprod_{\mathbf{i}_{[n]}}X(i),$

and so $|\Pi X|=NI\otimes X(i)$, which is equivalent to $X(i)$, and the map
$X(i)\to|\Pi X|$ in question is a homotopy inverse of the projection
$NI\otimes X(i)\to X(i)$. $\square$

###### Definition 3.11

For an opfibration ${\cal E}\to{\cal C}$, its _simplicial extension_ is a
_fibration_ $\mathbf{E}\to{\mathbb{C}}$ which is the pullback of a transpose
fibration ${\cal E}^{\top}\to{\cal C}^{\sf op}$ along $t_{\cal
C}:{\mathbb{C}}\to{\cal C}^{{\sf op}}$.

We stress that $\mathbf{E}$ is not a simplicial replacement of ${\cal E}$ or
${\cal E}^{\top}$. In particular, the fibre of $\mathbf{E}\to{\mathbb{C}}$
over an object $\mathbf{c}_{[n]}$ is equivalent to ${\cal E}(c_{n})$. If
${\cal E}\to{\cal C}$ comes from a functor ${\cal E}:{\cal C}\to{\rm\bf Cat}$,
then $\mathbf{E}\to{\mathbb{C}}$ corresponds to the functor viewed as a
contravariant functor on ${\mathbb{C}}$.

###### Remark 3.12

Given two functors $k_{1},k_{2}:K\to{\mathbb{C}}$ and a natural transformation
$\alpha:k_{1}\rightarrow k_{2}$ valued in structural maps $S_{\mathbb{C}}$, we
have that the induced Cartesian map of fibrations

$\alpha^{*}:k^{*}_{2}\mathbf{E}\to k^{*}_{1}\mathbf{E}$

is in fact an equivalence.

We can also pull back ${\cal E}\to{\cal C}$ to ${\mathbb{C}}$ by the means of
the functor $h_{\cal C}:{\mathbb{C}}\to{\cal C}$. The link between this
pullback and the fibration $\mathbf{E}\to{\mathbb{C}}$ is in the following:

###### Proposition 3.13

Given an opfibration $p:{\cal E}\to{\cal C}$, there is a morphism
$T:h^{*}_{\cal C}{\cal E}\to\mathbf{E}$ commuting with functors to
${\mathbb{C}}$ which sends opCartesian maps of $h^{*}_{\cal C}{\cal E}$ to
Cartesian maps of $\mathbf{E}$ and is universal, i.e. any other functor
$G:h^{*}_{\cal C}{\cal E}\to\mathbf{E}$ over ${\mathbb{C}}$ with such a
property factors through $T$ up to a natural isomorphism.

###### Proof.

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

  * •

An object of ${\cal X}$ 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 ${\cal E}$ which covers the
composition $c_{0}\to c_{n}$ in ${\cal C}$ (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 ${\cal X}\to{\mathbb{C}}$ is an
opfibration, and it is easy to see that the assignment
$(\mathbf{c},\alpha:x\to y)\mapsto(\mathbf{c},x)$ defines an equivalence over
${\mathbb{C}}$ of opfibrations ${\cal
X}\stackrel{{\scriptstyle\sim}}{{\to}}h_{\cal C}^{*}{\cal E}$.

On the other hand, consider the assignment $(\mathbf{c},\alpha:x\to
y)\mapsto(\mathbf{c},y)$. We claim that it defines a functor $\bar{T}:{\cal
X}\to\mathbf{E}$ 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 ${\cal
E}$:

$\begin{diagram}$ (3.3)

Suppose first that the map $t$ is fiberwise. Then by opCartesian property
there exists a map $t^{\prime}:y\to y^{\prime}$ rendering the diagram (3.3)
commutative. Remembering the description of arrows in Definition 2.13, 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
fiberwise map of ${\cal E}^{\top}$

Next, if $t$ is opCartesian, find an opCartesian map $k:y^{\prime}\to z$ in
${\cal E}$ 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 ${\cal E}\to{\cal C}$ 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 (fiberwise)
isomorphism $z\cong y$. This imples that the diagram (3.3) can be completed as
with all arrows opCartesian in ${\cal E}$. 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 ${\cal E}^{\top}$. Any other case of $(f,t)$ can be treated
by reducing to these two cases.

Inverting the equivalence ${\cal X}\stackrel{{\scriptstyle\sim}}{{\to}}h_{\cal
C}^{*}{\cal E}$ and composing with $\bar{T}$, we obtain the desired functor
$T:h_{\cal C}^{*}{\cal E}\to\mathbf{E}$, and one can verify its universal
property. $\square$

###### Lemma 3.14

Let ${\cal E}\to{\cal C}$ be a homotopical $\Delta$-opfibration. Then
$\mathbf{E}\to{\mathbb{C}}$ is a homotopical $\Delta$-fibration.

###### Proof.

Clear. $\square$

###  3.2 Homotopical category of derived sections

###### Definition 3.15

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

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

Recall the functors $h_{\cal C}$ and $T$ discussed before in Lemma 3.3 and
Proposition 3.13.

###### Proposition 3.16

The assignment $S\mapsto T\circ(h_{\cal C}^{*}S)$ defines a functor $i:{\sf
Sect}({\cal C},{\cal E})\to{\sf PSect}({\cal C},{\cal E})$. Its essential
image consists of the presections sending the anchor maps $A_{\mathbb{C}}$ to
Cartesian morphisms in $\mathbf{E}$.

###### Proof.

Note that for any anchor map $a:\mathbf{c}_{[n]}\to\mathbf{c}_{[k]}$ a map in
$h_{\cal C}^{*}{\cal E}$ is opCartesian over $a$ iff it is an isomorphism
$x\stackrel{{\scriptstyle\sim}}{{\to}}x$ in ${\cal E}(c_{0})$. On one hand,
the functor $T$ sends such maps to Cartesian maps in $\mathbf{E}$; on the
other hand, the pullback section $h^{*}_{\cal C}S:{\mathbb{C}}\to h^{*}_{\cal
C}{\cal E}$ sends anchor maps $A_{\mathbb{C}}$ precisely to identities in
${\cal E}$. Further details are then clear. $\square$

###### Remark 3.17

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 $S\in Sect({\cal
C},{\cal E})$ is sent by the functor above to $i(S)$ such that
$i(S)(\mathbf{c}_{[n]})\cong(f_{n}...f_{1})_{!}S(c_{0})$ where
$(f_{n}...f_{1})_{!}:{\cal E}(c_{0})\to{\cal
E}(c_{n})=\mathbf{E}(\mathbf{c}_{[n]})$ is a transition functor along the
composition of $f_{i}$.

Assume now that ${\cal E}\to{\cal C}$ has a homotopical structure ${\cal W}$.

###### Definition 3.18

The _standard homotopical structure_ on ${\sf PSect}({\cal C},{\cal E})$ is
defined by the subcategory of those morphisms $A\to A^{\prime}$ for which the
map $A(\mathbf{c}_{[n]})\to A^{\prime}(\mathbf{c}_{[n]})$ is in ${\cal W}$ for
each $\mathbf{c}_{[n]}\in{\mathbb{C}}$.

We henceforth assume this homotopical structure whenever dealing with ${\sf
PSect}({\cal C},{\cal E})$. We denote by ${\rm Ho}\,{\sf PSect}({\cal C},{\cal
E})$ the corresponding localisation.

###### Definition 3.19

A presection $A:{\mathbb{C}}\to\mathbf{E}$ is a _derived section_ iff $A$
sends anchor maps to weakly Cartesian morphisms in $\mathbf{E}$.

We denote by ${\sf\mathbb{R}Sect}({\cal C},{\cal E})$ the full subcategory of
${\sf PSect}({\cal C},{\cal E})$ spanned by derived sections. We restrict the
standard homotopical structure from ${\sf PSect}({\cal C},{\cal E})$ to
${\sf\mathbb{R}Sect}({\cal C},{\cal E})$ and denote by ${\rm
Ho}\,{\sf\mathbb{R}Sect}({\cal C},{\cal E})$ the corresponding localisation.

##  4 The pushforward functor

Given a functor $F:{\cal D}\to{\cal C}$, there is an induced pullback morphism

$\mathbb{F}^{*}:{\sf PSect}({\cal C},{\cal E})={\sf
Sect}({\mathbb{C}},\mathbf{E})\to{\sf Sect}({\mathbb{D}},F^{*}\mathbf{E})={\sf
PSect}({\cal D},F^{*}{\cal E})$

which restricts well to

$\mathbb{F}^{*}:{\sf\mathbb{R}Sect}({\cal C},{\cal
E})\to{\sf\mathbb{R}Sect}({\cal D},F^{*}{\cal E})$

and is moreover homotopical. It is natural to ask if such a functor may admit
a homotopy adjoint.

In this section, we provide a partial answer to this question, by constructing

$\mathbb{F}_{!}:{\sf PSect}({\cal D},F^{*}{\cal E})\to{\sf PSect}({\cal
C},{\cal E}),$

a homotopical functor in the other direction, together with natural spans
relating $\mathbb{F}_{!}\mathbb{F}^{*}$ and $\mathbb{F}^{*}\mathbb{F}_{!}$
with identity functors (Proposition 4.9). The functor $\mathbb{F}_{!}$ may be
viewed as an almost left adjoint to the pullback functor $\mathbb{F}^{*}$. If
$\mathbb{F}_{!}$ preserves derived sections, the spans mentioned above indeed
give well-defined maps $\mathbb{F}_{!}\mathbb{F}^{*}\to id$ and
$id\to\mathbb{F}^{*}\mathbb{F}_{!}$ on the level of homotopy categories of
derived sections, satisfying a triangle identity. Even under a weaker
assumption that the span relating $\mathbb{F}_{!}\mathbb{F}^{*}$ to the
identity functor consists of weak equivalences, Corollary 4.10 implies that
$\mathbb{F}^{*}$ is full and faithful on homotopy level.

###  4.1 Main construction

Fix a functor $F:{\cal D}\to{\cal C}$ and a homotopical $\Delta$-opfibration
${\cal E}\to{\cal C}$. Recall the diagram (3.1) for $G=id_{\cal C}$:

$\begin{diagram}$ (4.1)

The middle map is $pr_{\mathbb{F}/\\!\\!/{\mathbb{C}}}$. This diagram gives us
in particular the restriction morphism of Proposition 2.15

$R_{\mathbb{F}}:(\mathbb{F}pr_{\mathbb{D}})^{*}\mathbf{E}\to
pr^{*}_{\mathbb{F}/\\!\\!/{\mathbb{C}}}\mathbf{E}.$ (4.2)

This is a map of fibrations over $\mathbb{F}/\\!\\!/{\mathbb{C}}$.

Next, we observe there are equivalences

${\sf
Sect}(\mathbb{F}/\\!\\!/{\mathbb{C}},pr^{*}_{\mathbb{F}/\\!\\!/{\mathbb{C}}}\mathbf{E})\stackrel{{\scriptstyle\sim}}{{\leftarrow}}{\sf
Sect}(\mathbb{F}/\\!\\!/{\mathbb{C}},pr^{*}_{{\mathbb{C}}}\mathbf{E})\stackrel{{\scriptstyle\sim}}{{\to}}{\sf
Sect}({\mathbb{C}},\mathbf{E}^{\mathbb{F}/\\!\\!/{\mathbb{C}}})$ (4.3)

where the right equivalence is provided by the first assessment of Lemma 2.18
(keep in mind that $pr_{\mathbb{C}}$ is a small opfibration). The left map
comes from the equivalence

$pr^{*}_{{\mathbb{C}}}\mathbf{E}\stackrel{{\scriptstyle\sim}}{{\to}}pr^{*}_{\mathbb{F}/\\!\\!/{\mathbb{C}}}\mathbf{E}$

provided by Remark 3.12. We denote by

$D_{\mathbb{F}}:{\sf
Sect}(\mathbb{F}/\\!\\!/{\mathbb{C}},pr^{*}_{\mathbb{F}/\\!\\!/{\mathbb{C}}}\mathbf{E})\stackrel{{\scriptstyle\sim}}{{\to}}{\sf
Sect}({\mathbb{C}},\mathbf{E}^{\mathbb{F}/\\!\\!/{\mathbb{C}}})$ (4.4)

the resulting equivalence constructed from (4.3).

There is a natural ’projection’ functor $\Pi$ over ${\mathbb{C}}$, which acts
as $(\mathbf{d}_{[n]},\mathbf{c}_{[m]},\alpha:F(d_{n})\to
c_{0})\mapsto([n],\mathbf{c}_{[m]})$. Exponentiating and taking sections, we
obtain a functor $\Pi^{*}:{\sf Sect}({\mathbb{C}},\mathbf{E}^{\Delta^{\sf
op}})\to{\sf Sect}({\mathbb{C}},\mathbf{E}^{\mathbb{F}/\\!\\!/{\mathbb{C}}})$.

###### Proposition 4.1

The functor

$\Pi^{*}:{\sf Sect}({\mathbb{C}},\mathbf{E}^{\Delta^{\sf op}})\to{\sf
Sect}({\mathbb{C}},\mathbf{E}^{\mathbb{F}/\\!\\!/{\mathbb{C}}})$

admits a homotopical left adjoint

$\Pi_{!}:{\sf
Sect}({\mathbb{C}},\mathbf{E}^{\mathbb{F}/\\!\\!/{\mathbb{C}}})\to{\sf
Sect}({\mathbb{C}},\mathbf{E}^{\Delta^{\sf op}})$ (4.5)

###### Proof.

See Proposition 2.19 for the construction of $\Pi_{!}$. To observe that it is
homotopical, note that for each $\mathbf{c}$, the functor
$\mathbb{F}/\\!\\!/{\mathbb{C}}(\mathbf{c})\to\Delta^{\sf op}$ is a _discrete_
opfibration, and the pushforward along it amounts to taking coproducts, which
are homotopical. $\square$

Take a ${\cal D}$-presection $S:{\mathbb{D}}\to\mathbb{F}^{*}\mathbf{E}$. Then
apply functors (4.2), (4.4) and (4.5) to obtain

$B_{\bullet}(S):=\Pi_{!}D_{\mathbb{F}}(R_{\mathbb{F}}\circ
pr^{*}_{\mathbb{D}}S)\in Sect({\mathbb{C}},\mathbf{E}^{\Delta}).$ (4.6)

Lemma 3.14 implies that $\mathbf{E}\to{\mathbb{C}}$ is a homotopical
$\Delta$-fibration. Applying the realisation functor $|-|$ from Proposition
2.26, we get the following:

###### Definition 4.2

The _derived pushforward_ of a presection
$A:{\mathbb{D}}\to\mathbb{F}^{*}\mathbf{E}$ is defined as

$\mathbb{F}_{!}(S):=|B_{\bullet}(S)|=|\Pi_{!}D_{\mathbb{F}}(R_{\mathbb{F}}\circ
pr^{*}_{\mathbb{D}}S)|.$

this defines a homotopical functor $\mathbb{F}_{!}:{\sf PSect}({\cal D},{\cal
E})\to{\sf PSect}({\cal C},{\cal E})$.

###### Remark 4.3

Over an object $\mathbf{c}_{[m]}=c_{0}\stackrel{{\scriptstyle
f_{1}}}{{\to}}...\stackrel{{\scriptstyle f_{m}}}{{\to}}c_{m}$, we have

$B_{n}(S)(\mathbf{c}_{[m]})=\coprod_{\mathbf{d}_{[n]},\alpha:F(d_{n})\to
c_{0}}(f_{m}...f_{1}\alpha)_{!}S(\mathbf{d}_{[n]})$

where $(f_{m}...f_{1}\alpha)_{!}$ is the transition functor ${\cal
E}(F(d_{n}))\to{\cal E}(c_{m})$. This expression is very similar to the bar
construction (cf. [5, 23]); the value $\mathbb{F}_{!}S(\mathbf{c}_{[m]})$ is
just the realisation of this simplicial object $B_{n}(S)(\mathbf{c}_{[m]})$.

###  4.2 Unit and counit correspondences

Given a ${\cal C}$-presection $A:{\mathbb{C}}\to\mathbf{E}$, use
$pr_{\mathbb{F}/\\!\\!/{\mathbb{C}}}$ from the diagram (4.1) and functors
(4.2), (4.4) and (4.5) to obtain

$B^{\mathbb{F}}_{\bullet}(A):=\Pi_{!}D_{\mathbb{F}}(pr^{*}_{\mathbb{F}/\\!\\!/{\mathbb{C}}}A)\in
Sect({\mathbb{C}},\mathbf{E}^{\Delta}).$ (4.7)

Denote by $A^{\mathbb{F}}$ the realisation of $B_{\bullet}^{\mathbb{F}}(A)$.

###### Remark 4.4

Again, one can see that explicitly

$B^{\mathbb{F}}_{n}(A)(\mathbf{c}_{[m]})=\coprod_{\mathbf{d}_{[n]},\alpha:F(d_{n})\to
c_{0}}A(\mathbb{F}(\mathbf{d}_{[n]})*^{\alpha}\mathbf{c}_{[m]}).$

###### Lemma 4.5

There is a natural (in $A$) correspondence in ${\sf PSect}({\cal C},{\cal E})$

$\mathbb{F}_{!}\mathbb{F}^{*}A\leftarrow A^{\mathbb{F}}\rightarrow A$

coming from the realisation of the correspondence of simplicial presections

$B_{\bullet}(\mathbb{F}^{*}A)\leftarrow B_{\bullet}^{\mathbb{F}}(A)\rightarrow
A$

where the rightmost term is a constant simplicial object. When $A$ is a
derived section, the left morphsms in the correspondences above are weak
equivalences.

###### Proof.

First, the construction. Given a ${\cal C}$-presection
$A:{\mathbb{C}}\to\mathbf{E}$, Proposition 2.15 and the left triangle of the
diagram (4.1) gives us a map of
$pr_{\mathbb{F}/\\!\\!/{\mathbb{C}}}^{*}\mathbb{F}^{*}\mathbf{E}$-sections
over $\mathbb{F}/\\!\\!/{\mathbb{C}}$

$pr^{*}_{\mathbb{F}/\\!\\!/{\mathbb{C}}}A\to
R_{\mathbb{F}}(\mathbb{F}pr_{\mathbb{D}})^{*}A$

which is an equivalence when $A$ is a derived section. Indeed, over an object
$(\mathbf{d},\mathbf{c},\alpha)$ of $\mathbb{F}/\\!\\!/{\mathbb{C}}$ the map
looks like

$A(\mathbb{F}(\mathbf{d})*^{\alpha}\mathbf{c})\to(f_{n}...f_{1}\alpha)_{!}A(\mathbb{F}(\mathbf{d}))$

with $\mathbf{c}=c_{0}\stackrel{{\scriptstyle
f_{1}}}{{\to}}...\stackrel{{\scriptstyle f_{n}}}{{\to}}c_{n}$, and this map is
an equivalence precisely because of the derived section condition for $A$.
Applying the equivalence $D_{\mathbb{F}}$ of (4.4) and then $\Pi_{!}$ of
(4.5), we get the map

$B_{\bullet}^{\mathbb{F}}(A)\to B_{\bullet}(\mathbb{F}^{*}A)$

between (4.6) and (4.7) which is again a weak equivalence when $A$ is a
derived section.

Proposition 2.15 and the right triangle of the diagram (4.1) give us a map

$pr^{*}_{\mathbb{F}/\\!\\!/{\mathbb{C}}}A\to pr^{*}_{{\mathbb{C}}}A$

and we again apply $\Pi_{!}D_{\mathbb{F}}$. Observe that
$\Pi_{!}D_{\mathbb{F}}pr^{*}_{\mathbb{C}}A$ is the following simplicial
presection:

$(\Pi_{!}D_{\mathbb{F}}pr^{*}_{\mathbb{C}}A)_{n}(\mathbf{c})=\coprod_{\mathbf{d}_{[n]},\alpha:F(d_{n})\to
c_{0}}A(\mathbf{c})\cong N(F/c_{0})(n)\otimes A(\mathbf{c}).$

There is thus a natural map $\Pi_{!}D_{\mathbb{F}}pr^{*}_{\mathbb{C}}A\to A$
to the constant simplicial presection $A$.

The realisation of $\Pi_{!}D_{\mathbb{F}}pr^{*}_{\mathbb{C}}A$ is the
presection given by the assignment $\mathbf{c}\mapsto N(F/c_{0})\otimes
A(\mathbf{c})$. On this level as well, we get the map

$A^{\mathbb{F}}\to N(F/h_{\cal C}(-))\otimes A\to A$

which completes the construction. $\square$

###### Lemma 4.6

Let $A:{\mathbb{C}}\to\mathbf{E}$ be a derived section and
$s:\mathbf{c}_{[n]}\to\mathbf{c}^{\prime}_{[m]}$ be such a map in
${\mathbb{C}}$ that its underlying map $s:[m]\to[n]$ in $\Delta$ is the
surjective left inverse of the inclusion $i:[n]\to[m]$ of $[n]$ as last $n+1$
objects of $[m]$. Then $A(s)$ is weakly Cartesian in $\mathbf{E}$.

###### Proof.

Clear. $\square$

###### Lemma 4.7

For $F=id_{\cal C}$ and a derived section $A:{\mathbb{C}}\to\mathbf{E}$ both
morphisms in the span

$id_{{\mathbb{C}}!}id_{\mathbb{C}}^{*}A\leftarrow
A^{id_{\mathbb{C}}}\rightarrow A$

of Lemma 4.5 are weak equivalences.

###### Proof.

Fix $\mathbf{c}\in{\mathbb{C}}$. In the case of the identity functor, we see
that $B_{\bullet}^{id_{\mathbb{C}}}A(\mathbf{c})$ can calculated as the
realisation (cf. Definition 3.8) of the functor
$X:{\mathbb{C}}/\\!\\!/c_{0}\to\mathbf{E}(\mathbf{c})$ defined by the
assignment

$X((\mathbf{c}^{\prime}_{[k]},\alpha:c^{\prime}_{k}\to
c_{0}))=A(\mathbf{c}^{\prime}_{[k]}*^{\alpha}\mathbf{c}).$

The category ${\mathbb{C}}/\\!\\!/c_{0}$ is the simplicial replacement of the
category ${\cal C}/c_{0}$, and the latter has a terminal object. By Lemma 3.9,
the natural map
$A(c_{0}*^{id_{c_{0}}}\mathbf{c})=X(c_{0})\to|\Pi_{!}X|=A^{id_{\mathbb{C}}}(\mathbf{c})$
is an equivalence.

There is also an equivalence $A(\mathbf{c})\to
A(c_{0}*^{id_{c_{0}}}\mathbf{c})$ which comes from the degeneracy
$\mathbf{c}\to c_{0}*^{id_{c_{0}}}\mathbf{c}$ (cf. Lemma 4.6). One can then
see that the composition

$A(\mathbf{c})\to A(c_{0}*^{id_{c_{0}}}\mathbf{c})\to
A^{id_{\mathbb{C}}}(\mathbf{c})\to A(\mathbf{c})$

is the identity (it is such already on the level of corresponding simplicial
objects; also note that the composition $\mathbf{c}\to
c_{0}*^{id_{c_{0}}}\mathbf{c}\to\mathbf{c}$ in ${\mathbb{C}}$ is the identity
$id_{\mathbf{c}}$). Thus the $\mathbf{c}$-th component of the map
$A^{id_{\mathbb{C}}}\to A$ is an equivalence as a right inverse of an
equivalence $A(\mathbf{c})\to A(c_{0}*^{id_{c_{0}}}\mathbf{c})\to
A^{id_{\mathbb{C}}}(\mathbf{c})$. $\square$

###### Lemma 4.8

For a functor $F:{\cal D}\to{\cal C}$ and a ${\mathbb{D}}$-presection $A$,
there is a natural (in $A$) morphism

###### Proof.

By definition, $id_{{\mathbb{D}}!}id_{\mathbb{D}}^{*}A$ is the realisation of
the simplicially valued presection $X$ which at $d_{0}\stackrel{{\scriptstyle
g_{1}}}{{\to}}...\stackrel{{\scriptstyle g_{m}}}{{\to}}d_{m}$ takes the value

$[n]\mapsto X_{n}=\underset{\alpha:d^{\prime}_{n}\to
d_{0}}{\coprod_{d^{\prime}_{0}\to...\to
d^{\prime}_{n}}}(F(g_{m}...g_{1}\alpha))_{!}A(\mathbf{d}^{\prime}_{[n]}).$

In the case when we calculate $\mathbb{F}^{*}\mathbb{F}_{!}A$ at
$d_{0}\stackrel{{\scriptstyle g_{1}}}{{\to}}...\stackrel{{\scriptstyle
g_{m}}}{{\to}}d_{m}$, we have the following simplicial object $Y$:

$[n]\mapsto Y_{n}=\underset{\beta:F(d^{\prime}_{n})\to
F(d_{0})}{\coprod_{d^{\prime}_{0}\to...\to
d^{\prime}_{n}}}(F(g_{m}...g_{1})\beta)_{!}A(\mathbf{d}^{\prime}_{[n]}).$

The assignment of $\alpha:d^{\prime}_{n}\to d_{0}$ to
$F\alpha:F(d^{\prime}_{n})\to F(d_{0})$ induces the map of sets

$\\{d^{\prime}_{0}\to...\to d^{\prime}_{n},\alpha:d^{\prime}_{n}\to
d_{0}\\}\to\\{d^{\prime}_{0}\to...\to
d^{\prime}_{n},\beta:F(d^{\prime}_{n})\to F(d_{0})\\}$ (4.8)

and we obtain a map $X_{n}\to Y_{n}$ as $X_{n}$ and $Y_{n}$ are the coproducts
indexed by the sets in (4.8). Varying $[n]\in\Delta$, we assemble a map $X\to
Y$ of simplicial objects, which after realsiations gives the map in question,
$id_{{\mathbb{D}}!}id_{\mathbb{D}}^{*}A\longrightarrow\mathbb{F}^{*}\mathbb{F}_{!}A$.
$\square$

We finally prove the main proposition of this section:

###### Proposition 4.9

Let $F:{\cal D}\to{\cal C}$, $A\in{\sf\mathbb{R}Sect}({\cal C},{\cal E})$ and
$R\in{\sf\mathbb{R}Sect}({\cal D},F^{*}{\cal E})$.

  1. 1.

There is a natural (in $A$) span of presections

$\mathbb{F}_{!}\mathbb{F}^{*}A\leftarrow A^{\mathbb{F}}\rightarrow A$ (4.9)

which induces a natural transformation
$\epsilon:\mathbb{F}_{!}\mathbb{F}^{*}\to id$ of functors ${\rm
Ho}\,{\sf\mathbb{R}Sect}({\cal C},{\cal E})\to{\rm Ho}\,{\sf PSect}({\cal
C},{\cal E})$ (where $id$ is the inclusion functor).

  2. 2.

There is a natural (in $R$) sequence of morphisms

$R\leftarrow R^{id_{\mathbb{D}}}\rightarrow
id_{{\mathbb{D}}!}id^{*}_{{\mathbb{D}}}R\rightarrow\mathbb{F}^{*}\mathbb{F}_{!}R$
(4.10)

which induces a natural transformation
$\eta:id\to\mathbb{F}^{*}\mathbb{F}_{!}$ of functors ${\rm
Ho}\,{\sf\mathbb{R}Sect}({\cal D},F^{*}{\cal E})\to{\rm Ho}\,{\sf PSect}({\cal
D},F^{*}{\cal E})$.

  3. 3.

_(Triangle identity)_ For each $A\in{\rm Ho}\,{\sf\mathbb{R}Sect}({\cal
C},{\cal E})$, the composition in ${\rm Ho}\,{\sf PSect}({\cal D},F^{*}{\cal
E})$

$\begin{diagram}$ (4.11)

is the identity.

###### Proof.

We proved the first two claims in the preceding lemmas. Only the triangle
identity remains. Using the correspondences obtained before, we write a string
of morphisms

$\mathbb{F}^{*}A\stackrel{{\scriptstyle\sim}}{{\leftarrow}}(\mathbb{F}^{*}A)^{id_{\mathbb{D}}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}id_{{\mathbb{D}}!}id^{*}_{{\mathbb{D}}}\mathbb{F}^{*}A\rightarrow\mathbb{F}^{*}\mathbb{F}_{!}\mathbb{F}^{*}A\stackrel{{\scriptstyle\sim}}{{\leftarrow}}\mathbb{F}^{*}(A^{\mathbb{F}})\rightarrow\mathbb{F}^{*}A$

with all the weak equivalences drawn as
$\stackrel{{\scriptstyle\sim}}{{\rightarrow}}$ or
$\stackrel{{\scriptstyle\sim}}{{\leftarrow}}$. We can redraw this sequence,
obtaining the (potentially non-commutative) diagram The third claim is then
equivalent to the commutativity of this diagram. We proceed as follows:
writing down in components the simplicial object used to obtain
$(\mathbb{F}^{*}A)^{id_{\mathbb{D}}}$, we see

$(\mathbb{F}^{*}A)^{id_{\mathbb{D}}}\longleftrightarrow
B^{id_{\mathbb{D}}}_{n}(\mathbb{F}^{*}A)(\mathbf{d}_{[m]})=\coprod_{\mathbf{d}^{\prime}_{[n]},\alpha:d^{\prime}_{n}\to
d_{0}}A(\mathbb{F}(\mathbf{d}^{\prime}_{[n]}*^{\alpha}\mathbf{d}_{[m]})).$

In the same way,

$\mathbb{F}^{*}(A^{\mathbb{F}})\longleftrightarrow(\mathbb{F}^{*}B^{\mathbb{F}}_{n}(A))(\mathbf{d}_{[m]})=\coprod_{\mathbf{d}^{\prime}_{[n]},\beta:F(d^{\prime}_{n})\to
F(d_{0})}A(\mathbb{F}(\mathbf{d}^{\prime}_{[n]})*^{\beta}\mathbb{F}(\mathbf{d}_{[m]})).$

Assigning $\alpha\mapsto F(\alpha)$, we see that there is a natural in $A$ map
$(\mathbb{F}^{*}A)^{id_{\mathbb{D}}}\to\mathbb{F}^{*}(A^{\mathbb{F}})$.
Moreover, a comparison with the construction of Lemma 4.8 reveals that in the
resulting diagram both the left-hand triangle and the right-hand square
commute. $\square$

###### Corollary 4.10

Assume that for a functor $F:{\cal D}\to{\cal C}$, both maps in the span (4.9)
are weak equivalences. Then $\mathbb{F}^{*}:{\rm
Ho}\,{\sf\mathbb{R}Sect}({\cal C},{\cal E})\to{\rm
Ho}\,{\sf\mathbb{R}Sect}({\cal D},F^{*}{\cal E})$ is full and faithful.

###### Proof.

This result can be proven as a particular case of the following categorical
result:

Let $f:{\cal M}\rightleftarrows{\cal N}:u$ be two functors, $i:{\cal
N}_{0}\subset{\cal N}$ and $j:{\cal M}_{0}\subset{\cal M}$ are full
subcategories such that $ui$ is contained in ${\cal M}_{0}$. In other words,
there is a functor $u_{0}:{\cal N}_{0}\to{\cal M}_{0}$ with $ui=ju_{0}$.
Suppose furthermore that there are natural transformations
$\epsilon:fui\stackrel{{\scriptstyle\sim}}{{\to}}i$ and $\eta:j\to ufj$
defined over ${\cal N}_{0}$ and ${\cal M}_{0}$ respectively such that the
triangle identity is satisfied: $ui=ju_{0}\to ufju_{0}=ufui\to ui$ is the
identity. Then $u_{0}$, or equivalently $ui$, is full and faithful.

In turn, the categorical result is proven as follows. The functoriality of
$u_{0}$ supplies us with maps $u(x,y):{\cal N}_{0}(x,y)\to{\cal
M}_{0}(u_{0}x,u_{0}y)={\cal M}(uix,uiy)$. Given a map $\alpha:ux\to uy$, we
define $v(x,y)\alpha$ to be the map fitting in the commutative square (here we
use that $i$ is a full and faithful inclusion). This defines the map
$v(x,y):{\cal M}(uix,uiy)\to{\cal N}_{0}(x,y)$ which is inverse to $u(x,y)$.
$\square$

Note in particular that in the situation like above, for
$A\in{\sf\mathbb{R}Sect}({\cal C},{\cal E})$, $\mathbb{F}_{!}\mathbb{F}^{*}A$
is again a derived section.

##  5 Case of a resolution

In this section, we study the functors of $\mathbb{F}^{*}$ and
$\mathbb{F}_{!}$ for a particular class of functors $F$, which we call
resolutions:

###### Definition 5.1

A functor $F:{\cal D}\to{\cal C}$ is a _resolution_ if it is an opfibration
and each fiber ${\cal D}(c)$ is contractible (that is, its nerve $N{\cal
D}(c)$ is contractible).

###### Remark 5.2

Resolutions $F:{\cal D}\to{\cal C}$ should properly be called partial
resolutions since we will not make a claim about the smoothness of any
category of derived sections defined for ${\cal D}$, in any possible sense
(for what it may mean in the setting of enhanced triangulated or dg-
categories, the reader is invited to consult [26]).

Denote by ${\mathbb{D}}(c)$ the simplicial replacement of ${\cal D}(c)$.

###### Definition 5.3

Let $F:{\cal D}\to{\cal C}$ be a resolution. A presection
$A:{\mathbb{D}}\to\mathbb{F}^{*}\mathbf{E}$ is _locally constant_ if for any
fibre ${\cal D}(c)$ over $c\in{\cal C}$, the composite functor

${\mathbb{D}}(c)\to{\mathbb{D}}\stackrel{{\scriptstyle
A}}{{\to}}\mathbf{E}(c)={\cal E}(c)$

sends all morphisms of the domain to weak equivalences. A derived presection
is locally constant if it is locally constant as a presection.

We denote by ${\sf PSect}({\cal D},{\cal E})_{lc}$ and
${\sf\mathbb{R}Sect}({\cal D},{\cal E})_{lc}$ the corresponding full
homotopical subcategories of locally constant (pre)sections. It is clear that
any (pre)section of the form $\mathbb{F}^{*}A$ is locally constant.

When $F$ is a resolution, we can prove two general results concerning
$\mathbb{F}^{*}$. Here is the first result:

###### Theorem 5.4

Let ${\cal E}\to{\cal C}$ be a homotopical $\Delta$-opfibration and $F:{\cal
D}\to{\cal C}$ be a resolution (Definition 5.1). Then after passing to
localisations, the pullback functor $\mathbb{F}^{*}:{\rm
Ho}\,{\sf\mathbb{R}Sect}({\cal C},{\cal E})\to{\rm
Ho}\,{\sf\mathbb{R}Sect}({\cal D},{\cal E})$ is full and faithful.

The proof of this theorem relies on the machinery of pushforwards considered
in the previous chapter leading to Corollary 4.10 and the additional
manipulations are similar in the spirit to the proof of Cofinality Theorem in
[5]. Indeed, we shall prove that $\mathbb{F}_{!}S(\mathbf{c}_{[m]})$ which is
calculated as the realisation of

$[n]\mapsto
B_{n}(S)(\mathbf{c}_{[m]})=\coprod_{\mathbf{d}_{[n]}\subset{\mathbb{D}},\alpha:F(d_{n})\to
c_{0}}(f_{m}...f_{1}\alpha)_{!}S(\mathbf{d}_{[n]})$

can be also calculated, up to a coherent zigzag of equivalences, as the
realisation of

$[n]\mapsto\coprod_{\mathbf{d}_{[n]}\subset{\mathbb{D}}(c_{0})}(f_{m}...f_{1})_{!}S(\mathbf{d}_{[n]}).$

When $S=\mathbb{F}^{*}A$ for a derived section $A:{\mathbb{C}}\to\mathbf{E}$,
the second realisation is seen to be equivalent to $A(\mathbf{c}_{[m]})$. The
difficulty of the proof comes mostly from the complexity of objects involved,
and the necessity to make sure that the aforementioned zigzag of equivalences
is (equivalent, for a fixed $\mathbf{c}$, to) the counit correspondence (4.9)
of Proposition 4.9.

We can also characterise the homotopical essential image of $\mathbb{F}^{*}$.
Unfortunately, we only know how to do it for $F$-special (cf. Definition 5.15)
homotopical $\Delta$-opfibrations:

###### Theorem 5.5

Let $F:{\cal D}\to{\cal C}$ be a resolution and ${\cal E}\to{\cal C}$ be a
$F$-special homotopical $\Delta$-opfibration. Then the functor
$\mathbb{F}^{*}:{\rm Ho}\,{\sf\mathbb{R}Sect}({\cal C},{\cal E})\to{\rm
Ho}\,{\sf\mathbb{R}Sect}({\cal D},{\cal E})_{lc}$ is an equivalence.

However, while Definition 5.15 is fairly technical, the condition of
speciality is satisfied when, for example, each fiber of the fibration
$\mathbf{E}\to{\mathbb{C}}$ is a model category, and taking a realisation of
any simplicial object $X:\Delta^{\sf op}\to{\cal E}(c)$ amounts to calculating
its homotopy colimit (see [6] for the discussion of locally constant functors
in this setting). This includes examples like ${\rm\bf DVect}_{k}$ or any
other opfibrations which describe families of model categories with a
reasonable notion of geometric realisation leading up to a $\Delta$-structure.

###  5.1 Fullness and faithfullness

The main result of this section which we use to prove Theorem 5.4 is the
following proposition:

###### Proposition 5.6

Let $F:{\cal D}\to{\cal C}$ be a resolution. Then for any homotopical
$\Delta$-opfibration ${\cal E}\to{\cal C}$, the counit transformation

$\epsilon:\mathbb{F}_{!}\mathbb{F}^{*}A\to id_{{\rm Ho}\,{\sf PSect}({\cal
C},{\cal E})}A$

is an isomorphism in ${\rm Ho}\,{\sf PSect}({\cal C},{\cal E})$ for any
derived section $A$.

The proof will be carried out in several steps. Note that for an opfibration
$F:{\cal D}\to{\cal C}$ and an object $c\in{\cal C}$, we can take two
categories $F/c$, the comma category of $F$ and $c$ (viewed as a functor
$[0]\to{\cal C}$), and ${\cal D}(c)$, the fiber of $F$ at $c$. There is a
functor which sends $d\in{\cal D}(c)$ to
$(d,id_{c}:F(d)\stackrel{{\scriptstyle=}}{{\to}}c)\in F/c$ and it has a left
adjoint given by choosing, for each object $(d,f:F(d)\to c)\in F/c$, an
opCartesian morphism $d\to f_{!}d$ covering $f$. A similar pattern occurs a
few times in this section, and this motivates us to introduce the following
technical notion:

###### Definition 5.7

For a category ${\cal D}$, a functor $F:{\cal D}\to{\cal C}$ and an object
$c\in{\cal C}$, a _$(F,c)$ -transition structure_ consists of

  1. 1.

two categories $I,J$ and functors $\mathscr{I}:I\to{\cal D}$,
$\mathscr{J}:J\to{\cal D}$,

  2. 2.

a functor ${\mathrm{R}}:J\to I$ in ${\rm\bf Cat}/{\cal D}$.

These data are subject to the following conditions:

  * •

R admits a left adjoint ${\mathrm{L}}$ in ${\rm\bf Cat}$,

  * •

$\mathscr{J}$ maps $J$ to the fiber ${\cal D}(c)$, so that $F\mathscr{J}$
factors through $c$.

In the notation of this definition, we sometimes write
$({\mathscr{I}},{\mathscr{J}},{\textrm{R}})$ to denote a given
$(F,c)$-transition structure.

###### Example 5.8

The transition structures of importance for us are the following:

  1. 1.

For an opfibration $F:{\cal D}\to{\cal C}$ and an object $c\in{\cal C}$, there
is a $(F,c)$-transition structure given by $I=F/c$ and $J={\cal D}(c)$
outlined just before Definition 5.7.

  2. 2.

If $F:{\cal D}\to{\cal C}$ is an opfibration and $d\in{\cal D}$, one can have
the following $(F,F(d))$-transition structure: $I={\cal D}/d$ and $J={\cal
D}(F(d))/d$. The right adjoint $\mathrm{R}$ is the evident inclusion; the left
adjoint $\mathrm{L}$ is given by factoring any morphism $d^{\prime}\to d$ as
’opCartesian followed by fiberwise’ pair of morphisms.

  3. 3.

Any $(F,c)$ structure $(\mathscr{I},\mathscr{J},{\textrm{R}})$ induces a
$(F\circ\mathscr{I},c)$ structure $(id_{I},{\textrm{R}},{\textrm{R}})$ with
the same right adjoint R. Thus the first example gives us a
$(F_{c},c)$-structure where $F_{c}:{\cal D}/c\to{\cal C}$ is the functor
$(d,f:Fd\to c)\mapsto Fd$. For this structure, $I={\cal D}/c$, $J={\cal D}(c)$
and R acts in the same way as before.

###### Remark 5.9

Consider the unit map $\eta(i):i\to{\textrm{R}}{\textrm{L}}i$ for any $i\in
I$. Apply $F\circ\mathscr{I}$ to this map and obtain
$\bar{\eta}(i):F\mathscr{I}(i)\to c$. For any opfibration ${\cal E}\to{\cal
C}$ we then have a well-defined ’restriction’ functor

$R_{c}:\mathscr{I}^{*}\mathbb{F}^{*}\mathbf{E}\to\mathbf{E}(c)={\cal E}(c)$

where we denote by the same letter $\mathscr{I}$ the induced functor
${\mathbb{I}}\to{\mathbb{D}}$. Concretely, this functor sends
$(\mathbf{i}_{[n]},x\in{\cal E}(F\mathscr{I}(i_{n})))$ to
$\bar{\eta}(i)_{!}x$, using a (chosen) opCartesian lift
$x\to\bar{\eta}(i)_{!}x$ in ${\cal E}$ covering
$\bar{\eta}(i):F\mathscr{I}(i_{n})\to c$.

###### Construction 5.10

Assume given $c_{0}\to...\to c_{n}=\mathbf{c}\in{\mathbb{C}}$. Denote by
$\mathbf{c}_{!}$ the natural transition functor

$\mathbf{c}_{!}:\mathbf{E}(c_{0})\cong{\cal E}(c_{0})\to{\cal
E}(c_{n})\cong\mathbf{E}(c_{n}).$

Consider also the simplicial comma object (Definition 3.7)
${\mathbb{I}}/\\!\\!/{\mathbb{R}}$, where
${\mathbb{R}}:{\mathbb{J}}\to{\mathbb{I}}$ is the simplicial replacement of R.
Using the diagram (3.1) and postcomposing with functors to ${\mathbb{D}}$ we
obtain a new diagram

$\begin{diagram}$ (5.1)

and we henceforth denote the middle map again by
$pr_{{\mathbb{I}}/\\!\\!/{\mathbb{R}}}$.

For $B\in{\sf PSect}({\cal D},{\cal E})$ and a given $(F,c_{0})$-structure, we
get the following diagram with the middle map
$\mathbf{c}_{!}pr^{*}_{{\mathbb{I}}/\\!\\!/{\mathbb{R}}}B$. Thus we have the
span

$pr^{*}_{{\mathbb{I}}}\mathbf{c}_{!}R_{c_{0}}\mathscr{I}^{*}B\longleftarrow\mathbf{c}_{!}pr^{*}_{{\mathbb{I}}/\\!\\!/{\mathbb{R}}}B\longrightarrow
pr^{*}_{{\mathbb{J}}}\mathbf{c}_{!}\mathscr{J}^{*}B.$

Pushing this forward to the span given by projections,

$\Delta^{\sf op}\stackrel{{\scriptstyle\pi_{1}}}{{\longleftarrow}}\Delta^{{\sf
op}}\times\Delta^{{\sf
op}}\stackrel{{\scriptstyle\pi_{2}}}{{\longrightarrow}}\Delta^{\sf op},$

we obtain a span of bisimplicial objects in $\mathbf{E}(c)$:

$\pi^{*}_{1}\Pi(\mathbf{c}_{!}R_{c_{0}}\mathscr{I}^{*}B)\longleftarrow\Pi\mathbf{c}_{!}pr^{*}_{{\mathbb{I}}/\\!\\!/{\mathbb{R}}}B\longrightarrow\pi^{*}_{2}\Pi(\mathbf{c}_{!}\mathscr{J}^{*}B).$
(5.2)

Here $\Pi$ is the pushforward functor, simplicial (along
${\mathbb{I}}\to\Delta^{\sf op}$ and same for ${\mathbb{J}}$) or bisimplicial
(along ${\mathbb{I}}/\\!\\!/{\mathbb{R}}\to\Delta^{\sf op}\times\Delta^{\sf
op}$). We implicitly used the Beck-Chevalley morphisms, such as $\Pi
pr^{*}_{{\mathbb{I}}}\to\pi_{1}^{*}\Pi$, for pullbacks and pushforwards; they
arise from commutative squares like

by taking associated pullback functors on functor categories and then
replacing some of them by left adjoints.

###### Remark 5.11

Let us write the terms of the span (5.2) explicitly. For
$\mathbf{c}=c_{0}\stackrel{{\scriptstyle
f_{1}}}{{\to}}...\stackrel{{\scriptstyle f_{n}}}{{\to}}c_{n}$, we find that

$\Pi(\mathbf{c}_{!}R_{c_{0}}\mathscr{I}^{*}B)_{m}=\coprod_{\mathbf{i}_{[m]}}(f_{n}...f_{1}\bar{\eta}(i_{m}))_{!}B(\mathscr{I}\mathbf{i}_{[m]}),$

where $\bar{\eta}(i_{m})$ is induced from the unit of
${\textrm{L}}\dashv{\textrm{R}}$ (Remark 5.9). Next,

$\Pi(\mathbf{c}_{!}\mathscr{J}^{*}B)_{l}=\coprod_{\mathbf{j}_{[l]}}(f_{n}...f_{1})_{!}B(\mathscr{J}\mathbf{j}_{[l]}),$

and, finally,

$\displaystyle(\Pi\mathbf{c}_{!}pr^{*}_{{\mathbb{I}}/\\!\\!/{\mathbb{R}}}B)_{ml}=\coprod_{\mathbf{i}_{[m]},\,\,\mathbf{j}_{[l]},\,\,\alpha:i_{m}\to{\textrm{R}}j_{0}}(f_{n}...f_{1})_{!}B(\mathscr{I}(\mathbf{i}_{[m]})*^{\mathscr{I}\alpha}\mathscr{J}(\mathbf{j}_{[l]})).$

###### Proposition 5.12

For $c_{0}\stackrel{{\scriptstyle f_{1}}}{{\to}}...\stackrel{{\scriptstyle
f_{n}}}{{\to}}c_{n}=\mathbf{c}\in{\mathbb{C}}$, a $(F:{\cal D}\to{\cal
C},c_{0})$-transition structure $(\mathscr{I},\mathscr{J},{\textrm{R}})$, and
any $B\in{\sf\mathbb{R}Sect}({\cal D},F^{*}{\cal E})$ there is a natural (in
$B$) span of _weak equivalences_ in $\mathbf{E}(\mathbf{c})$

$|\Pi(\mathbf{c}_{!}R_{c_{0}}\mathscr{I}^{*}B)|\longleftarrow||\Pi\mathbf{c}_{!}pr^{*}_{{\mathbb{I}}/\\!\\!/{\mathbb{R}}}B||\longrightarrow|\Pi(\mathbf{c}_{!}\mathscr{J}^{*}B)|$
(5.3)

which comes from a natural (in $B$) span (5.2) of bisimplicial objects in
$\mathbf{E}(\mathbf{c})$.

###### Proof.

We need to prove that after realisations, both arrows become equivalences.
Consider the bisimplicial object

$\displaystyle\Pi\pi^{*}_{1}(\mathbf{c}_{!}R_{c_{0}}\mathscr{I}^{*}B)_{ml}=\coprod_{\mathbf{i}_{[m]},\,\,\mathbf{j}_{[l]},\,\,\alpha:i_{m}\to{\textrm{R}}j_{0}}(f_{n}...f_{1}\bar{\eta}(i_{m}))_{!}B(\mathscr{I}(\mathbf{i}_{[m]}))$

Our left hand side map in (5.2) passes through this object, as it is equal to
the composition

$\Pi\mathbf{c}_{!}pr^{*}_{{\mathbb{I}}/\\!\\!/{\mathbb{R}}}B\to\Pi\pi^{*}_{1}(\mathbf{c}_{!}R_{c_{0}}\mathscr{I}^{*}B)\to\pi^{*}_{1}\Pi(\mathbf{c}_{!}R_{c_{0}}\mathscr{I}^{*}B).$
(5.4)

Writing down the simplicial objects explicitly, we see that the first map in
(5.4) arises from the action of $B$ on anchor maps and is a termwise weak
equivalence of bisimplicial objects because $B$ is a derived section.
Realising the second map
$\Pi\pi^{*}_{1}(\mathbf{c}_{!}R_{c_{0}}\mathscr{I}^{*}B)\to\pi^{*}_{1}\Pi(\mathbf{c}_{!}R_{c_{0}}\mathscr{I}^{*}B)$
in (5.4) along the second simplicial argument, we obtain a map in $\Delta^{\sf
op}\mathbf{E}(\mathbf{c})$, whose $m$-th component is

$|\Pi\pi^{*}_{1}(\mathbf{c}_{!}R_{c_{0}}\mathscr{I}^{*}B)|_{m}\cong
N(i_{m}\backslash{\textrm{R}})\otimes\Pi(\mathbf{c}_{!}R_{c_{0}}\mathscr{I}^{*}B)_{m}\to\Pi(\mathbf{c}_{!}R_{c_{0}}\mathscr{I}^{*}B)_{m}.$
(5.5)

Observe that because of the adjunction ${\textrm{L}}\dashv{\textrm{R}}$, the
category $i_{m}\backslash{\textrm{R}}=({\textrm{R}}^{{\sf op}}/i_{m})^{{\sf
op}}$ has an initial object (the unit at $i_{m}$) and is thus contractible, so
the map (5.5) and thus (5.4) and the left-hand side map in (5.2) are all weak
equivalences.

We now have to prove that the right-hand side map
$\Pi\mathbf{c}_{!}pr^{*}_{{\mathbb{I}}/\\!\\!/{\mathbb{R}}}B\longrightarrow\pi^{*}_{2}\Pi(\mathbf{c}_{!}\mathscr{J}^{*}B)$
in (5.2) becomes an equivalence after realisations. For each fixed
$\mathbf{j}_{[l]}$, we have a map of simplicial objects, written in components
as

$\coprod_{\mathbf{i}_{[m]},\,\,\alpha:i_{m}\to{\textrm{R}}j_{0}}(f_{n}...f_{1})_{!}B(\mathscr{I}(\mathbf{i}_{[m]})*^{\mathscr{I}\alpha}\mathscr{J}(\mathbf{j}_{[l]}))\longrightarrow(f_{n}...f_{1})_{!}B(\mathscr{J}(\mathbf{j}_{[l]}));$
(5.6)

because ${\textrm{L}}/j_{0}$ has a terminal object, Lemma 3.9 and Lemma 4.6
imply that the map (5.6) is a weak equivalence after being realised. We
conclude that the morphism

$|\Pi\mathbf{c}_{!}pr^{*}_{{\mathbb{I}}/\\!\\!/{\mathbb{R}}}B|\longrightarrow|\pi^{*}_{2}\Pi(\mathbf{c}_{!}\mathscr{J}^{*}B)|\cong\Pi(\mathbf{c}_{!}\mathscr{J}^{*}B)$

is an equivalence of simplicial objects in $\mathbf{E}(\mathbf{c})$, where we
took the realisation of bisimplicial objects along the first argument.
Proposition 1.26 then implies that the double realisation

$||\Pi\mathbf{c}_{!}pr^{*}_{{\mathbb{I}}/\\!\\!/{\mathbb{R}}}B||\longrightarrow||\pi^{*}_{2}\Pi(\mathbf{c}_{!}\mathscr{J}^{*}B)|\cong|\Pi(\mathbf{c}_{!}\mathscr{J}^{*}B)|,$

taken in any order, is a weak equivalence. $\square$

We are now ready to prove Proposition 5.6. Fix
$\mathbf{c}_{[n]}\in{\mathbb{C}}$. For $A\in{\sf\mathbb{R}Sect}({\cal C},{\cal
E})$ there are functors $A^{aug}_{\mathbf{c}}$ and $A_{\mathbf{c}}$ (cf. the
proof of Lemma 4.5):

$\displaystyle
A^{aug}_{\mathbf{c}}:\mathbb{F}/\\!\\!/c_{0}\to\mathbf{E}(\mathbf{c}_{[n]}),\,\,\,\,(\mathbf{d}_{[m]},\alpha:Fd_{m}\to
c_{0})\mapsto A(\mathbb{F}(\mathbf{d}_{[m]})*^{\alpha}\mathbf{c}_{[n]}),$
$\displaystyle
A_{\mathbf{c}}:\mathbb{F}/\\!\\!/c_{0}\to\mathbf{E}(\mathbf{c}_{[n]}),\,\,\,\,(\mathbf{d}_{[m]},\alpha:Fd_{m}\to
c_{0})\mapsto A(\mathbf{c}_{[n]}).$

There is an obvious natural transformation $A^{aug}_{\mathbf{c}}\to
A_{\mathbf{c}}$. Pushing it forward to $\Delta^{\sf op}$ and realising gives
us a map $A^{\mathbb{F}}(\mathbf{c})\to N(F/c_{0})\otimes A(\mathbf{c})$ so
that the obvious composition

$A^{\mathbb{F}}(\mathbf{c})\to N(F/c_{0})\otimes A(\mathbf{c})\to
A(\mathbf{c})$

is the $\mathbf{c}$-th component of the right-hand map of the counit
correspondence (4.9).

###### Lemma 5.13

The morphism $N(F/c_{0})\otimes A(\mathbf{c})\to A(\mathbf{c})$ is a weak
equivalence.

###### Proof.

There is an adjunction $F/c_{0}\rightleftharpoons{\cal D}(c_{0})$ and ${\cal
D}(c_{0})$ is contractible, thus $F/c_{0}$ is contractible as well because
adjunctions of categories are known to induce homotopy equivalences between
the associated nerves [22]. $\square$

Now recall Example 5.8(3) where we work over $F/c_{0}$, with $I=F/c_{0}$,
$J={\cal D}(c_{0})$ and ${\textrm{R}}:{\cal D}(c_{0})\to F/c_{0}$ being the
evident functor. Also take the trivial opfibration ${\cal E}(c_{n})\times{\cal
C}\to{\cal C}$. Both $A_{\mathbf{c}}$ and $A_{\mathbf{c}}^{aug}$ are then
sections over $\mathbb{F}/\\!\\!/c_{0}$ of the trivial fibration
$\mathbf{E}(\mathbf{c}_{[n]})\times\mathbb{F}/\\!\\!/c_{0}\to\mathbb{F}/\\!\\!/c_{0}$.

###### Lemma 5.14

The map $A^{\mathbb{F}}(\mathbf{c})\to N(F/c_{0})\otimes A(\mathbf{c})$ is a
weak equivalence.

###### Proof.

The obvious natural transformation $A^{aug}_{\mathbf{c}}\to A_{\mathbf{c}}$,
when plugged in the left hand side of the span (5.3) for the transition
structure of the Example 5.8(3), gives us the map in question,
$A^{\mathbb{F}}(\mathbf{c})\to N(F/c_{0})\otimes A(\mathbf{c})$. The right-
hand side of span (5.3) gives the map

$|\Pi({\textrm{R}}^{*}A^{aug}_{\mathbf{c}})|\to N({\cal D}(c_{0}))\otimes
A(\mathbf{c})$ (5.7)

so by Proposition 5.12 we are done if the map (5.7) is a weak equivalence.
Observe however that

$\Pi({\textrm{R}}^{*}A^{aug}_{\mathbf{c}})_{m}=\coprod_{\mathbf{d}_{[m]}\in{\mathbb{D}}(c_{0})}A(\mathbb{F}(\mathbf{d}_{[m]})*\mathbf{c})=\coprod_{\mathbf{d}\in{\mathbb{D}}(c_{0})}A(id^{m}_{c_{0}}*\mathbf{c})$

with $id_{c_{0}}^{m}$ being the degenerate $m$-simplex
$c_{0}\stackrel{{\scriptstyle id_{c_{0}}}}{{\to}}...\stackrel{{\scriptstyle
id_{c_{0}}}}{{\to}}c_{0}$. Because $A$ is a derived section, Lemma 4.6 implies
that the obvious map $A(id_{c_{0}}^{m}*\mathbf{c})\to A(\mathbf{c})$ is a weak
equivalence, so that

$\Pi({\textrm{R}}^{*}A^{aug}_{\mathbf{c}})_{m}\to N({\cal
D}(c_{0}))_{m}\otimes A(\mathbf{c})=\Pi({\textrm{R}}^{*}A_{\mathbf{c}})_{m}$

is a weak equivalence as well. $\square$