# A motivic spectrum representing hermitian K-theory

Baptiste Calmès  Université d’Artois, Laboratoire de Mathématiques de Lens
(LML), UR 2462, Lens, France<EMAIL_ADDRESS>,  Yonatan Harpaz
Université de Paris 13; Institut Galilée; Villetaneuse, France
<EMAIL_ADDRESS>and  Denis Nardin  Universität Regensburg;
Mathematisches Institut; Regensburg, Germany<EMAIL_ADDRESS>regensburg.de

###### Abstract.

We establish fundamental motivic results about hermitian K-theory without
assuming that 2 is invertible on the base scheme. In particular, we prove that
both quadratic and symmetric Grothendieck-Witt theory satisfy Nisnevich
descent, and that symmetric Grothendieck-Witt theory further satisfies
dévissage and $\mathbb{A}^{1}$-invariance over a regular Noetherian base of
finite Krull dimension, as well as a projective bundle formula. We use this to
show that over a regular Noetherian base, symmetric Grothendieck-Witt theory
is represented by a motivic ${\mathrm{E}_{\infty}}$-ring spectrum, which we
then show is an absolutely pure spectrum, answering a question of Déglise. As
with algebraic K-theory, we show that over a general base, one can also
construct a hermitian K-theory motivic spectrum, representing this time a
suitable homotopy invariant and Karoubi-localising version of Grothendieck-
Witt theory.

###### Contents

  1. 1 Introduction
  2. 2 Recollections on hermitian K-theory
    1. 2.1 Poincaré $\infty$-categories
    2. 2.2 Hermitian K-theory
    3. 2.3 Karoubi-Grothendieck-Witt theory
  3. 3 Poincaré structures on derived categories of schemes
    1. 3.1 Poincaré $\infty$-categories associated to rigid symmetric monoidal structures
    2. 3.2 Derived Poincaré categories of schemes
    3. 3.3 Genuine structures
  4. 4 Nisnevich descent
    1. 4.1 Flat linear categories
    2. 4.2 Bounded Karoubi projections
    3. 4.3 Zariski descent
    4. 4.4 Nisnevich descent
    5. 4.5 The coniveau filtration
    6. 4.6 Genuine symmetric GW-theory of schemes
  5. 5 Dévissage
    1. 5.1 Push-forward as a Poincaré functor
    2. 5.2 Global dévissage
    3. 5.3 Local dévissage
    4. 5.4 End of the proof of global dévissage
  6. 6 The projective bundle formula and $\mathbb{A}^{1}$-invariance
    1. 6.1 The projective bundle formula
    2. 6.2 Genuine refinement for the projective line
    3. 6.3 Homotopy invariance of symmetric Grothendieck-Witt
  7. 7 Motivic realization of localising invariants
    1. 7.1 Recollections on motivic spectra
    2. 7.2 Bounded localising invariants
    3. 7.3 Motivic realization
    4. 7.4 The free delooping lemma
  8. 8 Hermitian K-theory as a motivic spectrum
    1. 8.1 Motivic hermitian K-theory spectra
    2. 8.2 Pullback invariance of hermitian K-theory
    3. 8.3 Thom isomorphism
    4. 8.4 Purity of the hermitian K-theory spectrum
  9. A Derived categories of schemes
    1. A.1 Grothendieck abelian categories and their derived counterparts
    2. A.2 Derived categories of $\mathcal{O}_{X}$-modules
    3. A.3 Quasi-coherent complexes
    4. A.4 Flat excision
    5. A.5 Perfect complexes
    6. A.6 Quasi-perfect maps
  10. B The Zariski and Nisnevich topologies
    1. B.1 Extensive $\infty$-categories and coverages
    2. B.2 The Zariski and Nisnevich coverages
    3. B.3 Characterization of Nisnevich sheaves

##  1\. Introduction

Originally justified by Voevodsky’s proof of Milnor’s conjecture, motivic
homotopy theory is now generally recognized as a deep and efficient framework
to study cohomological invariants of rings and schemes. A key insight of this
approach is that such invariants can be studied in a manner closely resembling
that taken up by classical algebraic topology in the study of cohomological
invariants of topological spaces. More precisely, the latter can be
axiomatised via the notion of generalized cohomology theories, which in turn
can be classified by the notion of a spectrum. Similarly, cohomological
invariants of schemes can be classified by a suitable notion of a motivic
spectrum. For example, motivic cohomology, algebraic K-theory, and algebraic
cobordism, which are algebraic analogues of singular cohomology, complex
K-theory, and cobordism, are each represented by a corresponding motivic
spectrum.

On the side of topological spaces, another generalized cohomology theory which
plays an important role in the study of smooth manifolds, is real K-theory.
Its analogue on the side of schemes is hermitian K-theory, also called
Grothendieck-Witt theory, which is the K-theory of unimodular quadratic forms.
This somewhat less obvious analogy is due to the inclusions
$\mathrm{O}_{n}(\mathbb{C})\supseteq\mathrm{O}_{n}(\mathbb{R})\subseteq\mathrm{GL}_{n}(\mathbb{R})$
both being homotopy equivalences, so that the $\operatorname{K}$-theory of
real vector bundles coincides with that of complex vector bundles equipped
with non-degenerate quadratic forms.

The behaviour of hermitian $\operatorname{K}$-theory is notoriously sensitive
to the prime 2. In particular, its behaviour simplifies considerably when
considering schemes over which 2 is invertible. As a result, until recently a
significant part of the relevant theory was developed under this assumption,
and in particular, the realization of hermitian K-theory as a motivic
spectrum, as was first done by Hornbostel [hornbostel-motivic], was limited to
$\mathbb{Z}[\tfrac{1}{2}]$-schemes.

In a recent series of papers [9-authors-I, 9-authors-II, 9-authors-III,
9-authors-IV, 9-authors-V] by a group of authors including the authors of the
present paper, the foundations of hermitian K-theory were revisited using the
powerful recent technology of higher category theory, in a manner which
enables one to dispense with the invertibility of 2 assumption. The goal of
the present paper is to harness these results in the service of the motivic
aspects of hermitian K-theory, allowing a base scheme $S$ in which 2 is not
assumed invertible. The first and primary outcome of our work is the following
result, which is itself a prerequisite to any further investigations in the
motivic direction:

######  Theorem 1.0.1.

Let $S$ be a regular Noetherian scheme. Then symmetric Grothendieck-Witt
theory is represented by a motivic spectrum over $S$.

Let us take a moment to clarify what the term “symmetric” means in this
context. First, note that when 2 is invertible, the notions of quadratic form
and of symmetric bilinear form are essentially equivalent: every symmetric
bilinear form admits a unique quadratic refinement. This is no longer true
when 2 is not invertible, so in general, one needs to specify with respect to
which flavour of forms Grothendieck-Witt theory is taken. For example, in the
above theorem, we make reference to symmetric bilinear forms.

There are, however, more then one variant of those as well. Recall that by the
Gillet-Waldhausen theorem, the algebraic K-theory of a ring can be calculated
either by considering finitely generated projective modules and then taking
group completion, or by considering all perfect complexes, in which case one
needs to enforce relations for every exact sequence of such. Passing to the
hermitian setting, projective modules can be endowed with symmetric bilinear
forms in the usual manner, while perfect complexes can be endowed with a
homotopical analogue of symmetric forms, where the symmetry property is
interpreted as a homotopy fixed point structure. When 2 is not invertible,
these two constructions lead to different Grothendieck-Witt groups (or
spectra) in general.

In the paper series mentioned above, we use the term symmetric Grothendieck-
Witt to refer to the latter construction using homotopy fixed points, and
introduce the term _genuine symmetric_ to refer to the former. While the
genuine symmetric is the one more directly related to classical constructions
in hermitian K-theory, the (non-genuine) symmetric admits better formal
properties. Most notably from the present viewpoint, it is
$\mathbb{A}^{1}$-invariant over regular Noetherian schemes, which is the main
reason why it is this flavour of Grothendieck-Witt theory that features in
Theorem 1.0.1. It also underlines a possible reason for why the motivic
hermitian K-theory spectrum was previously only constructed when 2 is
invertible: without this assumption, no flavour of Grothendieck-Witt groups
defined using a classical notion of forms is $\mathbb{A}^{1}$-invariant (even
over regular Noetherian bases); see Example 6.3.2.

While Theorem 1.0.1 is a necessary first step, it still leaves several obvious
questions unanswered:

  1. (1)

What happens if $S$ is not regular Noetherian?

  2. (2)

What about Witt theory, obtained from Grothendieck-Witt theory in principle by
quotienting out the hyperbolic forms?

  3. (3)

What about the multiplicative structure of Grothendieck-Witt theory?

  4. (4)

What about all the natural maps relating Grothendieck-Witt theory, Witt
theory, and algebraic K-theory?

  5. (5)

What happens to some of the expected properties of motivic Grothendieck-Witt
theory, such as purity or the projective bundle formula, when 2 is not
invertible?

In the present paper, we continue beyond Theorem 1.0.1 by giving complete
answers to all the above questions. To describe how this is done, let us
furthermore recall the theoretical framework of [9-authors-I] and its sequels.
The main point is that Grothendieck-Witt theory should be considered as an
invariant defined on _Poincaré $\infty$-categories_. These are stable
$\infty$-categories equipped with a spectrum valued functor
$\Qoppa\colon\mathcal{C}^{\mathrm{op}}\to{\mathcal{S}\mathrm{p}}$ satisfying
suitable axioms. These axioms guarantee in particular the existence of a
uniquely determined duality
$\mathrm{D}\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{C}$ such that
$\Qoppa(x\oplus y)=\Qoppa(x)\oplus\Qoppa(y)\oplus\mathrm{hom}(x,\mathrm{D}y)$.
The functor $\Qoppa$ is the called a _Poincaré structure_ on $\mathcal{C}$.
For an object $x\in\mathcal{C}$, the underlying infinite loop space
$\Omega^{\infty}\Qoppa(x)$ should be thought of as the space of forms on $x$.
In particular, we call a point $\beta\in\Omega^{\infty}\Qoppa(x)$ a _form_ on
$x$. Such a form determines a map $\beta_{\sharp}\colon x\to\mathrm{D}x$, and
we say that $\beta$ is a _Poincaré form_ if $\beta_{\sharp}$ is an
equivalence. Poincaré forms are what corresponds in this setting to non-
degenerate forms. A pair $(x,\beta)$ of an object equipped with a Poincaré
form $\beta\in\Omega^{\infty}\Qoppa(x)$ is called a Poincaré object.

The fact that $\Qoppa$ is an extra piece of structure (and not an inherent
property of $\mathcal{C}$) is what allows one to specify the flavour of forms
one is interested in. For example, if $R$ is a commutative ring, then we may
consider its perfect derived $\infty$-category $\mathcal{D}^{\mathrm{p}}(R)$,
whose objects are the perfect $R$-complexes, and consider the functor

$\Qoppa^{\mathrm{s}}_{R}(M):=\mathrm{hom}(M\otimes_{R}M,R)^{{\mathrm{h}\mathrm{C_{2}}}},$

where $(-)^{{\mathrm{h}\mathrm{C_{2}}}}$ denotes homotopy fixed points with
respect to the $\mathrm{C_{2}}$-action induced by swapping the two $M$
factors. Then $\Qoppa^{\mathrm{s}}$ sends $M$ to what can be considered as the
homotopy theoretical analogue of the spectrum of symmetric bilinear forms on
$M$. Alternatively, we may take the Poincaré structure
$\Qoppa^{\mathrm{q}}_{R}(M):=\mathrm{hom}(M\otimes_{R}M,R)_{{\mathrm{h}\mathrm{C_{2}}}}$
obtained by using homotopy orbits instead of fixed points. This is the
homotopy theoretical analogue of quadratic forms. As mentioned above, these
are not exactly the same as working with symmetric and quadratic forms on
projective modules. The latter notions can also be encoded via suitable
Poincaré structures $\Qoppa^{\mathrm{gs}}$ and $\Qoppa^{\mathrm{gq}}$, called
the genuine symmetric and genuine quadratic structures. All these Poincaré
structures are generally different from each other, and are related by a
sequence of natural transformations

$\Qoppa_{R}^{\mathrm{q}}\Rightarrow\Qoppa_{R}^{\mathrm{gq}}\Rightarrow\Qoppa_{R}^{\mathrm{gs}}\Rightarrow\Qoppa_{R}^{\mathrm{s}}.$

When $2$ is invertible in $R$, these maps are all equivalences.

To a Poincaré $\infty$-category $(\mathcal{C},\Qoppa)$, one may associate an
invariant $\operatorname{GW}(\mathcal{C},\Qoppa)$, called the Grothendieck-
Witt spectrum, whose homotopy groups are the Grothendieck-Witt groups of
$(\mathcal{C},\Qoppa)$. By [Hebestreit-Steimle], in the case of
$(\mathcal{D}^{\mathrm{p}}(R),\Qoppa^{\mathrm{gs}})$ and
$(\mathcal{D}^{\mathrm{p}}(R),\Qoppa^{\mathrm{gq}})$, these reproduce the
classical symmetric and quadratic Grothendieck-Witt groups of $R$, though, as
mentioned earlier, the Grothendieck-Witt groups of the Poincaré
$\infty$-category $(\mathcal{D}^{\mathrm{p}}(R),\Qoppa^{\mathrm{s}}_{R})$ have
better formal properties, and play a more important role in the current paper.
More generally, one may replace the commutative ring $R$ by a scheme $X$, and
consider the perfect derived $\infty$-category $\mathcal{D}^{\mathrm{p}}(X)$
of perfect complexes on $X$. As above, one can define the symmetric Poincaré
structure $\Qoppa^{\mathrm{s}}_{X}$ using the same formulas. More generally,
given a line bundle $L$ on $X$, we may consider the Poincaré structure
$\Qoppa^{\mathrm{s}}_{L}$ defined by
$\Qoppa^{\mathrm{s}}_{L}(M)=\mathrm{hom}(M\otimes_{\mathcal{O}_{X}}M,L)^{{\mathrm{h}\mathrm{C_{2}}}}$.
It is also useful to consider the slightly more general case where, on the one
hand, $L$ is endowed with an involution, and on the other hand is allowed to
be a shift of a line bundle (or more generally any tensor invertible perfect
complex). The Grothendieck-Witt groups of
$(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})$ are the main
invariants of schemes we consider in this paper. To keep the introduction
easily readable, we consider only line bundles with constant involution when
describing our main results, though we provide references to the body of the
paper where the reader can find the full version of each result.

The collection of Poincaré $\infty$-categories can be organized into a (large)
category $\mathrm{Cat}^{\mathrm{p}}$, which is presentable and compactly
generated (see [9-authors-IV]). The functor $\operatorname{GW}(-)$ is Verdier
localising, that is, it sends fibre-cofibre sequences of Poincaré
$\infty$-categories to exact sequences of spectra, and is furthermore the
universal such functor equipped with a natural transformation
$\mathrm{Pn}\Rightarrow\Omega^{\infty}\operatorname{GW}$, where $\mathrm{Pn}$
is the functor which associates to a Poincaré $\infty$-category its space of
Poincaré objects. It turns out that the property that is more relevant from
the motivic point of view is not being Verdier localising, but being Karoubi-
localising, which is equivalent to being Verdier localising and invariant
under idempotent completion. While the functor $\operatorname{GW}$ is not
Karoubi-localising, it can be turned into one in a universal manner, yielding
a functor $\mathbb{GW}$ we call the _Karoubi-Grothendieck-Witt_ functor. It
comes equipped with a canonical natural transformation
$\operatorname{GW}\Rightarrow\mathbb{GW}$, which is generally not equivalence.
This transformation is closely related to the transformation
$\operatorname{K}\Rightarrow\mathbb{K}$ between algebraic
$\operatorname{K}$-theory and Bass $\operatorname{K}$-theory (also known as
non-connective $\operatorname{K}$-theory). In particular, if $\mathcal{C}$ is
such that $\operatorname{K}(\mathcal{C})\to\mathbb{K}(\mathcal{C})$ is an
equivalence (e.g., $\mathcal{C}=\mathcal{D}^{\mathrm{p}}(X)$ for $X$ a regular
Noetherian scheme) then
$\operatorname{GW}(\mathcal{C},\Qoppa)\to\mathbb{GW}(\mathcal{C},\Qoppa)$ is
an equivalence, see [9-authors-V].

The importance of the Karoubi-localising property in the process of passing
from invariants of Poincaré $\infty$-categories to invariants of schemes is
expressed by the following key result of the present paper (see Corollary
4.4.2 below):

######  Proposition 1.0.2 (Nisnevich descent).

Let $S$ be a quasi-compact quasi-separated base scheme and
$\mathcal{F}\colon\mathrm{Cat}^{\mathrm{p}}\to{\mathcal{S}\mathrm{p}}$ a
Karoubi-localising spectrum valued functor. Then, the functor

$\mathcal{F}^{\mathrm{s}}_{S}\colon\mathrm{Sch}^{\mathrm{qq}}_{/S}\to{\mathcal{S}\mathrm{p}}\quad\quad
X\mapsto\mathcal{F}(X,\Qoppa^{\mathrm{s}}_{X})$

is a Nisnevich sheaf, where $\mathrm{Sch}^{\mathrm{qq}}_{/S}$ is the category
of quasi-compact quasi-separated $S$-schemes.

In particular, the functor
$\mathbb{GW}^{\mathrm{s}}_{S}(X)=\mathbb{GW}(X,\Qoppa^{\mathrm{s}}_{X})$ is a
Nisnevich sheaf on $\mathrm{Sch}^{\mathrm{qq}}_{/S}$. We may also restrict
attention to the full subcategory $\mathrm{Sm}_{S}\subseteq\mathrm{Sch}_{/S}$
spanned by the (quasi-compact quasi-separated) smooth $S$-schemes, and
consider $\mathbb{GW}^{\mathrm{s}}_{S}$ as a Nisnevich sheaf on that site. If
the base scheme $S$ itself is regular, then every smooth $S$-scheme is regular
as well, and, as mentioned above, for a regular Noetherian scheme $X$, the map
$\operatorname{GW}(X,\Qoppa^{\mathrm{s}}_{X})\to\mathbb{GW}(X,\Qoppa^{\mathrm{s}}_{X})$
is an equivalence. This means that for a regular Noetherian $S$, the functor
$\operatorname{GW}^{\mathrm{s}}_{S}(X)=\operatorname{GW}(X,\Qoppa^{\mathrm{s}})$
is a Nisnevich sheaf on $\mathrm{Sm}_{S}$. Alternatively, one may replace
$\operatorname{GW}$ by the functor $\operatorname{L}$ which associates to a
Poincaré $\infty$-category its $\operatorname{L}$-spectrum (whose homotopy
groups correspond to higher Witt groups in algebraic geometry), and obtain
using similar arguments that the symmetric $\operatorname{L}$-theory functor
$\operatorname{L}^{\mathrm{s}}_{S}$ is a Nisnevich sheaf on $\mathrm{Sm}_{S}$
for any regular Noetherian $S$.

Now a motivic spectrum over $S$ is not just a Nisnevich sheaf on
$\mathrm{Sm}_{S}$: it possess more structure (a tower of
$\mathbb{P}^{1}$-deloopings) and satisfies more axioms
($\mathbb{A}^{1}$-invariance). In order to incorporate the former into the
construction of the motivic Grothendieck-Witt spectrum, we need to understand
the behaviour of $\operatorname{GW}$ under taking products with
$\mathbb{P}^{1}$. More generally, we consider projective line bundles which
are not necessarily constant (and eventually also higher rank projective
bundles; see Theorem 6.1.6 below for the full version of this result, where
also the assumptions on $\mathcal{F}$ and $L$ are weakened):

######  Proposition 1.0.3 (Projective line formula).

Let $X$ be a qcqs scheme equipped with a line bundle $L$ and let $V$ be a
vector bundle over $X$ of rank $2$. Let
$\mathcal{F}\colon\mathrm{Cat}^{\mathrm{p}}\to{\mathcal{S}\mathrm{p}}$ be a
Karoubi-localising functor. Then we have a split fibre sequence

$\mathcal{F}\big{(}\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L}\big{)}\to\mathcal{F}\big{(}\mathcal{D}^{\mathrm{p}}(\mathbb{P}_{X}V),\Qoppa^{\mathrm{s}}_{p^{*}L}\big{)}\to\mathcal{F}\big{(}\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L\otimes\det
V^{\vee}[-r]}\big{)}\ ,$

where the first arrow is induced by pullback along $p$.

The last two propositions combined can be leveraged to obtain the following:

######  Proposition 1.0.4 (Bott periodicity).

Let $S$ be a quasi-compact quasi-separated base scheme. Then, the association
$\mathcal{F}\mapsto\mathcal{F}^{\mathrm{s}}_{S}$ described in Proposition
1.0.2 refines to a functor

$\operatorname{Fun}^{\mathrm{kloc}}(\mathrm{Cat}^{\mathrm{p}},{\mathcal{S}\mathrm{p}})\to{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(\mathrm{Sm}_{S},{\mathcal{S}\mathrm{p}}))\quad\quad\mathcal{F}\mapsto\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F}),$

where the left hand side is the $\infty$-category of (spectrum valued)
Karoubi-localising functors on $\mathrm{Cat}^{\mathrm{p}}$ and the right hand
side the $\infty$-category of $\mathbb{P}^{1}$-spectrum objects in Nisnevich
(spectrum valued) sheaves on $\mathrm{Sm}_{S}$. Furthermore, the
$\mathbb{P}^{1}$-deloopings of $\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F})$ are
governed by the rule

$\Sigma^{n}_{\mathbb{P}^{1}}\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F})\simeq\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F}^{[n]})$

where $\mathcal{F}^{[n]}$ is the functor
$\mathcal{F}^{[n]}(\mathcal{C},\Qoppa)=\mathcal{F}(\mathcal{C},\Qoppa^{[n]})=\mathcal{F}(\mathcal{C},\Sigma^{n}\Qoppa)$.

Applying the last result to $\mathbb{GW}$ yields a $\mathbb{P}^{1}$-spectrum
object $\mathcal{R}^{\mathrm{s}}_{S}(\mathbb{GW})$ (infinitely) delooping the
Nisnevich sheaf $\mathbb{GW}^{\mathrm{s}}_{S}$. In general
$\mathcal{R}^{\mathrm{s}}_{S}(\mathbb{GW})$ can fail to be a motivic spectrum,
as it might not be $\mathbb{A}^{1}$-invariant. However, if $S$ is regular
Noetherian, then the $\mathbb{A}^{1}$-invariance of
$\mathcal{R}^{\mathrm{s}}_{S}(\mathbb{GW})=\mathcal{R}^{\mathrm{s}}_{S}(\operatorname{GW})$
is assured, as we establish in the following key result (see Theorem 6.3.1):

######  Theorem 1.0.5.

Let $S$ be a regular Noetherian scheme. Then, the Nisnevich sheaf
$\operatorname{GW}^{\mathrm{s}}_{S}$ is $\mathbb{A}^{1}$-invariant.
Furthermore, all the components of its $\mathbb{P}^{1}$-delooping
$\mathcal{R}^{\mathrm{s}}_{S}(\operatorname{GW})$ are
$\mathbb{A}^{1}$-invariant, so that
$\mathcal{R}^{\mathrm{s}}_{S}(\operatorname{GW})$ is a motivic spectrum.

This means that for a regular Noetherian $S$, symmetric Grothendieck-Witt
theory is represented by a motivic spectrum, which we denote by
$\operatorname{KQ}_{S}\in\operatorname{SH}(S)$. This establishes Theorem 1.0.1
above. The proof of Theorem 1.0.5 is primarily based on another key property
of independent interest of symmetric Grothendieck-Witt theory - dévissage -
whose proof occupies §5 in its entirety (see Theorem 5.2.1 for the full
version of the result):

######  Theorem 1.0.6 (Dévissage).

Let $i\colon Z\hookrightarrow X$ be a closed embedding of finite dimensional
regular Noetherian schemes and $L$ a line bundle on $X$. Then the map

$\operatorname{GW}\big{(}\mathcal{D}^{\mathrm{p}}(Z),\Qoppa^{\mathrm{s}}_{i^{!}L}\big{)}\to\operatorname{GW}\big{(}\mathcal{D}^{\mathrm{p}}_{Z}(X),\Qoppa^{\mathrm{s}}_{L}|_{Z}\big{)}\
.$

induced by push-forward along $i$, is an equivalence.

If $S$ is only assumed quasi-compact and quasi-separated, then the
$\mathbb{P}^{1}$-spectrum $\mathcal{R}^{\mathrm{s}}_{S}(\mathbb{GW})$ is not
necessarily a motivic spectrum. We can however turn it into a motivic spectrum
in a universal manner by applying to it the $\mathbb{A}^{1}$-localisation
functor

$\mathrm{Loc}_{\mathbb{A}^{1}}\colon{\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}(\mathrm{Sm}_{/S},{\mathcal{S}\mathrm{p}}))\to\operatorname{SH}(S),$

which can be written explicitly as a suitable geometric realization. We hence
obtain a Grothendieck-Witt motivic spectrum
$\operatorname{KQ}_{S}\in\operatorname{SH}(S)$ for an arbitrary quasi-compact
quasi-separated $S$. These motivic spectra depend functorially on $S$, in the
sense that if $f\colon T\to S$ is a map of quasi-compact quasi-separated
schemes then we have an induced map

$\eta_{f}\colon f^{*}\operatorname{KQ}_{S}\to\operatorname{KQ}_{T}.$

We then verify the following property, which is of technical importance to
future applications:

######  Proposition 1.0.7 (Base change invariance).

The map $\eta_{f}$ is an equivalence. In particular, the motivic spectra
$\operatorname{KQ}_{S}$ are all pulled back from the absolute motivic spectrum
$\operatorname{KQ}:=\operatorname{KQ}_{\mathrm{spec}(\mathbb{Z})}\in\operatorname{SH}(\mathrm{spec}(\mathbb{Z}))$.

In addition to Grothendieck-Witt theory, we also apply our construction to
Karoubi $\operatorname{L}$-theory $\mathbb{L}$ (the Karoubi-localising
approximation of $\operatorname{L}$-theory) and to Bass
$\operatorname{K}$-theory $\mathbb{K}$, thus yielding motivic spectra
$\operatorname{KW}_{S}$ and $\operatorname{KGL}_{S}$ representing Karoubi Witt
theory and Bass $\operatorname{K}$-theory, respectively. Though the latter was
previously constructed in the literature, reproducing the construction in the
present setting allows one to use all the resulting functoriality. In
particular, all the structural maps relating Grothendieck-Witt theory,
$\operatorname{K}$-theory and $\operatorname{L}$-theory induce corresponding
maps on the level of motivic spectra, and we obtain for example, the Tate
fibre sequence

$(\operatorname{KGL}_{S})_{{\mathrm{h}\mathrm{C_{2}}}}\to\operatorname{KQ}_{S}\to\operatorname{KW}_{S}$

and the Wood sequence

$\Omega_{\mathbb{P}^{1}}\operatorname{KQ}_{S}\to\operatorname{KGL}_{S}\to\operatorname{KW}_{S}.$

Finally, we also consider the question of multiplicative structures on
$\operatorname{KQ},\operatorname{KW}$ and the maps relating them to each other
and to $\operatorname{KGL}$. In fact, this accounts for most of the more
technical parts of the paper, whose goal is to prove the following:

######  Proposition 1.0.8.

Let $S$ be a quasi-compact quasi-separated base scheme. Then, the association
$\mathcal{F}\mapsto\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F})$ described in
Proposition 1.0.4 refines to a lax symmetric monoidal functor. In particular,
if $\mathcal{F}$ is a Karoubi-localising lax symmetric monoidal functor then
$\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F})$ is a commutative algebra object in
the $\infty$-category
${\mathcal{S}\mathrm{p}}^{\mathbb{P}^{1}}(\mathrm{Sh}^{\mathrm{nis}}(\mathrm{Sm}_{S},{\mathcal{S}\mathrm{p}}))$
and $\mathrm{Loc}_{\mathbb{A}^{1}}\mathcal{R}^{\mathrm{s}}_{S}(\mathcal{F})$
is a motivic commutative ring spectrum.

This means in particular that $\operatorname{KQ}_{S}$ and
$\operatorname{KW}_{S}$ are motivic commutative ring spectra, and that the
canonical map $\operatorname{KQ}_{S}\to\operatorname{KW}_{S}$ is a commutative
ring map. It also implies that the forgetful map
$\operatorname{KQ}_{S}\to\operatorname{KGL}_{S}$ is one of motivic commutative
ring spectra.

### Acknowledgements

We thank Aravind Asok, Frédéric Déglise, Jens Hornbostel, Marc Hoyois, Marco
Schlichting and Mura Yakerson for fruitful discussions.

#### Financial Acknowledgements

The first author was partially supported by the french ANR project HQDiag
(ANR-21-CE40-0015). The second author was supported by the European Research
Council as part of the project “Foundations of Motivic Real K-Theory” (ERC
grant no. 949583).

##  2\. Recollections on hermitian K-theory

We use the framework developed in [9-authors-I, 9-authors-II]: hermitian
K-theory is a functor sending a Poincaré $\infty$-category, i.e. a stable
$\infty$-category endowed with a quadratic functor satisfying non-degeneracy
conditions, to a spectrum. Let us walk the reader through some of the
constructions and properties of this formalism.

###  2.1. Poincaré $\infty$-categories

In the framework developed in [9-authors-I] and [9-authors-II], we view
Grothendieck-Witt theory as an invariant of what we call a _Poincaré_
$\infty$-_category_ , a notion first introduced by Lurie [Lurie-L-theory] as a
novel framework for Ranicki’s L-theory. By definition, a Poincaré
$\infty$-category is a pair $(\mathcal{C},\Qoppa)$ consisting of a (small)
stable $\infty$-category $\mathcal{C}$ equipped with a _Poincaré structure_
$\Qoppa$, that is, a functor
$\Qoppa\colon\mathcal{C}^{\mathrm{op}}\to{\mathcal{S}\mathrm{p}}$ which is
reduced and $2$-excisive in the sense of Goodwillie’s functor calculus, and
whose symmetric cross-effect
$\mathrm{B}\colon\mathcal{C}^{\mathrm{op}}\times\mathcal{C}^{\mathrm{op}}\to{\mathcal{S}\mathrm{p}}$
is of the form $\mathrm{B}(X,Y)=\mathrm{hom}_{\mathcal{C}}(x,\mathrm{D}y)$ for
some equivalence
$\mathrm{D}\colon\mathcal{C}^{\mathrm{op}}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\simeq$}}\,}\mathcal{C}$.
In this case $\mathrm{D}$ is uniquely determined and endows $\mathcal{C}$ with
the structure of a _perfect duality_ , that is, a lift of $\mathcal{C}$ to a
$\mathrm{C_{2}}$-homotopy fixed point in $\mathrm{Cat}^{\mathrm{ex}}$ with
respect to the $\mathrm{op}$-action. The notion of a morphism of Poincaré
$\infty$-categories
$(\mathcal{C},\Qoppa)\to(\mathcal{C}^{\prime},\Qoppa^{\prime})$ is that of a
_Poincaré functor_ , which consists of a pair $(f,\eta)$ where
$f\colon\mathcal{C}\to\mathcal{C}^{\prime}$ is an exact functor and
$\eta\colon\Qoppa\Rightarrow\Qoppa^{\prime}\circ f^{\mathrm{op}}$ is a natural
transformation for which the induced arrow
$\mathrm{D}f^{\mathrm{op}}\Rightarrow f\mathrm{D}$ is an equivalence. In
particular, Poincaré functors can be considered as a refinement of duality
preserving functors. The collection of Poincaré $\infty$-categories and
Poincaré functors between then assembles to form an $\infty$-category we
denote by $\mathrm{Cat}^{\mathrm{p}}$.

We think of a Poincaré structure as providing an abstract notion of forms on
the objects of $\mathcal{C}$. More precisely, we call points
$q\in\Omega^{\infty}\Qoppa(x)$ hermitian forms on $x$, and we say that such a
form is Poincaré if the map $q_{\sharp}\colon x\to\mathrm{D}x$ determined by
$q$ is an equivalence.

######  Example 2.1.1.

  1. (1)

Let $(\mathcal{C},\mathrm{D})$ be a stable $\infty$-category with perfect
duality. Then the functors

$\Qoppa^{\mathrm{s}}(x)=\mathrm{hom}_{\mathcal{C}}(x,\mathrm{D}x)^{\mathrm{h}\mathrm{C_{2}}}\qquad\Qoppa^{\mathrm{q}}(x)=\mathrm{hom}_{\mathcal{C}}(x,\mathrm{D}x)_{\mathrm{h}\mathrm{C_{2}}}$

are Poincaré structures with underlying duality $\mathrm{D}$. We call them the
symmetric and quadratic Poincaré structures associated to the duality
$\mathcal{D}$, respectively. The corresponding notions of hermitian forms they
encode can be considered as the homotopy coherent avatars of the classical
notions of symmetric bilinear and quadratic forms, respectively.

  2. (2)

Let $X$ be a scheme and $L$ a line bundle. Then $L$ determines a duality on
the perfect derived $\infty$-category $\mathcal{D}^{\mathrm{p}}(X)$ of $X$,
whose underlying duality
$\mathcal{D}^{\mathrm{p}}(X)\to\mathcal{D}^{\mathrm{p}}(X)^{\mathrm{op}}$
sends $M$ to the internal mapping object
$\mathrm{D}_{L}(M):=\uline{\hom}_{X}(M,L)$. The symmetric and quadratic
Poincaré structures associated to this duality are then denoted by
$\Qoppa^{\mathrm{s}}_{L}$ and $\Qoppa^{\mathrm{q}}_{L}$. More generally, we
can introduce a $\mathrm{C_{2}}$-action on $L$, which affects the structure of
$\mathrm{D}_{L}$ as a duality (though not the underlying equivalence). The
Poincaré $\infty$-categories of the form
$(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})$ constitute the main
examples of interest for us in the present paper.

  3. (3)

For $\mathcal{C}$ a stable $\infty$-category consider
$\overline{\operatorname{Hyp}}(\mathcal{C}):=\mathcal{C}\times\mathcal{C}^{\mathrm{op}}$
equipped with the Poincaré structure

$\Qoppa_{\mathrm{hyp}}(X,Y):=\mathrm{hom}_{\mathcal{C}}(X,Y)\,.$

Then
$\operatorname{Hyp}(\mathcal{C}):=(\overline{\operatorname{Hyp}},\Qoppa_{\mathrm{hyp}})$
is a Poincaré $\infty$-category with underlying duality
$\mathrm{D}_{\mathrm{hyp}}(X,Y)=(Y,X)$. We refer to it as the hyperbolic
category of $\mathcal{C}$. We note that $\Qoppa_{\mathrm{hyp}}$ is both the
symmetric and the quadratic Poincaré structure associated to the duality
$\mathrm{D}_{\mathrm{hyp}}$ in the sense of ((1)).

  4. (4)

If $(\mathcal{C},\Qoppa)$ is a Poincaré $\infty$-category with underlying
duality $\mathrm{D}\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{C}$, then the
pair $(\mathcal{C},\Qoppa^{[n]}):=(\mathcal{C},\Sigma^{n}\Qoppa)$ is a
Poincaré $\infty$-category with underlying duality $\Sigma^{n}\mathrm{D}$.

  5. (5)

Let $R$ be a discrete commutative ring. Then, there exists a unique Poincaré
structure $\Qoppa^{\mathrm{gs}}$ on the perfect derived $\infty$-category
$\mathcal{D}^{\mathrm{p}}(R)$ of $R$ whose value on any finitely generated
projective module $P$ is the (Eilenberg-Mac Lane spectrum of the) abelian
group of symmetric bilinear forms on $P$. Similarly, there exists a unique
Poincaré structure $\Qoppa^{\mathrm{gq}}$ such that $\Qoppa^{\mathrm{gq}}(P)$
is the abelian group of quadratic forms on $P$ for such $P$. We call these the
_genuine symmetric_ and _genuine quadratic_ Poincaré structures, respectively.
Their underlying duality is the usual duality
$\mathrm{D}_{R}:=\underline{\hom}(-,R)$, but $\Qoppa^{\mathrm{gs}}$ and
$\Qoppa^{\mathrm{gq}}$ are generally _not_ the associated symmetric and
quadratic Poincaré structures in the sense of (1).

###  2.2. Hermitian K-theory

Let $(\mathcal{C},\Qoppa)$ be a Poincaré $\infty$-category.

A _Poincaré object_ in $(\mathcal{C},\Qoppa)$ is a pair $(x,q)$ where $x$ is
an object of $\mathcal{C}$ and $q\in\Omega^{\infty}\Qoppa(x)$ is a Poincaré
form on $x$, that is, a hermitian form such that the map $q_{\sharp}\colon
X\to\mathrm{D}X$ determined by the image of $q$ under
$\Qoppa(x)\to\mathrm{B}(x,x)=\mathrm{hom}_{\mathcal{C}}(x,\mathrm{D}x)$, is an
equivalence. We then write
$\mathrm{Pn}(\mathcal{C},\Qoppa)\subseteq\int_{\iota\mathcal{C}}\Omega^{\infty}$
for the full subgroupoid spanned by the Poincaré objects, where
$\iota\mathcal{C}$ denotes the core $\infty$-groupoid of $\mathcal{C}$. The
association $(\mathcal{C},\Qoppa)\mapsto\mathrm{Pn}(\mathcal{C},\Qoppa)$
assembles to a functor
$\mathrm{Pn}\colon\mathrm{Cat}^{\mathrm{p}}\to\mathcal{S}$ by
[9-authors-I]*Lemma 2.1.5, analogous to the groupoid core functor
$\mathrm{Cr}\colon\mathrm{Cat}^{\mathrm{ex}}\to\mathcal{S}$ yielding classes
in K-theory, where $\mathrm{Cat}^{\mathrm{ex}}$ is the $\infty$-category of
stable $\infty$-categories.

In order to describe the context in which hermitian K-theory naturally lives,
we need to recall the notions of _additive_ , _localising_ and _Karoubi-
localising_ functors.

Both the $\infty$-categories $\mathrm{Cat}^{\mathrm{ex}}$ and
$\mathrm{Cat}^{\mathrm{p}}$ are pointed (in fact, semi-additive), and we can
consider fiber and cofibre sequences in them. A _Verdier_ or _Poincaré-
Verdier_ sequence is a sequence

$\mathcal{C}\to\mathcal{D}\to\mathcal{E}\qquad\text{or}\qquad(\mathcal{C},\Qoppa)\to(\mathcal{D},\Phi)\to(\mathcal{E},\Psi)$

that is both a fiber and a cofibre sequence in $\mathrm{Cat}^{\mathrm{ex}}$ or
$\mathrm{Cat}^{\mathrm{p}}$ respectively. A Verdier sequence is furthermore
_split_ when both maps in the sequence have adjoints on both sides. Split
Verdier sequences are equivalently described as stable recollements (see
[9-authors-II]*Definition A.2.10 and Proposition A.2.11). A Poincaré-Verdier
sequence is _split_ when its underlying Verdier sequence is (see [9-authors-
II]*Definition 1.1.1).

A functor $\mathrm{Cat}^{\mathrm{p}}\to{\mathcal{S}\mathrm{p}}$ is _additive_
or _localising_ if it sends split Poincaré-Verdier or Poincaré-Verdier
respectively to fiber sequences of spectra. Analogous definitions apply to a
functor $\mathrm{Cat}^{\mathrm{ex}}\to{\mathcal{S}\mathrm{p}}$, removing
“Poincaré” everywhere (see [9-authors-II]*Definition 1.5.4 and Proposition
1.5.5).

Hermitian K-theory, also known as the _Grothendieck-Witt spectrum_ functor

$\operatorname{GW}\colon\mathrm{Cat}^{\mathrm{p}}\to{\mathcal{S}\mathrm{p}}$

is the universal (initial) _additive_ functor equipped with a transformation
of functors $\mathrm{Pn}\to\Omega^{\infty}\operatorname{GW}$ (see [9-authors-
II]*Definition 4.2.1 and Corollary 4.2.2). Analogously, the K-theory spectrum
functor is the universal additive functor
$\operatorname{K}\colon\mathrm{Cat}^{\mathrm{ex}}\to{\mathcal{S}\mathrm{p}}$
equipped with a map $\mathrm{Cr}\to\Omega^{\infty}\operatorname{K}$. A notable
consequence of additivity is the Bott-Genauer sequence

$\operatorname{GW}(\mathcal{C},\Qoppa)\to\operatorname{K}(\mathcal{C})\to\operatorname{GW}(\mathcal{C},\Qoppa^{[1]})$

which exhibits K-theory as an extension of GW-theory by GW-theory of a shifted
Poincaré structure.

Another important invariant of a Poincaré $\infty$-category is its L-theory
spectrum $\operatorname{L}(\mathcal{C},\Qoppa)$, which, in the setting of
Poincaré $\infty$-categories, was first defined in [Lurie-L-theory], but was
extensively studied in more classical contexts by Ranicki. For
$n\in\mathbb{Z}$ the group
$\operatorname{L}_{n}(\mathcal{C},\Qoppa):=\pi_{n}(\mathcal{C},\Qoppa)$ is the
quotient of the monoid of $\pi_{0}\mathrm{Pn}(\mathcal{C},\Qoppa^{[n]})$ of
equivalences classes of $n$-shifted Poincaré objects by the submonoid
consisting of the _metabolic Poincaré objects_ , that is, those admitting a
Lagrangian. As established in [9-authors-II], The Grothendieck-Witt, L-theory
and K-theory spectra are related via the _fundamental fibre sequence_

$\operatorname{K}_{{\mathrm{h}\mathrm{C_{2}}}}\to\operatorname{GW}\to\operatorname{L}.$

One can show that both $\operatorname{K}$ and $\operatorname{L}$ are in fact
Verdier-localising functors, and hence so is $\operatorname{GW}$ (see
[9-authors-II]*Corollary 4.4.15). In particular, $\operatorname{GW}$ is also
universal as a Verdier-localising invariant.

###  2.3. Karoubi-Grothendieck-Witt theory

In this article, we actually mostly use the Karoubi-Grothendieck-Witt functor,
i.e. a modified version of the above that is insensitive to idempotent
completion of the underlying stable $\infty$-category. Let us now describe it.

A _Karoubi_ or a _Poincaré-Karoubi_ sequence is a sequence in
$\mathrm{Cat}^{\mathrm{ex}}$ or $\mathrm{Cat}^{\mathrm{p}}$, respectively,
that becomes a fiber-cofibre sequence in idempotent complete stable
$\infty$-categories or idempotent complete Poincaré categories, respectively,
after idempotent completion of its terms (see [9-authors-II]*Definition
1.3.6). A functor $\mathrm{Cat}^{\mathrm{ex}}\to{\mathcal{S}\mathrm{p}}$ or
$\mathrm{Cat}^{\mathrm{p}}\to{\mathcal{S}\mathrm{p}}$ is called _Karoubi-
localising_ if it sends Karoubi or Poincaré-Karoubi sequences, respectively,
to fiber sequences (see [9-authors-II]*Definition 2.7.1 and Proposition
1.5.5). In [9-authors-V], we construct universal Karoubi-localising
approximations

$\operatorname{GW}\Rightarrow\mathbb{GW}\quad\text{and}\quad\operatorname{L}\Rightarrow\mathbb{L}$

of Grothendieck-Witt- and L-theory, respectively. These are the analogues of
the map $\operatorname{K}\Rightarrow\mathbb{K}$ from algebraic to Bass
K-theory, which is the universal Karoubi-localising approximation of K-theory,
see [BGT]. These Karoubi-localising variants fit to form analogues of the
Bott-Genauer sequence

(1) $\mathbb{GW}\Rightarrow\mathbb{K}\Rightarrow\mathbb{GW}((-)^{[1]})$

and fundamental fibre sequence

(2)
$\mathbb{K}_{{\mathrm{h}\mathrm{C_{2}}}}\Rightarrow\mathbb{GW}\Rightarrow\mathbb{L}.$

In addition, we prove in [9-authors-V] that the squares

(3)
${\operatorname{GW}(\mathcal{C},\Qoppa)}$${\mathbb{GW}(\mathcal{C},\Qoppa)}$${\operatorname{K}(\mathcal{C})^{\mathrm{h}\mathrm{C_{2}}}}$${\mathbb{K}(\mathcal{C})^{\mathrm{h}\mathrm{C_{2}}}}$
and
${\operatorname{L}(\mathcal{C})}$${\mathbb{L}(\mathcal{C},\Qoppa)}$${\operatorname{K}(\mathcal{C})^{\mathrm{t}\mathrm{C_{2}}}}$${\mathbb{K}(\mathcal{C})^{{\mathrm{t}\mathrm{C_{2}}}}\
.}$

are exact, a result we call _Karoubi cofinality_. This means, in particular,
that the map
$\operatorname{GW}(\mathcal{C},\Qoppa)\to\mathbb{GW}(\mathcal{C},\Qoppa)$ is
an equivalence on positive homotopy groups, and also on $\pi_{0}$ if
$\mathcal{C}$ is idempotent complete. In addition, if $\mathcal{C}$ is such
that the map $\operatorname{K}(\mathcal{C})\to\mathbb{K}(\mathcal{C})$ is an
equivalence then the maps
$\operatorname{GW}(\mathcal{C},\Qoppa)\to\mathbb{GW}(\mathcal{C},\Qoppa)$ and
$\operatorname{L}(\mathcal{C},\Qoppa)\to\mathbb{L}(\mathcal{C},\Qoppa)$ are
equivalences as well.

For a scheme $X$ with a line bundle $L$, the spectra

$\mathbb{GW}^{\mathrm{s}}(X,L):=\mathbb{GW}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})\quad\text{and}\quad\mathbb{L}^{\mathrm{s}}(X,L):=\mathbb{L}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})$

will be our principal object of interest in the present paper. When $X$ is
regular the map $\operatorname{K}(X)\to\mathbb{K}(X)$ is an equivalence and
hence by the above these coincide with

$\operatorname{GW}^{\mathrm{s}}(X,L)=\operatorname{GW}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})\quad\text{and}\quad\operatorname{L}^{\mathrm{s}}(X,L):=\operatorname{L}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L}),$

respectively. The specific role played by Poincaré-Karoubi sequences is due to
the fact that the localisation sequence of derived categories

$\mathcal{D}^{\mathrm{p}}_{Z}(X)\to\mathcal{D}^{\mathrm{p}}(X)\to\mathcal{D}^{\mathrm{p}}(U)$

associated to a closed subscheme $Z\subset X$ with open complement $U$ is only
a Karoubi sequence and not a Verdier sequence in general. As was emphasized in
[thomason-trobaugh], although all categories are idempotent complete, the map
$\mathcal{D}^{\mathrm{p}}(X)\to\mathcal{D}^{\mathrm{p}}(U)$ is not essentially
surjective in general: an object in the target is only a direct factor of an
object in the image.

##  3\. Poincaré structures on derived categories of schemes

For a scheme $X$, let ${\mathcal{P}ic}(X)$ denote the maximal subgroupoid of
$\mathcal{D}^{\mathrm{p}}(X)$ whose objects are tensor invertible. To an
invertible perfect complex with $\mathrm{C_{2}}$-action
$L\in{\mathcal{P}ic}(X)^{\mathrm{B}\mathrm{C_{2}}}$, one may associate a
Poincaré structure $\Qoppa^{\mathrm{s}}_{L}$ on the $\infty$-category
$\mathcal{D}^{\mathrm{p}}(X)$ of perfect complexes as follows:

######  Definition 3.0.1.

Let $X$ be a scheme and $L\in{\mathcal{P}ic}(X)^{\mathrm{B}\mathrm{C_{2}}}$ an
invertible object. We then define
$\Qoppa^{\mathrm{s}}_{L}\colon\mathcal{D}^{\mathrm{p}}(X)^{\mathrm{op}}\to{\mathcal{S}\mathrm{p}}$
to be the Poincaré structure given by

$\Qoppa^{\mathrm{s}}_{L}(M)=\mathrm{hom}_{\mathcal{D}^{\mathrm{p}}(X)}(M\otimes
M,L)^{{\mathrm{h}\mathrm{C_{2}}}}.$

The fact that this is a Poincaré structure will be proven below, see
Proposition 3.1.6. We refer to $\Qoppa^{\mathrm{s}}_{L}$ as the associated
_symmetric_ Poincaré structure. We then write $\mathcal{E}_{L}$ for the quasi-
coherent complex characterized by the existence of a natural equivalence

$\mathrm{hom}_{X}(P\otimes
P,L)^{\mathrm{t}\mathrm{C_{2}}}\cong\mathrm{hom}_{X}(P,\mathcal{E}_{L})$

for any perfect complex $P$ on $X$. In other words, $\mathcal{E}_{L}$ is the
object of
$\mathcal{D}^{\mathrm{qc}}(X)=\operatorname{Ind}(\mathcal{D}^{\mathrm{p}}(X))$
representing the linear part of $\Qoppa^{\mathrm{s}}_{L}$. Similarly for every
$m$ we may consider the truncated Poincaré structure $\Qoppa^{\geq m}_{L}$ on
$\mathcal{D}^{\mathrm{p}}(X)$, given as the fibre product $\Qoppa^{\geq
m}_{L}(M)=\Qoppa^{\mathrm{s}}_{L}(M)\times_{\mathrm{hom}_{X}(P,\mathcal{E}_{L})}\mathrm{hom}_{X}(P,\tau_{\geq
m}\mathcal{E}_{L})$

######  Remark 3.0.2.

The difference between the notion of an invertible perfect complex and that of
a line bundle is not huge. By the main result of [fausk], for any invertible
perfect complex $L$ on $X$, there exist a disjoint union decomposition
$X=\coprod_{n\in\mathbb{Z}}X_{n}$ (where each $X_{n}$ is potentially empty)
such that $L|_{X_{n}}$ is a degree $n$ shift of a line bundle. In particular,
if $X$ is connected then any invertible perfect complex $L$ is globally a
shift of a line bundle.

While the definition itself is simple, some effort is required in order to
formally set up all the relevant functoriality and multiplicativity of this
construction in $X$ and $L$, summarized by the commutative diagram (12) below,
and the present section is dedicated to these somewhat technical
considerations. Specifically, in §3.1, we analyse in detail the formation of
the symmetric Poincaré structure associated to a rigid stable symmetric
monoidal $\infty$-category, in §3.2, we focus on the dependency on $X$ and
$L$, and in §3.3, we consider the genuine variants. A reader willing to accept
all functorialities at faith is welcome to skip directly to Proposition 3.1.6.

###  3.1. Poincaré $\infty$-categories associated to rigid symmetric monoidal
structures

Recall that a symmetric monoidal $\infty$-category is said to be _rigid_ if
every object $x\in\mathcal{C}$ is dualisable, that is, admits a dual
$x^{\vee}\in\mathcal{C}$ equipped with a unit $1_{\mathcal{C}}\to x\otimes
x^{\vee}$ and a counit map $x^{\vee}\otimes x\to 1_{\mathcal{C}}$ satisfying
the triangle inequalities. If $\mathcal{C}$ is a rigid $\infty$-category then
the association $x\mapsto x^{\vee}$ assembles to form a duality
$(-)^{\vee}\colon\mathcal{C}\to\mathcal{C}^{\mathrm{op}}$ on $\mathcal{C}$
(that is, a $\mathrm{C_{2}}$-fixed structure on $\mathcal{C}$ with respect to
the $\mathrm{op}$-action of $\mathrm{C_{2}}$ on $\mathrm{Cat}$) which can be
encoded via the perfect symmetric bilinear functor
$(x,y)\mapsto\mathrm{hom}(x\otimes y,1_{\mathcal{C}})$. Furthermore, this
duality is symmetric monoidal, that is, it is a $\mathrm{C_{2}}$-fixed
structure also with respect to the $\mathrm{op}$-action of $\mathrm{C_{2}}$ on
$\operatorname{CAlg}(\mathrm{Cat})$, and is natural in
$\mathcal{C}\in\operatorname{CAlg}(\mathrm{Cat})$, so that symmetric monoidal
functors are canonically duality preserving. A proof of these statements can
be found in [HLAS]*Theorem 5.11; more precisely, in loc. cit. the authors show
that associating to each rigid $\infty$-category its duality determines a
section of the forgetful functor
$\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})^{{\mathrm{h}\mathrm{C_{2}}}}\to\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})$,
yielding a trivialization of the $\mathrm{op}$-action on the full subcategory
$\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})\subseteq\operatorname{CAlg}(\mathrm{Cat})$
spanned by the rigid $\infty$-categories. In fact, within the proof of
[HLAS]*Theorem 5.11, a slightly more general construction is considered: given
a tensor invertible object $L\in\mathcal{C}$, one may consider the twisted
duality $\mathrm{D}_{L}(x)=x^{\vee}\otimes L$, which is associated to the
perfect symmetric bilinear functor $(x,y)\mapsto\mathrm{hom}(x\otimes y,L)$.
This construction can be organized into a commutative square of product-
preserving functors

(4)
${(\mathcal{C},L)}$${(\mathcal{C},\mathrm{D}_{L})}$${\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})\times_{\mathrm{Cat}}\mathrm{Cat}_{*}}$${\mathrm{Inv}}$${\mathrm{Cat}^{{\mathrm{h}\mathrm{C_{2}}}}}$${\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})}$${\mathrm{Cat}\
,}$

$\in$

$\in$

${\supseteq}$

where $\mathrm{Inv}$ is the full subcategory of
$\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})\times_{\mathrm{Cat}}\mathrm{Cat}_{*}$
spanned by those $(\mathcal{C},L)$ such that $L$ is tensor invertible, where
$\mathrm{Cat}_{*}$ is the total category of the universal cocartesian
fibration [HTT]*3.3.2. Passing to commutative monoid objects one obtains a
functor

$\operatorname{CAlg}(\mathrm{Inv})\simeq\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})\to\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})\times_{\operatorname{CAlg}(\mathrm{Cat})}\operatorname{CAlg}(\mathrm{Cat})^{{\mathrm{h}\mathrm{C_{2}}}}=\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})^{{\mathrm{h}\mathrm{C_{2}}}},$

yielding a section of the forgetful functor
$\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})^{{\mathrm{h}\mathrm{C_{2}}}}\to\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})$.

We would like to adapt the above constructions to the setting of stable
$\infty$-categories and Poincaré structures. We first introduce a bit of
notation. In what follows we say that a symmetric monoidal functor

$p^{\otimes}\colon\mathcal{E}^{\otimes}\to\mathcal{B}^{\otimes}$

is a _$\mathrm{Fin}_{*}$ -cartesian fibration_ if the underlying functor
$p\colon\mathcal{E}\to\mathcal{B}$ is a cartesian fibration of
$\infty$-categories and for every $x\in\mathcal{E}$ the functor
$x\otimes(-)\colon\mathcal{E}\to\mathcal{E}$ preserves $p$-cartesian edges.
Dually, we say that $p^{\otimes}$ is a _$\mathrm{Fin}_{*}$ -cocartesian
fibration_ if the opposite symmetric monoidal functor
$(p^{\mathrm{op}})^{\otimes}\colon(\mathcal{E}^{\mathrm{op}})^{\otimes}\to(\mathcal{B}^{\mathrm{op}})^{\otimes}$
is a $\mathrm{Fin}_{*}$-cartesian fibration.

######  Lemma 3.1.1.

Let $p^{\otimes}\colon\mathcal{E}^{\otimes}\to\mathcal{B}^{\otimes}$ be a
symmetric monoidal functor. Then the following are equivalent:

  1. (1)

$p^{\otimes}$ is a $\mathrm{Fin}_{*}$-cartesian fibration in the above sense.

  2. (2)

$p\colon\mathcal{E}\to\mathcal{B}$ is a cartesian fibration and for every pair
of $p$-cartesian edges $x\to y$ and $z\to w$ in $\mathcal{E}$, the edge
$x\otimes z\to y\otimes w$ is again $p$-cartesian.

  3. (3)

For every $\langle n\rangle\in\mathrm{Fin}_{*}$ the functor $p_{\langle
n\rangle}^{\otimes}\colon\mathcal{E}^{\otimes}_{\langle
n\rangle}\to\mathcal{B}^{\otimes}_{\langle n\rangle}$ is a cartesian
fibration, and for every $\alpha\colon\langle n\rangle\to\langle m\rangle$ in
$\mathrm{Fin}_{*}$ the transition functor
$\alpha_{!}\colon\mathcal{E}^{\otimes}_{\langle
n\rangle}\to\mathcal{E}^{\otimes}_{\langle m\rangle}$ sends $p_{\langle
n\rangle}^{\otimes}$-cartesian arrows to $p^{\otimes}_{\langle
m\rangle}$-cartesian arrows.

  4. (4)

The opposite symmetric monoidal functor
$(p^{\mathrm{op}})^{\otimes}\to(\mathcal{E}^{\mathrm{op}})^{\otimes}\to(\mathcal{B}^{\mathrm{op}})^{\otimes}$
is a cocartesian fibration of $\infty$-operads (in the sense of
[HA]*Definition 2.1.2.13).

We note that, in particular, Lemma 3.1.1 says that the notion of a
$\mathrm{Fin}_{*}$-cocartesian fibration identifies with that of a cocartesian
fibration of $\infty$-operads (at least when the target is a symmetric
monoidal $\infty$-category). The former notion is however sometimes easier to
identify in practice.

###### Proof of Lemma 3.1.1.

In (2), writing $x\otimes z\to y\otimes w$ as a composite $x\otimes z\to
x\otimes w\to y\otimes w$ we see that one may as well assume that one of the
arrows is an identity arrow. We conclude that (1) and (2) are equivalent.

Since $p^{\otimes}$ is a map of $\infty$-operads we have that for every
$\langle n\rangle\in\mathrm{Fin}_{*}$ the map $\mathcal{E}^{\otimes}_{\langle
n\rangle}\to\mathcal{B}^{\otimes}_{\langle n\rangle}$ breaks as the cartesian
product
$\prod_{i=1}^{n}p\colon\prod_{i=1}^{n}\mathcal{E}\to\prod_{i=1}^{n}\mathcal{B}$
of $n$ copies of the underlying map $p\colon\mathcal{E}\to\mathcal{B}$, that
is, of the induced map on fibres over $\langle 1\rangle\in\mathrm{Fin}_{*}$.
We conclude that $p$ is a cartesian fibration if and only if
$p^{\otimes}_{\langle n\rangle}$ is a cartesian fibration for every $\langle
n\rangle\in\mathrm{Fin}_{*}$, in which case the transition functors
$\alpha_{!}\colon\mathcal{E}^{\otimes}_{\langle
n\rangle}\to\mathcal{E}^{\otimes}_{\langle m\rangle}$ automatically preserve
cartesian edges for every inert $\alpha$. Now any active $\alpha\colon\langle
n\rangle\to\langle m\rangle$ can be written as a composite of active maps
$\alpha=\alpha_{1}\circ\cdots\circ\alpha_{k}$ such that the fibres of each
$\alpha_{i}$ are of size at most $2$, and so we conclude that the transition
functors $\alpha_{!}\colon\mathcal{E}^{\otimes}_{\langle
n\rangle}\to\mathcal{E}^{\otimes}_{\langle m\rangle}$ all preserve cartesian
edges if and only if this holds for the active maps $\langle
2\rangle\to\langle 1\rangle$ and $\langle 0\rangle\to\langle 1\rangle$. The
latter case is automatic since $\mathcal{E}_{\langle
0\rangle}=\mathcal{B}_{\langle 0\rangle}=\ast$, and so we conclude that (2)
and (3) are equivalent.

Finally, let us show that (3) and (4) are equivalent. First if
$(p^{\mathrm{op}})^{\otimes}\to(\mathcal{E}^{\mathrm{op}})^{\otimes}\to(\mathcal{B}^{\mathrm{op}})^{\otimes}$
is a cocartesian fibration then each $(p^{\mathrm{op}})_{\langle
n\rangle}^{\otimes}\colon(\mathcal{E}^{\mathrm{op}})^{\otimes}_{\langle
n\rangle}\to(\mathcal{B}^{\mathrm{op}})^{\otimes}_{\langle n\rangle}$ is a
cocartesian fibration by base change, and on the other hand by
[HTT]*Proposition 2.4.2.11 we have that if each $(p^{\mathrm{op}})_{\langle
n\rangle}^{\otimes}$ is a cocartesian fibration then
$(p^{\mathrm{op}})^{\otimes}$ is at least a locally cocartesian fibration,
with the locally cocartesian edges of $(\mathcal{E}^{\mathrm{op}})^{\otimes}$
being exactly those of the form $e^{\prime}\circ e^{\prime\prime}$ with
$e^{\prime}$ a fibrewise cocartesian edge (lying over some $\langle
n\rangle\in\mathrm{Fin}_{*}$) and $e^{\prime\prime}$ an edge which is
cocartesian over $\mathrm{Fin}_{*}$. At the same time, by [HTT]*Proposition
2.4.2.8 a locally cocartesian fibration is cocartesian if and only if the
collection of locally cocartesian edges is closed under composition. To finish
the proof it will hence suffice to show that when all the
$(p^{\mathrm{op}})^{\otimes}_{\langle n\rangle}$ are cocartesian fibrations,
the collection of locally cocartesian edges in $(p^{\mathrm{op}})^{\otimes}$
is closed under composition if and only if each transition functor
$\alpha_{!}\colon(\mathcal{E}^{\mathrm{op}})^{\otimes}_{\langle
n\rangle}\to(\mathcal{E}^{\mathrm{op}})^{\otimes}_{\langle m\rangle}$
preserves cocartesian edges. Since the collection of cocartesian edges with
respect to any map is closed under composition the above explicit
characterization of the collection of locally
$(p^{\mathrm{op}})^{\otimes}$-cocartesian edges of
$(\mathcal{E}^{\mathrm{op}})^{\otimes}$ implies that this collection is closed
under composition if and only if any composite of the form
$e^{\prime\prime}\circ e^{\prime}$ is locally
$(p^{\mathrm{op}})^{\otimes}$-cocartesian whenever $e^{\prime}$ is a
$(p^{\mathrm{op}})^{\otimes}_{\langle n\rangle}$-cocartesian edge of
$(\mathcal{E}^{\mathrm{op}})^{\otimes}_{\langle n\rangle}$ for some $\langle
n\rangle\in\mathrm{Fin}_{*}$ and $e^{\prime\prime}$ is a cocartesian lift over
$\mathrm{Fin}_{*}$ of some $\alpha\colon\langle n\rangle\to\langle m\rangle$.
Now given such $e^{\prime}$ and $e^{\prime\prime}$ we may always lift the
identity square on the arrow $\alpha$ in $\mathrm{Fin}_{*}$ to a commutative
square in $\mathcal{E}$ of the form

${x}$${y}$${z}$${w\
,}$$\scriptstyle{f^{\prime\prime}}$$\scriptstyle{e^{\prime}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{e^{\prime\prime}}$

where $f^{\prime\prime}$ is a cocartesian lift of $\alpha$ starting at $x$ and
$f^{\prime}$ lies in the fibre $(\mathcal{E}^{\mathrm{op}})^{\otimes}_{\langle
m\rangle}$. In particular, such a square exhibits $f^{\prime}$ as the image
under $e^{\prime}$ by the transition functor
$\alpha_{!}\colon(\mathcal{E}^{\mathrm{op}})^{\otimes}_{\langle
n\rangle}\to(\mathcal{E}^{\mathrm{op}})^{\otimes}_{\langle m\rangle}$, and
$e^{\prime\prime}\circ e^{\prime}=f^{\prime}\circ f^{\prime\prime}$. To finish
the proof it will hence suffice to show that in this scenario,
$f^{\prime}\circ f^{\prime\prime}$ is locally
$(p^{\mathrm{op}})^{\otimes}$-cocartesian if and only if $f^{\prime}$ is
$(p^{\mathrm{op}})^{\otimes}_{\langle m\rangle}$-cocartesian. The “if”
direction follows from the explicit characterization of locally
$(p^{\mathrm{op}})^{\otimes}$-cocartesian edges above. Now assume that
$f^{\prime}\circ f^{\prime\prime}$ is locally
$(p^{\mathrm{op}})^{\otimes}$-cocartesian. By [HTT]*Proposition 2.4.1.3 and
the fact that $(p^{\mathrm{op}})^{\otimes}$ is symmetric monoidal we have that
the edge $f^{\prime\prime}$ is $(p^{\mathrm{op}})^{\otimes}$-cocartesian, and
hence by [HTT]*Lemma 2.4.2.7 we conclude that $f^{\prime}$ is locally
$(p^{\mathrm{op}})^{\otimes}$-cocartesian. It is consequently also locally
$(p^{\mathrm{op}})^{\otimes}_{\langle n\rangle}$-cocartesian. But
$(p^{\mathrm{op}})^{\otimes}_{\langle n\rangle}$ is a cocartesian fibration,
and hence $f^{\prime}$ is $(p^{\mathrm{op}})^{\otimes}_{\langle
n\rangle}$-cocartesian by [HTT]*Proposition 2.4.2.8. ∎

We now recall from [HA]*Remark 2.4.2.6 that for a symmetric monoidal
$\infty$-category $\mathcal{B}^{\otimes}$, lax symmetric monoidal functors
$\mathcal{B}^{\otimes}\to\mathrm{Cat}^{\times}$ correspond, via
unstraightening, to symmetric monoidal cocartesian fibrations
$\mathcal{E}^{\otimes}\to\mathcal{B}^{\otimes}$ of $\infty$-operads (where
$\mathcal{E}^{\otimes}$ is necessarily a symmetric monoidal $\infty$-category,
since $\mathcal{B}^{\otimes}$ is so). Passing to symmetric monoidal opposites,
we get from Lemma 3.1.1 that lax symmetric monoidal functors
$\chi\colon(\mathcal{B}^{\mathrm{op}})^{\otimes}\to\mathrm{Cat}^{\times}$
correspond via unstraightening to $\mathrm{Fin}_{*}$-cartesian fibrations
$\mathcal{E}^{\otimes}\to\mathcal{B}^{\otimes}$. The underlying cartesian
fibration $\mathcal{E}\to\mathcal{B}$ is then the one classified by the
underlying functor of $\chi$.

A prototypical example of a $\mathrm{Fin}_{*}$-cartesian fibration is the
projection $\mathrm{Cat}^{\otimes}_{//\mathcal{D}}\to\mathrm{Cat}^{\times}$
constructed in [9-authors-I]*Lemma 5.2.3 for a given symmetric monoidal
$\infty$-category $\mathcal{D}$. This is the $\mathrm{Fin}_{*}$-cartesian
fibration which classifies the lax symmetric monoidal functor
$(\mathrm{Cat}^{\mathrm{op}})^{\times}\to\mathrm{Cat}$ sending $\mathcal{C}$
to $\operatorname{Fun}(\mathcal{C},\mathcal{D})$. In particular, for a functor
$g\colon\mathcal{C}\to\mathcal{C}^{\prime}$, the associated cartesian
transition functor
$(\mathrm{Cat}_{//\mathcal{D}})_{\mathcal{C}^{\prime}}\to(\mathrm{Cat}_{//\mathcal{D}})_{\mathcal{C}}$
is given by the restriction functor
$g^{*}\colon\operatorname{Fun}(\mathcal{C}^{\prime},\mathcal{D})\to\operatorname{Fun}(\mathcal{C},\mathcal{D})$.
The underlying $\infty$-category $\mathrm{Cat}_{//\mathcal{D}}$ can be
described as having objects $(\mathcal{C},f)$ where $\mathcal{C}$ is an
$\infty$-category and $f\colon\mathcal{C}\to\mathcal{D}$ is a functor, and
morphisms are given by diagrams

${\mathcal{C}}$${\mathcal{D}}$${\mathcal{C}^{\prime}}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{q}$

filled by a non-invertible 2-cell $p\Rightarrow qf$. The tensor product in
$\mathrm{Cat}^{\otimes}_{//\mathcal{D}}$ is given by
$(\mathcal{C},f)\otimes(\mathcal{C}^{\prime},f^{\prime})=(\mathcal{C}\times\mathcal{C}^{\prime},f\otimes
f^{\prime})$, where $f\otimes f^{\prime}$ is the composite

$\mathcal{C}\times\mathcal{C}^{\prime}\xrightarrow{f\times
f^{\prime}}\mathcal{D}\times\mathcal{D}\xrightarrow{\otimes}\mathcal{D},$

obtained using the symmetric monoidal structure of $\mathcal{D}$. In
particular, one readily checks that for any $(\mathcal{C},f)$ the functor
$(\mathcal{C},f)\otimes(-)\colon\mathrm{Cat}_{//\mathcal{D}}\to\mathrm{Cat}_{//\mathcal{D}}$
preserves cartesian edges over $\mathrm{Cat}$. A closely related variant is
the $\mathrm{Fin}_{*}$-cartesian fibration

$\mathrm{Cat}^{\otimes}_{\mathrm{op}//\mathcal{D}}\to\mathrm{Cat}^{\times},$

which is obtained from
$\mathrm{Cat}^{\otimes}_{//\mathcal{D}}\to\mathrm{Cat}^{\times}$ by base
changing along the $\mathrm{op}$-equivalence
$(-)^{\mathrm{op}}\colon\mathrm{Cat}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\simeq$}}\,}\mathrm{Cat}$.
In particular, the underlying $\infty$-category
$\mathrm{Cat}_{\mathrm{op}//\mathcal{D}}$ can be described as having objects
$(\mathcal{C},f)$ where $\mathcal{C}$ is an $\infty$-category and
$f\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{D}$ is a functor.

We note that the construction of
$\mathrm{Cat}^{\otimes}_{\mathrm{op}//\mathcal{D}}$ (or of
$\mathrm{Cat}^{\otimes}_{//\mathcal{D}}$) works exactly the same if one takes
$\mathcal{D}$ to be a presentable (and in particular large) $\infty$-category,
while still considering only pairs
$(\mathcal{C},f\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{D})$ with
$\mathcal{C}$ small. In this case, each of the cartesian transition functors
$g^{*}\colon\operatorname{Fun}({\mathcal{C}^{\prime}}{{}^{\mathrm{op}}},\mathcal{D})\to\operatorname{Fun}(\mathcal{C}^{\mathrm{op}},\mathcal{D})$
admits a left adjoint
$g_{!}\colon\operatorname{Fun}(\mathcal{C}^{\mathrm{op}},\mathcal{D})\to\operatorname{Fun}(\mathcal{C}^{\prime}{{}^{\mathrm{op}}},\mathcal{D})$
in the form of left Kan extension, which means that the cartesian fibration
$\mathrm{Cat}_{\mathrm{op}//\mathcal{D}}\to\mathrm{Cat}$ is also a cocartesian
fibration. Furthermore, if the symmetric monoidal structure on $\mathcal{D}$
preserves colimits in each variable then the functor
$(\mathcal{C},f)\otimes(-)\colon\mathrm{Cat}_{\mathrm{op}//\mathcal{D}}\to\mathrm{Cat}_{\mathrm{op}//\mathcal{D}}$
preserves cocartesian edges for every
$(\mathcal{C},f)\in\mathrm{Cat}_{\mathrm{op}//\mathcal{D}}$, so that
$\mathrm{Cat}^{\otimes}_{\mathrm{op}//\mathcal{D}}\to\mathrm{Cat}^{\times}$ is
also a $\mathrm{Fin}_{*}$-cocartesian fibration, and hence a cocartesian
fibration of $\infty$-operads by Lemma 3.1.1. This presentable case is the one
relevant to us, as we will later on set $\mathcal{D}={\mathcal{S}\mathrm{p}}$,
as is also eventually done in [9-authors-I]*§5.2.

Let us now fix a presentably symmetric monoidal $\infty$-category
$\mathcal{D}$. Consider the $\infty$-category
$\operatorname{CAlg}(\mathrm{Cat})$ of symmetric monoidal $\infty$-categories
(and symmetric monoidal functors between them). We then set

$\mathcal{Z}^{\otimes}_{\mathcal{D}}=\operatorname{CAlg}(\mathrm{Cat})^{\times}\times_{\mathrm{Cat}^{\times}}(\mathrm{Cat}_{\mathrm{op}//\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}}})^{\otimes},$

so that $\mathcal{Z}^{\otimes}_{\mathcal{D}}$ is a symmetric monoidal
$\infty$-category sitting in a $\mathrm{Fin}_{*}$-cartesian fibration

$\mathcal{Z}^{\otimes}_{\mathcal{D}}\to\operatorname{CAlg}(\mathrm{Cat})^{\times}$

classified by the lax symmetric monoidal functor
$\mathcal{C}^{\otimes}\mapsto\operatorname{Fun}(\mathcal{C}^{\mathrm{op}},\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}})$.
In particular, the underlying $\infty$-category of
$\mathcal{Z}^{\otimes}_{\mathcal{D}}$ can be described as having objects pairs
$(\mathcal{C}^{\otimes},f)$ where $\mathcal{C}^{\otimes}$ is a symmetric
monoidal $\infty$-category and
$f\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}}$ is
a functor.

######  Construction 3.1.2.

Given a functor
$f\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}}$,
let us denote by $\overline{f}$ the composite

$\overline{f}\colon(\mathcal{C}^{\mathrm{op}})^{\mathrm{B}\mathrm{C_{2}}}\to\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}\times\mathrm{B}\mathrm{C_{2}}}\to\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}},$

where the first functor is induced by $f$ and the second is restriction along
the diagonal
$\mathrm{B}\mathrm{C_{2}}\to\mathrm{B}\mathrm{C_{2}}\times\mathrm{B}\mathrm{C_{2}}$.
Now, if $\mathcal{C}^{\otimes}$ is a symmetric monoidal $\infty$-category, we
can associate to
$f\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}}$
the composite functor

$T_{f}\colon\mathcal{C}^{\mathrm{op}}\xrightarrow{x\mapsto x\otimes
x}(\mathcal{C}^{\mathrm{op}})^{\mathrm{B}\mathrm{C_{2}}}\xrightarrow{\overline{f}}\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}},$

given by the formula $T_{f}(x)=\overline{f}(x\otimes x)$. Finally, if
$\mathcal{D}$ admits $\mathrm{B}\mathrm{C_{2}}$-indexed limits, we can take
homotopy fixed points of $\mathrm{C_{2}}$-objects, in which case we denote by
$\Qoppa^{\mathrm{s}}_{f}$ the functor
$T_{f}^{{\mathrm{h}\mathrm{C_{2}}}}\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{D}$,
given by

$\Qoppa^{\mathrm{s}}_{f}(x)=T_{f}^{{\mathrm{h}\mathrm{C_{2}}}}(x)=\overline{f}(x\otimes
x)^{{\mathrm{h}\mathrm{C_{2}}}}.$

######  Proposition 3.1.3.

Let $\mathcal{D}$ be a symmetric monoidal $\infty$-category which admits
$\mathrm{B}\mathrm{C_{2}}$-indexed limits. Then the association
$(\mathcal{C},f)\mapsto(\mathcal{C},\Qoppa^{\mathrm{s}}_{f})$ of Construction
3.1.2 assembles to form a lax symmetric monoidal functor

$\mathcal{Z}^{\otimes}_{\mathcal{D}}\to\mathrm{Cat}^{\otimes}_{\mathrm{op}//\mathcal{D}}$

sitting in a commutative diagram

(5)
${\mathcal{Z}^{\otimes}_{\mathcal{D}}}$${\mathrm{Cat}^{\otimes}_{\mathrm{op}//\mathcal{D}}}$${\operatorname{CAlg}(\mathrm{Cat})^{\times}}$${\mathrm{Cat}^{\times}\
,}$$\scriptstyle{\mathcal{C}^{\otimes}\mapsto\mathcal{C}}$

in which the vertical arrows are both $\mathrm{Fin}_{*}$-cartesian fibrations
and cocartesian fibrations of $\infty$-operads and the bottom horizontal arrow
is a symmetric monoidal functor.

The proof of Proposition 3.1.3 will make use of a small variant of the
$\mathrm{Fin}_{*}$-cartesian fibration
$(\mathrm{Cat}_{\mathrm{op}//\mathcal{D}})^{\otimes}\to\mathrm{Cat}^{\times}$,
where $\infty$-categories are replaced with $\infty$-categories with
$\mathrm{C_{2}}$-action. For this, note that $\mathrm{Fin}_{*}$-cartesian
fibrations are closed under cotensoring by an $\infty$-category. More
precisely, if $p^{\otimes}\colon\mathcal{E}^{\otimes}\to\mathcal{B}^{\otimes}$
is a $\mathrm{Fin}_{*}$-cartesian fibration and $\mathcal{I}$ is an
$\infty$-category, then

$(p^{\mathcal{I}})^{\otimes}\colon(\mathcal{E}^{\mathcal{I}})^{\otimes}\to(\mathcal{B}^{\mathcal{I}})^{\otimes}$

is again a $\mathrm{Fin}_{*}$-cartesian fibration, where
$(\mathcal{B}^{\mathcal{I}})^{\otimes}=(\mathcal{B}^{\otimes})^{\mathcal{I}}\times_{\mathrm{Fin}_{*}^{\mathcal{I}}}\mathrm{Fin}_{*}$
is the pointwise symmetric monoidal structure on $\mathcal{B}^{\mathcal{I}}$,
and similarly for $(\mathcal{E}^{\mathcal{I}})^{\otimes}$. If $p$ is
classified by a lax symmetric monoidal functor
$\chi\colon\mathcal{B}^{\mathrm{op}}\to\mathrm{Cat}$ then
$(p^{\mathcal{I}})^{\otimes}$ is classified by the functor
$(\mathcal{B}^{\mathcal{I}})^{\mathrm{op}}\to\mathrm{Cat}$ sending
$f\colon\mathcal{I}\to\mathcal{B}$ to $\lim_{i\in\mathcal{I}}\chi(f(i))$ (see
[HTT]*Corollary 3.3.3.2), with its induced lax symmetric monoidal structure.
In particular, taking $\mathcal{I}=\mathrm{B}\mathrm{C_{2}}$, we may consider,
for a symmetric monoidal $\infty$-category $\mathcal{D}$, the
$\mathrm{Fin}_{*}$-cartesian fibration

$\big{(}(\mathrm{Cat}_{\mathrm{op}//\mathcal{D}})^{\mathrm{B}\mathrm{C_{2}}}\big{)}^{\otimes}\to\big{(}\mathrm{Cat}^{\mathrm{B}\mathrm{C_{2}}}\big{)}^{\times},$

classified by the lax symmetric monoidal functor

(6)
$\big{(}\mathrm{Cat}^{\mathrm{B}\mathrm{C_{2}}}\big{)}^{\mathrm{op}}\to\mathrm{Cat}\quad\quad\mathcal{C}\mapsto\operatorname{Fun}(\mathcal{C},\mathcal{D})^{{\mathrm{h}\mathrm{C_{2}}}}=\operatorname{Fun}_{\mathrm{C_{2}}}(\mathcal{C},q^{*}\mathcal{D}),$

where $q^{*}\colon\mathrm{Cat}\to\mathrm{Cat}^{\mathrm{B}\mathrm{C_{2}}}$ is
induced by restriction along the terminal map
$q\colon\mathrm{B}\mathrm{C_{2}}\to\ast$ (otherwise put, $q^{*}\mathcal{D}$ is
$\mathcal{D}$ endowed with the trivial $\mathrm{C_{2}}$-action). Finally,
since
$\operatorname{Fun}_{\mathrm{C_{2}}}(q^{*}(-),q^{*}\mathcal{D})=\operatorname{Fun}(-,\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}})$
by adjunction and restriction corresponds via unstraightening to base change
we obtain a pullback square

(7)
${\mathrm{Cat}^{\otimes}_{\mathrm{op}//\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}}}}$${\big{(}(\mathrm{Cat}_{\mathrm{op}//\mathcal{D}})^{\mathrm{B}\mathrm{C_{2}}}\big{)}^{\otimes}}$${\mathrm{Cat}^{\times}}$${\big{(}\mathrm{Cat}^{\mathrm{B}\mathrm{C_{2}}}\big{)}^{\times}}$$\scriptstyle{(q^{*})^{\times}}$

of symmetric monoidal $\infty$-categories and symmetric monoidal functors,
where the vertical maps are $\mathrm{Fin}_{*}$-cartesian fibrations.

######  Construction 3.1.4.

Consider the composite functor
$p\colon\operatorname{CAlg}(\mathrm{Cat})\xrightarrow{\mathcal{C}^{\otimes}\mapsto\mathcal{C}}\mathrm{Cat}\xrightarrow{q^{*}}\mathrm{Cat}^{\mathrm{B}\mathrm{C_{2}}}$,
where the first functor just forgets the symmetric monoidal structure. We
construct a symmetric monoidal natural transformation $\tau\colon p\Rightarrow
p$ whose component at a given $\mathcal{C}^{\otimes}$ is the
$\mathrm{C_{2}}$-equivariant functor $q^{*}\mathcal{C}\xrightarrow{x\mapsto
x\otimes x}q^{*}\mathcal{C}$ as follows. Let
$\operatorname{Span}(\mathrm{Fin})$ be the span $\infty$-category of finite
sets. Then $\operatorname{Span}(\mathrm{Fin})$ is semi-additive, that is,
(finite) products and coproducts coincide, and are both given by disjoint
union. Furthermore, for an $\infty$-category with finite products there is a
natural equivalence

$\operatorname{Fun}^{\times}(\operatorname{Span}(\mathrm{Fin}),\mathcal{E})=\mathrm{CMon}(\mathcal{E})$

between product-preserving functors $\mathrm{Fin}\to\mathcal{E}$ and
commutative monoid objects in $\mathcal{E}$ (see, e.g.,
[ambidexterity]*Proposition 5.14 for $m=1$), where the underlying objects of a
given monoid corresponds to the value of the associated functor
$\operatorname{Span}(\mathrm{Fin}_{*})\to\mathcal{E}$ at
$\\{1\\}\in\operatorname{Span}(\mathrm{Fin}_{*})$. In particular, we have

$\operatorname{Fun}^{\times}(\operatorname{Span}(\mathrm{Fin}),\mathrm{Cat})=\mathrm{CMon}(\mathrm{Cat})=\operatorname{CAlg}(\mathrm{Cat})$

and we may identify the functor
$p\colon\operatorname{CAlg}(\mathrm{Cat})\to\mathrm{Cat}^{\mathrm{B}\mathrm{C_{2}}}$
above with the composite

$\operatorname{Fun}^{\times}(\operatorname{Span}(\mathrm{Fin}_{*}),\mathrm{Cat})\to\operatorname{Fun}(\operatorname{Span}(\mathrm{Fin}_{*}),\mathrm{Cat})\xrightarrow{g^{*}}\operatorname{Fun}(\mathrm{B}\mathrm{C_{2}},\mathrm{Cat}),$

where $g^{*}$ denotes restriction along the functor
$g\colon\mathrm{B}\mathrm{C_{2}}\to\Delta^{0}\xrightarrow{\\{1\\}}\operatorname{Span}(\mathrm{Fin}_{*})$.

Now the $\infty$-groupoid of maps $\\{1\\}\to\\{1\\}$ in
$\operatorname{Span}(\mathrm{Fin})$ is given by the core $\infty$-groupoid
$\iota\mathrm{Fin}$ of finite sets. In particular, this mapping space contains
a copy of $\mathrm{B}\mathrm{C_{2}}$, sitting in the form of the span
$\\{1\\}\leftarrow\\{1,2\\}\rightarrow\\{1\\}$ endowed with its order 2 group
of automorphisms (or self homotopies). This copy of $\mathrm{B}\mathrm{C_{2}}$
in the mapping space $\\{1\\}\to\\{1\\}$ determines a functor
$\Delta^{1}\times\mathrm{B}\mathrm{C_{2}}\to\operatorname{Span}(\mathrm{Fin})$
whose restriction to $\partial\Delta^{1}\times\mathrm{B}\mathrm{C_{2}}$ is
constant on $\\{1\\}$. Equivalently, this is the data of a natural
transformation $g\Rightarrow g$, which then determines a natural
transformation $g^{*}\Rightarrow g^{*}$ from the restriction functor
$g^{*}\colon\operatorname{Fun}(\operatorname{Span}(\mathrm{Fin}_{*}),\mathrm{Cat})\to\operatorname{Fun}(\mathrm{B}\mathrm{C_{2}},\mathrm{Cat})$
to itself. Horizontally pre-composing with the forgetful functor
$\operatorname{Fun}^{\times}(\operatorname{Span}(\mathrm{Fin}),\mathrm{Cat})\to\operatorname{Fun}(\operatorname{Span}(\mathrm{Fin}),\mathrm{Cat})$
we then obtain the desired natural transformation $\tau\colon p\Rightarrow p$.
Since $p$ is product-preserving we may consider it as a symmetric monoidal
functor
$p^{\times}\colon\operatorname{CAlg}(\mathrm{Cat})^{\times}\to(\mathrm{Cat}^{\mathrm{B}\mathrm{C_{2}}})^{\times}$,
in which case $\tau$ automatically refines to a symmetric monoidal natural
transformation $\tau^{\times}\colon p^{\times}\Rightarrow p^{\times}$.

###### Proof of Proposition 3.1.3.

Composing natural transformation $\tau^{\times}\colon p^{\times}\Rightarrow
p^{\times}$ horizontally with the contravariant lax symmetric monoidal functor
$\operatorname{Fun}_{\mathrm{C_{2}}}(-,q^{*}\mathcal{D})$ of (6) we obtain a
natural transformation $\sigma$ from the contravariant lax symmetric monoidal
functor
$\mathcal{C}^{\otimes}\mapsto\operatorname{Fun}_{\mathrm{C_{2}}}(q^{*}\mathcal{C},q^{*}\mathcal{D})$
to itself. After unstraightening, this provides an extension of the pullback
square (7) to a commutative rectangle

(8)
${\mathcal{Z}^{\otimes}_{\mathcal{D}}}$${\mathrm{Cat}_{\mathrm{op}//\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}}}^{\otimes}}$${((\mathrm{Cat}_{\mathrm{op}//\mathcal{D}})^{\mathrm{B}\mathrm{C_{2}}})^{\otimes}}$${\operatorname{CAlg}(\mathrm{Cat})^{\times}}$${\mathrm{Cat}^{\times}}$${(\mathrm{Cat}^{\mathrm{B}\mathrm{C_{2}}})^{\times}}$$\scriptstyle{\mathcal{C}^{\otimes}\mapsto\mathcal{C}}$$\scriptstyle{q^{*}}$

of symmetric monoidal $\infty$-categories and symmetric monoidal functors,
where the outer rectangle encodes the lax symmetric monoidal transformation
$\sigma$, and the left square is formed from the outer rectangle by means of
universal properties, since the right square is pullback.

If we now assume that the $\infty$-category $\mathcal{D}$ has
$\mathrm{B}\mathrm{C_{2}}$-indexed limits then the functor
$(-)^{{\mathrm{h}\mathrm{C_{2}}}}\colon\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}}\to\mathcal{D}$
is canonically lax symmetric monoidal, and hence induces a lax symmetric
monoidal functor

$h\colon\mathrm{Cat}^{\otimes}_{\mathrm{op}//\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}}}\to\mathrm{Cat}^{\otimes}_{\mathrm{op}//\mathcal{D}}\quad\quad(\mathcal{C},T\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}})\mapsto(\mathcal{C},T^{{\mathrm{h}\mathrm{C_{2}}}}\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{D})$

covering the identity $\mathrm{Cat}^{\times}\to\mathrm{Cat}^{\times}$ (indeed,
by its defining mapping property in [9-authors-I]*Lemma 5.2.3, the underlying
$\infty$-operad of $\mathrm{Cat}^{\otimes}_{//\mathcal{D}}$ is functorial in
the underlying $\infty$-operad of $\mathcal{D}^{\otimes}$). Composing this
with the left square of (8) we then obtain a commutative diagram

${(\mathcal{C}^{\otimes},f)}$${(\mathcal{C},T_{f})}$${(\mathcal{C},\Qoppa^{\mathrm{s}}_{f})}$${\mathcal{Z}^{\otimes}_{\mathcal{D}}}$${\mathrm{Cat}^{\otimes}_{\mathrm{op}//\mathcal{D}^{\mathrm{B}\mathrm{C_{2}}}}}$${\mathrm{Cat}^{\otimes}_{\mathrm{op}//\mathcal{D}}}$${\operatorname{CAlg}(\mathrm{Cat})^{\times}}$${\mathrm{Cat}^{\times}}$${\mathrm{Cat}^{\times}}$

$\in$

$\in$

$\in$

$\scriptstyle{h}$$\scriptstyle{\mathcal{C}^{\otimes}\mapsto\mathcal{C}}$

whose external rectangle is the desired square (5). (Here and later, we only
made reference above the top row to the individual objects of the
$\infty$-categories underlying the appearing symmetric monoidal
$\infty$-categories, avoiding to mention arrows, tuples and multi-operations.)
∎

For our present setting of Poincaré $\infty$-categories, we now replace
$\infty$-categories by stable $\infty$-categories and take $\mathcal{D}$ to be
the $\infty$-category of spectra ${\mathcal{S}\mathrm{p}}$. Recall that the
symmetric monoidal $\infty$-category $(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}$
can be defined as the suboperad of $\mathrm{Cat}^{\times}$ spanned by the
colors $\mathcal{C}$ which are stable $\infty$-categories, and by those multi-
operations whose representing functors
$\mathcal{C}_{1}\times\cdots\times\mathcal{C}_{n}\to\mathcal{C}$ are exact in
each variable separately. As in [9-authors-I]*§5, we define
$(\mathrm{Cat}^{\mathrm{h}})^{\otimes}$ as the suboperad of
$\mathrm{Cat}^{\times}_{\mathrm{op}//{\mathcal{S}\mathrm{p}}}$ spanned by the
colors $(\mathcal{C},\Qoppa)$ such that $\mathcal{C}$ is stable and
$\Qoppa\colon\mathcal{C}^{\mathrm{op}}\to{\mathcal{S}\mathrm{p}}$ is
quadratic, and those multi-operations whose images in $\mathrm{Cat}^{\times}$
are contained in $(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}$.

The symmetric monoidal structure $(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}$ on
$\mathrm{Cat}^{\mathrm{ex}}$ induces a symmetric monoidal structure
$\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}$ on
$\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})$ by means of internal mapping
objects in $\infty$-operads, such that the forgetful functor

$\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}\to(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}$

is symmetric monoidal. The lax symmetric monoidal functor
$(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}\to\mathrm{Cat}^{\times}$ then induces
a lax symmetric monoidal functor
$\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}\to\operatorname{CAlg}(\mathrm{Cat})^{\times}$,
and we define

$\mathcal{Z}^{\otimes}_{\mathrm{ex}}\subseteq\mathcal{Z}^{\otimes}_{{\mathcal{S}\mathrm{p}}}\times_{\operatorname{CAlg}(\mathrm{Cat})^{\times}}\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}$

to be the full suboperad spanned by the colors $(\mathcal{C}^{\otimes},L)$
such that
$L\colon\mathcal{C}^{\mathrm{op}}\to{\mathcal{S}\mathrm{p}}^{{\mathrm{h}\mathrm{C_{2}}}}$
is exact. The square (5) then restricts to a commutative square of
$\infty$-operads

(9)
${(\mathcal{C}^{\otimes},L)}$${(\mathcal{C},\Qoppa^{\mathrm{s}}_{L})}$${\mathcal{Z}^{\otimes}_{\mathrm{ex}}}$${(\mathrm{Cat}^{\mathrm{h}})^{\otimes}}$${\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}}$${(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}\
,}$

$\in$

$\in$

$\scriptstyle{\mathcal{C}^{\otimes}\mapsto\mathcal{C}}$

where the bottom horizontal arrow is a symmetric monoidal functor between
symmetric monoidal $\infty$-categories.

######  Lemma 3.1.5.

Both vertical maps in (9) are cocartesian fibrations of $\infty$-operads. In
particular, $\mathcal{Z}^{\otimes}_{\mathrm{ex}}$ and
$(\mathrm{Cat}^{\mathrm{h}})^{\otimes}$ are symmetric monoidal
$\infty$-categories.

To avoid confusion, let us point the top horizontal map in (9) does not
preserve cocartesian edges. In fact, it doesn’t even preserve cocartesian
edges over $\mathrm{Fin}_{*}$, namely, it is only a _lax_ symmetric monoidal
functor.

###### Proof of Lemma 3.1.5.

For the right vertical map the claim is proven in [9-authors-I]*Theorem 5.2.7.
The argument for the left vertical map is similar (and easier), let us spell
it out. First, note that the projection

$\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}\times_{\mathrm{Cat}^{\times}}\mathrm{Cat}^{\otimes}_{\mathrm{op}//{\mathcal{S}\mathrm{p}}}\to\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}$

is a cocartesian fibration of $\infty$-operads by base change, and that
$\mathcal{Z}^{\otimes}_{\mathrm{ex}}$ is a full suboperad of the fibre product
on the left, spanned by those colors $(\mathcal{C},L)$ such that
$L\colon\mathcal{C}^{\mathrm{op}}\to{\mathcal{S}\mathrm{p}}$ is exact. It will
hence suffice to check that $\mathcal{Z}^{\otimes}_{\mathrm{ex}}$ contains a
cocartesian arrow as soon as it contains its domain. Unwinding the
definitions, this amounts to checking that if $\mathcal{C}$ and
$\mathcal{C}_{1},\ldots,\mathcal{C}_{n}$ are stable $\infty$-categories,
$g\colon\mathcal{C}_{1}\times\cdots\times\mathcal{C}_{n}\to\mathcal{C}$ is a
functor which is exact in each variable, and
$L_{i}\colon\mathcal{C}_{i}^{\mathrm{op}}\to{\mathcal{S}\mathrm{p}}$ are exact
functors for $i=1,\ldots,n$, then the left Kan extension of the top composite

${\prod_{i}\mathcal{C}_{i}}$${\prod_{i}{\mathcal{S}\mathrm{p}}}$${{\mathcal{S}\mathrm{p}}}$${\mathcal{C}}$$\scriptstyle{g}$$\scriptstyle{\otimes}$

along $g$ is again exact. Indeed, since $g$ is exact in each variable its left
Kan extension along the universal multi-exact functor
$\prod_{i}\mathcal{C}_{i}\to\bigotimes_{i}\mathcal{C}$ yields an exact functor
$g^{\prime}\colon\bigotimes_{i}\mathcal{C}_{i}\to{\mathcal{S}\mathrm{p}}$, and
the left Kan extension of $g^{\prime}$ along the exact functor
$\bigotimes_{i}\mathcal{C}_{i}\to\mathcal{C}$ is again exact by
[9-authors-I]*Lemma 1.4.1. ∎

Now as we have canonical equivalences

$\operatorname{Fun^{ex}}(\mathcal{C}^{\mathrm{op}},{\mathcal{S}\mathrm{p}}^{\mathrm{B}\mathrm{C_{2}}})=\operatorname{Fun^{ex}}(\mathcal{C}^{\mathrm{op}},{\mathcal{S}\mathrm{p}})^{\mathrm{B}\mathrm{C_{2}}}=\operatorname{Fun}^{\mathrm{lex}}(\mathcal{C}^{\mathrm{op}},{\mathcal{S}\mathrm{p}})^{\mathrm{B}\mathrm{C_{2}}}=\operatorname{Ind}(\mathcal{C})^{\mathrm{B}\mathrm{C_{2}}}$

we will generally identify the objects of the underlying $\infty$-category of
$\mathcal{Z}^{\otimes}_{\mathrm{ex}}$ with pairs $(\mathcal{C}^{\otimes},L)$
where $\mathcal{C}^{\otimes}$ is a stably symmetric monoidal $\infty$-category
and $L$ is an object of
$\operatorname{Ind}(\mathcal{C})^{\mathrm{B}\mathrm{C_{2}}}$. A morphism
$(\mathcal{C},L)\to(\mathcal{C}^{\prime},L^{\prime})$ in this description then
corresponds to a pair $(g,\eta)$ where
$g\colon\mathcal{C}\to\mathcal{C}^{\prime}$ is an exact functor and
$\eta\colon g_{!}L\to L^{\prime}$ is a morphism in
$\operatorname{Ind}(\mathcal{C}^{\prime})$, where
$g_{!}\colon\operatorname{Ind}(\mathcal{C})\to\operatorname{Ind}(\mathcal{C}^{\prime})$
is the functor induced by $g$ on Ind objects (given by left Kan extension when
viewed as exact ${\mathcal{S}\mathrm{p}}$-valued presheaves). In this
description, the functor
$\Qoppa^{\mathrm{s}}_{L}\colon\mathcal{C}^{\mathrm{op}}\to{\mathcal{S}\mathrm{p}}$
can be written as

$\Qoppa^{\mathrm{s}}_{L}(x)=\mathrm{hom}_{\operatorname{Ind}(\mathcal{C})}(x\otimes
x,L)^{{\mathrm{h}\mathrm{C_{2}}}},$

where we have implicitly identified $x\otimes x$ with its image in
$\operatorname{Ind}(\mathcal{C})$ under the Yoneda embedding
$\mathcal{C}\to\operatorname{Ind}(\mathcal{C})$.

######  Proposition 3.1.6.

Suppose that $\mathcal{C}$ is a rigid stably symmetric monoidal
$\infty$-category and
$L\in\operatorname{Ind}(\mathcal{C})^{\mathrm{B}\mathrm{C_{2}}}$ a
$\mathrm{C_{2}}$-equivariant Ind-object. Then the hermitian structure
$\Qoppa^{\mathrm{s}}_{L}$ above is non-degenerate if and only if $L$ belongs
to
$\mathcal{C}^{\mathrm{B}\mathrm{C_{2}}}\subseteq\operatorname{Ind}(\mathcal{C})^{\mathrm{B}\mathrm{C_{2}}}$.
It is furthermore Poincaré if and only if the underlying object of $L$ is
tensor invertible in $\mathcal{C}$. In addition, if
$g\colon\mathcal{C}\to\mathcal{C}^{\prime}$ is an exact functor,
$L\in\mathcal{C}^{\mathrm{B}\mathrm{C_{2}}}$ and
$L^{\prime}\in(\mathcal{C}^{\prime})^{\mathrm{B}\mathrm{C_{2}}}$ tensor
invertible and $g(L)\to L^{\prime}$ an equivalence, then the induced hermitian
functor
$(\mathcal{C},\Qoppa^{\mathrm{s}}_{L})\to(\mathcal{C}^{\prime},\Qoppa^{\mathrm{s}}_{L^{\prime}})$
is a Poincaré functor.

###### Proof.

Unwinding the definitions, the cross effect of $\Qoppa^{\mathrm{s}}_{L}$ is
given by

$\mathrm{B}_{L}(x,y)=\mathrm{hom}_{\operatorname{Ind}(\mathcal{C})}(x\otimes
y,L).$

If $\mathrm{B}_{L}$ is represented by
$\mathrm{D}\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{C}$ then
$\mathrm{D}(1_{\mathcal{C}})=L$, so that $L$ belongs to $\mathcal{C}$. On the
other hand, if $L$ belongs to $\mathcal{C}$ then

$\mathrm{hom}_{\operatorname{Ind}(\mathcal{C})}(x\otimes
y,L)=\mathrm{hom}_{\mathcal{C}}(x\otimes
y,L)=\mathrm{hom}_{\mathcal{C}}(x,y^{\vee}\otimes L)$

by the rigidity of $\mathcal{C}$, so that $\mathrm{B}_{L}$ is represented by
$\mathrm{D}_{L}(y)=y^{\vee}\otimes L$. Moreover, the double duality map
$1_{\mathcal{C}}\to\mathrm{D}_{L}\mathrm{D}_{L}$ is

$y\to L\otimes L^{\vee}\otimes y$

induced by the adjoint of the $\mathrm{C_{2}}$-action $L\to L$, and so it is
an equivalence if and only if $L$ is invertible.

Finally, if $L$ and $L^{\prime}$ are invertible objects of $\mathcal{C}$ and
$\mathcal{C}^{\prime}$ respectively, the natural transformation
$g\mathrm{D}_{L}\to\mathrm{D}_{L^{\prime}}g$ is given by

$fL\otimes fx^{\vee}\to L^{\prime}\otimes fx^{\vee}$

and so it is an equivalence if and only if the map $fL\to L^{\prime}$ is an
equivalence. ∎

Let $\mathrm{Inv}^{\otimes}\subseteq\mathcal{Z}^{\otimes}_{\mathrm{ex}}$
denotes the suboperad spanned by those $(\mathcal{C}^{\otimes},L)$ such that
$\mathcal{C}^{\otimes}$ is rigid and $L$ is (the Yoneda image of) a tensor
invertible $\mathrm{C_{2}}$-equivariant object of $\mathcal{C}$, and those
multi-operations
$((\mathcal{C}_{1},L_{1}),\ldots,(\mathcal{C}_{n},L_{n}))\to(\mathcal{C},L)$
corresponding to a multi-exact functor
$g\colon\mathcal{C}_{1}\times\cdots\times\mathcal{C}_{n}\to\mathcal{C}$ and an
equivalence
$g(L_{1},\ldots,L_{n})\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\simeq$}}\,}L$.
Then the exact functor
$\mathcal{C}_{1}\otimes\cdots\otimes\mathcal{C}_{n}\to\mathcal{C}$ induced by
such a multi-operation sends the tensor invertible
$\mathrm{C_{2}}$-equivariant object $L_{1}\otimes\cdots\otimes L_{n}$ to $L$,
and hence by Proposition 3.1.6 the associated hermitian functor

$(\mathcal{C}_{1},\Qoppa^{\mathrm{s}}_{L_{1}})\otimes\cdots\otimes(\mathcal{C}_{n},\Qoppa^{\mathrm{s}}_{L_{n}})\to(\mathcal{C},\Qoppa^{\mathrm{s}}_{L})$

is a Poincaré functor between two Poincaré $\infty$-categories. As a
consequence, the square (9) restricts to a square of $\infty$-operads

(10)
${(\mathcal{C},L)}$${(\mathcal{C},\Qoppa^{\mathrm{s}}_{L})}$${\mathrm{Inv}^{\otimes}}$${(\mathrm{Cat}^{\mathrm{p}})^{\otimes}}$${\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}}$${(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}\
.}$

$\in$

$\in$

We also note that these are exactly the arrows in $\mathrm{Inv}^{\otimes}$
whose image in $\mathcal{Z}^{\otimes}_{\mathrm{ex}}$ is cocartesian over
$\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})$. In particular,

$\mathrm{Inv}^{\otimes}\to\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})^{\otimes}$

is a left fibration of $\infty$-operads and $\mathrm{Inv}^{\otimes}$ is a
symmetric monoidal $\infty$-category. By [9-authors-I]*Theorem 5.2.7(iii) the
$\infty$-operad $(\mathrm{Cat}^{\mathrm{p}})^{\otimes}$ is also a symmetric
monoidal $\infty$-category and the right vertical arrow is a symmetric
monoidal functor. We however point out that the top horizontal arrow in (10)
is only an $\infty$-operad map, that is, a lax symmetric monoidal functor.

Let $\mathrm{Cat}^{\mathrm{ps}}$ denote the $\infty$-category of stable
$\infty$-categories with duality. These can be defined in two equivalent
manners, see [9-authors-I]*Corollary 7.2.16. The first is as
$\mathrm{C_{2}}$-homotopy fixed points in $\mathrm{Cat}^{\mathrm{ex}}$ with
respect to the $\mathrm{C_{2}}$-action where the generator of $\mathrm{C_{2}}$
acts by $\mathcal{C}\mapsto\mathcal{C}^{\mathrm{op}}$. In particular, a stable
$\infty$-category is equipped with an equivalence
$\mathcal{D}\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{C}$, called the
duality, which satisfies all the coherence conditions of being a
$\mathrm{C_{2}}$-homotopy fixed point. The second is as stable
$\infty$-category $\mathcal{C}$ equipped with a symmetric bilinear functor
$\mathrm{B}\colon\mathcal{C}^{\mathrm{op}}\times\mathcal{C}^{\mathrm{op}}\to{\mathcal{S}\mathrm{p}}$
such that there exists an equivalence
$\mathcal{D}\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{C}$ satisfying
$\mathrm{B}(x,y)\simeq\mathrm{hom}(x,\mathrm{D}y)$ naturally in $x,y$. In this
case $\mathcal{D}$ is uniquely determined, and the symmetric structure on
$\mathrm{B}$ encodes the $\mathrm{C_{2}}$-homotopy fixed structure of
$\mathcal{D}$. There is a fully-faithful functor

$\mathrm{Cat}^{\mathrm{ps}}\hookrightarrow\mathrm{Cat}^{\mathrm{p}}\quad\quad(\mathcal{C},\mathrm{D})\mapsto(\mathcal{C},\Qoppa^{\mathrm{s}}_{\mathrm{D}}),$

where $\Qoppa^{\mathrm{s}}_{\mathrm{D}}$ is the Poincaré structure
$\Qoppa^{\mathrm{s}}_{\mathrm{D}}(x)=\mathrm{hom}(x,\mathrm{D}(x))^{{\mathrm{h}\mathrm{C_{2}}}}$.
This inclusion admits a left adjoint
$\mathrm{Cat}^{\mathrm{p}}\to\mathrm{Cat}^{\mathrm{ps}}$ sending
$(\mathcal{C},\Qoppa)$ to the underlying $\infty$-category with duality
$(\mathcal{C},\mathrm{D}_{\Qoppa})$ (see [9-authors-I]*Proposition 7.2.18). In
particular, $\mathrm{Cat}^{\mathrm{ps}}$ is a left localisation of
$\mathrm{Cat}^{\mathrm{p}}$. We refer the reader to [9-authors-I]*§7.2 for
further details.

We refer to Poincaré $\infty$-categories in the image of this functor as
_symmetric Poincaré $\infty$-categories_. In what follows, we sometimes abuse
notation and do not distinguish between $\mathrm{Cat}^{\mathrm{ps}}$ and its
image in $\mathrm{Cat}^{\mathrm{p}}$, that is, we identify an
$\infty$-category with duality with its associated symmetric Poincaré
$\infty$-category, and vice versa. We note that by construction, the lax
symmetric monoidal functor
$\mathrm{Inv}^{\otimes}\to(\mathrm{Cat}^{\mathrm{p}})^{\otimes}$ takes values
in the full sub-operad
$(\mathrm{Cat}^{\mathrm{ps}})^{\otimes}\subseteq(\mathrm{Cat}^{\mathrm{p}})^{\otimes}$
spanned by the symmetric Poincaré $\infty$-categories.

######  Proposition 3.1.7.

The localisation $\mathrm{Cat}^{\mathrm{p}}\to\mathrm{Cat}^{\mathrm{ps}}$ is
compatible with the symmetric monoidal structure. In particular, the
$\infty$-operad $(\mathrm{Cat}^{\mathrm{ps}})^{\otimes}$ is a symmetric
monoidal $\infty$-category and the left adjoint
$\mathrm{Cat}^{\mathrm{p}}\to\mathrm{Cat}^{\mathrm{ps}}$ canonically refines
to a symmetric monoidal functor.

###### Proof.

Unwinding the definitions, what we need to prove is that if
$(\mathcal{C},\Phi)$ is a Poincaré $\infty$-category and
$(\mathcal{D},\Phi)\to(\mathcal{E},\Psi)$ is a Poincaré functor such that the
induced functor
$(\mathcal{D},\mathrm{D}_{\Phi})\to(\mathcal{E},\mathrm{D}_{\Psi})$ is an
equivalence of stable $\infty$-categories with duality then the Poincaré
functor
$(\mathcal{C},\Qoppa)\otimes(\mathcal{D},\Phi)\to(\mathcal{C},\Qoppa)\otimes(\mathcal{D},\Phi)$
is also an equivalence on underlying stable $\infty$-categories with duality.
But this is clear, since an equivalence of stable $\infty$-categories with
duality is detected on underlying stable $\infty$-categories and the forgetful
functor $\mathrm{Cat}^{\mathrm{p}}\to\mathrm{Cat}^{\mathrm{ex}}$ is symmetric
monoidal, see [9-authors-I]*Theorem 5.2.7. ∎

Since the forgetful functor
$\mathrm{Cat}^{\mathrm{p}}\to\mathrm{Cat}^{\mathrm{ex}}$ factors through
$\mathrm{Cat}^{\mathrm{ps}}$ it follows from the universal property of
symmetric monoidal localisations that the symmetric monoidal functor
$(\mathrm{Cat}^{\mathrm{p}})^{\otimes}\to(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}$
factors through a symmetric monoidal functor
$(\mathrm{Cat}^{\mathrm{ps}})^{\otimes}\to(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}$,
and the diagram (10) determines a diagram of $\infty$-operads

(11)
${(\mathcal{C},L)}$${(\mathcal{C},\Qoppa^{\mathrm{s}}_{L})}$${\mathrm{Inv}^{\otimes}}$${(\mathrm{Cat}^{\mathrm{ps}})^{\otimes}}$${\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}}$${(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}\
.}$

$\in$

$\in$

######  Proposition 3.1.8.

In the square (11), all functors are symmetric monoidal functors between
symmetric monoidal $\infty$-categories. In addition, the underlying
$\infty$-categories all admit sifted colimits and these are preserved by all
the functors in the square.

The square (11) is the version of (4) relevant for our purposes.

###### Proof of Proposition 3.1.8.

By construction all functors are symmetric monoidal functors between symmetric
monoidal $\infty$-categories except possibly the top horizontal one, which is
a-priori only a lax symmetric monoidal functor. However, since the right
vertical functor is conservative by [9-authors-I]*Corollary 7.2.16 we have the
top horizontal arrow must also be a symmetric monoidal functor.

Now $\mathrm{Cat}^{\mathrm{ex}}$ has all colimits [9-authors-I]*Proposition
6.1.1 and so
$\mathrm{Cat}^{\mathrm{ps}}\simeq(\mathrm{Cat}^{\mathrm{ex}})^{{\mathrm{h}\mathrm{C_{2}}}}$
has all colimits as well, and these are preserved by the forgetful functor
$\mathrm{Cat}^{\mathrm{ps}}\to\mathrm{Cat}^{\mathrm{ex}}$. Now since the
tensor product on $\mathrm{Cat}^{\mathrm{ex}}$ preserves colimits in each
variable (indeed, the operation $\mathcal{C}\otimes(-)$ admits a right adjoint
$\operatorname{Fun^{ex}}(\mathcal{C},-)$ by its defining mapping property) we
have that $\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})$ has all colimits
as well and that the forgetful functor
$\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})\to\mathrm{Cat}^{\mathrm{ex}}$
preserves sifted colimits. We now claim that the full subcategory
$\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}})\subseteq\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})$
is closed under sifted colimits. For this, note that since the forgetful
functor
$\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})\to\mathrm{Cat}^{\mathrm{ex}}$
is conservative, it also detects sifted colimits, that is, sifted colimits in
$\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})$ are calculated in
$\mathrm{Cat}^{\mathrm{ex}}$. This means that if
$\chi\colon\mathcal{I}\to\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}})$ is a
sifted diagram with colimit
$\mathcal{C}^{\otimes}=\mathop{\mathrm{colim}}_{\mathcal{I}}\chi$, then
$\mathcal{C}$ is the colimit of the underlying
$\mathrm{Cat}^{\mathrm{ex}}$-valued diagram, and so the collection of
essential images $\mathrm{im}[\chi(i)\to\mathcal{C}]$ generates $\mathcal{C}$
as a stable $\infty$-category. Now if we assume that each $\chi(i)$ is rigid
then each of the objects in $\mathrm{im}[\chi(i)\to\mathcal{C}]$ is
dualisable, and hence any object in $\mathcal{C}$ is dualisable since in any
stable symmetric monoidal $\infty$-category the collection of dualisable
objects forms a stable subcategory. We hence conclude that
$\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}})$ admits sifted
colimits and that these are preserved by the forgetful functor
$\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}})\to\mathrm{Cat}^{\mathrm{ex}}$.

Finally, the left vertical arrow
$\mathrm{Inv}\to\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}})$
is the left fibration classified by the functor

$\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}})\to\mathcal{S}$

sending a rigid stably symmetric monoidal $\infty$-category
$\mathcal{C}^{\otimes}$ to the $\infty$-groupoid of tensor-invertible objects
in $\mathcal{C}$. Since any sifted category is weakly contractible and
groupoids admit all colimits indexed by weakly contractible
$\infty$-categories we conclude that $\infty$-groupoids admit sifted colimits,
and hence the domain $\mathrm{Inv}$ of the left fibration
$\mathrm{Inv}\to\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}})$
admit sifted colimits and these are preserved by the projection to
$\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}})$. Since the
forgetful functor $\mathrm{Cat}^{\mathrm{ps}}\to\mathrm{Cat}^{\mathrm{ex}}$ is
conservative it now follows that the functor
$\mathrm{Inv}\to\mathrm{Cat}^{\mathrm{ps}}$ preserves sifted colimits as well.
∎

###  3.2. Derived Poincaré categories of schemes

######  Definition 3.2.1.

Let $(\mathrm{Cat}^{\mathrm{ex}}_{\mathrm{t}})^{\otimes}$ be the symmetric
monoidal $\infty$-category of stable $\infty$-categories equipped with a
t-structure on its Ind-completion, that is,
$(\mathrm{Cat}^{\mathrm{ex}}_{\mathrm{t}})^{\otimes}$ is the $\infty$-category
sitting in the pullback square

${(\mathrm{Cat}^{\mathrm{ex}}_{\mathrm{t}})^{\otimes}}$${(\Pr^{L}_{\mathrm{st},\mathrm{t}})^{\otimes}}$${(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}}$${(\Pr^{L}_{\mathrm{st}})^{\otimes}\
.}$$\scriptstyle{\operatorname{Ind}}$

By (52) of Appendix A, we have a functor

$\mathrm{qSch}^{\mathrm{op}}\to\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}}_{\mathrm{t}})\quad\quad
X\mapsto\mathcal{D}^{\mathrm{p}}(X)\quad(=(\mathcal{D}^{\mathrm{p}}(X),\mathcal{D}^{\mathrm{qc}}(X)_{\geq
0},\mathcal{D}^{\mathrm{qc}}(X)_{\leq 0})),$

sending a quasi-compact quasi-separated (qcqs for short) scheme to its perfect
derived category, considered as a stably symmetric monoidal $\infty$-category
endowed with t-structure on its Ind-completion. By [integral]*Proposition 4.6
this last functor almost preserves pushouts (which on the right hand side
correspond to relative tensor products of stable $\infty$-categories): more
precisely, this functor is pushout preserving up to idempotent completion, and
hence becomes pushout preserving if instead of
$\mathrm{Cat}^{\mathrm{ex}}_{\mathrm{t}}$ we consider its localisation
$\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}}$ by the collection of
Karoubi equivalences (i.e. dense inclusions). This localisation is
multiplicative, so that $\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}}$
inherits from $\mathrm{Cat}^{\mathrm{ex}}_{\mathrm{t}}$ its symmetric monoidal
structure. Concretely, if we identify the localisation
$\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}}$ with the full subcategory
of $\mathrm{Cat}^{\mathrm{ex}}_{\mathrm{t}}$ spanned by those
$(\mathcal{C},t)$ such that $\mathcal{C}$ is idempotent complete, then the
localised tensor product is computed by first taking the tensor product in
$\mathrm{Cat}^{\mathrm{ex}}_{\mathrm{t}}$ and then taking idempotent
completion (an operation which does not affect the Ind-completion, so that the
t-structure remains unchanged). The localisation functor
$\mathrm{Cat}^{\mathrm{ex}}_{\mathrm{t}}\to\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}}$
then canonically refines to a symmetric monoidal functor, and induces a
functor

$\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}}_{\mathrm{t}})\to\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}}).$

By post-composition, we thus obtain a functor

$\mathrm{qSch}^{\mathrm{op}}\to\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})\quad\quad
X\mapsto\mathcal{D}^{\mathrm{p}}(X),$

which by [integral]*Proposition 4.6 now preserves pushouts. Since each
$\mathcal{D}^{\mathrm{p}}(X)$ is rigid as a symmetric monoidal
$\infty$-category the functor above factors through the full subcategory
$\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})\subseteq\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})$
spanned by those $(\mathcal{C},t)$ for which the idempotent complete
$\mathcal{C}$ is rigid. Since $\mathrm{spec}(\mathbb{Z})$ is initial in
$\mathrm{qSch}^{\mathrm{op}}$, this functor canonically refines to a functor

$\mathrm{qSch}^{\mathrm{op}}\to\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})_{\mathcal{D}^{\mathrm{p}}(\mathbb{Z})/}=\operatorname{CAlg}_{\mathrm{rig}}(\operatorname{Mod}_{\mathcal{D}^{\mathrm{p}}(\mathbb{Z})}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}}))$

to rigid idempotent complete $\mathbb{Z}$-linearly symmetric monoidal
$\infty$-categories, and this last functor now preserves both pushouts and
initial objects by construction, so that it preserves all finite colimits.
Similarly, for any base scheme $S\in\mathrm{qSch}^{\mathrm{op}}$ we obtain a
finite colimit preserving functor

$(\mathrm{qSch}_{S})^{\mathrm{op}}\to\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})_{\mathcal{D}^{\mathrm{p}}(S)/}=\operatorname{CAlg}_{\mathrm{rig}}(\operatorname{Mod}_{\mathcal{D}^{\mathrm{p}}(S)}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}}))$

from qcqs schemes over $S$ to idempotent complete
$\mathcal{D}^{\mathrm{p}}(S)$-linear symmetric monoidal $\infty$-categories.
In particular, the last functor preserves coproducts, and hence can be
considered as a symmetric monoidal functor between the cocartesian symmetric
monoidal structures on both sides. On the side of
$(\mathrm{qSch}_{S})^{\mathrm{op}}$ this corresponds to taking fibre product
over $S$, and on the side of
$\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})_{\mathcal{D}^{\mathrm{p}}(S)/}$,
this corresponds to tensoring over $\mathcal{D}^{\mathrm{p}}(S)$.

######  Construction 3.2.2.

We define the following $\infty$-categories by means of pullback:

$(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})^{\otimes}:=(\mathrm{Cat}^{\mathrm{ps}})^{\otimes}\times_{(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})^{\otimes}\quad\text{and}\quad\mathrm{Inv}_{\natural,\mathrm{t}}^{\otimes}:=\mathrm{Inv}^{\otimes}\times_{\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}})^{\otimes}}\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})^{\otimes}.$

By Proposition 3.1.8 the square (11) then induces a commutative square

${(\mathcal{C},L)}$${(\mathcal{C},\Qoppa^{\mathrm{s}}_{L})}$${\mathrm{Inv}_{\natural,\mathrm{t}}^{\otimes}}$${(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})^{\otimes}}$${\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})^{\otimes}}$${(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})^{\otimes}\
.}$

$\in$

$\in$

of symmetric monoidal $\infty$-categories and symmetric monoidal functors
between them. We note that the $\infty$-category
$\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}}$ admits all colimits and
hence $\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})$
admits all colimits and the forgetful functor
$\operatorname{CAlg}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})\to\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}}$
preserves and detects sifted colimits. By 3.1.8 all the $\infty$-categories in
the above square admit sifted colimits and all functors preserve and detect
them. In addition, since the monoidal structure on
$\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}}$ preserves colimits in each
variable separately it follows that the monoidal structures of all symmetric
monoidal $\infty$-categories in the above square preserve sifted colimits in
each variable. In particular, for any stable symmetric monoidal
$\infty$-category $\mathcal{A}^{\otimes}$ with unit $\mathbb{1}\in\mathcal{A}$
the square (11) induces a square of symmetric monoidal $\infty$-categories and
symmetric monoidal functors

${\operatorname{Mod}_{(\mathcal{A}^{\otimes},\mathbb{1})}(\mathrm{Inv}_{\natural,\mathrm{t}})^{\otimes}}$${\operatorname{Mod}_{(\mathcal{A}^{\otimes},\Qoppa^{\mathrm{s}}_{\mathbb{1}})}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})^{\otimes}}$${\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})^{\otimes}_{\mathcal{A}^{\otimes}/}}$${\operatorname{Mod}_{\mathcal{A}}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})^{\otimes}\
.}$

Pulling back the left vertical map of the square of Construction 3.2.2, we now
obtain a commutative square of symmetric monoidal $\infty$-categories and
symmetric monoidal functors

(12)
${(X,L)}$${(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})}$${(\mathrm{qSchPic}_{S}^{\mathrm{op}})^{\otimes}}$${\operatorname{Mod}_{(\mathcal{D}^{\mathrm{p}}(S),\Qoppa^{\mathrm{s}}_{S})}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})^{\otimes}}$${(\mathrm{qSch}_{S}^{\mathrm{op}})^{\otimes}}$${\operatorname{Mod}_{\mathcal{D}^{\mathrm{p}}(S)}(\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}})^{\otimes}\
,}$

$\in$

$\in$

where

$\mathrm{qSchPic}_{S}=\mathrm{qSch}_{S}^{\otimes}\times_{(\operatorname{CAlg}_{\mathrm{rig}}(\mathrm{Cat})^{\mathrm{op}})^{\otimes}}(\mathrm{Inv}^{\mathrm{op}})^{\otimes}$

is the symmetric monoidal $\infty$-category with objects $(X,L)$ where $X$ is
a qcqs scheme over $S$ and
$L\in{\mathcal{P}ic}(X)^{\mathrm{B}\mathrm{C_{2}}}={\mathcal{P}ic}(\mathcal{D}^{\mathrm{p}}(X))^{\mathrm{B}\mathrm{C_{2}}}$
is a tensor invertible perfect complex on $X$ with $\mathrm{C_{2}}$-action.
Here, the left vertical arrow is again a left fibration of $\infty$-operads by
base change, the top horizontal arrow is a lax symmetric monoidal functor
between symmetric monoidal $\infty$-categories and the lower horizontal arrow
is symmetric monoidal. The square (12) then encodes the principal
functoriality and multiplicativity enjoyed by the association
$(X,L)\mapsto(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{L})$.

Let us now spell out a variant of the above construction which will be useful
as well. For this, consider the left fibration of $\infty$-operads
$\pi\colon(\mathrm{qSchPic}_{S}^{\mathrm{op}})^{\otimes}\to(\mathrm{qSch}_{S}^{\mathrm{op}})^{\otimes}$.
Let

$\mathcal{N}^{\otimes}=\mathcal{N}\mathrm{m}_{\mathrm{qSch}_{S}^{\mathrm{op}}}(\mathrm{qSchPic}_{S}^{\mathrm{op}})\to\mathrm{Com}^{\otimes}$

be the norm of $\pi$ along the cocartesian fibration of $\infty$-operads
$(\mathrm{qSch}_{S}^{\mathrm{op}})^{\otimes}\to\mathrm{Com}^{\otimes}$, see
[HA]*Theorem 2.2.6.2. Then $\mathcal{N}^{\otimes}$ is an $\infty$-operad whose
underlying $\infty$-category is the $\infty$-category of sections
$\mathrm{Sch}_{S}^{\mathrm{op}}\to\mathrm{qSchPic}_{S}^{\mathrm{op}}$ of
$\pi$. Now the unit $\mathrm{id}_{S}\in\mathrm{Sch}_{S}^{\mathrm{op}}$ can be
viewed as a map of $\infty$-operads
$\mathrm{Com}^{\otimes}\to(\mathrm{Sch}_{S}^{\mathrm{op}})^{\mathrm{op}}$, and
as such it induces a map

$\mathcal{N}^{\otimes}=\mathcal{N}\mathrm{m}_{\mathrm{Sch}_{S}^{\mathrm{op}}}(\mathrm{qSchPic}_{S}^{\mathrm{op}})\to\mathcal{N}\mathrm{m}_{\mathrm{Com}^{\otimes}}((\mathrm{qSchPic}_{S}^{\mathrm{op}})^{\otimes}\times_{(\mathrm{Sch}_{S}^{\mathrm{op}})^{\otimes}}\mathrm{Com}^{\otimes})=\big{(}\mathrm{qSchPic}_{S}^{\mathrm{op}}\times_{\mathrm{Sch}_{S}^{\mathrm{op}}}\\{\mathrm{id}_{S}\\}\big{)}^{\otimes}={\mathcal{P}ic}(S)^{\mathrm{B}\mathrm{C_{2}}},$

and since $\mathrm{id}_{S}$ is initial in $\mathrm{Sch}_{S}^{\mathrm{op}}$ and
$\pi$ is a left fibration we see that this map is actually an equivalence of
$\infty$-operads. By the universal property of the norm construction this
means that for every $\infty$-operad $\mathcal{O}^{\otimes}$ we have a natural
equivalence between $\mathcal{O}$-algebras in
${\mathcal{P}ic}(S)^{\mathrm{B}\mathrm{C_{2}}}$ and lax $\mathcal{O}$-monoidal
sections $\mathrm{Sch}_{S}^{\mathrm{op}}\to\mathrm{qSchPic}_{S}^{\mathrm{op}}$
of $\pi$. Better yet, since $\pi$ is a left fibration any lax
$\mathcal{O}$-monoidal section is automatically $\mathcal{O}$-monoidal. In
particular, taking the unit algebra object
$\mathcal{O}_{X}\in{\mathcal{P}ic}(X)^{\mathrm{B}\mathrm{C_{2}}}$ yields a
symmetric monoidal section
$s_{0}\colon\mathrm{Sch}_{S}^{\mathrm{op}}\to\mathrm{qSchPic}_{S}^{\mathrm{op}}$,
given by the formula $[p\colon X\to S]\mapsto(X,\mathcal{O}_{X})$, and any
object $L\in{\mathcal{P}ic}(S)^{\mathrm{B}\mathrm{C_{2}}}$ (which can be
canonically considered as a module over the unit $\mathcal{O}_{X}$) determines
a section
$s_{L}\colon\mathrm{Sch}_{S}^{\mathrm{op}}\to\mathrm{qSchPic}_{S}^{\mathrm{op}}$
of $\mathrm{Sch}_{S}^{\mathrm{op}}$-module $\infty$-categories, where the
target is considered as an $\mathrm{Sch}_{S}^{\mathrm{op}}$-module via
$s_{0}$. Composing with the formation of derived categories we obtain a
composable pair of symmetric monoidal functors

(13)
${X}$${(X,\mathcal{O}_{X})}$${(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{\mathcal{O}_{X}})}$${(\mathrm{Sch}_{S}^{\mathrm{op}})^{\otimes}}$${(\mathrm{qSchPic}_{S}^{\mathrm{op}})^{\otimes}}$${\operatorname{Mod}_{(\mathcal{D}^{\mathrm{p}}(S),\Qoppa^{\mathrm{s}}_{S})}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})^{\otimes}\
,}$

$\in$

$\in$

$\in$

and similarly, for every $L\in{\mathcal{P}ic}(S)^{\mathrm{B}\mathrm{C_{2}}}$,
we obtain a composable pair of $\mathrm{Sch}_{S}^{\mathrm{op}}$-module
functors

(14) ${[p\colon X\to
S]}$${(X,p^{*}L)}$${(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{p^{*}L})}$${\mathrm{Sch}_{S}^{\mathrm{op}}}$${\mathrm{qSchPic}_{S}^{\mathrm{op}}}$${\operatorname{Mod}_{(\mathcal{D}^{\mathrm{p}}(S),\Qoppa^{\mathrm{s}}_{S})}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})\
,}$

$\in$

$\in$

$\in$

where the $\mathrm{Sch}_{S}$-action on the last two terms is determined by the
symmetric monoidal functors in (13). The diagrams (12), (13) and (14)
summarize the key types of functoriality relating schemes and Poincaré
$\infty$-categories, and will be exploited systematically throughout the
present paper.

######  Notation 3.2.3.

For simplicity, we now suppress the mention of the Poincaré structure on the
base and set

$\operatorname{Mod}_{S}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}})=\operatorname{Mod}_{(\mathcal{D}^{\mathrm{p}}(S),\Qoppa^{\mathrm{s}}_{S})}(\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}).$

We also use similar notation when
$\mathrm{Cat}^{\mathrm{ps}}_{\natural,\mathrm{t}}$ is replaced by other
variants. Note that this always means that the Poincaré structure considered
on the base is the _symmetric_ one.

###  3.3. Genuine structures

In this section, we explain the general formalism for truncating the linear
part of a Poincaré structure, in terms of an arbitrary t-structure, although
we’ll only use the standard one on derived categories of schemes in the rest
of the paper.

Let $(\mathcal{C},\Qoppa)$ be a Poincaré $\infty$-category. Then the first
excisive approximation
$\mathrm{P}_{1}\Qoppa=\mathop{\mathrm{colim}}_{n}\Omega^{n}\Qoppa\Sigma^{n}$,
is an exact functor $\mathcal{C}^{\mathrm{op}}\to{\mathcal{S}\mathrm{p}}$ and
so can be identified with an element of $\operatorname{Ind}\mathcal{C}$.
Suppose now that $\operatorname{Ind}\mathcal{C}$ is equipped with a
t-structure. Then we say that $\Qoppa$ is _$m$ -connective_ if
$\mathrm{P}_{1}\Qoppa$ is $m$-connective when seen as an element of
$\operatorname{Ind}\mathcal{C}$.

######  Lemma 3.3.1.

Let $\mathcal{C}$ be a stable $\infty$-category and $t$ be a t-structure on
$\operatorname{Ind}\mathcal{C}$. Then the inclusion of the subcategory
$\operatorname{Fun^{q}}_{\geq
m}(\mathcal{C})\subseteq\operatorname{Fun^{q}}(\mathcal{C})$ of $m$-connective
quadratic functors has a right adjoint sending $\Qoppa$ to $\Qoppa^{\geq
m}=\Qoppa\times_{\mathrm{P}_{1}\Qoppa}(\tau_{\geq m}\mathrm{P}_{1}\Qoppa)$,
where the connective cover of $\mathrm{P}_{1}\mathcal{C}$ is taken in
$\operatorname{Ind}\mathcal{C}$.

###### Proof.

It suffices to show that the map $\Qoppa^{\geq m}\to\Qoppa$ induces an
equivalence

$\operatorname{Map}(\Qoppa^{\prime},\Qoppa^{\geq
m})\to\operatorname{Map}(\Qoppa^{\prime},\Qoppa)$

for every $m$-connected quadratic functor $\Qoppa^{\prime}$. But

$\displaystyle\operatorname{Map}(\Qoppa^{\prime},\Qoppa^{\geq m})$
$\displaystyle\cong\operatorname{Map}(\Qoppa^{\prime},\Qoppa)\times_{\operatorname{Map}(\Qoppa^{\prime},\mathrm{P}_{1}\Qoppa)}\operatorname{Map}(\Qoppa^{\prime},\tau_{\geq
m}\mathrm{P}_{1}\Qoppa)$
$\displaystyle\cong\operatorname{Map}(\Qoppa^{\prime},\Qoppa)\times_{\operatorname{Map}(\mathrm{P}_{1}\Qoppa^{\prime},\mathrm{P}_{1}\Qoppa)}\operatorname{Map}(\mathrm{P}_{1}\Qoppa^{\prime},\tau_{\geq
m}\mathrm{P}_{1}\Qoppa)$

and the map

$\operatorname{Map}(\mathrm{P}_{1}\Qoppa^{\prime},\tau_{\geq
m}\mathrm{P}_{1}\Qoppa)\to\operatorname{Map}(\mathrm{P}_{1}\Qoppa^{\prime},\Qoppa)$

is an equivalence since $\mathrm{P}_{1}\Qoppa^{\prime}$ is $m$-connected. ∎

Recall from Definition 3.2.1 that we write
$\mathrm{Cat}^{\mathrm{ex}}_{\mathrm{t}}=\mathrm{Cat}^{\mathrm{ex}}\times_{\Pr^{L}}\Pr^{L}_{t}$
for $\infty$-category of pairs $(\mathcal{C},t)$ where $\mathcal{C}$ is a
stable $\infty$-category and
$\mathrm{t}=(\operatorname{Ind}(\mathcal{C})_{\geq
0},\operatorname{Ind}(\mathcal{C})_{\leq 0})$ is a t-structure on
$\operatorname{Ind}(\mathcal{C})$. Let
$\mathrm{Cat}^{\mathrm{p}}_{\mathrm{t}}=\mathrm{Cat}^{\mathrm{p}}\times_{\mathrm{Cat}^{\mathrm{ex}}}\mathrm{Cat}^{\mathrm{ex}}_{\mathrm{t}}$.
We then write objects in $\mathrm{Cat}^{\mathrm{p}}_{\mathrm{t}}$ as triples
$(\mathcal{C},\Qoppa,t)$ where $(\mathcal{C},\Qoppa)$ is a Poincaré
$\infty$-category $t$ is a t-structure on $\operatorname{Ind}(\mathcal{C})$.

######  Proposition 3.3.2.

For every $m$ the inclusion $\mathrm{Cat}^{\mathrm{p}}_{t\geq
m}\subseteq\mathrm{Cat}^{\mathrm{p}}_{\mathrm{t}}$ of the full subcategory of
the Poincaré $\infty$-categories whose Poincaré structure is $m$-connective in
the above sense has a right adjoint, sending $(\mathcal{C},\Qoppa,t)$ to the
triple $(\mathcal{C},\Qoppa^{\geq m},t)$ where $\mathcal{C}$ and $t$ are the
same and $\Qoppa^{\geq m}$ is the functor of Lemma 3.3.1.

###### Proof.

Note that $\Qoppa^{\geq m}$ has the same cross-effect as $\Qoppa$, so it is a
perfect quadratic functor on $\mathcal{C}$ with the same underlying duality.
In particular $(\mathcal{C},\Qoppa^{\geq m})$ is a Poincaré $\infty$-category.
Moreover the excisive approximation of $\Qoppa^{\geq m}$ is $m$-connective by
definition. So it suffices to show that if
$(\mathcal{C}^{\prime},\Qoppa^{\prime},t^{\prime})$ is a Poincaré
$\infty$-category such that $\Qoppa^{\prime}$ is $m$-connective, the map

$\operatorname{Map}_{\mathrm{Cat}^{\mathrm{p}}_{\mathrm{t}}}((\mathcal{C}^{\prime},\Qoppa^{\prime},t^{\prime}),(\mathcal{C},\Qoppa^{\geq
m},t))\to\operatorname{Map}_{\mathrm{Cat}^{\mathrm{p}}_{\mathrm{t}}}((\mathcal{C}^{\prime},\Qoppa^{\prime},t^{\prime}),(\mathcal{C},\Qoppa,t))$

is an equivalence. Since both of these spaces lie above the space
$\operatorname{Map}_{\mathrm{Cat}^{\mathrm{ex}}_{\natural,\mathrm{t}}}((\mathcal{C}^{\prime},t^{\prime}),(\mathcal{C},t))$
of t-right exact functors, it suffices to show that the above map induces an
equivalence on the fibre over every
$F\colon\mathcal{C}^{\prime}\to\mathcal{C}$. The map on the fibre is the
restriction of the map

$\operatorname{Map}_{\operatorname{Fun^{q}}(\mathcal{C}^{\prime})}\left(\Qoppa^{\prime},\Qoppa^{\geq
m}\circ
F\right)\to\operatorname{Map}_{\operatorname{Fun^{q}}(\mathcal{C}^{\prime})}\left(\Qoppa^{\prime},\Qoppa\circ
F\right)$

to the subspaces of those maps that induce an equivalence $F\to\mathrm{D}\circ
F\circ\mathrm{D}^{\prime}$. Since the underlying duality of $\Qoppa$ and of
$\Qoppa^{\geq m}$ is the same, it suffices to show that the above map is an
equivalence. Finally we can rewrite it as the map

$\operatorname{Map}_{\operatorname{Fun^{q}}(\mathcal{C})}\left(F_{!}\Qoppa^{\prime},\Qoppa^{\geq
m}\right)\to\operatorname{Map}_{\operatorname{Fun^{q}}(\mathcal{C})}\left(F_{!}\Qoppa^{\prime},\Qoppa\right)\,$

so by Lemma 3.3.1 it suffices to show that $F_{!}\Qoppa^{\prime}$ is
$m$-connective. But
$\mathrm{P}_{1}(F_{!}\Qoppa^{\prime})=F_{!}(\mathrm{P}_{1}\Qoppa^{\prime})$
and $F_{!}$ is right t-exact by definition. So, since
$\mathrm{P}_{1}\Qoppa^{\prime}$ is $m$-connective, so is
$F_{!}\Qoppa^{\prime}$. ∎

Note that if the t-structure is left complete,
$t_{\geq\infty}\mathrm{P}_{1}\Qoppa=0$ and so $\Qoppa^{\geq\infty}$ coincides
with the quadratic Poincaré structure associated with the duality of $\Qoppa$.

Specializing to derived categories of schemes, this has the following
consequences. Using the unit of the adjunction of Proposition 3.3.2 the square
(12) induces for any line bundle with $\mathrm{C_{2}}$-action $L$ on $S$ and
any integer $m\in\mathbb{Z}$ a functor

$\mathrm{qSch}_{S}^{\mathrm{op}}\to\mathrm{Cat}^{\mathrm{p}}_{\natural,\mathrm{t}}\subset\mathrm{Cat}^{\mathrm{p}}_{\mathrm{t}}\quad\quad[p\colon
X\to S]\mapsto(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\geq m}_{p^{*}L}),$

fitting together in a sequence

$(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\geq\infty}_{p^{*}L})=(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{q}}_{p^{*}L})\to\cdots\to(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\geq
m}_{p^{*}L})\to\cdots\to(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\geq-\infty}_{p^{*}L})=(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\mathrm{s}}_{p^{*}L})\,.$

######  Definition 3.3.3.

Given a scheme $X$ and $L\in{\mathcal{P}ic}(X)^{{\mathrm{h}\mathrm{C_{2}}}}$,
we call the Poincaré structures $\Qoppa^{\geq 0}_{L}$, $\Qoppa^{\geq 1}_{L}$
and $\Qoppa^{\geq 2}_{L}$ on $\mathcal{D}^{\mathrm{p}}(X)$ the _genuine
symmetric_ , _genuine even_ and _genuine quadratic_ Poincaré structures,
respectively, and use the notation $\Qoppa^{\mathrm{gs}}_{L}$,
$\Qoppa^{\mathrm{ge}}_{L}$ and $\Qoppa^{\mathrm{gq}}_{L}$ to refer to them.

By [Hebestreit-Steimle]*Theorem A, when $X=\mathrm{spec}R$ is affine the
Grothendieck-Witt space of $(\mathcal{D}^{\mathrm{p}}(R),\Qoppa^{r}_{L})$ for
$r=\mathrm{gs},\mathrm{ge},\mathrm{gq}$ coincides with the group completion of
the symmetric monoidal groupoid of projective $R$-modules equipped with a
perfect $L$-valued symmetric bilinear/even/quadratic form, respectively. In
other words, the non-negative genuine symmetric/even/quadratic
$\operatorname{GW}$-groups of $\mathrm{spec}(R)$ coincide with the classical
symmetric/even/quadratic Grothendieck-Witt groups of $R$, as defined by
Karoubi-Villamayor [karoubi-villamayor]. For the genuine symmetric case we
will extend this comparison to all divisorial schemes in §4.6.

######  Notation 3.3.4.

For a functor $\mathcal{F}\colon\mathrm{Cat}^{\mathrm{p}}\to\mathcal{E}$ and a
scheme $X$ and an invertible perfect complex with $\mathrm{C_{2}}$-action
$L\in{\mathcal{P}ic}(X)^{{\mathrm{h}\mathrm{C_{2}}}}$ we write

$\mathcal{F}^{\geq
m}(X,L):=\mathcal{F}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\geq m}_{X})$

for every $m\in\mathbb{Z}\cup\\{-\infty,\infty\\}$. For shorthand we will also
write

$\mathcal{F}^{\geq m}(X):=\mathcal{F}^{\geq m}(X,\mathcal{O}_{X})$

and if $p\colon X\to S$ is a $S$-scheme and
$L\in{\mathcal{P}ic}(X)^{{\mathrm{h}\mathrm{C_{2}}}}$ is an invertible perfect
complex with $\mathrm{C_{2}}$-action on $S$ then we write

$\mathcal{F}^{\geq m}_{L}(X):=\mathcal{F}^{\geq m}(X,p^{*}L).$

More generally, for $n,m\in\mathbb{Z}$ we define

$\mathcal{F}^{\geq
m,[n]}_{L}(X)=\mathcal{F}\left(\mathcal{D}^{\mathrm{p}}(X),(\Qoppa^{\geq
m}_{L})^{[n]}\right).$

When $n=0$, we drop the shift in the notation, which becomes
$\mathcal{F}^{\geq m}_{L}$. Similarly, for $m=\pm\infty$ we replace the
superscripts $(-)^{\geq-\infty}$ and $(-)^{\geq\infty}$ with
$(-)^{\mathrm{s}}$ and $(-)^{\mathrm{q}}$, respectively. and for
$L=\mathcal{O}_{S}$ we use the subscript $S$ instead of $\mathcal{O}_{S}$.
Using the functoriality of (14) we may view $\mathcal{F}^{\geq m}_{L}$ as a
functor on $\mathrm{Sch}_{S}$, or on the full subcategory
$\mathrm{Sm}_{S}\subseteq\mathrm{Sch}_{S}$ spanned by the smooth $S$-schemes.

##  4\. Nisnevich descent

Our main goal in this section is to show that for a qcqs scheme $S$, a perfect
invertible complex with $\mathrm{C_{2}}$-action
$L\in{\mathcal{P}ic}(S)^{\mathrm{B}\mathrm{C_{2}}}$, and a Karoubi-localising
functor $\mathcal{F}\colon\mathrm{Cat}^{\mathrm{p}}\to\mathcal{E}$, the
functor

$\mathcal{F}^{\geq
m}_{L}\colon(\mathrm{Sch}^{\mathrm{qq}}_{S})^{\mathrm{op}}\to\mathcal{E}\quad\quad(X,L)\mapsto\mathcal{F}(\mathcal{D}^{\mathrm{p}}(X),\Qoppa^{\geq
m}_{X})$

of Construction 3.3.4 satisfies Nisnevich descent, that is, it is a sheaf with
respect to the Nisnevich topology for every
$m\in\mathbb{Z}\cup\\{\pm\infty\\}$. To arrive to this, we first dedicate §4.1
and 4.2 to introduce and study the notions of flat $S$-linear
$\infty$-categories (with duality) and bounded Karoubi sequences between them.
The main reason for this setting is that it will eventually allow us in §7 to
bootstrap Nisnevich descent into the construction of motivic realization
functors with suitable multiplicative properties. In §4.3 we then prove
Zariski descent for $\mathcal{F}^{\geq m}_{L}$, followed by Nisnevich descent
in §4.4. Finally, §4.5 and §4.6 contain applications of these descent results,
the former to establishing a version of the coniveau filtration in the present
setting, and the second for the study of genuine symmetric
$\operatorname{GW}$-theory of schemes.

###  4.1. Flat linear categories

As discussed in §3 and in Appendix §A, for every scheme $X$, the stable
$\infty$-category $\mathcal{D}^{\mathrm{p}}(X)$ comes equipped with a
t-structure on its Ind-completion $\mathcal{D}^{\mathrm{qc}}(X)$. To
facilitate terminology, let us refer to the data of a t-structure on the Ind-
completion of a given stable $\infty$-category $\mathcal{A}$ as an
_orientation_ on $\mathcal{A}$. If $\mathcal{A}$ carries a symmetric monoidal
structure (e.g., if $\mathcal{A}=\mathcal{D}^{\mathrm{p}}(X)$) then
$\operatorname{Ind}(\mathcal{A})$ acquires an induced symmetric monoidal
structure, in which case we use the term _multiplicative orientation_ to mean
a t-structure $(\operatorname{Ind}(\mathcal{A})_{\geq
0},\operatorname{Ind}(\mathcal{A})_{\leq 0})$ on
$\operatorname{Ind}(\mathcal{A})$ which is compatible with the induced
symmetric monoidal structure, that is, $\operatorname{Ind}(\mathcal{A})_{\geq
0}$ contains $\mathbb{1}_{\mathcal{A}}$ and is closed under tensor product.

######  Definition 4.1.1.

Let $\mathcal{A}$ be a stably symmetric monoidal $\infty$-category equipped
with a multiplicative orientation. Given
$x\in\operatorname{Ind}(\mathcal{A})$, we say that $x$ _has Tor amplitude
$\leq n$_ for a given $n\in\mathbb{Z}$ if the operation $x\otimes(-)$ sends
coconnective objects to $n$-coconnective objects. We say that $x$ has _bounded
Tor amplitude_ if it has Tor amplitude $\leq n$ for some $n$.

######  Remark 4.1.2.

If $x\in\operatorname{Ind}(\mathcal{A})$ has Tor amplitude $\leq n$ and
$\operatorname{Ind}(y)\in\mathcal{A}$ has Tor amplitude $\leq m$ then
$x\otimes y$ has Tor amplitude $\leq n+m$.

######  Lemma 4.1.3.

Let $\mathcal{A}$ be a rigid stably symmetric monoidal $\infty$-category
equipped with a multiplicative orientation. Then for a given $n\in\mathbb{Z}$,
an object $x\in\mathcal{A}$ has Tor amplitude $\leq n$ if and only if
$\mathrm{D}x$ is $(-n)$-connective.

###### Proof.

The dual $\mathrm{D}x$ has in particular the property that the functor
$\mathrm{D}x\otimes(-)$ is left adjoint to $x\otimes(-)$. In addition, as in
any t-structure, we have for every $i$ the equalities

$\operatorname{Ind}(\mathcal{A})_{\leq
i}=\operatorname{Ind}(\mathcal{A})_{\geq
i+1}^{\perp}=\\{z\in\mathcal{A}\;|\;\mathrm{hom}(y,z)=0,\;\forall
y\in\operatorname{Ind}(\mathcal{A})_{\geq i+1}\\}$

and

$\operatorname{Ind}(\mathcal{A})_{\geq
i+1}={}^{\perp}\operatorname{Ind}(\mathcal{A})_{\leq
i}=\\{x\in\mathcal{A}\;|\;\mathrm{hom}(x,y)=0,\;\forall
y\in\operatorname{Ind}(\mathcal{A})_{\leq i}\\}.$

We conclude that $x\otimes(-)$ sends $\operatorname{Ind}(\mathcal{A})_{\leq
0}$ to $\operatorname{Ind}(\mathcal{A})_{\leq n}$ if and only if
$\mathrm{D}x\otimes(-)$ sends $\operatorname{Ind}(\mathcal{A})_{\geq n+1}$ to
$\operatorname{Ind}(\mathcal{A})_{\geq 1}$, hence if and only if
$\mathrm{D}x\otimes(-)$ sends $\operatorname{Ind}(\mathcal{A})_{\geq 0}$ to
$\operatorname{Ind}(\mathcal{A})_{\geq-n}$. Since the t-structure is
multiplicative, this is equivalent to $\mathrm{D}x$ belonging to
$\operatorname{Ind}(\mathcal{A})_{\geq-n}$. ∎

Let $\mathcal{A}$ be a stably symmetric monoidal $\infty$-category. By an
$\mathcal{A}$-linear $\infty$-category $\mathcal{C}$, we mean an
$\mathcal{A}$-module object in $\mathrm{Cat}^{\mathrm{ex}}$. Such an
$\mathcal{A}$-module structure determines a refinement of the mapping spectra
in $\mathcal{C}$ to $\operatorname{Ind}(\mathcal{A})$-valued mapping objects,
which we denote by
$\uline{\mathrm{hom}}_{\mathcal{C}}(x,y)\in\operatorname{Ind}(\mathcal{A})$.
They are uniquely characterized by the property that

$\mathrm{hom}_{\operatorname{Ind}(\mathcal{A})}(j(a),\uline{\mathrm{hom}}_{\mathcal{C}}(x,y))=\mathrm{hom}_{\mathcal{C}}(a\otimes
x,y)$

for any $a\in\mathcal{A}$, where $j\colon A\to\operatorname{Ind}(\mathcal{A})$
is the Yoneda embedding. We refer to morphisms of $\mathcal{A}$-modules in
$\mathrm{Cat}^{\mathrm{ex}}$ as $\mathcal{A}$-linear functors, and write
$\operatorname{Fun}^{\mathcal{A}}(\mathcal{C},\mathcal{D})$ for the
$\infty$-category of $\mathcal{A}$-linear functors from $\mathcal{C}$ to
$\mathcal{D}$. Note that our terminology is such that an $\mathcal{A}$-linear
functor is by definition exact.

######  Definition 4.1.4.

Let $\mathcal{A}$ be a stably symmetric monoidal $\infty$-category equipped
with a multiplicative orientation. We say that an $\mathcal{A}$-linear
$\infty$-category $\mathcal{E}$ is _flat_ over $\mathcal{A}$ if the enriched
mapping object $\uline{\mathrm{hom}}(x,y)\in\operatorname{Ind}(\mathcal{A})$
has bounded Tor amplitude for every $x,y\in\mathcal{E}$.

######  Example 4.1.5.

Let $X\to S$ be a flat morphism of qcqs schemes. Then
$\mathcal{D}^{\mathrm{p}}(X)$ is flat over
$\mathcal{A}=\mathcal{D}^{\mathrm{p}}(S)$.

###### Proof.

The assumption that $f$ is flat implies that
$f^{*}\mathcal{D}^{\mathrm{qc}}(S)\to\mathcal{D}^{\mathrm{qc}}(X)$ is exact
and hence its right adjoint
$f_{*}\colon\mathcal{D}^{\mathrm{qc}}(X)\to\mathcal{D}^{\mathrm{qc}}(S)$
preserves objects with bounded Tor amplitude $\leq n$ by the projection
formula. We may hence assume without loss of generality that $f$ is the
identity. In this case the result follows directly from the characterization
of Lemma 4.1.3. ∎

Recall that the tensor product $(-)\otimes_{\mathcal{A}}(-)$ in
$\operatorname{Mod}_{\mathcal{A}}(\mathrm{Cat}^{\mathrm{ex}})$ is determined
by the following universal property: there exists a natural map
$i\colon\mathcal{C}\times\mathcal{D}\to\mathcal{C}\otimes_{\mathcal{A}}\mathcal{D}$
which is $\mathcal{A}$-linear in each variable separately and such that for
every $\mathcal{A}$-linear $\infty$-category $\mathcal{E}$ restriction along
$i$ determines an equivalence between $\mathcal{A}$-linear functors
$\mathcal{C}\otimes_{\mathcal{A}}\mathcal{D}\to\mathcal{E}$ and functors
$\mathcal{C}\times\mathcal{D}\to\mathcal{E}$ which are $\mathcal{A}$-linear in
each variable separately. In particular, we have a canonical equivalence

$\operatorname{Fun}^{\mathcal{A}}(\mathcal{C}\otimes_{\mathcal{A}}\mathcal{D},\mathcal{E})=\operatorname{Fun}^{\mathcal{A}}(\mathcal{C},\operatorname{Fun}^{\mathcal{A}}(\mathcal{D},\mathcal{E})),$

so that the functor
$(-)\otimes_{\mathcal{A}}\mathcal{D}\colon\operatorname{Mod}_{\mathcal{A}}(\mathrm{Cat}^{\mathrm{ex}})\to\operatorname{Mod}_{\mathcal{A}}(\mathrm{Cat}^{\mathrm{ex}})$
admits a right adjoint given by
$\mathcal{E}\mapsto\operatorname{Fun}^{\mathcal{A}}(\mathcal{D},\mathcal{E})$.
To obtain a more explicit description of the underlying stable
$\infty$-category of $\mathcal{C}\otimes_{\mathcal{A}}\mathcal{D}$, we may
ignore for the moment its $\mathcal{A}$-action and view the association
$\mathcal{C}\mapsto\mathcal{C}\otimes_{\mathcal{A}}\mathcal{D}$ as a functor
$\operatorname{Mod}_{\mathcal{A}}(\mathrm{Cat}^{\mathrm{ex}})\to\mathrm{Cat}^{\mathrm{ex}}$.
As such, it also admits a right adjoint, given by the formula
$\mathcal{E}\mapsto\operatorname{Fun^{ex}}(\mathcal{D},\mathcal{E})$, where
$\operatorname{Fun^{ex}}(\mathcal{D},\mathcal{E})$ is considered as an
$\mathcal{A}$-module via the pre-composition action determined by its action
on $\mathcal{D}$. We may hence describe
$\mathcal{C}\otimes_{\mathcal{A}}\mathcal{D}$ by embedding it in its Ind-
completion, which by the last adjunction can be described as

(15)
$\displaystyle\operatorname{Ind}(\mathcal{C}\otimes_{\mathcal{A}}\mathcal{D})$
$\displaystyle=\operatorname{Fun^{ex}}((\mathcal{C}\otimes_{\mathcal{A}}\mathcal{D})^{\mathrm{op}},{\mathcal{S}\mathrm{p}})$
$\displaystyle=\operatorname{Fun^{ex}}(\mathcal{C}^{\mathrm{op}}\otimes_{\mathcal{A}^{\mathrm{op}}}\mathcal{D}^{\mathrm{op}},{\mathcal{S}\mathrm{p}})$
$\displaystyle=\operatorname{Fun}^{\mathcal{A}^{\mathrm{op}}}(\mathcal{C}^{\mathrm{op}},\operatorname{Fun^{ex}}(\mathcal{D},{\mathcal{S}\mathrm{p}}))$
$\displaystyle=\operatorname{Fun}^{\mathcal{A}^{\mathrm{op}}}(\mathcal{C}^{\mathrm{op}},\operatorname{Ind}(\mathcal{D}))\
,$

where the $\mathcal{A}^{\mathrm{op}}$-action on
$\operatorname{Ind}(\mathcal{D})=\operatorname{Fun^{ex}}(\mathcal{D}^{\mathrm{op}},{\mathcal{S}\mathrm{p}})$
is the pre-composition action induced by the
$\mathcal{A}^{\mathrm{op}}$-action on $\mathcal{D}^{\mathrm{op}}$. To identify
the full subcategory of
$\operatorname{Fun}^{\mathcal{A}^{\mathrm{op}}}(\mathcal{C}^{\mathrm{op}},\operatorname{Ind}(\mathcal{D}))$
corresponding to $\mathcal{C}\otimes_{\mathcal{A}}\mathcal{D}$, we need to
track the composite functor

$\mathcal{C}\times\mathcal{D}\to\mathcal{C}\otimes_{\mathcal{A}}\mathcal{D}\to\operatorname{Ind}(\mathcal{C}\otimes_{\mathcal{A}}\mathcal{D})=\operatorname{Fun}^{\mathcal{A}^{\mathrm{op}}}(\mathcal{C}^{\mathrm{op}},\operatorname{Ind}(\mathcal{D})).$

For this, one needs to embed
$\operatorname{Fun}^{\mathcal{A}^{\mathrm{op}}}(\mathcal{C}^{\mathrm{op}},\operatorname{Ind}(\mathcal{D}))$
as a reflective subcategory (and accessible localisation) of the
$\infty$-category
$\operatorname{Fun}^{\mathcal{A}^{\mathrm{op}}-\mathrm{oplax}}(\mathcal{C}^{\mathrm{op}},\operatorname{Ind}(\mathcal{D}))$
of oplax $\mathcal{A}^{\mathrm{op}}$-linear functors
$\mathcal{C}^{\mathrm{op}}\to\operatorname{Ind}(\mathcal{D})$. Then, to every
$(c,d)\in\mathcal{C}\times\mathcal{D}$ we may associate the oplax