# On the Equivariant Derived Category of Perverse Sheaves

Geoff Vooys

###### Abstract.

In this paper we extend Beilinson’s realization formalism for triangulated
categories and filtered triangulated categories to a pseudofunctorial and
pseudonatural setting. As a consequence we prove an equivariant version of
Beilinson’s Theorem: for any algebraic group $G$ over an algebraically closed
field $K$ and for any $G$-variety $X$, there is an equivalence of categories
$D_{G}^{b}(X;\overline{\mathbb{Q}}_{\ell})\simeq
D_{G}^{b}(\mathbf{Perv}(X;\overline{\mathbb{Q}}_{\ell}))$ where $\ell$ is an
integer prime coprime to the characteristic of $K$. We also show that the
equivariant analogues of the other non-$D$-module aspects of Beilinson’s
Theorem hold in the equivariant case.

###### Key words and phrases:

Pseudocones, Pseudolimit, Descent, Derived Descent, Equivariant Descent,
Equivariant Derived Category, Equivariant Homological Algebra, Equivariant
Sheaves, Equivariant Perverse Sheaves, Beilinson’s Theorem, Equivariant
Derived Category of Perverse Sheaves

###### 1991 Mathematics Subject Classification:

Primary 14A30, 18N10; Secondary 18G15, 14F08, 14F43, 14M17

###### Contents

  1. 1 Introduction
    1. 1.1 Main Results and Structure of the Paper
    2. 1.2 Acknowledgments
  2. 2 Equivariant Categories, Descent, and Pseudocones
  3. 3 Pseudocone Realization Functors
  4. 4 Equivariant Beilinson’s Theorem and Ext Functors

##  1\. Introduction

In the 1980’s Beilinson proved in [3] a result of crucial importance in both
representation theory and arithmetic algebraic geometry: there is an
equivalence of categories

$D^{b}(\operatorname{\mathbf{Perv}}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell}))\simeq
D_{c}^{b}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})$

for all separated finite type schemes over an algebraically closed field $K$
with $\ell$ coprime to the characteristic of $K$. This celebrated result has
had myriad ramifications and led to applications which allow us to compute
extensions of perverse sheaves in the derived category as opposed to in the
derived category of perverse sheaves, which is often a crucial step in
actually being able to calculate things because we have access to the usual
(derived) functors which are well-understood. This is also important for
$K$-theoretic results, as it gives rise to an isomorphism of Grothendieck
groups
$\mathsf{K}_{0}D^{b}(\operatorname{\mathbf{Perv}}(X))\cong\mathsf{K}_{0}D^{b}(X)$,
which is helpful in testing conjectures which appear in representation theory.

As representation theory and the Langlands Programme have proceeded over time,
there has been a need to develop more of the theory of equivariant derived
categories and equivariant algebraic geometry which is applicable to the study
of $p$-adic groups. In [6] the authors introduced a microlocal approach
towards the study of A-packets motivated by [2] which use, in fundamental
ways, the theory of equivariant derived categories, equivariant perverse
sheaves, and equivariant nearby and vanishing cycles. These techniques have
been further used in [7] and related work. These insights highlight the
importance of the equivariant derived category and equivariant perverse
sheaves in the study of $p$-adic groups and the Local Langlands Programme for
$p$-adic groups.

Alternatively, the study of much of the pure homological algebra which goes
into proving Beilinson’s Theorem [3, Theorem 1.3] is of independent interest.
The techniques used require a careful manipulation of triangulated and
truncation structures, and so admit applications beyond algebro-geometric and
representation-theoretic circumstances. It is shown in [14], with an eye
towards these further applications, that much of the category theory which
equivariant descent comes down to the language of pseudofunctors, pseudocones,
and higher-category theory. In fact, the equivariant derived category
$D_{G}^{b}(X)$ is the pseudolimit in $\operatorname{\mathfrak{Cat}}$ of a
certain pseudofunctor. The various properties of the equivariant derived
category we care about in practice (such as its triangulation, standard
$t$-structure, perverse $t$-structure, six functor formalism, etc.) follow
directly from various properties of the pseudofunctor indexing the diagram for
which $D_{G}^{b}(X)$ is a pseudolimit. Because the pseudocone and pseudolimit
language is so fruitful in working with the equivariant derived category, we
show this perspective is well-suited towards generalizing the filtered
triangulated categories of [3] to certain classes of pseudofunctors and also
towards using this general framework to study equivariant algebraic geometry.
It is also worth noting that this pseudofunctorial approach is an alternative
to the averaging functors used in [1], which showed that this equivalence
occurs when $G$ is a finite complex algebraic group and $X$ is a $G$-variety
over $\operatorname{\mathbb{C}}$.

###  1.1. Main Results and Structure of the Paper

The main result of the paper is Theorem 4.1 below, which shows many cases in
which the equivariant bounded derived category of perverse sheaves is
equivalent to the equivariant bounded derived category.

######  Theorem A (Cf. Theorem 4.1).

The following equivalences of categories hold:

  1. (1)

Let $K$ be a field, $G$ a smooth algebraic group over $K$, and $X$ a left
$G$-variety. If $R$ is any finite ring with characteristic coprime to the
characteristic of $K$ then there is an equivalence of categories

$D^{b}_{G}(X;\operatorname{\mathnormal{R}-\mathbf{Mod}})\simeq
D^{b}_{G}(\operatorname{\mathbf{Perv}}(X;\operatorname{\mathnormal{R}-\mathbf{Mod}})$

where
$\operatorname{\mathbf{Perv}}(X;\operatorname{\mathnormal{R}-\mathbf{Mod}})$
denotes the category of perverse sheaves with coefficients in
$\operatorname{\mathnormal{R}-\mathbf{Mod}}$,
$D^{b}_{G}(\operatorname{\mathbf{Perv}}(X;\operatorname{\mathnormal{R}-\mathbf{Mod}}))$
is the equivariant derived category of perverse sheaves of $R$-modules.

  2. (2)

Let $K$ be an algebraically closed field, $G$ an algebraic group over $K$, and
$X$ a left $G$-variety. Then

$D_{G}^{b}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})\simeq
D_{G}^{b}(\operatorname{\mathbf{Perv}}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})).$

  3. (3)

If $G$ is a complex algebraic group and $X$ is a left $G$-variety over
$\operatorname{\mathbb{C}}$ then for any field $K$ there is an equivalence

$D_{G}^{b}(X(\operatorname{\mathbb{C}}),\operatorname{\mathnormal{K}-\mathbf{Vect}})\simeq
D_{G}^{b}(\operatorname{\mathbf{Perv}}(X(\operatorname{\mathbb{C}}),\operatorname{\mathnormal{K}-\mathbf{Vect}})).$

  4. (4)

If $K=\operatorname{\mathbb{F}}_{q}$ is a finite field, $G$ is an algebraic
group over $K$, and $X$ is a left $G$-variety then for any prime $\ell$
coprime to $q$,

$D_{G,m}^{b}(X,\overline{\operatorname{\mathbb{Q}}}_{\ell})\simeq
D_{G}^{b}(\operatorname{\mathbf{Perv}}_{m}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})),$

where $D_{G,m}^{b}(X,\overline{\operatorname{\mathbb{Q}}}_{\ell})$ is the
equivariant derived category of mixed $\ell$-adic sheaves (and similarly in
the perverse case).

######  Corollary B (Cf. Corollary 4.5).

Let $K$ be a field, $G$ a smooth algebraic group over $K$, and let $X$ be a
left $G$-variety. Then there is an equivalence of categories

$D_{G}^{b}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})\simeq
D^{b}_{G}\left(\operatorname{\mathbf{Perv}}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})\right).$

We prove this primarily by extending the language of Beilinson’s filtered
triangulated categories and realization functor to pseudofunctors and
pseudocone categories. In Section 2 we give a review of pseudocone categories
and highlight some terminology, results, and other background material used
throughout this paper. Afterwards, in Section 3, we begin the meat of our work
and develop the extension of Beilinson’s realization functors to the
pseudofunctor setting. This long section culminates in the two main theorems
below.

######  Theorem C (Cf. Theorem 3.27).

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and assume that $T$ and $F$ are truncated with compatible $t$-structures. Then
if $Q$ is the quasi-isomorphism subpseudofunctor of
$\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})$, there is a unique
pseudonatural transformation
$\underline{{\operatorname{real}}}:Q^{-1}\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})\to
T$ for which the diagram

${\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})}$${T}$${Q^{-1}\operatorname{\mathbf{Ch}}^{b}\left(T^{\heartsuit}\right)}$$\scriptstyle{\widetilde{\operatorname{real}}}$$\scriptstyle{\underline{{\lambda}}_{Q}}$$\scriptstyle{\underline{{\operatorname{real}}}}$

commutes in
$\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})$.
Furthermore, each object functor $\,{}_{X}{\underline{{\operatorname{real}}}}$
is $t$-exact for all obects $X$ of $\operatorname{\mathscr{C}}$.

######  Theorem D (Cf. Theorem 3.28).

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and assume that $T$ and $F$ are truncated with compatible $t$-structures. Then
if $Q$ is the quasi-isomorphism subpseudofunctor of
$\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})$, there is a unique
pseudonatural transformation
$\underline{{\operatorname{real}}}:Q^{-1}\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})\to
T$ for which the diagram

${\operatorname{\mathbf{PC}}(\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit}))}$${\operatorname{\mathbf{PC}}(T)}$${\operatorname{\mathbf{PC}}(D^{b}\left(T^{\heartsuit}\right))}$$\scriptstyle{\operatorname{\mathbf{PC}}(\widetilde{\operatorname{real}})}$$\scriptstyle{\operatorname{\mathbf{PC}}(\underline{{\lambda}}_{Q})}$$\scriptstyle{\operatorname{\mathbf{PC}}(\underline{{\operatorname{real}}})}$

commutes in
$\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})$
with $\operatorname{\mathbf{PC}}(\underline{{\operatorname{real}}})$
$t$-exact. Furthermore, for any obejct $A$ of
$\operatorname{\mathbf{PC}}(\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit}))$,

$\operatorname{\mathbf{PC}}(\underline{{\operatorname{real}}})(A)=\left\\{\,{}_{X}{\underline{{\operatorname{real}}}}(\,_{X}{A})\;:\;X\in\operatorname{\mathscr{C}}_{0}\right\\}$

where $\,{}_{X}{\underline{{\operatorname{real}}}}$ is the realization
$D^{b}(T(X)^{\heartsuit})\to T(X).$

Finally we conclude the paper with our main theorem and corollary before
giving a result on extension groups of equivariant perverse sheaves.

######  Proposition E (Cf. Proposition 4.6).

Let $\operatorname{\mathscr{F}}$ and $\operatorname{\mathscr{G}}$ be two
equivariant perverse sehaves and let $n\in\operatorname{\mathbb{Z}}$. Then
there is an isomorphism

$\operatorname{Ext}^{n}_{\operatorname{\mathbf{Perv}}_{G}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})}(\operatorname{\mathscr{F}},\operatorname{\mathscr{G}})\cong\operatorname{Ext}^{n}_{D_{G}^{b}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})}(\operatorname{\mathscr{F}},\operatorname{\mathscr{G}}).$

###  1.2. Acknowledgments

I would like to thank Clifton Cunningham and James Steele for many stimulating
conversations and posing the questions “is it true that there is an
equivalence

$D_{G}^{b}\left(\operatorname{\mathbf{Perv}}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})\right)\simeq
D_{G}^{b}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})$

for a complex algebraic group $G$ and $G$-variety $X$?” and “for any
$n\in\operatorname{\mathbb{Z}}$ is it true that

$\operatorname{Ext}^{n}_{\operatorname{\mathbf{Perv}}_{G}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})}(\operatorname{\mathscr{F}},\operatorname{\mathscr{G}})\cong
D^{b}_{G}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})(\operatorname{\mathscr{F}},\operatorname{\mathscr{G}}[n])$

for equivariant perverse sheaves
$\operatorname{\mathscr{F}},\operatorname{\mathscr{G}}$?” to me which
motivated writing this paper. I am additionally grateful to the entire
Voganish research group for helpful discussions and input.

##  2\. Equivariant Categories, Descent, and Pseudocones

In this section we recall the basics of the language of categories of
pseudocones, some of their properties which we will need, and how they apply
to the categories we study. A systematic and in-depth study of these
categories, their properties, and how they apply to equivariant descent for
varieties and topological spaces is done in [14]. What is important for this
paper is that the work there allows us to construct an equivariant derived
category $D_{G}^{b}(\operatorname{\mathbf{Perv}}(X))$ of perverse sheaves on
$X$ which is a triangulated category with standard $t$-structure for which
$D^{b}_{G}(\operatorname{\mathbf{Perv}}(X))^{\heartsuit_{\operatorname{stand}}}\simeq\operatorname{\mathbf{Perv}}_{G}(X)$.

Before diving in directly, let us set some conventions. In this paper all the
schemes we consider are quasi-projective and by varieties we mean separated
schemes of finite type over a field. We also assume some familarity with
$2$-category theory, the theory of pseudofunctors, pseudonatural
transformations, modifications, and some of the basics involving the theory of
bicategories. A textbook introduction to the higher category theory we use in
this paper can be found in [10].

We will begin by discussing some of the basics of equivariant descent we use
to define the equivariant derived category and equivariant derived category of
perverse sheaves. The main technique used to define the equivariant category
on a variety follows the definitions of Lusztig and Achar: we must descend by
principal $G$-varieties (which, in case $G$ is not affine, are locally
isotrivial). An in-depth discussion of this condition is given in [14, Section
6.1] but for the reader who is content with working over affine algebraic
groups this poses no issue nor distinction.

######  Definition 2.1.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a pseudofunctor. The category of pseudocones of shape $F$ (with apex
$\operatorname{\mathbbm{1}}$) is defined to be the category

$\operatorname{\mathbf{PC}}(F):=\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})(\operatorname{cnst}(\operatorname{\mathbbm{1}}),F)$

where
$\operatorname{cnst}(\operatorname{\mathbbm{1}}):\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
is the constant pseudofunctor at the terminal category
$\operatorname{\mathbbm{1}}$.

A large list of these pseudocone categories is given in [14, Example 2.3.12]
and many of the categories given come from algebraic-geometric contexts. We
now recall from [14, Remark 2.3.8] a more explicit combinatorial description
of the objects and morphisms in a pseudocone category.

######  Remark 2.2.

Let us unwrap the definition of $\operatorname{\mathbf{PC}}(F)$ in order to
give an explicit description of the objects and morphisms of
$\operatorname{\mathbf{PC}}(F)$, as working with
$\operatorname{\mathbf{PC}}(F)$ is much more straightforward with this
knowledge. To this end recall that to give a functor
$\,{}_{X}{\alpha}:\operatorname{\mathbbm{1}}\to F(X)$ is to pick out an object
of $F(X)$ (and its identity morphism, of course). Similarly, if $f:X\to Y$ is
a morphism in $\operatorname{\mathscr{C}}$ then the $2$-cell

$\,{}_{f}{\alpha}:(\,_{X}{\alpha}\circ\operatorname{id}_{\operatorname{\mathbbm{1}}})(\ast)\xrightarrow{\cong}(F(f)\circ\,_{Y}{\alpha})(\ast)$

corresponds to giving an isomorphism $\,{}_{X}{\alpha}(\ast)\cong
F(f)(\,_{Y}{\alpha}(\ast))$; we write

$\tau_{f}:F(f)(\,_{Y}{\alpha}(\ast))\xrightarrow{\cong}\,_{X}{\alpha}$

for the corresponding inverse isomorphism and call these the transition
isomorphisms (of the pseudocone) — note that since the inverse of an
isomorphism is unique, determining the $\,{}_{f}{\alpha}$ is equivalent to
determining the $\tau_{f}$. Asking the pasting diagrams to coincide is
equivalent to asking that for a composable pair
$X\xrightarrow{f}Y\xrightarrow{g}Z$ of morphisms in
$\operatorname{\mathscr{C}}$, the diagram

${F(f)\left(F(g)\,_{Z}{\alpha}(\ast)\right)}$${F(f)(\,_{Y}{\alpha}(\ast))}$${F(g\circ
f)\,_{Z}{\alpha}(\ast)}$${\,{}_{X}{\alpha}(\ast)}$$\scriptstyle{\phi_{f,g}^{\,{}_{Z}{\alpha}(\ast)}}$$\scriptstyle{F(f)(\tau_{g})}$$\scriptstyle{\tau_{f}}$$\scriptstyle{\tau_{g\circ
f}}$

must commute. That is, the pseudonaturality condition gives rise to the
cocycle condition

$\tau_{g\circ f}\circ\phi_{f,g}^{\,{}_{Z}{\alpha}(\ast)}=\tau_{f}\circ
F(f)(\tau_{g}).$

Similarly, if we have pseudocones
$\alpha,\beta:\operatorname{cnst}(\operatorname{\mathbbm{1}})\Rightarrow F$ a
modification $\rho:\alpha\Rightarrow\beta$ is given by a family of natural
transformations $\,{}_{X}{\rho}:\,_{X}{\alpha}\Rightarrow\,_{X}{\beta}$, one
for each $X\in\operatorname{\mathscr{C}}_{0}$, such that for all morphisms
$f:X\to Y$ in $\operatorname{\mathscr{C}}$ the diagrams of functors and
natural transformations

${\,{}_{X}{\alpha}}$${\,{}_{X}{\beta}}$${F(f)\circ\,_{Y}{\alpha}}$${F(f)\circ\,_{Y}{\beta}}$$\scriptstyle{\,{}_{f}{\alpha}}$$\scriptstyle{\,{}_{X}{\rho}}$$\scriptstyle{\,{}_{f}{\beta}}$$\scriptstyle{F(f)\ast\,_{Y}{\rho}}$

commute. However, this implies that we can replace the isomorphisms
$\,{}_{f}{\alpha}$ with the corresponding transition isomorphisms to deduce
that the diagrams

${F(f)\circ\,_{Y}{\alpha}}$${F(f)\circ\,_{Y}{\beta}}$${\,{}_{X}{\alpha}}$${\,{}_{X}{\beta}}$$\scriptstyle{F(f)\ast\,_{Y}{\rho}}$$\scriptstyle{\tau_{f}}$$\scriptstyle{\sigma_{f}}$$\scriptstyle{\,{}_{X}{\rho}}$

also must commute. This allows us to give the following explicit description
for $\operatorname{\mathbf{PC}}(F)$:

  * •

Objects: Pairs $(A,T_{A})$ where

$A=\\{\,_{X}{A}\;|\;X\in\operatorname{\mathscr{C}}_{0},\,_{X}{A}\in
F(X)_{0}\\}$

is a collection of objects and transition isomorphisms

$T_{A}=\\{\tau_{f}^{A}:F(F)(\,_{Y}{A})\xrightarrow{\cong}\,_{X}{A}\;|\;f:X\to
Y,f\in\operatorname{\mathscr{C}}_{1}\\}$

which satisfy the cocycle condition

$\tau_{g\circ f}^{A}\circ\phi_{f,g}=\tau_{f}^{A}\circ
F(f)\left(\tau_{g}^{A}\right)$

for any composable morphisms $X\xrightarrow{f}Y\xrightarrow{g}Z$ in
$\operatorname{\mathscr{C}}$.

  * •

Morphisms: A morphism $P:(A,T_{A})\to(B,T_{B})$ is a collection of morphisms

$P=\\{\,_{X}{\rho}:\,_{X}{A}\to\,_{X}{B}\;|\;X\in\operatorname{\mathscr{C}}_{0}\\}$

such that for any morphisms $f:X\to Y$ the diagram

${F(F)(\,_{X}{A})}$${F(f)(\,_{Y}{B})}$${\,{}_{X}{A}}$${\,{}_{X}{B}}$$\scriptstyle{F(f)(\,_{X}{\rho})}$$\scriptstyle{\tau_{f}^{A}}$$\scriptstyle{\tau_{f}^{B}}$$\scriptstyle{\,{}_{X}{\rho}}$

commutes.

Let us now define the pseudofunctors which will induce the equivariant bounded
derived category of perverse sheaves we consider in this paper; the core ideas
we use go back to the perspectives pioneered and championed by [5], [11], [1].
We first need to define the category which controls the equivariant descent
which we use to define our equivariant derived categories. While the
discussion here is quite brief, a detailed and in-depth discussion is given in
[14, Section 2.1, 6.1]. As a result, we recall what it means to be a smooth
free $G$-variety, a smooth free $G$-resolution of a $G$-variety, and then
describe the pseudofunctors which induce the equivariant derived categories of
perverse sheaves and the usual equivariant derived category.

######  Definition 2.3 ([11, Section 1.9]).

A left $G$-variety $\Gamma$ is a smooth free $G$-variety if:

  * •

$\Gamma$ is smooth, pure dimensional, and admits a geometric quotient
$q:\Gamma\to G\backslash\Gamma$ (cf. [12, Definition 0.6]).

  * •

The geometric quotient

$\operatorname{\mathsf{quot}}:\Gamma\to G\backslash\Gamma$

is (étale) locally isotrivial with fibre $G$, i.e., there is an étale cover
$\\{\varphi_{i}:U_{i}\to G\backslash\Gamma\;|\;i\in I\\}$ of
$G\backslash\Gamma$ such that in each pullback diagram

${\operatorname{\mathsf{quot}}^{-1}(U_{i})}$${\Gamma}$${U_{i}}$${G\backslash\Gamma}$$\scriptstyle{q}$$\scriptstyle{\varphi_{i}}$

there is an isomorphism $\operatorname{\mathsf{quot}}^{-1}(U_{i})\cong G\times
U_{i}$, the corresponding map $G\times U_{i}\to U_{i}$ acts via projection
$\pi_{2}:G\times U_{i}\to U_{i}$, and the étale map $\varphi_{i}:U_{i}\to
G\backslash\Gamma$ can be taken to be finite étale.

######  Remark 2.4.

The reason we ask for the quotients above to be locally isotrivial is so that
we can apply [14, Proposition 6.1.4], which states that for any smooth free
$G$-variety $\Gamma$ and any left $G$-variety $X$ there is a geometric
quotient $G\backslash(\Gamma\times X)$. When $G$ is affine this poses no
issue; by [14, Remark 6.1.5] (cf. also [1, Proposition 6.1.13] and [13, Lemme
XIV 1.4]) $\operatorname{\mathbf{Sf}}(G)$-varieties are exactly those pure
dimensional smooth free $G$-varieties which admit a geometric quotient.

######  Definition 2.5 ([11, Section 1.9]).

A morphism of smooth free $G$-varieties is a smooth $G$-equivariant morphism
$f:\Gamma\to\Gamma^{\prime}$ where $\Gamma$ and $\Gamma^{\prime}$ are smooth
free $G$-varieties and $f$ is a morphism of constant fibre dimension.
Explicitly, for every $\gamma_{0},\gamma_{1}\in\lvert\Gamma\rvert$ we have
$\dim f^{-1}(\gamma_{0})=\dim f^{-1}(\gamma_{1})$.

######  Definition 2.6.

Define the category of smooth free $G$-varieties
$\operatorname{\mathbf{Sf}}(G)$ as follows:

  * •

Objects: Smooth free $G$-varieties $\Gamma$ (cf. Definition 2.3).

  * •

Morphisms: Morphisms of smooth free $G$-varieties (cf. Definition 2.5).

  * •

Composition and Identities: As in $\operatorname{\mathbf{Var}}_{/K}$.

The category of smooth free $G$-resolutions of a $G$-variety is then the
category “$\operatorname{\mathbf{Sf}}(G)\times X$.” We’ll justify this as a
good category with which to do equivariant descent shortly, but we’ll get to
know the category of resolutions itself first.

######  Definition 2.7.

Let $X$ be a left $G$-variety. We define the category
$\operatorname{\mathbf{SfResl}}_{G}(X)$ of smooth free $G$-resolutions of $X$
as follows.

  * •

Objects: Maps $\pi_{2}:\Gamma\times X\to X$ for
$\Gamma\in\operatorname{\mathbf{Sf}}(G)_{0}$.

  * •

Morphisms: Maps $f:\Gamma\times X\to\Gamma^{\prime}\times X$ of the form
$f=h\times\operatorname{id}_{X}$ for
$h\in\operatorname{\mathbf{Sf}}(G)(\Gamma,\Gamma^{\prime})$.

  * •

Composition and Identities: As in $\operatorname{\mathbf{Var}}_{/K}$.

The categories $\operatorname{\mathbf{Sf}}(G)$ and
$\operatorname{\mathbf{SfResl}}_{G}(X)$ are studied in more detail in [14,
Chapter 6]. The important aspects we need to know about the category
$\operatorname{\mathbf{SfResl}}_{G}(X)$ are the following:

  1. (1)

For every smooth free $G$-variety $\Gamma\in\operatorname{\mathbf{Sf}}(G)_{0}$
and every $G$-variety $X$, the variety $\Gamma\times X$ admits a geometric
quotient variety $G\backslash(\Gamma\times X)$ which is functorial in
$\operatorname{\mathbf{SfResl}}_{G}(X)$ (cf. [14, Proposition 6.1.5] for the
existence of the quotient functor; note that the local isotriviality of the
resolving varieties $\Gamma$ and quasi-projectivity of $X$ are both
essential). If $f\times\operatorname{id}_{X}:\Gamma\times
X\to\Gamma^{\prime}\times X$ is a morphism in
$\operatorname{\mathbf{SfResl}}_{G}(X)$ then we write
$\operatorname{\overline{\mathnormal{f}}}:G\backslash(\Gamma\times X)\to
G\backslash(\Gamma^{\prime}\times X)$

  2. (2)

If $X$ is a free $G$-variety then there is an equivalence of categories

$D_{G}^{b}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})\simeq
D_{c}^{b}(G\backslash X;\overline{\operatorname{\mathbb{Q}}}_{\ell}).$

Note that this follows from [14, Proposition 2.3.13]; cf. also Proposition
2.11 below.

######  Definition 2.8.

Consider the following flavours of pseudofunctors of derived categories of
perverse sheaves:

  1. (1)

Let $K$ be a field, $G$ a smooth algebraic group over $K$, $X$ a left
$G$-variety, and let $R$ be a finite ring for which the characteristic of $R$
is coprime to the characteristic of $K$. Then we define the pseudofunctor

$D^{b}(\operatorname{\mathbf{Perv}}(G\backslash(-));\operatorname{\mathnormal{R}-\mathbf{Mod}}):\operatorname{\mathbf{SfResl}}_{G}(X)^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$

as follows: for each object $\Gamma\times X$ of
$\operatorname{\mathbf{SfResl}}_{G}(X)$ we define the fibre category to be
$D^{b}(\operatorname{\mathbf{Perv}}(G\backslash(\Gamma\times
X);\operatorname{\mathnormal{R}-\mathbf{Mod}}))$, where
$\operatorname{\mathbf{Perv}}(G\backslash(\Gamma\times
X);\operatorname{\mathnormal{R}-\mathbf{Mod}})$ is the category of perverse
(étale) sheaves of $R$-modules on the quotient variety
$G\backslash(\Gamma\times X)$, and for each morphism
$f\times\operatorname{id}_{X}$ we define the fibre functor to be given by
${}^{p}(G\backslash(f\times\operatorname{id}_{X}))^{\ast}$.

  2. (2)

Let $K$ be a field, $G$ a smooth algebraic group over $K$, $X$ a left
$G$-variety, and let $\ell$ be an integer prime which is coprime to the
characteristic of $K$. Then we define the pseudofunctor

$D^{b}(\operatorname{\mathbf{Perv}}(G\backslash(-));\overline{\operatorname{\mathbb{Q}}}_{\ell}):\operatorname{\mathbf{SfResl}}_{G}(X)^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$

as follows: for each object $\Gamma\times X$ of
$\operatorname{\mathbf{SfResl}}_{G}(X)$ we define the fibre category to be
$D^{b}(\operatorname{\mathbf{Perv}}(G\backslash(\Gamma\times
X);\overline{\operatorname{\mathbb{Q}}}_{\ell}))$, where
$\operatorname{\mathbf{Perv}}(G\backslash(\Gamma\times
X);\overline{\operatorname{\mathbb{Q}}}_{\ell})$ is the category of perverse
$\ell$-adic sheaves on the quotient variety $G\backslash(\Gamma\times X)$, and
for each morphism $f\times\operatorname{id}_{X}$ we define the fibre functor
to be given by ${}^{p}(G\backslash(f\times\operatorname{id}_{X}))^{\ast}$.

  3. (3)

Let $G$ be a complex algebraic group, let $X$ be a left $G$-variety (over
$\operatorname{\mathbb{C}}$), and let $K$ be a field. We then define the
psuedofunctor

$D^{b}\left(\operatorname{\mathbf{Perv}}((G\backslash(-))(\operatorname{\mathbb{C}})),\operatorname{\mathnormal{K}-\mathbf{Vect}}\right):\operatorname{\mathbf{SfResl}}_{G}(X)^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$

as follows: for each object $\Gamma\times X$ of
$\operatorname{\mathbf{SfResl}}_{G}(X)$ we define the fibre category to be
$D^{b}(\operatorname{\mathbf{Perv}}((G\backslash(\Gamma\times
X))(\operatorname{\mathbb{C}}),\operatorname{\mathnormal{K}-\mathbf{Vect}}))$,
where the category $\operatorname{\mathbf{Perv}}((G\backslash(\Gamma\times
X))(\operatorname{\mathbb{C}}),\operatorname{\mathnormal{K}-\mathbf{Vect}})$
is the category of perverse sheaves of $K$-vector spaces in the classical
topology on the quotient variety $G\backslash(\Gamma\times X)$, and for each
morphism $f\times\operatorname{id}_{X}$ we define the fibre functor to be
given by ${}^{p}(G\backslash(f\times\operatorname{id}_{X}))^{\ast}$.

  4. (4)

Let $\operatorname{\mathbb{F}}_{q}$ be a finite field, let $G$ be an algebraic
group over $\operatorname{\mathbb{F}}_{q}$, $X$ a left $G$-variety, and let
$\ell$ be an integer prime with $\gcd(q,\ell)=1$. Then we define the
pseudofunctor

$D^{b}(\operatorname{\mathbf{Perv}}_{m}(G\backslash(-));\overline{\operatorname{\mathbb{Q}}}_{\ell}):\operatorname{\mathbf{SfResl}}_{G}(X)^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$

as follows: for each object $\Gamma\times X$ of
$\operatorname{\mathbf{SfResl}}_{G}(X)$ we define the fibre category to be
$D^{b}(\operatorname{\mathbf{Perv}}_{m}(G\backslash(\Gamma\times
X);\overline{\operatorname{\mathbb{Q}}}_{\ell}))$, where
$\operatorname{\mathbf{Perv}}_{m}(G\backslash(\Gamma\times
X);\operatorname{\mathnormal{R}-\mathbf{Mod}})$ is the category of mixed
perverse $\ell$-adic sheaves on the quotient variety $G\backslash(\Gamma\times
X)$, and for each morphism $f\times\operatorname{id}_{X}$ we define the fibre
functor to be given by
${}^{p}(G\backslash(f\times\operatorname{id}_{X}))^{\ast}$.

######  Definition 2.9.

The four main flavours of equivariant derived category we consider in this
paper are given as follows:

  1. (1)

Let $K$ be a field, $G$ a smooth algebraic group over $K$, $X$ a left
$G$-variety, and let $R$ be a finite ring with characteristic coprime to the
characteristic of $K$. The equivariant bounded derived category of perverse
sheaves of $R$-modules is given by

$D_{G}^{b}\left(\operatorname{\mathbf{Perv}}\left(X,\operatorname{\mathnormal{R}-\mathbf{Mod}}\right)\right):=\operatorname{\mathbf{PC}}\left(D^{b}(\operatorname{\mathbf{Perv}}(G\backslash(-),\operatorname{\mathnormal{R}-\mathbf{Mod}}))\right)$

where
$D^{b}(\operatorname{\mathbf{Perv}}(G\backslash(-),\operatorname{\mathnormal{R}-\mathbf{Mod}}))$
is the pseudofunctor described in Flavour $(1)$ of Definition 2.8.

  2. (2)

Let $K$ be a field, $G$ a smooth algebraic group over $K$, $X$ a left
$G$-variety, and let $\ell$ be an integer prime which is coprime to the
characteristic of $K$. Then the equivariant bounded derived category of
$\ell$-adic perverse sheaves is defined to be

$D_{G}^{b}\left(\operatorname{\mathbf{Perv}}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})\right):=\operatorname{\mathbf{PC}}\left(D^{b}\left(\operatorname{\mathbf{Perv}}\left(G\backslash(-);\overline{\operatorname{\mathbb{Q}}}_{\ell}\right)\right)\right)$

where
$D^{b}(\operatorname{\mathbf{Perv}}(G\backslash(-);\overline{\operatorname{\mathbb{Q}}}_{\ell}))$
is the pseudofunctor described in Flavour $(2)$ of 2.8.

  3. (3)

Let $G$ be a complex algebraic group, let $X$ be a left $G$-variety (over
$\operatorname{\mathbb{C}}$), and let $K$ be a field. We then define the
equivariant bounded derived category of perverse $K$-sheaves in the classical
topology on $X$ by

$D_{G}^{b}\left(X(\operatorname{\mathbb{C}}),\operatorname{\mathnormal{K}-\mathbf{Vect}}\right):=\operatorname{\mathbf{PC}}\left(D^{b}\left(\operatorname{\mathbf{Perv}}((G\backslash(-))(\operatorname{\mathbb{C}})),\operatorname{\mathnormal{K}-\mathbf{Vect}}\right)\right)$

where
$D^{b}(\operatorname{\mathbf{Perv}}((G\backslash(-))(\operatorname{\mathbb{C}}),\operatorname{\mathnormal{K}-\mathbf{Vect}}))$
is the pseudofunctor described in Flavour $(3)$ of Definition 2.8.

  4. (4)

Let $\operatorname{\mathbb{F}}_{q}$ be a finite field, let $G$ be an algebraic
group over $\operatorname{\mathbb{F}}_{q}$, $X$ a left $G$-variety, and let
$\ell$ be an integer prime with $\gcd(q,\ell)=1$. Then we define the
equivariant bounded derived category of mixed $\ell$-adic perverse sheaves to
be

$D_{G}^{b}\left(\operatorname{\mathbf{Perv}}_{m}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})\right):=\operatorname{\mathbf{PC}}\left(D^{b}\left(\operatorname{\mathbf{Perv}}_{m}\left(G\backslash(-);\overline{\operatorname{\mathbb{Q}}}_{\ell}\right)\right)\right)$

where
$D^{b}(\operatorname{\mathbf{Perv}}_{m}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell}))$
is the pseudofunctor described in Flavour $(4)$ of Definition 2.8.

######  Definition 2.10.

Consider the following different flavours of equivariant derived categories:

  1. (1)

Let $K$ be a field, $G$ a smooth algebraic group over $K$, and $X$ a left
$G$-variety. Then the equivariant derived category of étale sheaves on $X$,
$D_{G}^{b}(X)$, is the category $\operatorname{\mathbf{PC}}(D)$ where
$D:\operatorname{\mathbf{SfResl}}_{G}(X)^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
is the pseudofunctor which sends each variety $\Gamma\times X$ to the
constructible derived category $D_{c}^{b}(G\backslash(\Gamma\times X))$, sends
a morphism $f\times\operatorname{id}_{X}:\Gamma\times
X\to\Gamma^{\prime}\times X$ to the pullback functor
$(G\backslash(f\times\operatorname{id}_{X}))^{\ast}$, and has any fixed choice
of compositor isomorphisms.

  2. (2)

Let $K$ be a field, $G$ a smooth algebraic group over $K$, and $X$ a left
$G$-variety. Then the equivariant derived category of étale sheaves of
$R$-modules on $X$, $D_{G}^{b}(X;\operatorname{\mathnormal{R}-\mathbf{Mod}})$,
is the category $\operatorname{\mathbf{PC}}(D)$ where
$D:\operatorname{\mathbf{SfResl}}_{G}(X)^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
is the pseudofunctor which sends each variety $\Gamma\times X$ to the
constructible derived category $D_{c}^{b}(G\backslash(\Gamma\times
X);\operatorname{\mathnormal{R}-\mathbf{Mod}})$ of $R$-modules, sends a
morphism $f\times\operatorname{id}_{X}:\Gamma\times X\to\Gamma^{\prime}\times
X$ to the pullback functor
$(G\backslash(f\times\operatorname{id}_{X}))^{\ast}$, and has any fixed choice
of compositor isomorphisms.

  3. (3)

Let $K$ be an algebraically closed field, $G$ an algebraic group over $K$, and
$X$ a left $G$-variety. Then the equivariant derived category of $\ell$-adic
sheaves on $X$ for a prime $\ell$ coprime to the characteristic of $K$,
$D_{G}^{b}(X;\overline{\operatorname{\mathbb{Q}}}_{\ell})$, is the category
$\operatorname{\mathbf{PC}}(D)$ where
$D:\operatorname{\mathbf{SfResl}}_{G}(X)^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
is the pseudofunctor which sends each variety $\Gamma\times X$ to the
constructible $\ell$-adic derived category $D_{c}^{b}(G\backslash(\Gamma\times
X);\overline{\operatorname{\mathbb{Q}}}_{\ell})$, sends a morphism
$f\times\operatorname{id}_{X}:\Gamma\times X\to\Gamma^{\prime}\times X$ to the
pullback functor $(G\backslash(f\times\operatorname{id}_{X}))^{\ast}$, and has
any fixed choice of compositor isomorphisms.

  4. (4)

Let $G$ be a complex algebraic group, $X$ a left $G$-variety, and let $K$ be a
field. Then the equivariant derived category of $K$-sheaves in the classical
topology on $X$,
$D_{G}^{b}(X(\operatorname{\mathbb{C}}),\operatorname{\mathnormal{K}-\mathbf{Vect}})$
is given by $\operatorname{\mathbf{PC}}(D)$ where
$D:\operatorname{\mathbf{SfResl}}_{G}(X)^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
is the pseudofunctor given by sending each variety $\Gamma\times X$ to the
bounded derived category of $K$-sheaves

$D_{c}^{b}\left(\left(G\backslash(\Gamma\times
X)\right)(\operatorname{\mathbb{C}}),\operatorname{\mathnormal{K}-\mathbf{Vect}}\right)$

with algebriacally constructible cohomology and sends each morphism
$f\times\operatorname{id}_{X}$ to the corresponding fibre functor
$(G\backslash(f\times\operatorname{id}_{X}))^{\ast}$.

As an immediate consequence of the definition of the equivariant derived
category of perverse sheaves in all cases, we get the following equivalences
of categories.

######  Proposition 2.11.

Let $D^{b}_{G}(\operatorname{\mathbf{Perv}}(X))$ be any of the four flavours
of equivariant derived cateogries of perverse sheaves given in Definition 2.9.
Then if $X$ is a free $G$-variety in the sense that the $G$-action on $X$ is
locally isotrivial there is an equivalence of categories

$D_{G}^{b}(\operatorname{\mathbf{Perv}}(X))\simeq
D^{b}(\operatorname{\mathbf{Perv}}(G\backslash X)).$

###### Proof.

Because the action of $G$ on $X$ is locally isotrivial, $X$ is a terminal
object in $\operatorname{\mathbf{SfResl}}_{G}(X)$. The proposition now follows
from [14, Proposition 2.3.13] and the fact that if $D$ is the corresponding
pseudofunctor for which
$D_{G}^{b}(\operatorname{\mathbf{Perv}}(X))=\operatorname{\mathbf{PC}}(D)$,
$D(X)=D^{b}(\operatorname{\mathbf{Perv}}(G\backslash X)).$ ∎

We now recall the definitions of what it means to be a triangulated and
truncated pseudofunctor as given in [14]. These describe the structure on a
pseudofunctor sufficient to determine when the category
$\operatorname{\mathbf{PC}}(F)$ is triangulated (with triangulation induced by
triangulations on the fibre categories $F(X)$) and admits $t$-structures which
are given by $t$-structures on the fibre categories $F(X)$.

######  Definition 2.12 ([14, Definition 5.1.9]).

A pseudofunctor
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
is triangulated if for every object $X\in\operatorname{\mathscr{C}}_{0}$ the
category $F(X)$ is triangulated and if for every morphism
$f\in\operatorname{\mathscr{C}}_{1}$ the functor $F(f)$ is triangulated.

######  Definition 2.13 ([14, Definition 5.1.18]).

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a triangulated pseudofunctor. Assume each fibre category has $t$-structure
with truncation functors $(\,_{X}{\tau^{\leq 0}},\,_{X}{\tau^{\geq 0}})$ such
that these give rise to pseudofunctors $F^{\leq
0}:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and $F^{\geq
0}:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
for which the functors $F^{\geq 0}(f)$ and $F^{\leq 0}(f)$ are $t$-exact
functors which coincide on all common subcategories. Then if for all morphisms
$f:X\to Y$ in $\operatorname{\mathscr{C}}$ there are natural isomorphisms

${F(Y)}$${F(X)}$${F(Y)^{\leq 0}}$${F(X)^{\leq
0}}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{Y}{\tau^{\leq
0}}}$$\scriptstyle{\,{}_{X}{\tau^{\leq 0}}}$$\scriptstyle{F^{\leq
0}(f)}$$\scriptstyle{\theta_{f}^{\leq 0}}$

and

${F(Y)}$${F(X)}$${F(Y)^{\geq 0}}$${F(X)^{\geq
0}}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{Y}{\tau^{\geq
0}}}$$\scriptstyle{\,{}_{X}{\tau^{\geq 0}}}$$\scriptstyle{F^{\geq
0}(f)}$$\scriptstyle{\theta_{f}^{\geq 0}}$

which satisfy the coherence diagrams presented below, we say that the
pseudofunctor is truncated. The pasting diagram

${F(Z)}$${F(Y)}$${F(X)}$${F(Z)^{\leq 0}}$${F(Y)^{\leq 0}}$${F(X)^{\leq
0}}$$\scriptstyle{F(g)}$$\scriptstyle{\,{}_{Z}{\tau^{\leq
0}}}$$\scriptstyle{F(g\circ f)}$$\scriptstyle{\,{}_{Y}{\tau^{\leq
0}}}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{X}{\tau^{\leq
0}}}$$\scriptstyle{F^{\leq 0}(g)}$$\scriptstyle{F^{\leq 0}(g\circ
f)}$$\scriptstyle{F^{\leq
0}(f)}$$\scriptstyle{\,{}_{F}{\phi_{f,g}}^{-1}}$$\scriptstyle{\theta_{f}^{\leq
0}}$$\scriptstyle{\theta_{g}^{\leq 0}}$$\scriptstyle{\,{}_{F^{\leq
0}}{\phi_{f,g}}}$

is equal to the $2$-cell

${F(Z)}$${F(X)}$${F(Z)^{\leq 0}}$${F(X)^{\leq 0}}$$\scriptstyle{F(g\circ
f)}$$\scriptstyle{\,{}_{Z}{\tau^{\leq 0}}}$$\scriptstyle{\,{}_{X}{\tau^{\leq
0}}}$$\scriptstyle{F^{\leq 0}(g\circ f)}$$\scriptstyle{\theta_{g\circ f}^{\leq
0}}$

and dually the pasting diagram

${F(Z)}$${F(Y)}$${F(X)}$${F(Z)^{\geq 0}}$${F(Y)^{\geq 0}}$${F(X)^{\geq
0}}$$\scriptstyle{F(g)}$$\scriptstyle{\,{}_{Z}{\tau^{\geq
0}}}$$\scriptstyle{F(g\circ f)}$$\scriptstyle{\,{}_{Y}{\tau^{\geq
0}}}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{X}{\tau^{\geq
0}}}$$\scriptstyle{F^{\geq 0}(g)}$$\scriptstyle{F^{\geq 0}(g\circ
f)}$$\scriptstyle{F^{\geq
0}(f)}$$\scriptstyle{\,{}_{F}{\phi_{f,g}}^{-1}}$$\scriptstyle{\theta_{f}^{\geq
0}}$$\scriptstyle{\theta_{g}^{\geq 0}}$$\scriptstyle{\,{}_{F^{\geq
0}}{\phi_{f,g}}}$

is equal to the $2$-cell:

${F(Z)}$${F(X)}$${F(Z)^{\geq 0}}$${F(X)^{\geq 0}}$$\scriptstyle{F(g\circ
f)}$$\scriptstyle{\,{}_{Z}{\tau^{\geq 0}}}$$\scriptstyle{\,{}_{X}{\tau^{\geq
0}}}$$\scriptstyle{F^{\geq 0}(g\circ f)}$$\scriptstyle{\theta_{g\circ f}^{\geq
0}}$

WWith these definitions and [14], we now get that each flavour of
$D_{G}^{b}(\operatorname{\mathbf{Perv}}(X))$ is a triangulated category with a
$t$-structure equivalent to the category of equivariant perverse sheaves.

######  Proposition 2.14.

Let $D_{G}^{b}(\operatorname{\mathbf{Perv}}(X))$ be any of the flavours of
equivariant bounded derived category of perverse sheaves in Definition 2.8.
Then $D_{G}^{b}(\operatorname{\mathbf{Perv}}(X))$ is a triangulated category
with $t$-structure whose heart satisfies

$D^{b}_{G}(\operatorname{\mathbf{Perv}}(X))^{\heartsuit}\simeq\operatorname{\mathbf{Perv}}_{G}(X)$

where $\operatorname{\mathbf{Perv}}_{G}(X)$ is the corresponding flavour of
equivariant perverse sheaves.

###### Proof.

That $D_{G}^{b}(\operatorname{\mathbf{Perv}}(X))$ is triangulated follows from
[14, Theorem 5.1.10] the fact that each category
$D^{b}(\operatorname{\mathbf{Perv}}_{G}(G\backslash(\Gamma\times X)))$ is
triangulated and each fibre functor ${}^{p}(G\backslash(\Gamma\times
X))^{\ast}$ is triangulated exact. The $t$-structure on
$D_{G}^{b}(\operatorname{\mathbf{Perv}}(X))$ is induced by [14, Theorem
5.1.21], the fact that each bounded derived category
$D^{b}(\operatorname{\mathbf{Perv}}_{G}(G\backslash(\Gamma\times X)))$ has a
standard $t$-structure, each fibre functor is exact for this $t$-structure,
and the collection of truncation/inclusion adjunctions gives rise to a
truncation structure for the corresponding pseudofunctor. Finally the Change
of Heart Theorem (cf. [14, Theorem 5.1.24]) gives that

$\displaystyle D_{G}^{b}(\operatorname{\mathbf{Perv}}(X))^{\heartsuit}$
$\displaystyle=\operatorname{\mathbf{PC}}\left(D^{b}(\operatorname{\mathbf{Perv}}_{G}\backslash(-)))\right)^{\heartsuit}=\operatorname{\mathbf{PC}}\left(D^{b}(\operatorname{\mathbf{Perv}}(G\backslash(-)))^{\heartsuit}\right)$
$\displaystyle\simeq\operatorname{\mathbf{PC}}(\operatorname{\mathbf{Perv}}(G\backslash(-)))$
$\displaystyle\simeq\operatorname{\mathbf{Perv}}_{G}(X);$

note the last equivalence follows from techinques similar to those used in
[14, Corollary 5.1.28]. ∎

##  3\. Pseudocone Realization Functors

In this section we show how to construct realization functors
$\operatorname{\mathbf{PC}}(F)\to\operatorname{\mathbf{PC}}(T)$ which are
given in terms of local realization functors $F(X)\to T(X)$. Our strategy is
to use these pseudoconical realizations to prove the equivariant version of
Beilinson’s Theorem; as such, we develop them in a relatively high level of
generality. This section is, in essence, a pseudoconical version of [3,
Appendix A] and gives a notion of pseudocone filtered categories and tools for
descent filtrations.

######  Definition 3.1 ([3, Definiiton A.1.a]).

Let $\operatorname{\mathscr{T}}$ be a triangulated category. We say that
$\operatorname{\mathscr{T}}$ is filtered (or that $\operatorname{\mathscr{T}}$
is an $f$-category) if:

  1. (1)

$\operatorname{\mathscr{T}}$ has two strictly full subcategories
$\operatorname{\mathscr{T}}^{f\geq 0}$ and $\operatorname{\mathscr{T}}^{f\leq
0}$.

  2. (2)

$\operatorname{\mathscr{T}}$ has a triangulated automorphism
$s:\operatorname{\mathscr{T}}\to\operatorname{\mathscr{T}}$ (which we think of
as a shift of filtration) together with a natural transformation
$\alpha:\operatorname{id}_{\operatorname{\mathscr{T}}}\Rightarrow s$.

  3. (3)

If we define, for any $n\in\operatorname{\mathbb{Z}}$,

$\operatorname{\mathscr{T}}^{f\leq
n}:=s^{n}\left(\operatorname{\mathscr{T}}^{f\leq
0}\right),\quad\operatorname{\mathscr{T}}^{f\geq
n}:=s^{n}\left(\operatorname{\mathscr{T}}^{f\geq 0}\right)$

then

$\bigcup_{n\in\operatorname{\mathbb{Z}}}\operatorname{\mathscr{T}}^{f\geq
n}=\operatorname{\mathscr{T}}=\bigcup_{n\in\operatorname{\mathbb{Z}}}\operatorname{\mathscr{T}}^{f\leq
n}.$

  4. (4)

The identity $\alpha=s\ast\alpha\ast s^{-1}$ holds.

  5. (5)

For all $X,Y\in\operatorname{\mathscr{T}}_{0}$ if
$X\in(\operatorname{\mathscr{T}}^{f\geq 1})_{0}$ and if
$Y\in(\operatorname{\mathscr{T}}^{f\leq 0})_{0}$ then
$\operatorname{\mathscr{T}}(X,Y)=0$. Furthermore, $\alpha$ induces
isomorphisms

$\operatorname{\mathscr{T}}(Y,s^{-1}X)\xrightarrow{\operatorname{\mathscr{T}}\left(Y,(\alpha\ast
s^{-1})_{X}\right)}\operatorname{\mathscr{T}}(Y,X)$

and

$\operatorname{\mathscr{T}}(sY,X)\xrightarrow{\operatorname{\mathscr{T}}(\alpha_{Y},X)}\operatorname{\mathscr{T}}(Y,X).$

We will call the information $(\operatorname{\mathscr{T}}^{f\leq
0},\operatorname{\mathscr{T}}^{f\geq 0},s,\alpha)$ the corresponding
filtration on $\operatorname{\mathscr{T}}$.

######  Definition 3.2 ([3, Definition A.1.b]).

Let $\operatorname{\mathscr{C}},\operatorname{\mathscr{T}}$ be filtered
triangulated categories. We say that a functor
$F:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{T}}$ is filtered if $F$
is triangulated, $F\circ
s_{\operatorname{\mathscr{C}}}=s_{\operatorname{\mathscr{T}}}\circ F$,
$\alpha_{\operatorname{\mathscr{T}}}\ast
F=F\ast\alpha_{\operatorname{\mathscr{C}}}$, and if $F$ preserves
$\operatorname{\mathscr{C}}^{f\leq 0}$ and $\operatorname{\mathscr{C}}^{f\geq
0}$ in the sense that the diagrams

${\operatorname{\mathscr{C}}^{f\leq
0}}$${\operatorname{\mathscr{C}}}$${\operatorname{\mathscr{T}}^{f\leq
0}}$${\operatorname{\mathscr{T}}}$$\scriptstyle{\operatorname{incl}^{f\leq
0}_{\operatorname{\mathscr{C}}}}$$\scriptstyle{F|_{\operatorname{\mathscr{C}}^{f\leq
0}}}$$\scriptstyle{F}$$\scriptstyle{\operatorname{incl}_{\operatorname{\mathscr{T}}}^{f\leq
0}}$

and

${\operatorname{\mathscr{C}}^{f\geq
0}}$${\operatorname{\mathscr{C}}}$${\operatorname{\mathscr{T}}^{f\geq
0}}$${\operatorname{\mathscr{T}}}$$\scriptstyle{\operatorname{incl}^{f\geq
0}_{\operatorname{\mathscr{C}}}}$$\scriptstyle{F|_{\operatorname{\mathscr{C}}^{f\geq
0}}}$$\scriptstyle{F}$$\scriptstyle{\operatorname{incl}_{\operatorname{\mathscr{T}}}^{f\geq
0}}$

commute.

Our first result in this section is that if we have a pseudofunctor $F$ which
is coherently filtered, then the pseudocone category
$\operatorname{\mathbf{PC}}(F)$ is filtered.

######  Proposition 3.3.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a triangulated pseudofunctor for which:

  * •

For all $X\in\operatorname{\mathscr{C}}_{0}$ the category $F(X)$ is filtered
with filtration given by the quadruple $(F(X)^{f\leq 0},F(X)^{f\geq
0},\,_{X}{s},\,_{X}{\alpha})$.

  * •

For all morphisms $f:X\to Y$ in $\operatorname{\mathscr{C}}$ the functor
$F(f)$ is a filtered functor.

Then $\operatorname{\mathbf{PC}}(F)$ is an $f$-category.

###### Proof.

We define the strictly full subcategories
$\operatorname{\mathbf{PC}}(F)^{f\leq 0}$ and
$\operatorname{\mathbf{PC}}(F)^{f\geq 0}$ by taking the objects to be given by

$\left(\operatorname{\mathbf{PC}}(F)^{f\leq
0}\right)_{0}:=\left\\{A\in\operatorname{\mathbf{PC}}(F)_{0}\;:\;\forall\,X\in\operatorname{\mathscr{C}}_{0}.\,_{X}{A}\in\left(F(X)^{f\leq
0}\right)_{0}\right\\}$

and

$\left(\operatorname{\mathbf{PC}}(F)^{f\geq
0}\right)_{0}:=\left\\{A\in\operatorname{\mathbf{PC}}(F)_{0}\;:\;\forall\,X\in\operatorname{\mathscr{C}}_{0}.\,_{X}{A}\in\left(F(X)^{f\geq
0}\right)_{0}\right\\}$

and generating the morphisms appropriately in both cases. The assumptions
regarding the fibre functors $F(f)$ and the shift functors $\,{}_{X}{s}$ imply
that if for any morphism $f:X\to Y$ in $\operatorname{\mathscr{C}}$ we set
$\,{}_{f}{s}$ to be the commutativity witness

${F(Y)}$${F(Y)}$${F(X)}$${F(X)}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{Y}{s}}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{X}{s}}$$\scriptstyle{\,{}_{f}{s}}$

then the collection
$s=(\,_{X}{s},\,_{f}{s})_{X\in\operatorname{\mathscr{C}}_{0},f\in\operatorname{\mathscr{C}}_{1}}$
determines a pseudonatural transformation $\underline{{s}}:F\Rightarrow F$.
Consequently, setting
$s:=\operatorname{\mathbf{PC}}(\underline{{s}}):\operatorname{\mathbf{PC}}(F)\to\operatorname{\mathbf{PC}}(F)$,
where $\operatorname{\mathbf{PC}}$ is the (strict) pseudoconification
$2$-functor of [14, Lemma 4.1.13], defines our shift automorphism. Similarly,
we define our remaining component is done by first noticing that if we define
the collection

$\underline{{\alpha}}:=\left\\{\,{}_{X}{\alpha}:\operatorname{id}_{F(X)}\Rightarrow\,_{X}{s}\;|\;X\in\operatorname{\mathscr{C}}_{0}\right\\}$

then for any map $f:X\to Y$ in $\operatorname{\mathscr{C}}$ the diagram of
functors and natural transformations

${F(f)}$${F(f)\circ\,_{Y}{s}}$${F(f)}$${\,{}_{X}{s}\circ
F(f)}$$\scriptstyle{F(f)\ast\,_{Y}{\alpha}}$$\scriptstyle{\,{}_{f}{s}}$$\scriptstyle{\,{}_{Y}{\alpha}\ast
F(f)}$

commutes exactly because $F(f)$ is an $f$-functor. Consequently
$\underline{{\alpha}}$ determines a modification:

${\operatorname{\mathscr{C}}^{\operatorname{op}}}$${\operatorname{\mathfrak{Cat}}}$$\scriptstyle{F}$$\scriptstyle{F}$$\scriptstyle{\underline{{s}}}$$\scriptstyle{\operatorname{id}_{F}}$$\underset{\underline{{\alpha}}}{\Rrightarrow}$

We thus define $\alpha:\operatorname{id}_{\operatorname{\mathbf{PC}}(F)}\to s$
by $\alpha:=\operatorname{\mathbf{PC}}(\underline{{\alpha}})$. That this
satisfies the required axioms of a filtered category then follows from the
fact that for all $X\in\operatorname{\mathscr{C}}_{0}$ the corresponding
identity holds. For example, that $\alpha=s\ast\alpha\ast s$ holds follows
from the fact that for all $X\in\operatorname{\mathscr{C}}_{0}$,
$\,{}_{X}{\alpha}=\,_{X}{s}\ast\,_{X}{\alpha}\ast\,_{X}{s^{-1}}.$ ∎

######  Definition 3.4.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a triangulated pseudofunctor. If each category $F(X)$ is filtered for all
$X\in\operatorname{\mathscr{C}}_{0}$ and if $F(f)$ is an $f$-functor for all
$f\in\operatorname{\mathscr{C}}_{1}$ then we say that $F$ is a triangulated
filtered pseudofunctor.

######  Remark 3.5.

If $F$ is a filtered triangulated pseudofunctor then the assumptions imply
that there are pseudofunctors $F^{\operatorname{fil}\leq 0},$
$F^{\operatorname{fil}\geq 0},$ and $F^{\operatorname{fil}\leq 0}\cap
F^{\operatorname{fil}\geq 0}$ which are defined by

$F^{\operatorname{fil}\geq 0}(X):=F(X)^{\operatorname{fil}\geq 0},\quad
F^{\operatorname{fil}\geq 0}(\varphi:X\to
Y):=F(\varphi)|_{F(X)^{\operatorname{fil}\geq 0}},$

and similarly for each of the other cases. These pseudofunctors have induced
inclusion pseudonatural transformations which fit into a commuting diagram of
triangulated pseudofunctors

${F^{\operatorname{fil}\geq 0}\cap F^{\operatorname{fil}\leq
0}}$${F^{\operatorname{fil}\leq 0}}$${F^{\operatorname{fil}\geq
0}}$${F}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\geq
0,\operatorname{fil}\leq
0}}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\geq
0,\operatorname{fil}\leq 0}^{\operatorname{fil}\leq
0}}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\geq
0,\operatorname{fil}\leq 0}^{\operatorname{fil}\geq
0}}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\leq
0}}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\geq 0}}$

and satisfy

$\operatorname{\mathbf{PC}}\left(F^{\operatorname{fil}\leq
0}\right)=\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\leq
0},\quad\operatorname{\mathbf{PC}}\left(F^{\operatorname{fil}\geq
0}\right)=\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\geq 0},$

and

$\operatorname{\mathbf{PC}}\left(F^{\operatorname{fil}\geq 0}\cap
F^{\operatorname{fil}\leq
0}\right)=\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\geq
0}\cap\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\leq 0}.$

Note also that each of the inclusion pseudonatural transformations are
triangulated in the sense that each functor $\,{}_{X}{\operatorname{incl}}$ is
triangulated for all $X\in\operatorname{\mathscr{C}}_{0}$. These then
consequently fit into a commuting diagram of triangulated categories

${\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\geq
0}\cap\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\leq
0}}$${\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\leq
0}}$${\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\geq
0}}$${\operatorname{\mathbf{PC}}(F)}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\geq
0,\operatorname{fil}\leq
0}}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\geq
0,\operatorname{fil}\leq 0}^{\operatorname{fil}\leq
0}}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\geq
0,\operatorname{fil}\leq 0}^{\operatorname{fil}\geq
0}}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\leq
0}}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\geq 0}}$

with, of course, $\operatorname{\mathbf{PC}}(F)$ a filtered triangulated
category.

We now recall what it means to have a filtered category of a triangulated
category.

######  Definition 3.6 ([3, Definition A.1.c]).

Let $\operatorname{\mathscr{C}}$ be a triangulated category. A filtered
category over $\operatorname{\mathscr{C}}$ (also known as an $f$-category over
$\operatorname{\mathscr{C}}$) is a filtered category
$\operatorname{\mathscr{T}}$ together with a triangulated equivalence of
triangulated categories

$i:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{T}}^{\operatorname{fil}\geq
0}\cap\operatorname{\mathscr{T}}^{\operatorname{fil}\leq 0}.$

We now show that if we have a filtered triangulated pseudofunctor $F$ and a
triangulated pseudofunctor $T$ for which each fibre category $F(X)$ is an
$f$-category over $T(X)$ for which the requisite equivalences vary
pseudonaturally, then $\operatorname{\mathbf{PC}}(F)$ is an $f$-category over
$\operatorname{\mathbf{PC}}(T)$.

######  Proposition 3.7.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor and let
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a triangulated. If:

  1. (1)

For each $X\in\operatorname{\mathscr{C}}_{0}$ we have triangulated
equivalences of categories

$\,{}_{X}{i}:T(X)\xrightarrow{\simeq}F(X)^{\operatorname{fil}\geq 0}\cap
F(X)^{\operatorname{fil}\leq 0};$

  2. (2)

The equivalences $\,{}_{X}{i}$ vary pseudonaturally in
$\operatorname{\mathscr{C}}^{\operatorname{op}}$;

then $\operatorname{\mathbf{PC}}(F)$ is an $f$-category over
$\operatorname{\mathbf{PC}}(T)$.

###### Proof.

The assumptions of the proposition imply that there is a pseudonatural
equivalence

$i:T\xRightarrow{\simeq}F^{\operatorname{fil}\geq 0}\cap
F^{\operatorname{fil}\leq 0}$

which extends to a triangulated equivalence of categories

$\operatorname{\mathbf{PC}}(i):\operatorname{\mathbf{PC}}(T)\xrightarrow{\simeq}\operatorname{\mathbf{PC}}\left(F^{\operatorname{fil}\geq
0}\cap F^{\operatorname{fil}\leq
0}\right)=\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\geq
0}\cap\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\leq 0};$

that $\operatorname{\mathbf{PC}}(i)$ is triangulated follows from the fact
that each functor $\,{}_{X}{i}$ is triangulated and the triangulations on
$\operatorname{\mathbf{PC}}(T)$ and
$\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\geq
0}\cap\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\leq 0}$ are locally
induced by the triangulations on the categories $T(X)$ and
$F(X)^{\operatorname{fil}\geq 0}\cap F(X)^{\operatorname{fil}\leq 0}$ by [14,
Theorem 5.1.10]. ∎

This motivates the definition of what we call an $f$-pseudofunctor over
another triangulated pseudofunctor.

######  Definition 3.8.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor and let
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a triangulated. If:

  1. (1)

For each $X\in\operatorname{\mathscr{C}}_{0}$ we have triangulated
equivalences of categories

$\,{}_{X}{i}:T(X)\xrightarrow{\simeq}F(X)^{\operatorname{fil}\geq 0}\cap
F(X)^{\operatorname{fil}\leq 0};$

  2. (2)

The equivalences $\,{}_{X}{i}$ vary pseudonaturally in
$\operatorname{\mathscr{C}}^{\operatorname{op}}$;

then we say that $F$ is a filtered triangulated pseudofunctor (or an
$f$-pseudofunctor) over $T$.

In this new terminology, Proposition 3.7 can be restated as saying that if $F$
is an $f$-pseudofunctor over a triangulated pseudofunctor $T$, then
$\operatorname{\mathbf{PC}}(F)$ is an $f$-category over the triangulated
category $\operatorname{\mathbf{PC}}(T)$.

Continuing to follow [3, Appendix A], we now assume that we have an
$f$-pseudofunctor
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
over a triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$.
Because by Proposition 3.7 $\operatorname{\mathbf{PC}}(F)$ is an $f$-category
over $\operatorname{\mathbf{PC}}(T)$, for every
$n\in\operatorname{\mathbb{Z}}$ there are adjunctions

${\operatorname{\mathbf{PC}}(F)}$${\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\geq
n}}$$\scriptstyle{\sigma_{\operatorname{fil}\geq
n}}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\geq n}}$$\dashv$

and

${\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\leq
n}}$${\operatorname{\mathbf{PC}}(F)}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\leq
n}}$$\scriptstyle{\sigma_{\operatorname{fil}\leq n}}$$\dashv$

for which each of the functors $\sigma_{\operatorname{fil}\geq n}$ and
$\sigma_{\operatorname{fil}\leq n}$ are both triangulated by [3, Proposition
A.3.i]. Furthermore, by construction and the fact that $F$ is an
$f$-pseudofunctor over $T$, we have local adjunctions

${F(X)}$${F(X)^{\operatorname{fil}\geq
n}}$$\scriptstyle{\,{}_{X}{\sigma_{\operatorname{fil}\geq
n}}}$$\scriptstyle{\,{}_{X}{\operatorname{incl}_{\operatorname{fil}\geq
n}}}$$\dashv$

and:

${F(X)^{\operatorname{fil}\leq
n}}$${F(X)}$$\scriptstyle{\,{}_{X}{\operatorname{incl}_{\operatorname{fil}\leq
n}}}$$\scriptstyle{\,{}_{X}{\sigma_{\operatorname{fil}\leq n}}}$$\dashv$

Furthermore, because the functors
$\,{}_{X}{\operatorname{incl}_{\operatorname{fil}\geq n}}$ and
$\,{}_{X}{\operatorname{incl}_{\operatorname{fil}\leq n}}$ are the object
functors of pseudonatural transformations
$\underline{{\operatorname{incl}}}_{\operatorname{fil}\geq
n}:F^{\operatorname{fil}\geq n}\Rightarrow F$ and
$\underline{{\operatorname{incl}}}_{\operatorname{fil}\leq
n}:F^{\operatorname{fil}\leq n}\Rightarrow F$, it follows that the
$\,{}_{X}{\sigma_{\operatorname{fil}\geq n}}$ and
$\,{}_{X}{\sigma_{\operatorname{fil}\geq n}}$ are the object functors of
pseudonatural transformations

$\underline{{\sigma}}_{\operatorname{fil}\geq n}:F\Rightarrow
F^{\operatorname{fil}\geq n},\quad\underline{{\sigma}}_{\operatorname{fil}\leq
n}:F\Rightarrow F^{\operatorname{fil}\leq n}.$

It then follows from [14, Theorem 4.1.17] that we have adjunctions

$\underline{{\operatorname{incl}}}_{\operatorname{fil}\geq
n}\dashv\underline{{\sigma}}_{\operatorname{fil}\geq
n}:F^{\operatorname{fil}\geq n}\Rightarrow F$

and

$\underline{{\sigma}}_{\operatorname{fil}\leq
n}\dashv\underline{{\operatorname{incl}}}_{\operatorname{fil}\leq
n}:F\Rightarrow F^{\operatorname{fil}\leq n}$

in the $2$-category
$\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})$.
Consequently we deduce that the adjoints

$\operatorname{incl}_{\operatorname{fil}\geq
n}\dashv\sigma_{\operatorname{fil}\geq n}:PC(F)^{\operatorname{fil}\geq
n}\to\operatorname{\mathbf{PC}}(F)$

and

$\sigma_{\operatorname{fil}\leq
n}\dashv\operatorname{incl}_{\operatorname{fil}\leq
n}:\operatorname{\mathbf{PC}}(F)\to\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\leq
n}$

are induced as the psuedoconification of the corresponding adjoints in
$\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})$.
This allows us to in turn deduce the complete pseudocone analogue of [3,
Proposition A.3.i].

######  Proposition 3.9.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor. Then for every
$n\in\operatorname{\mathbb{Z}}$ there are adjunctions

$\operatorname{incl}_{\operatorname{fil}\geq
n}\dashv\sigma_{\operatorname{fil}\geq n}:PC(F)^{\operatorname{fil}\geq
n}\to\operatorname{\mathbf{PC}}(F)$

and

$\sigma_{\operatorname{fil}\leq
n}\dashv\operatorname{incl}_{\operatorname{fil}\leq
n}:\operatorname{\mathbf{PC}}(F)\to\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\leq
n}$

for which:

  1. (1)

Each of the functors $\sigma_{\operatorname{fil}\leq n}$ and
$\sigma_{\operatorname{fil}\geq n}$ are triangulated.

  2. (2)

The functors $\sigma_{\operatorname{fil}\leq n}$ and
$\sigma_{\operatorname{fil}\geq n}$ preserve the categories
$\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\leq n}$ and
$\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\geq n}$, respectively.

  3. (3)

For all $m,n\in\operatorname{\mathbb{Z}}$ there is a natural isomorphism
$\theta_{\operatorname{fil}\leq n}^{\operatorname{fil}\geq m}$ which makes the
diagram

${\operatorname{incl}_{\operatorname{fil}\geq
m}\circ\sigma_{\operatorname{fil}\geq
m}}$${\operatorname{id}_{\operatorname{\mathbf{PC}}(F)}}$${\operatorname{incl}_{\operatorname{fil}\leq
n}\circ\sigma_{\operatorname{fil}\leq
n}}$${\operatorname{incl}_{\operatorname{fil}\geq
m}\circ\sigma_{\operatorname{fil}\geq
m}\circ\operatorname{incl}_{\operatorname{fil}\leq
n}\circ\sigma_{\operatorname{fil}\leq
n}}$${\operatorname{incl}_{\operatorname{fil}\leq
n}\circ\sigma_{\operatorname{fil}\leq
n}\circ\operatorname{incl}_{\operatorname{fil}\geq
m}\circ\sigma_{\operatorname{fil}\geq
m}}$$\scriptstyle{\eta^{\operatorname{fil}\leq
n}\ast(\operatorname{incl}_{\operatorname{fil}\geq
m}\circ\sigma_{\operatorname{fil}\geq
m})}$$\scriptstyle{\varepsilon^{\operatorname{fil}\geq
m}}$$\scriptstyle{\eta^{\operatorname{fil}\leq
n}}$$\scriptstyle{\varepsilon^{\operatorname{fil}\geq
m}\ast(\operatorname{incl}_{\operatorname{fil}\leq
n}\circ\sigma_{\operatorname{fil}\leq
n})}$$\scriptstyle{\theta_{\operatorname{fil}\leq n}^{\operatorname{fil}\geq
m}}$

of functors and natural transformations commute.

###### Proof.

(1): The triangulation of $\sigma_{\operatorname{fil}\leq n}$ and
$\sigma_{\operatorname{fil}\geq n}$ are immediate from [3, Proposition A.3.i]
or, alternatively, the fact that all the object functors of
$\sigma_{\operatorname{fil}\leq
n}=\operatorname{\mathbf{PC}}(\underline{{\sigma}}_{\operatorname{fil}\leq
n})$ and $\sigma_{\operatorname{fil}\geq
n}=\operatorname{\mathbf{PC}}(\underline{{\sigma}}_{\operatorname{fil}\geq
n})$ are triangulated by [3, Proposition A.3.i].

(2): This follows from the fact that for every
$X\in\operatorname{\mathscr{C}}_{0}$, by [3, Proposition A.3.i] the functors
$\,{}_{X}{\sigma_{\operatorname{fil}\leq n}}$ and
$\,{}_{X}{\sigma_{\operatorname{fil}\geq n}}$ preserve
$F(X)^{\operatorname{fil}\leq n}$ and $F(X)^{\operatorname{fil}\geq n}$.

(3): By [3, Proposition A.3.i] for every $X\in\operatorname{\mathscr{C}}_{0}$
there is a unique natural isomorphism $\,{}_{X}{\theta_{\operatorname{fil}\leq
n}^{\operatorname{fil}\geq m}}$ making the diagram

${\,{}_{X}{\operatorname{incl}}_{\operatorname{fil}\geq
m}\circ\,_{X}{\sigma}_{\operatorname{fil}\geq
m}}$${\operatorname{id}_{\operatorname{\mathbf{PC}}(F)}}$${\,{}_{X}{\operatorname{incl}}_{\operatorname{fil}\leq
n}\circ\,_{X}{\sigma}_{\operatorname{fil}\leq
n}}$${\,{}_{X}{\operatorname{incl}}_{\operatorname{fil}\leq
n}\circ\,_{X}{\sigma}_{\operatorname{fil}\leq
n}\circ\,_{X}{\operatorname{incl}}_{\operatorname{fil}\geq
m}\circ\,_{X}{\sigma}_{\operatorname{fil}\geq
m}}$${\,{}_{X}{\operatorname{incl}}_{\operatorname{fil}\geq
m}\circ\,_{X}{\sigma}_{\operatorname{fil}\geq
m}\circ\,_{X}{\operatorname{incl}}_{\operatorname{fil}\leq
n}\circ\,_{X}{\sigma}_{\operatorname{fil}\leq
n}}$$\scriptstyle{\,{}_{X}{\eta}^{\operatorname{fil}\leq
n}\ast(\,_{X}{\operatorname{incl}}_{\operatorname{fil}\geq
m}\circ\,_{X}{\sigma}_{\operatorname{fil}\geq
m})}$$\scriptstyle{\,{}_{X}{\varepsilon}^{\operatorname{fil}\geq
m}}$$\scriptstyle{\,{}_{X}{\eta}^{\operatorname{fil}\leq
n}}$$\scriptstyle{\,{}_{X}{\theta}_{\operatorname{fil}\leq
n}^{\operatorname{fil}\geq
m}}$$\scriptstyle{\,{}_{X}{\varepsilon}^{\operatorname{fil}\geq
m}\ast(\,_{X}{\operatorname{incl}}_{\operatorname{fil}\leq
n}\circ\,_{X}{\sigma}_{\operatorname{fil}\leq n})}$

commute; consequently, if we can show that the collection

$\Theta_{\operatorname{fil}\leq n}^{\operatorname{fil}\geq
m}=\left\\{\,{}_{X}{\theta_{\operatorname{fil}\leq n}^{\operatorname{fil}\geq
m}}\;|\;X\in\operatorname{\mathscr{C}}_{0}\right\\}$

is a modification then setting $\theta_{\operatorname{fil}\leq
n}^{\operatorname{fil}\geq
m}:=\operatorname{\mathbf{PC}}(\Theta_{\operatorname{fil}\leq
n}^{\operatorname{fil}\geq m})$ completes the proof. To this end note that
showing for any $f:X\to Y$ in $\operatorname{\mathscr{C}}$,
$\Theta_{\operatorname{fil}\leq n}^{\operatorname{fil}\geq n}$ being a
modification is equivalent to proving that the equation

$\displaystyle\left(\,{}_{X}{{\theta}_{\operatorname{fil}\leq
n}^{\operatorname{fil}\geq m}}\ast
F(f)\right)\circ\,_{f}{\left(\operatorname{incl}_{\operatorname{fil}\leq
n}\circ\sigma_{\operatorname{fil}\leq
n}\circ\operatorname{incl}_{\operatorname{fil}\geq
m}\circ\sigma_{\operatorname{fil}\geq m}\right)}$
$\displaystyle=\,_{f}{\left(\operatorname{incl}_{\operatorname{fil}\geq
m}\circ\sigma_{\operatorname{fil}\geq
m}\circ\operatorname{incl}_{\operatorname{fil}\leq
n}\circ\sigma_{\operatorname{fil}\leq
n}\right)}\circ\left(F(f)\ast\,_{Y}{\theta_{\operatorname{fil}\leq
n}^{\operatorname{fil}\geq m}}\right)$

commutes follows from the naturality and uniqueness of
$\,{}_{Y}{\theta_{\operatorname{fil}\leq n}^{\operatorname{fil}\geq m}}$ and
$\,{}_{X}{\theta_{\operatorname{fil}\leq n}^{\operatorname{fil}\geq m}}$. ∎

These natural transformations and functors allow us to define a pseudonatural
transformation $F\Rightarrow T$ which object-locally gives the $n$-th graded
component of $F(X)$ as a filtered category over $T(X)$.

######  Lemma 3.10.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over a triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and let $\underline{{j}}:F^{\operatorname{fil}\geq 0}\cap
F^{\operatorname{fil}\leq 0}\Rightarrow T$ be an inverse pseudonatural
equivalence to $\underline{{i}}:T\Rightarrow F^{\operatorname{fil}\geq 0}\cap
F^{\operatorname{fil}\leq 0}$. For any $n\in\operatorname{\mathbb{Z}}$ there
is an $n$-th graded component pseudonatural transformation:

${\operatorname{\mathscr{C}}^{\operatorname{op}}}$${\operatorname{\mathfrak{Cat}}}$$\scriptstyle{F}$$\scriptstyle{T}$$\scriptstyle{\underline{{\operatorname{gr}}}_{f}^{n}}$

In particular this gives rise to an $n$-th graded component functor
$\operatorname{gr}_{\operatorname{fil}}^{n}:\operatorname{\mathbf{PC}}(F)\to\operatorname{\mathbf{PC}}(T)$.

###### Proof.

We define $\underline{{\operatorname{gr}}}_{f}^{n}$ by

$\underline{{\operatorname{gr}}}_{f}^{n}:=\underline{{j}}\circ\underline{{s}}^{-n}\circ\underline{{\operatorname{incl}}}_{\operatorname{fil}\leq
n}\circ\underline{{\sigma}}_{\operatorname{fil}\leq
n}\circ\underline{{\operatorname{incl}}}_{\operatorname{fil}\geq
n}\circ\underline{{\sigma}}_{\operatorname{fil}\geq n}.$

That this is justified as the $n$-th graded component pseudonatural
transformation follows both from the fact that for all
$X\in\operatorname{\mathscr{C}}_{0}$, the $X$-local component is given by the
functor

$\,{}_{X}{\underline{{\operatorname{gr}}}_{f}^{n}}=\,_{X}{j}\circ\,_{X}{{s}^{-n}}\circ\,_{X}{{\operatorname{incl}}_{\operatorname{fil}\leq
n}}\circ\,_{X}{{\sigma}_{\operatorname{fil}\leq
n}}\circ\,_{X}{{\operatorname{incl}}_{\operatorname{fil}\geq
n}}\circ\underline{{\sigma}}_{\operatorname{fil}\geq n},$

which is the $n$-th graded functor $F(X)\to T(X)$ described in [3, Proposition
A.3.i]. The final statement of the lemma follows by taking
$\operatorname{gr}_{\operatorname{fil}}^{n}:=\operatorname{\mathbf{PC}}(\operatorname{gr}_{\operatorname{fil}}^{n})$.
∎

######  Remark 3.11.

As we have seen, it is frequently important to use the functors
$\sigma_{\operatorname{fil}\leq n}$ and $\sigma_{\operatorname{fil}\geq n}$
followed by composition with the inclusions
$\operatorname{incl}_{\operatorname{fil}\leq n}$ and
$\operatorname{incl}_{\operatorname{fil}\geq n}$. Because of this, we let
$\Sigma_{\operatorname{fil}\geq
k}:=\operatorname{incl}_{\operatorname{fil}\geq
k}\circ\sigma_{\operatorname{fil}\geq k}$ and $\Sigma_{\operatorname{fil}\leq
k}:=\operatorname{incl}_{\operatorname{fil}\leq
k}\circ\sigma_{\operatorname{fil}\geq k}$ for all
$k\in\operatorname{\mathbb{Z}}$ in order to save space while also being
categorically rigorous. Note that this also has the benefit of allowing us to
rewrite $\operatorname{gr}_{\operatorname{fil}}^{n}$ as $j\circ
s^{-n}\circ\Sigma_{\operatorname{fil}\leq
n}\circ\Sigma_{\operatorname{fil}\geq n}.$

We now give a pseudoconical version of [3, Proposition A.3.ii]; however, we
don’t _just_ want to know that there is a distinguished triangle

${\left(\Sigma_{\operatorname{fil}\geq
1}\right)(A)}$${A}$${\left(\Sigma_{\operatorname{fil}\leq
0}\right)(A)}$${\left(\Sigma_{\operatorname{fil}\geq
1}\right)(A)[1]}$$\scriptstyle{\varepsilon^{\operatorname{fil}\geq
1}_{A}}$$\scriptstyle{\eta_{A}^{\operatorname{fil}\leq 0}}$$\scriptstyle{d}$

in $\operatorname{\mathbf{PC}}(F)$ for any object $A$. Instead, we want to
know that this distinguished triangle comes from a family of distinguished
triangles in each category $F(X)$.

######  Proposition 3.12.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over a triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$.
If $A$ is any object in $\operatorname{\mathbf{PC}}(F)_{0}$ then there is a
unique morphism $d:(\Sigma_{\operatorname{fil}\leq
0})(A)\to(\Sigma_{\operatorname{fil}\geq 1})(A)[1]$ for which there is a
distinguished triangle:

${(\Sigma_{\operatorname{fil}\geq
1})(A)}$${A}$${(\Sigma_{\operatorname{fil}\leq
0})(A)}$${(\Sigma_{\operatorname{fil}\geq
1})(A)[1]}$$\scriptstyle{\varepsilon^{\operatorname{fil}\geq
1}_{A}}$$\scriptstyle{\eta_{A}^{\operatorname{fil}\leq 0}}$$\scriptstyle{d}$

Furthermore, for any $X\in\operatorname{\mathscr{C}}_{0}$ there are
distinguished triangles

${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\geq
1}}\right)\left(\,{}_{X}{A}\right)}$${\,{}_{X}{A}}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\leq
0}}\right)\left(\,{}_{X}{A}\right)}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\geq
1}}\right)\left(\,{}_{X}{A}\right)[1]}$$\scriptstyle{\,{}_{X}{\varepsilon^{\operatorname{fil}\geq
1}}}$$\scriptstyle{\,{}_{X}{\eta^{\operatorname{fil}\leq
0}}}$$\scriptstyle{\,{}_{X}{d}}$

and if $f:X\to Y$ is any morphism in $\operatorname{\mathscr{C}}$ the diagram

${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\geq
1}}\right)\left(F(f)(\,_{Y}{A})\right)}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\geq
1}}\right)\left(\,{}_{X}{A}\right)}$${\,{}_{Y}{A}}$${\,{}_{X}{A}}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\leq
0}}\right)\left(F(f)\,_{Y}{A}\right)}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\leq
0}}\right)\left(\,{}_{X}{A}\right)}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\geq
1}}\right)\left(F(f)(\,_{Y}{A})\right)}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\geq
1}}\right)\left(\,{}_{X}{A}\right)[1]}$$\scriptstyle{\,{}_{X}{\varepsilon^{\operatorname{fil}\geq
1}_{F(f)A}}}$$\scriptstyle{\,{}_{X}{\varepsilon_{A}^{\operatorname{fil}\geq
1}}}$$\scriptstyle{\,{}_{X}{\eta_{F(f)A}^{\operatorname{fil}\leq
0}}}$$\scriptstyle{\tau_{f}^{A}}$$\scriptstyle{\,{}_{X}{\eta^{\operatorname{fil}\leq
0}}}$$\scriptstyle{F(f)\left(\,{}_{Y}{d}\right)}$$\scriptstyle{\,{}_{X}{d}}$

commutes with every horizontal map the corresponding functor applied to
$\tau_{f}^{A}$ or $\tau_{f}^{A[1]}$.

###### Proof.

Applying [3, Proposition A.3.ii] gives a unique morphism
$d:(\Sigma_{\operatorname{fil}\leq 0})(A)\to(\Sigma_{\operatorname{fil}\geq
1})(A)[1]$ for which there is a distinguished triangle

${(\Sigma_{\operatorname{fil}\geq
1})(A)}$${A}$${(\Sigma_{\operatorname{fil}\leq
0})(A)}$${(\Sigma_{\operatorname{fil}\geq
1})(A)[1]}$$\scriptstyle{\varepsilon^{\operatorname{fil}\geq
1}_{A}}$$\scriptstyle{\eta_{A}^{\operatorname{fil}\leq 0}}$$\scriptstyle{d}$

in $\operatorname{\mathbf{PC}}(F)$. By the construction of the triangulation
on $\operatorname{\mathbf{PC}}(F)$ given in [14, Theorem 5.1.10] determined by
saying a triangle is distinguished if and only if the corresponding object-
local triangles are distinguished, we see that for every object $X$ of
$\operatorname{\mathscr{C}}$ there is then a distinguished triangle:

${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\geq
1}}\right)\left(\,{}_{X}{A}\right)}$${\,{}_{X}{A}}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\leq
0}}\right)\left(\,{}_{X}{A}\right)}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\geq
1}}\right)\left(\,{}_{X}{A}\right)[1]}$$\scriptstyle{\,{}_{X}{\varepsilon^{\operatorname{fil}\geq
1}}}$$\scriptstyle{\,{}_{X}{\eta^{\operatorname{fil}\leq
0}}}$$\scriptstyle{\,{}_{X}{d}}$

Note that the map $\,{}_{X}{d}$ is then the $X$-local version of the unique
morphism of [3, Proposition A.3.ii]. Now because $A$ is an object of
$\operatorname{\mathbf{PC}}(F)$, for any morphism $f:X\to Y$ in
$\operatorname{\mathscr{C}}$ the diagram

${F(f)\left(\left(\,{}_{Y}{\Sigma_{\operatorname{fil}\geq
1}}\right)\left(\,{}_{Y}{A}\right)\right)}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\geq
1}}\right)\left(\,{}_{X}{A}\right)}$${\,{}_{Y}{A}}$${\,{}_{X}{A}}$${F(f)\left(\,{}_{Y}{\Sigma_{\operatorname{fil}\leq
0}}\right)\left(\,{}_{Y}{A}\right)}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\leq
0}}\right)\left(\,{}_{X}{A}\right)}$${F(f)\left(\left(\,{}_{Y}{\Sigma_{\operatorname{fil}\geq
1}}\right)\left((\,_{Y}{A})\right)\right)}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\geq
1}}\right)\left(\,{}_{X}{A}\right)[1]}$$\scriptstyle{F(f)\left(\,{}_{Y}{\varepsilon^{\operatorname{fil}\geq
1}_{A}}\right)}$$\scriptstyle{\,{}_{X}{\varepsilon_{A}^{\operatorname{fil}\geq
1}}}$$\scriptstyle{F(f)\left(\,{}_{Y}{\eta_{A}^{\operatorname{fil}\leq
0}}\right)}$$\scriptstyle{\tau_{f}^{A}}$$\scriptstyle{\,{}_{X}{\eta^{\operatorname{fil}\leq
0}}}$$\scriptstyle{\,{}_{X}{d_{F(f)A}}}$$\scriptstyle{\,{}_{X}{d}}$

commutes with each horizontal arrow a transition isomorphism. Finally the
commutativity of the last stated diagram in the lemma follows by using
pseudonatural transformation commutativity witness isomorphisms. ∎

We now want to build a pseudonatural transformation

${\operatorname{\mathscr{C}}^{\operatorname{op}}}$${\operatorname{\mathfrak{Cat}}}$$\scriptstyle{F}$$\scriptstyle{T}$$\scriptstyle{\underline{{\omega}}}$

which interacts well with each inclusion functor, as this transformation is
crucial to construction of the pseudocone realization functor in the same way
that the corresponding functor $\omega$ allows us to build the realization
functor in [3]. For this we recall the construction and description of the
$\omega$ functors in the conventional case.

######  Proposition 3.13 ([3, Proposition A.3.iii]).

Let $\operatorname{\mathscr{C}}$ be a filtered triangulated category over a
triangulated category $\operatorname{\mathscr{A}}$. There is then a functor
$\omega:\operatorname{\mathscr{C}}\to\operatorname{\mathscr{A}}$ such that:

  1. (1)

There is an adjunction:

${\operatorname{\mathscr{C}}^{\operatorname{fil}\leq
0}}$${\operatorname{\mathscr{A}}}$$\scriptstyle{\omega\circ\operatorname{incl}_{\operatorname{fil}\leq
0}}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\geq
0,\operatorname{fil}\leq 0}^{\operatorname{fil}\leq 0}\circ i}$$\dashv$

  2. (2)

There is an adjunction:

${\operatorname{\mathscr{A}}}$${\operatorname{\mathscr{C}}^{\operatorname{fil}\geq
0}}$$\scriptstyle{\operatorname{incl}_{\operatorname{fil}\geq
0,\operatorname{fil}\leq 0}^{\operatorname{fil}\geq 0}\circ
i}$$\scriptstyle{\omega\circ\operatorname{incl}_{\operatorname{fil}\geq
0}}$$\dashv$

  3. (3)

For any object $X$ of $\operatorname{\mathscr{A}}$, the morphism
$\omega(\alpha_{iX}):\omega(iX)\to\omega(s(iX))$ is an isomorphism.

  4. (4)

For all objects $X$ of $\operatorname{\mathscr{C}}^{\operatorname{fil}\leq 0}$
and all objects $Y$ of $F^{\operatorname{fil}\geq 0}$, the map

$\operatorname{\mathscr{C}}(A,B)\to\operatorname{\mathscr{A}}(\omega(A),\omega(B))$

given by $\varphi\mapsto\omega(\varphi)$ is an isomorphism.

Because of this proposition, to construct the pseudonatural version of
$\omega$ (and hence the pseudoconical version by applying the $2$-functor
$\operatorname{\mathbf{PC}}$) we need to prove that these functors $\omega$
vary when in the presence of appropriate structure. With this in mind, assume
that we have a filtered triangulated pseudofunctor
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
over a triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$.
Asking that the $\omega$ vary suitably asks that if we write
$\,{}_{Z}{\omega}:F(Z)\to T(Z)$ for each object $Z$ in
$\operatorname{\mathscr{C}}$ as the functor of Proposition 3.13, then for any
morphism $f:X\to Y$ in $\operatorname{\mathscr{C}}$, there be a natural
isomorphism $\,{}_{f}{\omega}$ fitting into the $2$-cell

${F(Y)}$${F(X)}$${T(Y)}$${T(X)}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{Y}{\omega}}$$\scriptstyle{\,{}_{X}{\omega}}$$\scriptstyle{T(f)}$$\scriptstyle{\,{}_{f}{\omega}}$

which varies pseudonaturally in
$\operatorname{\mathscr{C}}^{\operatorname{op}}$. If they all exist, ensuring
the adjoints are defined so as to be pseudonatural is no major issue; adjoints
are unique up to unique isomorphism, so we can always redefine them if
necessary to satisfy the coherence requirements. As such we simply need to
ensure their existence. For this it is worth noting that since the functors
$\omega$ are uniquely determined by either properties (1) and (3) or (2) and
(3) in Proposition 3.13, it suffices in this case to show the existence of a
natural isomorphism $\,{}_{f}{\omega^{\operatorname{fil}\leq 0}}$ or
$\,{}_{f}{\omega^{\operatorname{fil}\geq 0}}$ and then deduce the existence of
$\,{}_{f}{\omega}$ by extending. To this end note that since each of
$\operatorname{incl}_{\operatorname{fil}\geq 0,\operatorname{fil}\leq
0}^{\operatorname{fil}\leq 0}$, $\operatorname{incl}_{\operatorname{fil}\leq
0}$, and $i$ are psueodnatural transformations we have invertible $2$-cells

${T(Y)}$${T(X)}$${F(Y)^{\operatorname{fil}\leq
0}}$${F(X)^{\operatorname{fil}\leq
0}}$$\scriptstyle{T(f)}$$\scriptstyle{\,{}_{Y}{\operatorname{incl}_{\operatorname{fil}\geq
0,\operatorname{fil}\leq 0}^{\operatorname{fil}\leq
0}}\circ\,_{Y}{i}}$$\scriptstyle{\,{}_{X}{\operatorname{incl}_{\operatorname{fil}\geq
0,\operatorname{fil}\leq 0}^{\operatorname{fil}\leq
0}}\circ\,_{X}{i}}$$\scriptstyle{F^{\operatorname{fil}\leq
0}(f)}$$\scriptstyle{\,{}_{f}{\left(\operatorname{incl}_{\operatorname{fil}\geq
0,\operatorname{fil}\leq 0}^{\operatorname{fil}\leq 0}\circ i\right)}}$

for every morphism $f:X\to Y$ in $\operatorname{\mathscr{C}}$. Because the
vertical arrows are left adjoints to the functors
$\,{}_{Y}{\omega}\circ\,_{Y}{\operatorname{incl}_{\operatorname{fil}\leq 0}}$
and $\,{}_{X}{\omega}\circ\,_{X}{\operatorname{incl}_{\operatorname{fil}\leq
0}}$, respectively, we then can also deduce the existence of invertible
$2$-cells

${F(Y)^{\operatorname{fil}\leq 0}}$${F(X)^{\operatorname{fil}\leq
0}}$${T(Y)}$${T(X)}$$\scriptstyle{\,{}_{Y}{\omega}\circ\,_{Y}{\operatorname{incl}_{\operatorname{fil}\leq
0}}}$$\scriptstyle{F^{\operatorname{fil}\leq
0}(f)}$$\scriptstyle{\,{}_{X}{\omega}\circ\,_{X}{\operatorname{incl}_{\operatorname{fil}\leq
0}}}$$\scriptstyle{T(f)}$$\scriptstyle{\,{}_{f}{\omega^{\operatorname{fil}\leq
0}}}$

Extending this to the existence of the natural isomorphisms $\,{}_{f}{\omega}$
then allow us to deduce the proposition below.

######  Proposition 3.14.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over a triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$.
Then there is a pseudonatural transformation

${\operatorname{\mathscr{C}}^{\operatorname{op}}}$${\operatorname{\mathfrak{Cat}}}$$\scriptstyle{F}$$\scriptstyle{T}$$\scriptstyle{\underline{{\omega}}}$

such that:

  1. (1)

There is an adjunction

${\operatorname{\mathscr{C}}^{\operatorname{op}}}$${\operatorname{\mathfrak{Cat}}}$$\scriptstyle{F}$$\scriptstyle{F}$$\scriptstyle{\underline{{\omega}}\circ\underline{{\operatorname{incl}}}_{\operatorname{fil}\leq
0}}$$\scriptstyle{\underline{{\operatorname{incl}}}_{\operatorname{fil}\geq
0,\operatorname{fil}\geq 0}^{\operatorname{fil}\leq
0}\circ\underline{{i}}}$$\dashv$

in
$\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})$.

  2. (2)

There is an adjunction

${\operatorname{\mathscr{C}}^{\operatorname{op}}}$${\operatorname{\mathfrak{Cat}}}$$\scriptstyle{F}$$\scriptstyle{F}$$\scriptstyle{\underline{{\operatorname{incl}}}_{\operatorname{fil}\geq
0,\operatorname{fil}\geq 0}^{\operatorname{fil}\geq
0}\circ\underline{{i}}}$$\scriptstyle{\underline{{\omega}}\circ\underline{{\operatorname{incl}}}_{\operatorname{fil}\geq
0}}$$\dashv$

in
$\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})$.

  3. (3)

The modification

${\operatorname{\mathscr{C}}^{\operatorname{op}}}$${\operatorname{\mathfrak{Cat}}}$$\scriptstyle{F}$$\scriptstyle{T}$$\scriptstyle{\underline{{\omega}}\circ\underline{{s}}}$$\scriptstyle{\underline{{\omega}}}$$\underset{\underline{{\omega}}\ast\underline{{\alpha}}}{\Rrightarrow}$

is an isomorphism in
$\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})(F,F)$.

  4. (4)

For every object $A$ of $\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\leq
0}$ and $B$ of $\operatorname{\mathbf{PC}}(F)^{\operatorname{fil}\geq 0}$, the
map

$\operatorname{\mathbf{PC}}(F)(A,B)\to\operatorname{\mathbf{PC}}(T)(\omega(A),\omega(B))$

given object-locally by $\varphi\mapsto\,_{X}{\omega}(\varphi)$ is an
isomorphism.

###### Proof.

The existence of $\underline{{\omega}}$ follows from the discussion prior ot
the statement of the proposition. From here both (1) and (2) follow from the
object-local adjunctions asserted by by Proposition 3.13 and an appeal to [14,
Theorem 4.1.17]. Item (3) holds because each natural map
$(\omega\ast\alpha)_{X}=\,_{X}{\omega}(\,_{X}{\alpha}_{-})$ is an isomorphism.
Finally, (4) follows from the fact that each corresponding object-local map is
an isomorphism. ∎

We now extend these constructions to the case when the filtered triangulated
category $\operatorname{\mathbf{PC}}(F)$ and the triangulated category
$\operatorname{\mathbf{PC}}(T)$ come equipped with $t$-structures which are
compatible with the relevant pseudofunctorial constructions. This requires us
to both recall what it means to have compatible $t$-structures on filtered
triangulated categories as well as extend this to the pseudofunctorial and
pseudoconical case of this paper.

######  Definition 3.15 ([3, Definition A.4]).

Let $\operatorname{\mathscr{A}}$ be a filtered triangulated category over a
triangulated category $\operatorname{\mathscr{T}}$ and assume that both
$\operatorname{\mathscr{A}}$ and $\operatorname{\mathscr{T}}$ come equipped
with $t$-structures. We say that the $t$-structures and filtrations are
compatible if the functor

$\operatorname{incl}\circ
i:\operatorname{\mathscr{T}}\to\operatorname{\mathscr{A}}$

is $t$-exact and if there is an equivalence

$s(\operatorname{\mathscr{A}}^{t\leq
0})\simeq\operatorname{\mathscr{A}}^{t\leq-1}$

which commutes with the corresponding inclusion functors into
$\operatorname{\mathscr{A}}$ in the sense that there is an invertible
$2$-cell:

${s(\operatorname{\mathscr{A}}^{t\leq
0})}$${\operatorname{\mathscr{A}}}$${\operatorname{\mathscr{A}}^{t\leq-1}}$$\scriptstyle{\operatorname{incl}}$$\scriptstyle{\simeq}$$\scriptstyle{\operatorname{incl}}$$\scriptstyle{\cong}$

######  Remark 3.16.

In [3, Definition A.4] it is only required that
$\operatorname{\mathscr{A}}^{t\leq-1}=s(\operatorname{\mathscr{A}}^{t\leq
0})$; however, we have asked a weaker condition because the strict equality is
too, well, strict for the higher categorical formalism we use in this paper.
It is also worth noting that the invertible $2$-cell above is not an
additional ask; it exists by virtue of the given equivalence.

Note that in the above definition we make the convention that we write
$\operatorname{\mathscr{A}}^{t\leq n}$ and $\operatorname{\mathscr{A}}^{t\geq
n}$ for the $t$-structure graded subcategories of $\operatorname{\mathscr{A}}$
and $\operatorname{\mathscr{A}}^{\operatorname{fil}\leq n}$ and
$\operatorname{\mathscr{A}}^{\operatorname{fil}\geq n}$ for the filtration
induced subcategories of the filtered triangulated category
$\operatorname{\mathscr{A}}$. For the pseudofunctorial convention, we need to
replace filtered triangulated categories equipped with a $t$-structure with
truncated pseudofunctors (cf. Definition 2.13) which are also filtered
triangulated categories (cf. Definition 3.8).

Let us discuss some of this structure which will be interacting. Assume that
we have a filtered triangulated pseudofunctor
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
over a triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
for which both $F$ and $T$ are also truncated pseuodofunctors. Now if we want
to extend the first condition, namely asking for the functor
$\operatorname{\mathscr{T}}\to\operatorname{\mathscr{A}}$ to be $t$-exact, we
need not do any non-naïve work. Because both $F$ and $T$ are assumed to be
truncated, if we assume that the functors
$\,{}_{X}{\operatorname{incl}}\circ\,_{X}{i}:T(X)\to F(X)$ are $t$-exact for
every object $X\in\operatorname{\mathscr{C}}_{0}$, then for every morphism
$f:X\to Y$, each functor in the diagram

${T(Y)}$${F(Y)}$${T(X)}$${F(X)}$$\scriptstyle{\,{}_{Y}{\operatorname{incl}}\circ\,_{Y}{i}}$$\scriptstyle{T(f)}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{X}{\operatorname{incl}}\circ\,_{X}{i}}$

is $t$-exact. Consequently this gives us that the functor
$\operatorname{incl}\circ
i:\operatorname{\mathbf{PC}}(T)\to\operatorname{\mathbf{PC}}(F)$ is $t$-exact
and so provides us with the correct notion, at least as far as this paper is
concerned, of being inclusion compatible. The more difficult aspect lies in
understanding and unwinding the equivalence condition.

We begin our extension of the equivalence condition by first seeing if we can
define a subpseudofunctor $S$ of $F$ determined by object-locally taking
$S(X)$ to be the image of $F(X)^{t\leq 0}$ under the filtration shift functor
$\,{}_{X}{s}$. On objects the definition of $S$ is essentially forced: we
define, for every object $X$ in $\operatorname{\mathscr{C}}$,

$S(X):=\,_{X}{s}\left(F(X)^{t\leq 0}\right).$

To define the fibre functors $S(f):S(Y)\to S(X)$ we need to induce some map
$\,{}_{Y}{s}(F(Y)^{t\leq 0})\to\,_{X}{s}(F(X)^{t\leq 0})$. Here we first use
the fact that since $F$ is a filtered triangulated functor, for every morphism
$f:X\to Y$ we have that $\,{}_{X}{s}\circ F(f)=F(f)\circ\,_{Y}{s}$ so for
every object $A$ of $F(Y)^{t\leq 0}$,

$F(f)\left(\,{}_{Y}{s}(A)\right)=\left(F(f)\,_{Y}{s}\right)(A)=\left(\,{}_{X}{s}\circ
F(f)\right)(A)=\,_{X}{s}\left(F(f)\left(A\right)\right);$

as such if we know that $F(f):F(Y)\to F(X)$ restricts to a functor

$F(f)^{t\leq 0}:F(Y)^{t\leq 0}\to F(X)^{t\leq 0}$

we can define $S(f)$. However, because $F$ is also a truncated pseudofunctor,
$F(f)$ _does_ in fact restrict so for every object $A$ of $F(Y)^{t\leq 0}$,
$\,{}_{X}{s}(F(f)(A))$ is an object of $S(X)$. Thus we define the functor
components by

$S(f):=F(f)|_{S(Y)}$

Composition then follows the rule, if $X\xrightarrow{f}Y\xrightarrow{g}Z$ is a
pair of composable arrows,

$S(f)\circ S(g)=F(f)|_{S(Y)}\circ F(g)|_{S(Z)}=\left(F(f)\circ
F(g)\right)|_{S(Z)}$

so we define the compositor isomorphisms by simply restricting the
$\phi_{f,g}$ as appropriate. With $S$ constructed we note already that by
construction that restricting $s$ to the subpseudofunctor $F^{t\leq 0}$ of $F$
gives a pseudonatural isomorphism

${\operatorname{\mathscr{C}}^{\operatorname{op}}}$${\operatorname{\mathfrak{Cat}}}$$\scriptstyle{F^{t\leq
0}}$$\scriptstyle{S}$$\scriptstyle{\underline{{s}}|_{F^{t\leq 0}}}$

with strict commutativity witnesses. Note that the commutativity witnesses are
strict identities because of the definition of $S(f)$:

${F(Y)^{t\leq
0}}$${S(Y)}$${\,{}_{Y}{s}(F(Y))}$${F(X)}$${S(X)}$${\,{}_{X}{s}(F(X)^{t\leq
0})}$$\scriptstyle{\,{}_{Y}{s}}$$\scriptstyle{F(f)^{t\leq
0}}$$\scriptstyle{S(f)}$$\scriptstyle{F(f)^{t\leq
0}|_{S(Y)}}$$\scriptstyle{\,{}_{X}{s}}$

Now that we have built $S$, the most straightforward aspect of extending the
equivalence condition in Definition 3.15 is to simply require that we have a
pseudonatural equivalence $\underline{{e}}:S\to F^{t\leq-1}$. What remains is
determining the coherence(s) we need to ask $\underline{{e}}$ to satisfy.
However, this is not too rough to determine. First, because both $S$ and
$F^{t\leq-1}$ are subpseudofunctors of $F$ and $\underline{{e}}$ is a
pseudonatural equivalence, is an invertible $2$-cell

${S}$${F}$${F^{t\leq-1}}$$\scriptstyle{\underline{{e}}}$$\scriptstyle{\cong}$

by the existence of $\underline{{e}}$. Consequently, our primary concern lies
in ensuring that the inclusions of $S$ vary compatibly with those of
$F^{t\leq-1}$. This can be done by asking that, for every morphism $f:X\to Y$
in $\operatorname{\mathscr{C}}$ the pasting diagram

${S(Y)}$${F(Y)^{t\leq-1}}$${F(Y)}$${S(X)}$${F(X)^{t\leq-1}}$${F(X)}$$\scriptstyle{\,{}_{Y}{e}}$$\scriptstyle{S(f)}$$\scriptstyle{F(f)^{t\leq-1}}$$\scriptstyle{\,{}_{Y}{\operatorname{incl}_{t\leq-1}}}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{X}{e}}$$\scriptstyle{\,{}_{X}{\operatorname{incl}_{t\leq-1}}}$$\scriptstyle{\,{}_{f}{\operatorname{incl}_{t\leq-1}}}$$\scriptstyle{\,{}_{f}{e}}$

is equivalent to the $2$-cell:

${S(Y)}$${F(Y)}$${S(X)}$${F(X)}$$\scriptstyle{\,{}_{Y}{\operatorname{incl}_{S}}}$$\scriptstyle{S(f)}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{X}{\operatorname{incl}_{S}}}$$\scriptstyle{\,{}_{f}{\operatorname{incl}_{S}}}$

This is the condition we seek, as we see below.

######  Definition 3.17.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over a triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and assume that $F$ and $T$ are both truncated pseudofunctors as well. We say
the truncation structure is compatible with the filtered structure if:

  * •

For all objects $X$ in $\operatorname{\mathscr{C}}$ the functor

$\,{}_{X}{\operatorname{incl}}\circ\,_{X}{i}:T(X)\to F(X)$

is $t$-exact;

  * •

With $S$ the pseudofunctor defined prior to the definition, there is a
pseudonatural equivalence $\underline{{e}}:S\to F^{t\leq-1}$ such that for any
morphism $f:X\to Y$ in $\operatorname{\mathscr{C}}$, the pasting diagram

${S(Y)}$${F(Y)^{t\leq-1}}$${F(Y)}$${S(X)}$${F(X)^{t\leq-1}}$${F(X)}$$\scriptstyle{\,{}_{Y}{e}}$$\scriptstyle{S(f)}$$\scriptstyle{F(f)^{t\leq-1}}$$\scriptstyle{\,{}_{Y}{\operatorname{incl}_{t\leq-1}}}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{X}{e}}$$\scriptstyle{\,{}_{X}{\operatorname{incl}_{t\leq-1}}}$$\scriptstyle{\,{}_{f}{\operatorname{incl}_{t\leq-1}}}$$\scriptstyle{\,{}_{f}{e}}$

is equivalent to the $2$-cell:

${S(Y)}$${F(Y)}$${S(X)}$${F(X)}$$\scriptstyle{\,{}_{Y}{\operatorname{incl}_{S}}}$$\scriptstyle{S(f)}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{X}{\operatorname{incl}_{S}}}$$\scriptstyle{\,{}_{f}{\operatorname{incl}_{S}}}$

######  Lemma 3.18.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over a triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and assume that $F$ and $T$ are truncated pseudofunctors as well. If the
truncation structures on $F$ and $T$ are compatible with the filtered
structure then the $t$-structures on $\operatorname{\mathbf{PC}}(F)$ and
$\operatorname{\mathbf{PC}}(T)$ are compatible in the sense of Definition
3.15.

###### Proof.

The first property we must prove is trivial to show, as that proving the
functor $\operatorname{incl}\circ i:\operatorname{\mathbf{PC}}(T)\to F(X)$ is
$t$-exact is equivalent to checking that for every
$X\in\operatorname{\mathscr{C}}_{0}$ the functor
$\,{}_{X}{\operatorname{incl}}\circ\,_{X}{i}$ is $t$-exact.

In the second case note that because $\underline{{e}}:S\to F^{t\leq 1}$ is a
pseudonatural equivalence,
$e:\operatorname{\mathbf{PC}}(S)\to\operatorname{\mathbf{PC}}(F)^{t\leq-1}$ is
an equivalence. That it commutes with the relevant inclusion functors is then
immediate. ∎

Assume that we have a filtered triangulated pseudofunctor
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
over a triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and that $T$ is truncated. Since each category $T(X)$, for every object
$X\in\operatorname{\mathscr{C}}_{0}$, is equipped with a $t$-structure by
applying [3, Proposition A.5.i] we get that there is a unique $t$-structure on
$F(X)$ for which the $t$-structures on $F(X)$ and $T(X)$ are compatible with
the filtered structure. This $t$-structure is given by declaring that the
objects of the $t\geq 0$ subcategory take the form

$\left(F(X)^{t\geq 0}\right)_{0}=\left\\{A\in
F(X)_{0}\;:\;\,_{X}{\underline{{\operatorname{gr}}}}_{f}^{n}(A)\in\left(T(X)^{t\geq
n}\right)_{0}\right\\}$

and similarly the objects of the $t\leq 0$ subcategory take the form

$\left(F(X)^{t\geq 0}\right)_{0}=\left\\{A\in
F(X)_{0}\;:\;\,_{Y}{\underline{{\operatorname{gr}}}}_{f}^{n}(A)\in\left(T(X)^{t\leq
n}\right)_{0}\right\\}.$

Our goal now is to show that these definitions make $F$ into a truncated
pseudofunctor, as this will allow us to not only control the unique
$t$-structure on $\operatorname{\mathbf{PC}}(F)$ induced by [3, Proposition
A.5.i], but also help us translate the $t$-structure on
$\operatorname{\mathbf{PC}}(F)$ through the filtration and be important for
the realization functor we desire to build. We will do this in a series of
compartmentalized lemmas.

######  Lemma 3.19.

With $F$ and $T$ as above, for any morphism $f:X\to Y$ in
$\operatorname{\mathscr{C}}$ there are functors $F(f)^{t\leq 0}:F(Y)^{t\leq
0}\to F(X)^{t\leq 0}$ and $F(f)^{t\geq 0}:F(Y)^{t\geq 0}\to F(X)^{t\geq 0}$
and natural isomorphisms which fit into invertible $2$-cells

${F(Y)}$${F(X)}$${F(Y)^{t\geq 0}}$${F(X)^{t\geq
0}}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{Y}{\tau_{\geq
0}}}$$\scriptstyle{\,{}_{X}{\tau_{\geq 0}}}$$\scriptstyle{F(f)^{t\geq
0}}$$\scriptstyle{\,{}_{f}{\theta^{t\geq 0}}}$

and:

${F(Y)}$${F(X)}$${F(Y)^{t\leq 0}}$${F(X)^{t\leq
0}}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{Y}{\tau_{\leq
0}}}$$\scriptstyle{\,{}_{X}{\tau_{\leq 0}}}$$\scriptstyle{F(f)^{t\leq
0}}$$\scriptstyle{\,{}_{f}{\theta^{t\leq 0}}}$

###### Proof.

The existence of the truncation functors $\,{}_{Z}{\tau_{\leq 0}}:F(Z)\to
F(Z)^{t\leq 0}$ and $\,{}_{Z}{\tau_{\geq 0}}:F(Z)\to F(Z)^{t\geq 0}$ for every
object $Z$ of $\operatorname{\mathscr{C}}$ follow from the fact that $F(Z)$
has a $t$-structure (cf. [4, Proposition 1.3.3.i]).

To induce the functors $F(f)^{t\leq 0}$ and $F(f)^{t\geq 0}$ we must show that
if $A$ is an object of $F(Y)$ for which $A$ is an object of $F(Y)^{t\leq 0}$
then $F(f)(A)$ is an object of $F(X)^{t\leq 0}$, and similarly for the $t\geq
0$ case. This in turn amounts to showing that if
$\,{}_{Y}{\underline{{\operatorname{gr}}}_{f}^{n}}(A)\in(T(Y)^{t\leq n})_{0}$
then $F(f)(A)$ has the property that
$\,{}_{X}{\underline{{\operatorname{gr}}}_{f}^{n}}(F(f)A)$ is an object of
$T(X)^{t\leq n}$. For this first note that because
$\underline{{\operatorname{gr}}}_{f}^{n}:F\Rightarrow T$ is a pseudonatural
transformation we have isomorphisms

$\left(\,{}_{X}{\underline{{\operatorname{gr}}}_{f}^{n}}\circ
F(f)\right)(A)\cong\left(T(f)\circ\,_{Y}{\underline{{\operatorname{gr}}}_{f}^{n}}\right)(A)=T(f)\left(\left(\,{}_{Y}{\underline{{\operatorname{gr}}}_{f}^{n}}\right)(A)\right).$

Because $T$ is a truncated pseudofunctor, $T(f)$ is left and right $t$-exact
and so necessarily preserves truncation degrees. Thus
$\,{}_{Y}{\underline{{\operatorname{gr}}}_{f}^{n}}(A)$ is an object of
$T(X)^{t\leq n}$. However, because $T(X)^{t\leq n}$ is a strictly full
subcategory of $T(X)$, this implies that
$\,{}_{X}{\underline{{\operatorname{gr}}}_{f}^{n}}\left(F(f)(A)\right)$ is an
object of $T(X)^{t\leq n}$ as well and hence that we have our desired functor
$F(f)^{t\leq 0}$. Finally, the existence of the isomorphism
$\,{}_{f}{\theta^{t\leq 0}}$ follows from the fact that each diagram

${F(Y)^{t\leq 0}}$${F(X)^{t\leq 0}}$${F(Y)}$${F(X)}$$\scriptstyle{F(f)^{t\leq
0}}$$\scriptstyle{\,{}_{Y}{\operatorname{incl}_{t\leq
0}}}$$\scriptstyle{\,{}_{X}{\operatorname{incl}_{t\leq
0}}}$$\scriptstyle{F(f)}$

commutes strictly and because we have adjunctions
$\,{}_{Y}{\operatorname{incl}_{t\leq 0}}\dashv\,_{Y}{\tau_{\leq 0}}$ and
$\,{}_{X}{\operatorname{incl}_{t\leq 0}}\dashv\,_{X}{\tau_{\leq 0}}$. The
existence of the functors $F(f)^{t\geq 0}$ and the commutativity witnesses
$\,{}_{f}{\theta^{t\geq 0}}$ follows similarly save that we use the
adjunctions $\,{}_{Y}{\tau_{\geq 0}}\dashv\,_{Y}{\operatorname{incl}_{t\geq
0}}$ and $\,{}_{X}{\tau_{\geq 0}}\dashv\,_{X}{\operatorname{incl}_{t\geq 0}}$
instead. ∎

######  Lemma 3.20.

The natural isomorphisms $\,{}_{f}{\theta^{t\geq 0}}$ and
$\,{}_{f}{\theta^{t\leq 0}}$ vary pseudonaturally in
$\operatorname{\mathscr{C}}^{\operatorname{op}}$.

###### Proof.

We only prove this for the $t\geq 0$ case, as the $t\leq 0$ case holds
similarly. Begin by letting $X\xrightarrow{f}Y\xrightarrow{g}Z$ be a
composable pair of morphisms in $\operatorname{\mathscr{C}}$ and consider the
pasting diagram:

(3.1) ${F(Z)}$${F(Y)}$${F(X)}$${F(Z)^{t\geq 0}}$${F(Y)}$${F(X)^{t\geq
0}}$$\scriptstyle{F(g)}$$\scriptstyle{\,{}_{Z}{\tau^{\geq
0}}}$$\scriptstyle{\,{}_{Y}{\tau^{\geq
0}}}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{X}{\tau^{\geq
0}}}$$\scriptstyle{F(g)^{t\geq 0}}$$\scriptstyle{F(g\circ f)^{t\geq
0}}$$\scriptstyle{F(f)^{t\geq 0}}$$\scriptstyle{\,{}_{g}{\theta^{t\geq
0}}}$$\scriptstyle{\,{}_{f}{\theta^{t\geq 0}}}$$\scriptstyle{\,{}_{t\geq
0}{\phi_{f,g}}}$

Because the isomorphisms $\,{}_{f}{\theta^{t\geq 0}}$ and
$\,{}_{g}{\theta^{t\geq 0}}$ are constructed from the adjunctions
$\,{}_{X}{\tau_{\geq 0}}\dashv\,_{X}{\operatorname{incl}_{t\geq 0}}$ and
$\,{}_{Y}{\tau_{\geq 0}}\dashv\,_{Y}{\operatorname{incl}_{t\geq 0}}$ together
with the pseudonaturality of $\operatorname{incl}_{t\geq 0}$, it suffices to
show that the pseudonaturality pasting requoriements come from the same
diagrams on the adjoint side, as then the corresponding isomorphisms are
members of a contractible groupoid of isomorphisms. For this, however, we
simply note that because the inclusion functors vary pseudonaturally (and in
fact strictly in this case) we have that the pasting diagrazm

${F(Z)^{t\geq 0}}$${F(Y)^{t\geq 0}}$${F(X)^{t\geq
0}}$${F(Z)}$${F(Y)}$${F(X)}$$\scriptstyle{F(g)^{t\geq
0}}$$\scriptstyle{\,{}_{Z}{\operatorname{incl}_{t\geq
0}}}$$\scriptstyle{F(f)^{t\geq
0}}$$\scriptstyle{\,{}_{Y}{\operatorname{incl}_{t\geq
0}}}$$\scriptstyle{\,{}_{X}{\operatorname{incl}_{t\geq
0}}}$$\scriptstyle{F(g)}$$\scriptstyle{F(g\circ
f)}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{F}{\phi_{f,g}}}$

is equal to the pasting diagram:

${F(Y)^{t\geq 0}}$${F(Z)^{t\geq 0}}$${F(X)^{t\geq
0}}$${F(Z)}$${F(X)}$$\scriptstyle{F(f)^{t\geq
0}}$$\scriptstyle{\,{}_{Z}{\operatorname{incl}_{t\geq
0}}}$$\scriptstyle{F(g\circ f)^{t\geq 0}}$$\scriptstyle{F(g)^{t\geq
0}}$$\scriptstyle{\,{}_{X}{\operatorname{incl}_{t\geq
0}}}$$\scriptstyle{F(g\circ f)}$$\scriptstyle{\,{}_{t\geq 0}{\phi_{f,g}}}$

Thus, after potentially redefining the $\theta$ along a choice of contraction
of the groupoid of isomorphisms, we find that the pasting diagram expressed in
Diagram 3.1 conicides with the pasting diagram:

${F(Y)}$${F(Z)}$${F(X)}$${F(Z)^{t\geq 0}}$${F(X)^{t\geq
0}}$$\scriptstyle{F(f)}$$\scriptstyle{\,{}_{Z}{\tau_{\geq
0}}}$$\scriptstyle{F(g)}$$\scriptstyle{F(g\circ
f)}$$\scriptstyle{\,{}_{X}{\tau_{\geq 0}}}$$\scriptstyle{F(g\circ f)^{t\geq
0}}$$\scriptstyle{\,{}_{F}{\phi_{f,g}}}$$\scriptstyle{\,{}_{g\circ
f}{\theta^{t\geq 0}}}$

∎

######  Corollary 3.21.

Assume that we have a filtered triangulated pseudofunctor
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
over a triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and that $T$ is truncated. The assignments $F^{t\leq 0}$ and $F^{t\geq 0}$
induce pseudofunctors with adjoints

${F^{t\geq 0}}$${F}$$\scriptstyle{\underline{{\operatorname{incl}}}_{t\geq
0}}$$\scriptstyle{\underline{{\tau}}_{\geq 0}}$$\dashv$

and

${F}$${F^{t\leq 0}}$$\scriptstyle{\underline{{\tau}}_{\leq
0}}$$\scriptstyle{\underline{{\operatorname{incl}}}_{t\leq 0}}$$\dashv$

in the $2$-category
$\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})$.
In particular, $F$ is a truncated pseudofunctor with each $t$-structure on
$F(X)$ the unique $t$-structure compatible with the $t$-structure on $T(X)$.

###### Proof.

Lemma 3.19 shows that $F^{t\leq 0}$ and $F^{t\geq 0}$ induce pseudofunctors
and Lemma 3.20 shows that the collections $(\,_{X}{\tau_{\geq
0}},\,_{f}{\theta^{t\geq
0}})_{X\in\operatorname{\mathscr{C}}_{0},f\in\operatorname{\mathscr{C}}_{1}}$
and $(\,_{X}{\tau_{\leq 0}},\,_{f}{\theta^{t\leq
0}})_{X\in\operatorname{\mathscr{C}}_{0},f\in\operatorname{\mathscr{C}}_{1}}$
are pseudonatural transformations. From here the adjunctions follow by virtue
of [14, Theorem 4.1.17] and the facts that $\,{}_{X}{\tau_{\geq
0}}\dashv\,_{X}{\operatorname{incl}_{t\geq 0}}$ and
$\,{}_{X}{\operatorname{incl}_{t\leq 0}}\dashv\,_{X}{\tau_{\leq 0}}$ for all
objects $X$ in $\operatorname{\mathscr{C}}_{0}$. Finally that $F$ is truncated
is immediate from the pseudonaturality $\underline{{\tau_{\leq 0}}}$ and
$\underline{{\tau_{\geq 0}}}$ while the statement regarding compatibility
follows by construction. ∎

This allows us to deduce the following proposition regarding the $t$-structure
on the filtered triangulated category $\operatorname{\mathbf{PC}}(F)$ over
$\operatorname{\mathbf{PC}}(T)$ when $T$ is truncated.

######  Proposition 3.22.

Assume that we have a filtered triangulated pseudofunctor
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
over a triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and that $T$ is truncated. Then the unique $t$-structure on
$\operatorname{\mathbf{PC}}(T)$ compatible with the $t$-structure on
$\operatorname{\mathbf{PC}}(T)$ satisfies

$\displaystyle\left(\operatorname{\mathbf{PC}}(F)^{t\geq 0}\right)_{0}$
$\displaystyle=\left\\{A\;:\;\operatorname{gr}_{\operatorname{fil}}^{n}(A)\in\left(\operatorname{\mathbf{PC}}(T)^{t\geq
n}\right)_{0}\right\\}=\left\\{A\;:\;\,_{X}{\underline{{\operatorname{gr}}}_{f}^{n}}\left(\,{}_{X}{A}\right)\in\left(T(X)^{t\geq
n}\right)_{0}\right\\}$
$\displaystyle=\left\\{A\;:\;\,_{X}{\underline{{\operatorname{gr}}}_{f}^{n}}\left(\,{}_{X}{A}\right)\in\left(F(X)^{t\geq
0}\right)_{0}\right\\}$

and

$\displaystyle\left(\operatorname{\mathbf{PC}}(F)^{t\leq 0}\right)_{0}$
$\displaystyle=\left\\{A\;:\;\operatorname{gr}_{\operatorname{fil}}^{n}(A)\in\left(\operatorname{\mathbf{PC}}(T)^{t\leq
n}\right)_{0}\right\\}=\left\\{A\;:\;\,_{X}{\underline{{\operatorname{gr}}}_{f}^{n}}\left(\,{}_{X}{A}\right)\in\left(T(X)^{t\leq
n}\right)_{0}\right\\}$
$\displaystyle=\left\\{A\;:\;\,_{X}{\underline{{\operatorname{gr}}}_{f}^{n}}\left(\,{}_{X}{A}\right)\in\left(F(X)^{t\leq
0}\right)_{0}\right\\}$

###### Proof.

In both $t\geq 0$ and $t\leq 0$ cases, the first two equalities are immediate
from the fact that $T$ is truncated and [3, Proposition A.5.i]. The last
equality follows from the fact that $F$ is truncated by Corollary 3.21 with
truncation induced object-locally by the unique $t$-structure on $F(X)$
compatible with the $t$-structure on $T(X)$. ∎

We now want to build a cohomological functor
$H_{\operatorname{fil}}:\operatorname{\mathbf{PC}}(F)\to\operatorname{\mathbf{PC}}(\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit}))$
by pseudoconifying the construction of the functor $H_{\operatorname{fil}}$ of
[3, Proposition A.5.b]. Because those functors allow us to construct
cohomological functors

$\,{}_{X}{H_{\operatorname{fil}}}:F(X)\to\operatorname{\mathbf{Ch}}^{b}\left(T(X)^{\heartsuit}\right)$

for all objects $X$ of $\operatorname{\mathscr{C}}$, our primary obstructions
in defining the functor
$H_{\operatorname{fil}}:\operatorname{\mathbf{PC}}(F)\to\operatorname{\mathbf{PC}}\left(\operatorname{\mathbf{Ch}}^{b}(T)^{\heartsuit}\right)$
comes down to showing that the differentials of the
$\,{}_{X}{H_{\operatorname{fil}}}$ are compatible with pseudonatural
information, as well as that they are suitably natural. To this end we first
recall the construction of the cohomological functors

$\,{}_{X}{H_{\operatorname{fil}}}:F(X)\to\operatorname{\mathbf{Ch}}^{b}\left(T(X)^{\heartsuit}\right).$

######  Proposition 3.23 ([3, Proposition A.5.b]).

Assume that $\operatorname{\mathscr{C}}$ is a filtered triangulated category
over a triangulated category $\operatorname{\mathscr{A}}$ and assume that they
have compatible $t$-structures with hearts
$\operatorname{\mathscr{C}}^{\heartsuit}$ and
$\operatorname{\mathscr{A}}^{\heartsuit}$, respectively. Then there is a
cohomological functor
$H_{\operatorname{fil}}:\operatorname{\mathscr{C}}\to\operatorname{\mathbf{Ch}}^{b}(\operatorname{\mathscr{A}}^{\heartsuit})$
such that $H_{\operatorname{fil}}|_{\operatorname{\mathscr{C}}^{\heartsuit}}$
is an equivalence of categories.

###### Sketch.

We provide a sketch of the proposition because we will need to use it later.
If $X$ is an object of $\operatorname{\mathscr{C}}$ then the objects of the
cochain complex $(H_{\operatorname{fil}}(X),\partial)$ are given by defining

$H_{\operatorname{fil}}(X)^{n}:=H^{n}\left(\operatorname{gr}_{\operatorname{fil}}^{n}(X)\right)$

where $H:\operatorname{\mathscr{A}}\to\operatorname{\mathscr{A}}^{\heartsuit}$
is the cohomology functor induced by the $t$-structure on
$\operatorname{\mathscr{A}}$. The differentials
$\partial_{n}:H_{\operatorname{fil}}(X)^{n}\to
H_{\operatorname{fil}}(X)^{n+1}$ are induced as follows. For each
$n\in\operatorname{\mathbb{Z}}$, consider the object
$\left(\Sigma_{\operatorname{fil}\geq n+1}\circ\Sigma_{\operatorname{fil}\leq
n}\right)(X);$ cf. Remark 3.11 for the definition of the $\Sigma$ functors. By
proceeding as in the usual $t$-structure case by using the adjoint calculus as
in [4], we can show that for all natural numbers
$m,n\in\operatorname{\mathbb{N}}$ there are natural isomorphisms

(3.2) $\displaystyle\Sigma_{\operatorname{fil}\leq n}$
$\displaystyle\cong\Sigma_{\operatorname{fil}\leq
n}\circ\Sigma_{\operatorname{fil}\leq m}$ (3.3)
$\displaystyle\Sigma_{\operatorname{fil}\geq m}$
$\displaystyle\cong\Sigma_{\operatorname{fil}\geq m}\
\circ\Sigma_{\operatorname{fil}\geq n}$ (3.4)
$\displaystyle\Sigma_{\operatorname{fil}\geq
n}\circ\Sigma_{\operatorname{fil}\leq m}$
$\displaystyle\cong\Sigma_{\operatorname{fil}\leq
m}\circ\Sigma_{\operatorname{fil}\geq n}.$

whenever $n\leq m$. Now, apply [3, Proposition A.3.ii] to give a distinguished
triangle

${\left(\Sigma_{\operatorname{fil}\leq n+1}\circ\Sigma_{\operatorname{fil}\geq
n+1}\right)(X)}$${\left(\Sigma_{\operatorname{fil}\leq
n+1}\circ\Sigma_{\operatorname{fil}\geq
n}\right)(X)}$${\left(\Sigma_{\operatorname{fil}\leq
n+1}\circ\Sigma_{\operatorname{fil}\geq
n+1}\right)(X)[1]}$${\left(\Sigma_{\operatorname{fil}\leq
n}\circ\Sigma_{\operatorname{fil}\geq n}\right)(X)}$$\scriptstyle{d}$

where we have implicitly used the various natural isomorphisms in Equations
3.2, 3.3, 3.4 in various places. Now applying the functor $\omega$ gives a
distinguished triangle

${\left(\omega\circ\Sigma_{\operatorname{fil}\leq
n+1}\circ\Sigma_{\operatorname{fil}\geq
n+1}\right)(X)}$${\left(\omega\circ\Sigma_{\operatorname{fil}\leq
n+1}\circ\Sigma_{\operatorname{fil}\geq
n}\right)(X)}$${\left(\omega\circ\Sigma_{\operatorname{fil}\leq
n+1}\circ\Sigma_{\operatorname{fil}\geq
n+1}\right)(X)[1]}$${\left(\omega\circ\Sigma_{\operatorname{fil}\leq
n}\circ\Sigma_{\operatorname{fil}\geq n}\right)(X)}$$\scriptstyle{\omega(d)}$

which, after manipulating the adjunctions $\omega$ satisfies with the various
inclusion functors, gives rise to a distinguished triangle

${\left(j\circ s^{-n}\circ\Sigma_{\operatorname{fil}\leq
n+1}\circ\Sigma_{\operatorname{fil}\geq n+1}\right)(X)}$${\left(j\circ
s^{-n}\circ\Sigma_{\operatorname{fil}\leq
n+1}\circ\Sigma_{\operatorname{fil}\geq n}\right)(X)}$${\left(j\circ
s^{-n}\circ\Sigma_{\operatorname{fil}\leq
n+1}\circ\Sigma_{\operatorname{fil}\geq n+1}\right)(X)[1]}$${\left(j\circ
s^{-n}\circ\Sigma_{\operatorname{fil}\leq
n}\circ\Sigma_{\operatorname{fil}\geq
n}\right)(X)}$$\scriptstyle{\widetilde{d}}$

are in $\operatorname{\mathscr{A}}$. Consequently, applying the cohomology
functor $H^{n}$ gives rise to the differential

$\partial_{n}=H^{n}(\widetilde{d}):H_{\operatorname{fil}}(X)^{n}\to
H_{\operatorname{fil}}(X)^{n+1}$

as the connecting morphism of the corresponding long exact cohomology
sequence. That these maps are differentials is then routine as they come from
the long exact sequence. Finally the statement regarding
$H_{\operatorname{fil}}|_{\operatorname{\mathscr{C}}^{\heartsuit}}$ being an
equivalence is routine and omitted. ∎

Note that by Lemma 3.10 and Propositions 3.12 and 3.14, we know that the
constructions used above can extended to the pseudoconical situation. In
particular, we can use the functors $H_{\operatorname{fil}}$ to construct a
pseudonatural transformation that lifts to a cohomological functor
$\operatorname{\mathbf{PC}}(F)\to\operatorname{\mathbf{PC}}(\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit}))$.
Note that technically speaking we should check that
$\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})$ determines a pseudofunctor,
but this is routine by virtue of the fact that $T^{\heartsuit}$ is a
pseudofunctor and the $\operatorname{\mathbf{Ch}}^{b}(-)$ construction is
(strictly) functorial with its assignment on morphisms given by prolongation.

######  Proposition 3.24.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and assume that $T$ and $F$ are truncated with compatible $t$-structures. Then
there is a pseudonatural transformation

${\operatorname{\mathscr{C}}^{\operatorname{op}}}$${\operatorname{\mathfrak{Cat}}}$$\scriptstyle{F}$$\scriptstyle{\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})}$$\scriptstyle{\underline{{H}}_{\operatorname{fil}}}$

such that each object functor $\,{}_{X}{\underline{{H}}_{\operatorname{fil}}}$
is a cohomological functor.

###### Proof.

For each object $X$ of $\operatorname{\mathscr{C}}$ define the functor
$\,{}_{X}{\underline{{H}}_{\operatorname{fil}}}$ to be the cohomology functor

$\,{}_{X}{H_{\operatorname{fil}}}:F(X)\to\operatorname{\mathbf{Ch}}^{b}\left(T(X)^{\heartsuit}\right)$

of Proposition 3.23. To define the witness isomorphisms
$\,{}_{f}{\underline{{H}}_{\operatorname{fil}}}$ for a morphism $f:X\to Y$ in
$\operatorname{\mathscr{C}}$, we note that for every object $A$ of $T(Y)$, our
desired natural isomorphism
$(\,_{f}{\underline{{H}}_{\operatorname{fil}}})_{A}$ is an isomorphism in the
category $\operatorname{\mathbf{Ch}}^{b}(T(X)^{\heartsuit})$. As such, we must
define isomorphisms $(\,_{f}{\underline{{H}}_{\operatorname{fil}}})_{A}^{n}$
for all $n\in\operatorname{\mathbb{Z}}$ and then show that these commute with
the differentials. To this end fix an $n\in\operatorname{\mathbb{Z}}$ and
define $(\,_{f}{\underline{{H}}_{\operatorname{fil}}})_{-}^{n}$ as follows.

Consider that since $F$ is filtered over $T$, there is an invertible $2$-cell

${F(Y)}$${T(Y)}$${F(X)}$${T(X)}$$\scriptstyle{\,{}_{Y}{\operatorname{gr}_{\operatorname{fil}}^{n}}}$$\scriptstyle{F(f)}$$\scriptstyle{T(f)}$$\scriptstyle{\,{}_{X}{\operatorname{gr}_{\operatorname{fil}}^{n}}}$$\scriptstyle{\,{}_{f}{\underline{{\operatorname{gr}}}_{\operatorname{fil}}^{n}}}$

by Lemma 3.10. Furthermore, since $T$ is truncated, there is an invertible
$2$-cell

${T(Y)}$${T(Y)^{\heartsuit}}$${T(X)}$${T(X)^{\heartsuit}}$$\scriptstyle{\,{}_{Y}{H_{T}^{n}}}$$\scriptstyle{T(f)}$$\scriptstyle{T(f)^{\heartsuit}}$$\scriptstyle{\,{}_{X}{H_{T}^{n}}}$$\scriptstyle{\,{}_{f}{H_{T}^{n}}}$

because $T$ is truncated. Pasting the $2$-cells together gives the invertible
$2$-cell:

${F(Y)}$${T(Y)^{\heartsuit}}$${F(X)}$${T(X)^{\heartsuit}}$$\scriptstyle{\,{}_{Y}{H_{T}^{n}}\circ\,_{Y}{\operatorname{gr}_{\operatorname{fil}}^{n}}}$$\scriptstyle{F(f)}$$\scriptstyle{T(f)^{\heartsuit}}$$\scriptstyle{\,{}_{X}{H_{T}^{n}}\circ\,_{X}{\operatorname{gr}_{\operatorname{fil}}^{n}}}$$\scriptstyle{\,{}_{f}{H_{T}^{n}}\ast\,_{f}{\operatorname{gr}_{\operatorname{fil}}^{n}}}$

Now observe that we can rewrite the above $2$-cell as

${F(Y)}$${T(Y)^{\heartsuit}}$${F(X)}$${T(X)^{\heartsuit}}$$\scriptstyle{(\,_{Y}{H_{\operatorname{fil}}})^{n}}$$\scriptstyle{F(f)}$$\scriptstyle{T(f)^{\heartsuit}}$$\scriptstyle{(\,_{X}{H_{\operatorname{fil}}})^{n}}$$\scriptstyle{\,{}_{f}{H_{T}^{n}}\ast\,_{f}{\operatorname{gr}_{\operatorname{fil}}^{n}}}$

which we then use to define the degree $n$-component of our candidate
isomorphism:

$(\,_{f}{\underline{{H}}_{\operatorname{fil}}})_{-}^{n}:=\,_{f}{H_{T}^{n}}\ast\,_{f}{\operatorname{gr}_{\operatorname{fil}}^{n}}.$

Fix an object $A$ of $F(Y)$. Observe that by the construction of the
distinguished triangles in the proof of Proposition 3.12 used in the proof of
Proposition 3.23, the triangles

${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\leq
n+1}}\circ\,_{X}{\Sigma_{\operatorname{fil}\geq
n+1}}\right)\left(F(f)(A)\right)}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\leq
n+1}}\circ\,_{X}{\Sigma_{\operatorname{fil}\geq
n}}\right)\left(F(f)(A)\right)}$${\left(\,{}_{X}{\Sigma_{\operatorname{fil}\leq
n+1}}\circ\,_{X}{\Sigma_{\operatorname{fil}\geq
n+1}}\right)\left(F(f)(A)\right)[1]}$${\left(\,{}_{X}{\Sigma}_{\operatorname{fil}\leq
n}\circ\,_{X}{\Sigma}_{\operatorname{fil}\geq
n}\right)\left(F(f)(A)\right)}$$\scriptstyle{\,{}_{X}{d_{F(f)A}}}$

and

${F(f)\left(\,{}_{Y}{\Sigma}_{\operatorname{fil}\leq
n+1}\circ\,_{Y}{\Sigma}_{\operatorname{fil}\geq
n+1}\right)(A)}$${\left(\,{}_{Y}{\Sigma}_{\operatorname{fil}\leq
n+1}\circ\,_{Y}{\Sigma}_{\operatorname{fil}\geq
n}\right)(A)}$${\left(\,{}_{Y}{\Sigma}_{\operatorname{fil}\leq
n+1}\circ\,_{Y}{\Sigma}_{\operatorname{fil}\geq
n+1}\right)(A)[1]}$${\left(\,{}_{Y}{\Sigma}_{\operatorname{fil}\leq
n}\circ\,_{Y}{\Sigma}_{\operatorname{fil}\geq
n}\right)(A)}$$\scriptstyle{F(f)\,_{Y}{d_{A}}}$

are object-wise isomorphic with non-labeled arrows induced by the
$\operatorname{incl}$ and $\sigma$ adjunctions, it follows that the
corresponding distinguished triangles are homotopic in $T(X)$ after applying
the $\operatorname{gr}_{\operatorname{fil}}^{n}$ functors via
$\,{}_{f}{\operatorname{gr}_{\operatorname{fil}}^{n}}$. However, this implies
that because the functor $H_{T}^{n}$ is is a cohomology functor that we induce
an isomorphism of long exact sequences and hence a commuting square:

${\left(T(f)\circ\,_{Y}{H_{\operatorname{fil}}}\right)(A)^{n}}$${\left(T(f)\circ\,_{Y}{H_{\operatorname{fil}}}\right)(A)^{n}}$${\left(\,{}_{X}{H_{\operatorname{fil}}}\circ
F(f)\right)(A)^{n}}$${\left(\,{}_{X}{H_{\operatorname{fil}}}\circ
F(f)\right)(A)^{n+1}}$$\scriptstyle{\partial}$$\scriptstyle{(\,_{f}{\underline{{H}}_{\operatorname{fil}}})_{A}^{n}}$$\scriptstyle{(\,_{f}{\underline{{H}}_{\operatorname{fil}}})^{n+1}_{A}}$$\scriptstyle{\partial}$

It thus follows that $\underline{{H_{\operatorname{fil}}}}$ is a pseudonatural
transformation. That each corresponding functor
$\,{}_{X}{\underline{{H}}_{\operatorname{fil}}}$ is a cohomological functor is
the content of Proposition 3.23. ∎

######  Corollary 3.25.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and assume that $T$ and $F$ are truncated with compatible $t$-structures.
There is a pseudonatural equivalence

$\underline{{H}}_{\operatorname{fil}}|_{F^{\heartsuit}}:F^{\heartsuit}\Rightarrow\operatorname{\mathbf{Ch}}^{b}\left(T^{\heartsuit}\right).$

###### Proof.

Proposition 3.23 shows that for every object $X$ of
$\operatorname{\mathscr{C}}$, the functor

$\,{}_{X}{\underline{{H}}_{\operatorname{fil}}}|_{F(X)^{\heartsuit}}:F(X)^{\heartsuit}\to\operatorname{\mathbf{Ch}}^{b}\left(T(X)^{\heartsuit}\right)$

is an equivalence of categories. However, since $F$ is truncated and hence
each fibre functor $F(f)$ is $t$-exact, it follows that the pseudonatural
transformation $\underline{{H}}_{\operatorname{fil}}$ restricts to a
pseudonatural transformation

$\underline{{H}}_{\operatorname{fil}}|_{F^{\heartsuit}}:F^{\heartsuit}\Rightarrow\operatorname{\mathbf{Ch}}^{b}\left(T^{\heartsuit}\right).$

From the given equivalences of categories, the result follows. ∎

######  Corollary 3.26.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and assume that $T$ and $F$ are truncated with compatible $t$-structures.
There is a cohomological functor

$H_{\operatorname{fil}}:\operatorname{\mathbf{PC}}(F)\to\operatorname{\mathbf{PC}}\left(\operatorname{\mathbf{Ch}}^{b}\left(T^{\heartsuit}\right)\right)$

which restricts to an equivalence of categories
$\operatorname{\mathbf{PC}}(F)^{\heartsuit}\simeq\operatorname{\mathbf{PC}}(\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit}))$.

###### Proof.

The existence of the functor $H_{\operatorname{fil}}$ follows by setting
$H_{\operatorname{fil}}:=\operatorname{\mathbf{PC}}(\underline{{H}}_{\operatorname{fil}})$,
where $\underline{{H}}_{\operatorname{fil}}$ is the pseudonatural
transformation of Proposition 3.24. That this is a cohomological functor
follows from the fact that each $\,{}_{X}{H_{\operatorname{fil}}}$ is a
cohomological functor. Finally, that this restricts to an equivalence is
routine from the fact that
$\underline{{H}}_{\operatorname{fil}}|_{F^{\heartsuit}}$ is an equivalence by
Corollary 3.25 and from the Change of Heart Theorem (cf. [14, Theorem 5.1.24])
giving that
$\operatorname{\mathbf{PC}}(F^{\heartsuit})=\operatorname{\mathbf{PC}}(F)^{\heartsuit}$.
∎

We now can proceed as in the remainder of [3] to construct the pseudocone
realization functor. Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and assume that $T$ and $F$ are truncated with compatible $t$-structures. Let
$I:\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})\to F^{\heartsuit}$ be an
inverse pseudonatural equivalence to
$H_{\operatorname{fil}}|_{F^{\heartsuit}}$. Define the pseudonatural
transformation

${\operatorname{\mathbf{Ch}}^{b}\left(T^{\heartsuit}\right)}$${F^{\heartsuit}}$${F}$${T}$$\scriptstyle{\widetilde{\operatorname{real}}}$$\scriptstyle{I}$$\scriptstyle{\operatorname{incl}}$$\scriptstyle{\omega}$

and consider the composite

${\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})}$${T}$${T^{\heartsuit}}$$\scriptstyle{\widetilde{\operatorname{real}}}$$\scriptstyle{H_{T}}$

in
$\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})$.
Now define the subpseudofunctor $Q$ of
$\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})$ defined as follows:

  * •

For each object $X$ of $\operatorname{\mathscr{C}}$, the category
$Q(X):={}^{t}\mathsf{QIso}(\operatorname{\mathbf{Ch}}^{b}(T(X)^{\heartsuit}))$
is the category of quasi-isomorphisms of
$\operatorname{\mathbf{Ch}}^{b}(T(X)^{\heartsuit})$, i.e., the category whose
objects are the objects of $\operatorname{\mathbf{Ch}}^{b}(T(X)^{\heartsuit})$
but whose morphisms are precisely the quasi-isomorphisms of
$\operatorname{\mathbf{Ch}}^{b}(T(X)^{\heartsuit})$.

  * •

For each morphism $f:X\to Y$ of $\operatorname{\mathscr{C}}$, the fibre
functor $Q(f)$ is defined to be the prolongation
$\operatorname{\mathbf{Ch}}^{b}(T(f))$ as restricted to $Q(X)$. Because $T$ is
truncated and so each functor $T(f)$ is $t$-exact, it follows that
$\operatorname{\mathbf{Ch}}^{b}(T(f))$ preserves quasi-isomorphisms.

  * •

For each pair of composable morphisms $f:X\to Y$ and $g:Y\to Z$, define the
compositor $\,{}_{Q}{\phi_{f,g}}$ to simply be the restriction of the
compositor of $\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})$ to $Q$; because
it is comprised of isomorphisms it is stable under the formation of $Q$.

Because each functor $T(f)$ is exact, we can also apply [14, Proposition
4.1.27] to deduce that there is a localization pseudofunctor
$Q^{-1}\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit}).$ It is routine to check
that this is pseudonaturally isomorphic to the pseudofunctor
$D^{b}(T^{\heartsuit})$.

Observe that for each object $X$ of $\operatorname{\mathscr{C}}_{0}$ the
functor $H_{T}\circ\widetilde{\operatorname{real}}$ carries quasi-isomorphisms
of $\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})$ to isomorphisms in $T$.
Consequently by [14, Theorem 4.1.29] there is a unique pseudonatural
transformation

$\underline{{\operatorname{real}}}:Q^{-1}\operatorname{\mathbf{Ch}}^{b}\left(T^{\heartsuit}\right)\Rightarrow
T$

for which the diagram

${\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})}$${T}$${Q^{-1}\operatorname{\mathbf{Ch}}^{b}\left(T^{\heartsuit}\right)}$$\scriptstyle{\widetilde{\operatorname{real}}}$$\scriptstyle{\underline{{\lambda}}_{Q}}$$\scriptstyle{\underline{{\operatorname{real}}}}$

commutes strictly in
$\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})$.
This produces a pseudofunctor realization pseudonatural translation and
completes our pseudofunctorification of [3, Appendix A], but not our
justification of calling it a realization pseudonatural transformation or the
pseudoconification of [3, Appendix A].

######  Theorem 3.27.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and assume that $T$ and $F$ are truncated with compatible $t$-structures. Then
if $Q$ is the quasi-isomorphism subpseudofunctor of
$\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})$, there is a unique
pseudonatural transformation
$\underline{{\operatorname{real}}}:Q^{-1}\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})\to
T$ for which the diagram

${\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})}$${T}$${Q^{-1}\operatorname{\mathbf{Ch}}^{b}\left(T^{\heartsuit}\right)}$$\scriptstyle{\widetilde{\operatorname{real}}}$$\scriptstyle{\underline{{\lambda}}_{Q}}$$\scriptstyle{\underline{{\operatorname{real}}}}$

commutes in
$\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})$.
Furthermore, each object functor $\,{}_{X}{\underline{{\operatorname{real}}}}$
is $t$-exact for all obects $X$ of $\operatorname{\mathscr{C}}$.

###### Proof.

Everything in the statement of the theorem prior to the word “furthermore” was
shown prior to stating the theorem. For the final statment, because each
category $\,{}_{X}{Q}^{-1}\operatorname{\mathbf{Ch}}^{b}(T(X)^{\heartsuit})$
is uniquely isomorphic to $D^{b}(T(X)^{\heartsuit})$,
$\,{}_{X}{\operatorname{real}}$ is uniquely isomorphic to the realization
functor and so the result follows from [3, Section A.6]. ∎

Note that because for each object $X$ in $\operatorname{\mathscr{C}}$ the
$X$-local component of the diagram in Theorem 3.27 is induced by the diagram

${\operatorname{\mathbf{Ch}}^{b}(T(X)^{\heartsuit})}$${T(X)}$${\,{}_{X}{Q^{-1}}\operatorname{\mathbf{Ch}}^{b}(T(X)^{\heartsuit})}$$\scriptstyle{\,{}_{X}{\widetilde{\operatorname{real}}}}$$\scriptstyle{\,{}_{X}{\lambda}}$$\scriptstyle{\,{}_{X}{\operatorname{real}}}$

where $\,{}_{X}{\operatorname{real}}$ is the realization functor constructed
in [3, Section A.6] and $\,{}_{X}{\widetilde{\operatorname{real}}}$ is the
corresponding chain complex realization. Thus upon taking psuedocones of the
diagram in Theorem 3.27 we get the pseudocone realization functor.

######  Theorem 3.28.

Let
$F:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
be a filtered triangulated pseudofunctor over triangulated pseudofunctor
$T:\operatorname{\mathscr{C}}^{\operatorname{op}}\to\operatorname{\mathfrak{Cat}}$
and assume that $T$ and $F$ are truncated with compatible $t$-structures. Then
if $Q$ is the quasi-isomorphism subpseudofunctor of
$\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})$, there is a unique
pseudonatural transformation
$\underline{{\operatorname{real}}}:Q^{-1}\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})\to
T$ for which the diagram

${\operatorname{\mathbf{PC}}(\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit}))}$${\operatorname{\mathbf{PC}}(T)}$${\operatorname{\mathbf{PC}}(D^{b}\left(T^{\heartsuit}\right))}$$\scriptstyle{\operatorname{\mathbf{PC}}(\widetilde{\operatorname{real}})}$$\scriptstyle{\operatorname{\mathbf{PC}}(\underline{{\lambda}}_{Q})}$$\scriptstyle{\operatorname{\mathbf{PC}}(\underline{{\operatorname{real}}})}$

commutes in
$\operatorname{\mathsf{Bicat}}(\operatorname{\mathscr{C}}^{\operatorname{op}},\operatorname{\mathfrak{Cat}})$
with $\operatorname{\mathbf{PC}}(\underline{{\operatorname{real}}})$
$t$-exact. Furthermore, for any obejct $A$ of
$\operatorname{\mathbf{PC}}(\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit}))$,

$\operatorname{\mathbf{PC}}(\underline{{\operatorname{real}}})(A)=\left\\{\,{}_{X}{\underline{{\operatorname{real}}}}(\,_{X}{A})\;:\;X\in\operatorname{\mathscr{C}}_{0}\right\\}$

where $\,{}_{X}{\underline{{\operatorname{real}}}}$ is the realization
$D^{b}(T(X)^{\heartsuit})\to T(X).$

###### Proof.

Because for every object $X$ in $\operatorname{\mathscr{C}}$ there is a unique
natural isomorphism
$\,{}_{X}{Q^{-1}}\operatorname{\mathbf{Ch}}^{b}(T(X)^{\heartsuit})\cong
D^{b}(T(X)^{\heartsuit})$ we get a corresponding pseudonatural isomorphism
$D^{b}(T^{\heartsuit})\cong
Q^{-1}\operatorname{\mathbf{Ch}}^{b}(T^{\heartsuit})$. This allows us to
construct the listed pseudocone realization functor as well as prove that it
is $t$-exact, as each functor $\,{}_{X}{\underline{{\operatorname{real}}}}$ is
$t$-exact where

$\,{}_{X}{\operatorname{real}}:D^{b}(T^{\heartsuit})\to T$

is the realization functor of [3]. Finally the statement regarding
$\operatorname{\mathbf{PC}}(\underline{{\operatorname{real}}})$ follows from
[14, Theorem 4.1.1]. ∎

######  Corollary 3.29.

For all $X$ in $\operatorname{\mathscr{C}}$,

$\,{}_{X}{\underline{{\operatorname{real}}}}|_{T(X)^{\heartsuit}}=\operatorname{id}_{T(X)^{\heartsuit}}$

and

$\operatorname{\mathbf{PC}}(\underline{{\operatorname{real}}})(A)=A$

for all objects $A$ of $\operatorname{\mathbf{PC}}(T^{\heartsuit})$ as
embedded in $\operatorname{\mathbf{PC}}(D^{b}(T^{\heartsuit}))$.

###### Proof.