# Towards $\mathbb{A}^{1}$-homotopy theory of rigid analytic spaces

Christian Dahlhausen and Can Yaylali

###### Abstract

To any rigid analytic space (in the sense of Fujiwara-Kato) we assign an
$\mathbb{A}^{1}$-invariant rigid analytic homotopy category with coefficients
in any presentable category. We show some functorial properties of this
assignment as a functor on the category of rigid analytic spaces. Moreover, we
show that there exists a full six functor formalism for the precomposition
with the analytification functor by evoking Ayoub’s thesis. As an application,
we identify connective analytic K-theory in the unstable homotopy category
with both $\mathbb{Z}\times\textup{BGL}$ and the analytification of connective
algebraic K-theory. As a consequence, we get a representability statement for
coefficients in light condensed spectra.

###### Contents

  1. 1 Introduction
  2. 2 Background in rigid analytic geometry
  3. 3 Unstable $\mathbb{A}^{1}$-homotopy theory in rigid geometry
  4. 4 Stable $\mathbb{A}^{1}$-homotopy theory in rigid geometry
  5. 5 Applications to the K-theory of rigid spaces
  6. A Condensed objects
  7. B Enriched Yoneda

##  1 Introduction

The $\mathbb{A}^{1}$-homotopy category of schemes was constructed and studied
by Morel-Voevodsky [MV99]. In modern language, their homotopy category
$\textup{H}(S)$ is the category of $\mathbb{A}^{1}$-invariant Nisnevich
sheaves on smooth schemes over a base scheme $S$ with values in spaces. Under
this construction they show, under some regularity assumptions, that
connective algebraic K-theory can be represented as
$\mathbb{Z}\times\operatorname{BGL}\in\textup{H}(S)$. Moreover, after
$\otimes$-inverting $\mathbb{P}^{1}$, we obtain the stable homotopy category
$\textup{SH}(S)$. In his thesis, Ayoub showed that the assignment
$S\mapsto\textup{SH}(S)$ admits a full six functor formalism [Ayo07]. Ayoub
deduces this from properties of the functor $\textup{SH}(-)$ such as
$\mathbb{A}^{1}$-invariance, $\mathbb{P}^{1}$-stability, and the existence of
the localisation sequence. In this article, we are going to extend these
results to the analytic setting, using the rigid analytic line
$\mathbb{A}^{1}$ as an interval.

Motives on rigid analytic varieties have been constructed and studied by
Ayoub[Ayo15]. The idea is to follow Morel-Voevodsky’s approach in constructing
the stable homotopy category $\textup{RigSH}^{\mathbb{B}^{1}}(X)$ for a rigid
analytic variety $X$. Ayoub defines the category of rigid motives by Nisnevich
localisation, contracting the closed unit disc $\mathbb{B}^{1}$, and then
$\otimes$-inverting the unit disc without the origin. In op. cit. it is
shown that this definition yields a coefficient system in the sense that any
morphism $f$ of rigid analytic varieties induces a colimit preserving functor
$f^{*}$ on $\textup{RigSH}^{\mathbb{B}^{1}}(-)$ with the following properties.

  1. (PF)

If $f$ is smooth, then $f$ admits a left adjoint satisfying base change and
projection formula.

  2. (Loc)

Any closed immersion $i\colon Z\hookrightarrow X$ of rigid analytic varieties
with open complement $j\colon U\hookrightarrow X$ induces a fibre sequence
$j_{\\#}j^{*}\overset{}{\rightarrow}\operatorname{id}\overset{}{\rightarrow}i_{*}i^{*}$
of endofunctors on $\textup{RigSH}^{\mathbb{B}^{1}}(X)$.

While the proof of (PF) is a formality, the proof of (Loc) needs more care and
follows the idea of Voevodsky. These two properties and the definition
immediately imply that on algebraic maps $X\rightarrow Y$ of rigid analytic
varieties there exists a six functor formalism [Ayo07, Scholie 1.4.2]. More
recently, in joint work Ayoub-Gallauer-Vezzani generalise this result to
arbitrary finite type maps of rigid analytic varieties [AGV22].

A drawback of this definition of $\textup{RigSH}^{\mathbb{B}^{1}}$ is that not
all cohomology theories on rigid analytic varieties satisfy
$\mathbb{B}^{1}$-homotopy invariance. For instance, continuous K-theory
$\textup{K}^{\textup{cont}}$ defined by Morrow [Mor16] and analytic K-theory
defined by Kerz-Saito-Tamme [KST19a] are not $\mathbb{B}^{1}$-invariant
(Remark 3.5). Hence, these theories cannot be represented in RigDA. On the
other hand, letting $\mathbb{A}^{1}$ be the rigid affine line, analytic
K-theory is $\mathbb{A}^{1}$-invariant [KST23, Cor. 2.7] and – assuming
resolution of singularities – continuous K-theory is
$\mathbb{A}^{1}$-invariant for regular rigid analytic varieties [KST19a, Prop.
5.14]. This suggests that one should define the category of motives on a rigid
analytic variety $X$ as

$\textup{RigSH}(X)\coloneqq\textup{Sh}_{\textup{Nis}}^{\mathbb{A}^{1}}(\textup{SmRig}_{X})[\textup{Th}(\mathbb{B}^{1})^{-1}],$

where $\textup{Th}(\mathbb{B}^{1})$ denotes the Thom motive of the rigid
closed unit disc. This is equivalent to $\otimes$-inverting $\mathbb{P}^{1}$,
see section 4.3.

Eventually, we apply our theory to K-theory of rigid analytic spaces, namely
analytic K-theory as defined and studied by Kerz-Saito-Tamme [KST19a, KST23].
Since this K-theory takes values in pro-categories, it makes sense to consider
motives with coefficients in an arbitrary symmetric monoidal stable
presentable category ${\mathcal{V}}_{\textup{st}}$. Then we can apply the
theory to the categories of pro-spectra and, since pro-categories do not
behave that nicely, (light or $\kappa$-small) condensed spectra.

Let us fix a symmetric monoidal presentable category ${\mathcal{V}}$ and its
stabilisation ${\mathcal{V}}_{\textup{st}}$. The motivating examples include
the category $\textup{Cond}_{\kappa}(\textup{Spc})$ of $\kappa$-small
condensed spaces and the category $\textup{Cond}_{\kappa}(\operatorname{Sp})$
of $\kappa$-small condensed spectra, for a fixed uncountable strong limit
cardinal cardinal $\kappa$. The category $\textup{RigSH}(-,{\mathcal{V}})$
keeps the property (PF) above and perhaps more surprisingly also (Loc), even
without stabilising.

######  Theorem 1 (3.18, 3.29).

The assignment
$X\mapsto\textup{Sh}^{\mathbb{A}^{1}}_{\textup{Nis}}(X,{\mathcal{V}}),\
f\mapsto f^{*}$, from rigid analytic varieties to symmetric monoidal
presentable ${\mathcal{V}}$-linear categories satisfies (PF) and (Loc).

As an immediate consequence, we get a partial six functor formalism on RigSH
over a nonarchimedean field $k$.

######  Corollary 2 (4.14).

The assignment
$X\mapsto\textup{RigSH}(X^{\textup{an}},{\mathcal{V}}_{\textup{st}}),\
f\mapsto f^{*}$, from separated finite type $k$-schemes to symmetric monoidal
stable presentable ${\mathcal{V}}_{\textup{st}}$-linear categories admits a
six functor formalism satisfying base change, purity, and projection formula.

An extension of the six functor formalism to non-algebraic maps of rigid
analytic varieties is not clear. In the classical rigid motivic theory of
Ayoub, the rigid affine line and the closed unit disc are
$\mathbb{B}^{1}$-homotopic [Ayo15, Prop. 1.3.4]. This will not be the case,
when working with $\mathbb{A}^{1}$-homotopy instead. This shows that we need
more careful treatment to obtain a full six functor formalism for rigid spaces
and cannot rely purely on the results of [AGV22]. Nevertheless, in future work
we aim to establish a such full six functor formalism.

Let us pass to an application of our theory. By design, analytic K-theory of
rigid spaces is an $\mathbb{A}^{1}$-invariant Nisnevich sheaf, hence it yields
an object of the unstable $\mathbb{A}^{1}$-homotopy category. Assuming
resolution of singularities, we get the following identification (which we
state in the introduction in less generality than in the main text).

######  Theorem 3 (5.7).

Let $k$ be a discretely valued nonarchimedean field and assume
$(\dagger)_{k}$, see section 5.1. Then there is a canonical equivalence

$\Omega^{\infty}\textup{K}^{\textup{an}}_{\geq
0}\simeq\mathbb{Z}\times\operatorname{BGL}$

in the category $\textup{RigH}(k)$. In particular, for every rigid space $X$
there is a functorial equivalence

$\Omega^{\infty}\textup{K}^{\textup{an}}_{\geq
0}(X)\simeq\underline{\operatorname{Hom}}_{\textup{Cond}^{\omega}(\textup{Spc})}(\textup{L}_{\textup{mot}}X,\textup{L}_{\textup{mot}}(\mathbb{Z}\times\operatorname{BGL}))$

in the category $\textup{Cond}^{\omega}(\textup{Spc})$; here the right-hand
side denotes the enriched Hom-space as a left
$\textup{Cond}^{\omega}(\textup{Spc})$-module, see Appendix B.

This is analogous to the identification $\textup{K}_{\geq
0}\simeq\mathbb{Z}\times\operatorname{BGL}$ of algebraic K-theory in the
Morel-Voevodsky category $\textup{H}(k)$ [MV99, Prop. 3.10]. In particular,
assuming $(\dagger)_{k}$, the analytification functor
$\textup{H}(k)\overset{}{\rightarrow}\textup{RigH}(k)$ maps connective
algebraic K-theory to connective analytic K-theory. See also [Ayo15, p. ix].

We expect Theorem 3 to be helpful in the study of analytic K-theory whose
underlying spectrum is related to Efimov’s K-theory of dualisable categories
and Clausen-Scholze’s nuclear modules on analytic spaces, see Remark 5.4.

Furthermore, we prove an $\mathbb{A}^{1}$-analytic Bass Fundamental Theorem
for analytic K-theory. Its proof relies on the analogous result for
$\mathbb{B}^{1}$ by Kerz-Saito-Tamme [KST23, Cor. 2.6].

######  Theorem 4 (5.16).

For every $n\in\mathbb{Z}$ and every $X\in\textup{Rig}_{k}$ there is an exact
sequence

$0\overset{}{\longrightarrow}\textup{K}^{\textup{an}}_{n}(X)\overset{}{\longrightarrow}\textup{K}^{\textup{an}}_{n}(X\times\mathbb{A}^{1})\oplus\textup{K}^{\textup{an}}_{n}(X\times\mathbb{A}^{-1})\overset{\pm}{\longrightarrow}\textup{K}^{\textup{an}}(X\times\operatorname{\mathbb{G}_{m}})\overset{\partial}{\longrightarrow}\textup{K}^{\textup{an}}_{n-1}(X)\overset{}{\rightarrow}0$

of pro-abelian groups where the map $\partial$ has a split.

Using the Bass Fundamental Theorem together with Theorem 3 we will see that
the analytification functor maps the $\mathbb{P}^{1}$-spectrum representing
algebraic K-theory to a $\mathbb{P}^{1}$-spectrum representing analytic
K-theory under assumption $(\dagger)_{k}$.

######  Corollary 5 (5.17).

There exists a $\mathbb{P}^{1}$-action on
$\mathbb{Z}\times\operatorname{BGL}\in\textup{RigH}(k)$ inducing a
$\mathbb{P}^{1}$-spectrum $\textup{KGL}^{\textup{an}}\in\textup{RigSH}(k)$.
Under assumption $(\dagger)_{k}$, the $\mathbb{P}^{1}$-spectrum
$\textup{KGL}^{\textup{an}}$ represents (non-connective) analytic $K$-theory.

Weibel defined homotopy K-theory as the $\mathbb{A}^{1}$-localisation of
algebraic K-theory. In this spirit, we define _continuous homotopy K-theory_
$\textup{KH}^{\textup{cont}}$ (Defintion 5.19) and identify it with analytic
K-theory (Theorem 5.18) by evoking a result of Kerz-Saito-Tamme.

Assumptions and notations. Throughout, we fix some inaccessible regular
cardinal ${\bar{\kappa}}$. By small, we will mean ${\bar{\kappa}}$-small. A
_category_ always means an $\infty$-category, i.e. an $(\infty,1)$-category in
the sense of [Lur09], and we identify 1-categories with their image under the
nerve functor. Let us fix some notation for the categories appearing in this
article.

  1. $\bullet$

Spc, the category of small spaces, or equivalently small $\infty$-groupoids or
small $\infty$-sets, or small anima (depending on your taste).

  2. $\bullet$

$\operatorname{Sp}$, the category of spectra.

  3. $\bullet$

$\textup{Pr}^{L}$, the category of presentable categories with colimit
preserving functors.

  4. $\bullet$

$\textup{Pr}^{L,\otimes}$, the category $\textup{Pr}^{L}$ endowed with the
Lurie tensor product [Lur17, §4.8.1].

The term _essentially unique_ is short hand for the expression _unique up to
contractible choice_.

Acknowledgements. We heartily thank Joseph Ayoub for guiding us through his
work on six functor formalisms [Ayo07] and rigid analytic motives [Ayo15] and
Martin Gallauer for insightful explanations about his work [AGV22] and Brad
Drew’s work on coefficient systems [Dre18]. We thank Peter Scholze for
communicating to us the proof of Theorem A.12 and Vlaidimir Hinich for
answering our questions about the enriched Yoneda lemma. Furthermore, we thank
Ryomei Iwasa and Florian Strunk for helpful discussions around this paper’s
content. The first named author thanks Georg Tamme for continuous discussions
about K-theory of rigid spaces.

Both authors were supported by the Deutsche Forschungsgemeinschaft
(DFG)through the Collaborative Research Centre TRR 326 Geometry and Arithmetic
of Uniformized Structures, project number 444845124. The second named author
was furthermore supported through the Walter Benjamin Programme (DFG, project
number 524431573) during his stay at the Laboratoire de Mathématiques d’Orsay.
He wants to thank Vincent Pilloni for hosting him and all the other people
giving a friendly and mathematically interesting environment.

##  2 Background in rigid analytic geometry

We place ourselves in the same setting as Ayoub-Gallauer-Vezzani [AGV22, §1]
and use the notion of _rigid spaces_ 111For the sake of brevity, we mostly use
the terminology ‘rigid space’ instead of the more accurate ‘rigid analytic
space’. as defined by Fujiwara-Kato [FK18] which goes back to Raynaud [Ray74]
and was also developed by Abbes [Abb10].

Rigid spaces. Let FSch be the category of formal schemes and let
$\textup{FSch}^{\textup{qcqs}}$ be its full subcategory of quasi-compact and
quasi-separated formal schemes. The category$\textup{Rig}^{\textup{qcqs}}$ of
quasi-compact and quasi-separated _rigid spaces_ is defined to be the
1-categorical localisation of $\textup{FSch}^{\textup{qcqs}}$ by the set of
admissible formal blow-ups. The category Rig is defined by gluing quasi-
compact and quasi-separated rigid spaces along open immersions, see [FK18, ch.
II, §2.2 (c)]. For a formal scheme ${\mathcal{X}}$ denote by
${\mathcal{X}}^{\textup{rig}}$ its associated rigid space.

Relation to adic spaces. There exists a fully faithful functor
$\textup{Adic}^{\textup{unif}}\overset{}{\hookrightarrow}\textup{Rig}$ from
the category of uniform adic spaces into the category of rigid spaces sending
$\operatorname{Spa}(R,R^{+})$ to $\operatorname{Spf}(R^{+})^{\textup{rig}}$
which is compatible with gluing along open immersions [AGV22, §1.2]. We call a
rigid space _adic_ if it lies in the essential image of this functor.

Morphisms locally of finite type. A morphism $f\colon
Y\overset{}{\rightarrow}X$ of rigid spaces is said to be _locally of finite
type_ if there exists an open cover $(X_{i})_{i}$ of $X$ such that the
restricted morphisms $f^{-1}(X_{i})\overset{}{\rightarrow}X$ admit formal
models which are locally of finite type, i.e. locally given by algebras of
topologically finite type, see [FK18, ch. II, sec. 2.3]. Note that an algebra
of topologically finite type over a uniform Huber ring, is again a uniform
Huber ring. It follows that, if $X$ is adic, then $Y$ is adic as well.

Smooth morphisms. A morphism $f\colon Y\overset{}{\rightarrow}X$ of rigid
spaces is called _smooth_ if, locally on $Y$ and $X$, there exists a formal
formal model ${\mathcal{Y}}\overset{}{\rightarrow}{\mathcal{X}}$ which,
locally on rings, is given as a composition $A\overset{}{\rightarrow}A\langle
t_{1},\ldots,t_{n}\rangle\overset{\phi}{\rightarrow}B$ where $\phi$ is rig-
étale, see [AGV22, 1.3.3., 1.3.13]. It holds true that a morphism of adic
rigid spaces is smooth in this sense if and only if it smooth in the sense of
Huber, see [AGV22, 1.3.14] and the references there.

The Nisnevich topology. A family
$({\mathcal{Y}}_{i}\overset{}{\rightarrow}{\mathcal{X}})_{i}$ in FSch is a
_Nisnevich cover_ if the induced family
$({\mathcal{Y}}_{i,\sigma}\overset{}{\rightarrow}{\mathcal{X}})_{i}$ on
special fibres is a Nisnevich cover of schemes. A family
$(Y_{i}\overset{}{\rightarrow}X)_{i}$ in Rig is a _Nisnevich cover_ if locally
on $X$ the family admits a refinement that has a formal model which is a
Nisnevich cover of formal schemes.

The rigid affine line. Let us quickly recall some background concerning the
rigid affine line, for more details consider the lecture notes of Scholze-
Weinstein [SW20, Lecture 4] or Hübner [Hü24, Ex. 1.4.3, Ex. 1.10.4]. Let
$(R,R^{+})$ be a uniform Huber pair and set $S=\operatorname{Spa}(R,R^{+})$.
Throughout this article, we work with the rigid affine line, seen as a rigid
space222We will mostly work in the setting of rigid spaces and therefore omit
a superscript indicating that $\mathbb{A}^{1}$ is not the schematic affine
line.

$\mathbb{A}^{1}_{S}\coloneqq\operatorname{Spa}(\mathbb{Z}[T],\mathbb{Z})\times
S$

using the fully faithful embedding
$\textup{Adic}^{\textup{unif}}\overset{}{\hookrightarrow}\textup{Rig}$ from
above.333The ring $\mathbb{Z}[T]$ is equipped with the discrete topology. For
the existence of fibre products we refer to [FK18, ch. II, sec. 2.4]. This
description of the affine line will make it easy to construct explicit
homotopies. All of the computations (e.g. descriptions of homotopies) can be
reduced to computations on $\operatorname{Spa}(\mathbb{Z}[T],\mathbb{Z})$.

Analogously, we define the closed rigid unit disc as

$\mathbb{B}^{1}_{S}\coloneqq\operatorname{Spa}(\mathbb{Z}[T],\mathbb{Z}[T])\times
S=\operatorname{Spa}(R\langle T\rangle,R^{+}\langle T\rangle)$

where $R\langle T\rangle$ denotes the ring of those power series in $T$ whose
coefficients tend to $0$.

Now assume that $R$ is a Tate ring and let $\pi$ be a pseudo-uniformiser. Then
we can describe the affine line as the union of closed units discs with larger
and larger radius

$\mathbb{A}^{1}_{S}=\bigcup_{n\geq
0}\operatorname{Spa}(R\langle\pi^{n}T\rangle,R^{+}\langle\pi^{n}T\rangle).$
(2.0.1)

##  3 Unstable $\mathbb{A}^{1}$-homotopy theory in rigid geometry

In this section, we want to analyse $\mathbb{A}^{1}$-invariant Nisnevich
sheaves on smooth rigid spaces, also called unstable rigid homotopy category.
The upshot is that this category will define a pullback formalism in the sense
that for any smooth morphism of rigid spaces $f$ the pullback $f^{*}$ admits a
left adjoint satisfying base change and projection formula. But we will go
even a bit further and prove that we even have a localisation sequence in this
more general setting.

Notation. Throughout this section, let $(R,R^{+})$ be a uniform Huber pair
with associated rigid space $S=\operatorname{Spa}(R,R^{+})$ and let
${\mathcal{D}}$ be a presentable category of coefficients.

###  3.1 Homotopy invariant presheaves on rigid spaces

As mentioned earlier, there are several reasonable choices of an interval
object to do homotopy theory with rigid spaces such as the closed unit disc
$\mathbb{B}^{1}$ and the rigid affine line $\mathbb{A}^{1}$. If the target
category is a pro-category, one can also speak about
pro-$\mathbb{B}^{1}$-invariance (Lemma 5.3).

######  Definition 3.1.

We say that a presheaf
$F\in\operatorname{Fun}(\textup{Rig}_{S}^{\operatorname{op}},{\mathcal{D}})$
is …

  1. 1.

$\mathbb{B}^{1}$_-invariant_ if for every $X\in\textup{Rig}_{S}$ the canonical
map $F(X)\overset{}{\rightarrow}F(X\times\mathbb{B}^{1})$ is an equivalence in
${\mathcal{D}}$.

  2. 2.

$\mathbb{A}^{1}$_-invariant_ if for every $X\in\textup{Rig}_{S}$ the canonical
map $F(X)\overset{}{\rightarrow}F(X\times\mathbb{A}^{1})$ is an equivalence in
${\mathcal{D}}$.

######  Definition 3.2.

Let ${\mathcal{C}}$ be a site. A presheaf
$F\in\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{D}})$ is
said to be a _sheaf_ if for every object $X$ of ${\mathcal{C}}$ and every
covering sieve $U\overset{}{\hookrightarrow}X$ the canonical map
$F(X)\overset{}{\rightarrow}F(U)$ is an equivalence in ${\mathcal{D}}$; here
$F(U)$ is defined via the equivalence

$\operatorname{Fun}^{\mathrm{lim}}(\textup{PSh}({\mathcal{C}})^{\operatorname{op}},{\mathcal{D}})\overset{}{\longrightarrow}\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{D}}).$

Thus if we write $U\simeq\mathop{\textup{colim}}_{i}U_{i}$ as a colimit of
representables, then $F(U)\simeq\lim_{i}F(U_{i})$.

######  Lemma 3.3.

Every $\mathbb{B}^{1}$-invariant sheaf on $\textup{Rig}_{S}$ is
$\mathbb{A}^{1}$-invariant.

###### Proof.

We can identify $\mathbb{A}^{1}$ with the colimit
$\mathop{\textup{colim}}_{t\mapsto\pi t}\mathbb{B}^{1}$ in the category
$\textup{Rig}_{S}$ where $t$ is a parameter for
$\mathbb{B}^{1}=\operatorname{Spa}(k\langle t\rangle),k^{\circ}\langle
t\rangle)$. Since $F$ is a sheaf (and hence preserves this colimit) and since
a colimit of open immersions commutes with fibre products, we have for every
sheaf $F$ equivalences

$\displaystyle F(X\times\mathbb{A}^{1})\simeq
F(X\times\mathop{\textup{colim}}_{t\mapsto\pi t}\mathbb{B}^{1})\simeq
F(\mathop{\textup{colim}}_{t\mapsto\pi
t}(X\times\mathbb{B}^{1}))\simeq\lim_{t\mapsto\pi t}F(X\times\mathbb{B}^{1}).$

Hence the map $F(X)\overset{}{\rightarrow}F(X\times\mathbb{A}^{1})$ is an
equivalence if the map $F(X)\overset{}{\rightarrow}F(X\times\mathbb{B}^{1})$
is an equivalence. ∎

######  Definition 3.4.

Denote by $\textup{PSh}^{\mathbb{A}^{1}}(\textup{Rig}_{k},{\mathcal{D}})$ the
full subcategory of the category ofpresheaves
$\textup{PSh}(\textup{Rig}_{k},{\mathcal{D}})$ spanned by
$\mathbb{A}^{1}$-invariant presheaves.

######  Remark 3.5.

There are $\mathbb{A}^{1}$-invariant sheaves which are not
$\mathbb{B}^{1}$-invariant. For instance, analytic K-theory
$\textup{K}^{\textup{an}}\colon\textup{Rig}_{k}^{\operatorname{op}}\overset{}{\rightarrow}\textup{Pro}(\operatorname{Sp}^{+})$
is a sheaf [KST19a, 6.15] [KST23, 4.3] which is $\mathbb{A}^{1}$-invariant
[KST23, 4.4]. However, assuming resolutions of singularities, it is not
$\mathbb{B}^{1}$-invariant as it agrees with continuous K-theory [KST19a,
6.19] which on $\pi_{0}$ is equivalent to $\textup{K}_{0}$ [KST19a, 5.10]
which has the non-$\mathbb{B}^{1}$-invariant Picard group as a direct summand
[KST19b, Example 2]. For more details about continuous K-theory and analytic
K-theory we refer to subsection 5.1.

######  Lemma 3.6.

For every cocomplete category ${\mathcal{D}}$ there exists a localisation
functor

$\textup{L}_{\mathbb{A}^{1}}\colon\textup{PSh}(\textup{Rig}_{S},{\mathcal{D}})\overset{}{\longrightarrow}\textup{PSh}^{\mathbb{A}^{1}}(\textup{Rig}_{S},{\mathcal{D}})$

which is left-adjoint to the inclusion. It has the following explicit
description: for every $F\in\textup{PSh}(\textup{Rig}_{k},{\mathcal{C}})$ and
every $X\in\textup{Rig}_{S}$ there is a functorial equivalence

$(\textup{L}_{\mathbb{A}}^{1}F)(X)\simeq\mathop{\textup{colim}}_{\Delta^{\operatorname{op}}}F(X\times\Delta^{\mathrm{an},\bullet})$

where $\Delta^{\mathrm{an},\bullet}$ is the analytification of its algebraic
analogue.

###### Proof.

For the proof, denote by $\textup{H}F$ the presheaf whose values are the
right-hand side of the desired identification. This yields a natural
transformation
$\alpha\colon\operatorname{id}\overset{}{\rightarrow}\textup{H}$ of presheaves
with values in ${\mathcal{D}}$. Assume that the following conditions hold:

  1. (i)

For every presheaf $F$, the presheaf $\textup{H}F$ is
$\mathbb{A}^{1}$-invariant.

  2. (ii)

For every $\mathbb{A}^{1}$-invariant presheaf $F$, the map $\alpha_{F}\colon
F\overset{}{\rightarrow}\textup{H}F$ is an equivalence.

These conditions imply for every presheaf $F$ that both maps
$\textup{H}(\alpha_{F}),\alpha_{\textup{H}F}\colon\textup{H}F\overset{}{\rightarrow}\textup{H}\textup{H}F$
are equivalences. Then the desired claim follows from [Lur09, 5.2.7.4]. Hence
it suffices to check conditions (i) and (ii).

(i) Let $X\in\textup{Rig}_{k}$ and let $p\colon
X\times\mathbb{A}^{1}\overset{}{\rightarrow}X$ the projection and
$\sigma\colon X\overset{}{\rightarrow}X\times\mathbb{A}^{1}$ the zero section.
As $p\circ\sigma=\operatorname{id}$ it remains to show that the induced map
$p^{*}\circ\sigma^{*}$ on $(\textup{H}F)(X\times\mathbb{A}^{1})$ is equivalent
to the identity. For this purpose, we claim that the maps

$p^{*}\circ\sigma^{*},\operatorname{id}\colon
X\times\mathbb{A}^{1}\times\Delta^{\mathrm{an},\bullet}\rightrightarrows
X\times\mathbb{A}^{1}\times\Delta^{\mathrm{an},\bullet}$

are simplicial homotopic (see Remark 3.7 below). In degree $n$, a homotopy
$h_{n}$ on $X\times\mathbb{A}^{1}\times\Delta^{\mathrm{an},n}$ is given by
$h_{n}(f)(t)=(\sum_{j\in f^{-1}(1)}t_{j})\cdot t$ for a map
$f\colon[n]\overset{}{\rightarrow}{}[1]$. We have $i_{0}^{*}h(t)=0$ and
$i_{1}^{*}h(t)=t$.

(ii) We follow the classical reasoning [Wei13, ch. IV, Lemma 11.5.1]. Let $F$
be an $\mathbb{A}^{1}$-invariant presheaf. Then for every $X\in\textup{Rig}$
there is a morphism
$X\times\Delta^{\textup{an},\bullet}\overset{}{\rightarrow}X$ of simplicial
rigid spaces which is induces by the projections. For every
$n\in\Delta^{\operatorname{op}}$ the induced map
$F(X)\overset{}{\rightarrow}F(X\times\Delta^{\textup{an},n})$ is an
equivalence. Hence the map

$F(X)=\mathop{\textup{colim}}_{\Delta^{\operatorname{op}}}F(X)\overset{}{\longrightarrow}\mathop{\textup{colim}}_{\Delta^{\operatorname{op}}}F(X\times\Delta^{\textup{an},n})=\textup{H}F(X)$

is an equivalence.

∎

######  Remark 3.7.

Let ${\mathcal{C}}$ be an category and $\mathrm{s}{\mathcal{C}}$ the
simplicial objects in ${\mathcal{C}}$. We have a diagram

$\Delta\overset{i_{0}}{\underset{i_{1}}{\rightrightarrows}}\Delta_{/[1]}\overset{\delta}{\rightarrow}\Delta$

where
$i_{k}([n])=([n]\overset{}{\rightarrow}{}[0]\overset{k}{\rightarrow}[1])$ and
$\delta([n]\overset{}{\rightarrow}{}[1])=[n]$. A _simplicial homotopy_ between
two maps $f,g\colon X_{\bullet}\overset{}{\rightarrow}Y_{\bullet}$ in
$\mathrm{s}{\mathcal{C}}$ is a map
$h\colon\delta^{*}(X_{\bullet})\overset{}{\rightarrow}\delta^{*}(Y_{\bullet})$
in $\operatorname{Fun}((\Delta_{/[1]})^{\operatorname{op}},{\mathcal{C}})$
such that $i_{0}^{*}(h)=f$ and $i_{1}^{*}(h)=g$. In this case, if
${\mathcal{C}}$ is cocomplete, the induces maps
$|f|,|g|\colon|X_{\bullet}|\overset{}{\rightarrow}|Y_{\bullet}|$ on geometric
realisations are homotopic in ${\mathcal{C}}$. This follows since the map
$\mathop{\textup{colim}}_{(\Delta_{/[1]})^{\operatorname{op}}}\delta^{*}(X_{\bullet})\overset{}{\rightarrow}|X_{\bullet}|=\mathop{\textup{colim}}_{\Delta^{\operatorname{op}}}X_{\bullet}$
is an equivalence.

###  3.2 The unstable analytic motivic category

######  Definition 3.8.

Let $S$ be a rigid space. The _unstable rigid motivic homotopy category with
coefficients in ${\mathcal{D}}$_ is the reflective subcategory

$\textup{RigH}(S,{\mathcal{D}}):=\textup{Sh}^{\mathbb{A}^{1}}_{\textup{Nis}}(\textup{SmRig}_{S},{\mathcal{D}})$

of $\textup{Sh}_{\textup{Nis}}(\textup{SmRig}_{S},{\mathcal{D}})$ which is
spanned by $\mathbb{A}^{1}$-invariant sheaves. We have the following diagram
of localisation functors:

${\textup{Sh}_{\textup{Nis}}(\textup{SmRig}_{S},{\mathcal{D}})}$${\textup{PSh}(\textup{SmRig}_{S},{\mathcal{D}})}$${\textup{RigH}(S,{\mathcal{D}})}$${\textup{PSh}^{\mathbb{A}^{1}}(\textup{SmRig}_{S},{\mathcal{D}})}$$\scriptstyle{\textup{L}_{\mathbb{A}^{1}}^{\\#}}$$\scriptstyle{\textup{L}_{\textup{Nis}}}$$\scriptstyle{\textup{L}_{\mathbb{A}^{1}}}$$\scriptstyle{\textup{L}_{\textup{mot}}}$$\scriptstyle{\textup{L}_{\textup{Nis}}^{\\#}}$

We note that the motivic localisation $\textup{L}_{\textup{mot}}$ is given by
the colimit of functors

$\textup{L}_{\textup{mot}}\simeq\mathop{\textup{colim}}(\textup{L}_{\textup{Nis}}\textup{L}_{\mathbb{A}^{1}}\overset{}{\rightarrow}\textup{L}_{\textup{Nis}}\textup{L}_{\mathbb{A}^{1}}\textup{L}_{\textup{Nis}}\textup{L}_{\mathbb{A}^{1}}\overset{}{\rightarrow}\ldots).$

###  3.3 Analytification for motivic spaces

Let $k$ be a nonarchimedean field and let $S$ be a $k$-scheme of finite type.
In this section, we construct an analytification functor from the algebraic
stable homotopy category $\textup{H}(S,{\mathcal{D}})$ to the rigid analytic
$\mathbb{A}^{1}$-homotopy category
$\textup{RigH}(S^{\textup{an}},{\mathcal{E}})$ depending on a _change of
coefficients functor_ ${\mathcal{D}}\overset{}{\rightarrow}{\mathcal{E}}$
between symmetric monoidal presentable categories. Later on, we apply this
with the functors
$\textup{Spc}\overset{}{\hookrightarrow}\textup{Cond}(\textup{Spc})$ and
$\operatorname{Sp}\overset{}{\hookrightarrow}\textup{Cond}(\operatorname{Sp})$.

Notation. In this section, let $R_{0}$ be an adic ring with ideal of
definition $I$. We set $S:=\operatorname{Spec}(R_{0})\setminus\textup{V}(I)$
and $S^{\textup{an}}:=\operatorname{Spf}(R_{0})^{\textup{rig}}$. In the
special case where $R_{0}$ is a ring of definition of a Huber pair
$(R,R^{+})$, we have $S=\operatorname{Spec}(R)$ and
$S^{\textup{an}}=\operatorname{Spa}(R)=\operatorname{Spa}(R,R^{+})$. For any
$R$-algebra $A$ we set $A^{+}$ as the integral closure of (the image of)
$R^{+}$ in $A$ and write $\operatorname{Spa}(A):=\operatorname{Spa}(A,A^{+})$
for simplicity.

######  Lemma 3.9.

There exists an analytification functor
$(-)^{\textup{an}}\colon\textup{Sch}_{S}^{\textup{lft}}\overset{}{\rightarrow}\textup{Rig}_{S^{\textup{an}}}^{\textup{lft}}$
sending the algebraic affine line $\mathbb{A}^{1}_{S}$ to the rigid analytic
affine line $\mathbb{A}^{1}_{S^{\textup{an}}}$. In particular, for every
$X\in\textup{Sch}^{\textup{lft}}_{S}$ we get an analytification functor
$(-)^{\textup{an}}\colon\textup{Sch}_{X}^{\textup{lft}}\overset{}{\rightarrow}\textup{Rig}_{X^{\textup{an}}}^{\textup{lft}}$.
By restriction, we get a functor
$(-)^{\textup{an}}\colon\textup{Sm}_{X}\overset{}{\rightarrow}\textup{SmRig}_{X^{\textup{an}}}$.

###### Proof.

For general rigid spaces the analytification functor has been constructed
byFujiwara-Kato [FK18, ch. 2, sec. 9.1.]. We give a sketch of the construction
in the adic setting; for a full proof we refer to Hübner’s lecture notes
[Hü24, §1.10].

So, assuming that $S=\operatorname{Spec}(R)$ for a Huber pair $(R,R^{+})$, the
analytification functor is locally given by the assignment

$R[t_{1},\ldots,t_{n}]/(f_{1},\ldots,f_{k})\mapsto\bigcup_{r\geq
1}\operatorname{Spa}\bigl{(}R\langle
t_{1},\ldots,t_{n}\rangle_{r}/(f_{1},\ldots,f_{k})\bigr{)}$

where $R\langle t_{1},\ldots,t_{n}\rangle_{r}$ denote the subring of
$R\llbracket t_{1},\ldots,t_{n}\rrbracket$ consisting of power
series$\sum_{i\in\mathbb{N}^{n}}a_{i}t^{i}$ such that $a_{i}r^{i}$ converges
to zero as $|i|\overset{}{\rightarrow}\infty$. This assignment is compatible
with gluing and extends to the desired functor.

Alternatively, one can associate with any $X\in\textup{Sch}_{S}^{\textup{ft}}$
a natural presheaf $h_{\textup{an}}(X)$ on $\textup{Rig}_{k}$ which turns out
to be representable by a rigid space $X^{\textup{an}}$ yielding the
analytification functor, see [Ayo15, §1.1.3].

∎

######  Lemma 3.10.

The analytification functor
$(-)^{\textup{an}}\colon\textup{Sch}_{S}^{\textup{lft}}\overset{}{\longrightarrow}\textup{Rig}_{S^{\textup{an}}}^{\textup{lft}}$
extends to an essentially unique colimit-preserving functor

$\textup{PSh}(\textup{Sch}_{S}^{\textup{lft}})\overset{}{\longrightarrow}\textup{PSh}(\textup{Rig}_{S^{\textup{an}}}^{\textup{lft}})$

compatible with the respective Yoneda embeddings. Furthermore, this functor is
monoidal with respect to the cartesian monoidal structure. The analogous
statement for the functor
$(-)^{\textup{an}}\colon\textup{Sm}_{S}\overset{}{\rightarrow}\textup{SmRig}_{S^{\textup{an}}}$
holds true, too.

###### Proof.

Restricting along the Yoneda embedding
$\textup{Sch}_{S}^{\textup{ft}}\overset{}{\hookrightarrow}\operatorname{Fun}(\textup{Sch}_{S}^{\textup{ft},\operatorname{op}},\textup{Spc})$
induces for any category ${\mathcal{C}}$ which admits small colimits an
equivalence [Lur09, Prop. 5.1.5.6]

$\operatorname{Fun}^{\mathrm{L}}\bigl{(}\textup{PSh}(\textup{Sch}_{S}^{\textup{ft}}),{\mathcal{C}}\bigr{)}\overset{\simeq}{\longrightarrow}\operatorname{Fun}(\textup{Sch}_{S}^{\textup{ft}},{\mathcal{C}})$

where $\operatorname{Fun}^{\mathrm{L}}$ denotes the full subcategory of all
functors that preserve small colimits. Hence the composition
$\textup{Sch}_{S}^{\textup{lft}}\overset{(-)\textup{an}}{\longrightarrow}\textup{Rig}_{S^{\textup{an}}}^{\textup{lft}}\overset{}{\hookrightarrow}\textup{PSh}(\textup{Rig}_{S^{\textup{an}}}^{\textup{lft}})$
extends to an essentially unique functor

$\textup{PSh}(\textup{Sch}_{S}^{\textup{lft}})\overset{}{\longrightarrow}\textup{PSh}(\textup{Rig}_{S^{\textup{an}}}^{\textup{lft}})$

via left Kan extension. Note that the analytification functor preserves finite
products and hence also the left Kan extension of this functor along the
Yoneda embedding. ∎

######  Lemma 3.11.

The functor from Lemma 3.10 fits into a commutative square

$\textstyle{\operatorname{Fun}(\textup{Sch}_{S}^{\textup{lft},\operatorname{op}},\textup{Spc})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textup{L}_{\mathbb{A}^{1}}}$$\textstyle{\operatorname{Fun}(\textup{Rig}_{S^{\textup{an}}}^{\textup{lft},\operatorname{op}},\textup{Spc})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textup{L}_{\mathbb{A}^{1}}}$$\textstyle{\operatorname{Fun}^{\mathbb{A}^{1}}(\textup{Sch}_{S}^{\textup{lft},\operatorname{op}},\textup{Spc})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fun}^{\mathbb{A}^{1}}(\textup{Rig}_{S^{\textup{an}}}^{\textup{lft},\operatorname{op}},\textup{Spc})}$

The analogous statement for the functor
$(-)^{\textup{an}}\colon\textup{Sm}_{S}\overset{}{\rightarrow}\textup{SmRig}_{S^{\textup{an}}}$
holds true, too.

###### Proof.

The functor
$\operatorname{Fun}(\textup{Sch}_{S}^{\textup{lft},\operatorname{op}},\textup{Spc})\overset{}{\longrightarrow}\operatorname{Fun}(\textup{Rig}_{S^{\textup{an}}}^{\textup{lft},\operatorname{op}},\textup{Spc})$
sends a morphism $\mathrm{pr}_{1}\colon
X\times\mathbb{A}^{1}_{S}\overset{}{\rightarrow}X$ to the morphism
$\mathrm{pr}_{1}\colon
X^{\textup{an}}\times\mathbb{A}^{1}_{S^{\textup{an}}}\overset{}{\rightarrow}X^{\textup{an}}$
which becomes an equivalence after applying $\textup{L}_{\mathbb{A}^{1}}$,
hence we get the desired functor from the universal property of localisation.
∎

######  Lemma 3.12.

The functor from Lemma 3.10 fits into a commutative square

$\textstyle{\operatorname{Fun}(\textup{Sch}_{S}^{\textup{lft},\operatorname{op}},\textup{Spc})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textup{L}_{\textup{Nis}}}$$\textstyle{\operatorname{Fun}(\textup{Rig}_{S^{\textup{an}}}^{\textup{lft},\operatorname{op}},\textup{Spc})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textup{L}_{\textup{Nis}}}$$\textstyle{\operatorname{Fun}^{\textup{Nis}}(\textup{Sch}_{S}^{\textup{lft},\operatorname{op}},\textup{Spc})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fun}^{\textup{Nis}}(\textup{Rig}_{S^{\textup{an}}}^{\textup{lft},\operatorname{op}},\textup{Spc})}$

The analogous statement for the functor
$(-)^{\textup{an}}\colon\textup{Sm}_{S}\overset{}{\rightarrow}\textup{SmRig}_{S^{\textup{an}}}$
holds true, too.

###### Proof.

The functor
$\operatorname{Fun}(\textup{Sch}_{k}^{\textup{lft},\operatorname{op}},\textup{Spc})\overset{}{\longrightarrow}\operatorname{Fun}(\textup{Rig}_{S^{\textup{an}}}^{\textup{lft},\operatorname{op}},\textup{Spc})$
sends Nisnevich covers to Nisnevich covers so that we get the desired functor
from the universal property of localisation. ∎

######  Lemma 3.13.

There exists an analytification functor
$\textup{H}(S)\overset{}{\rightarrow}\textup{RigH}(S^{\textup{an}})$ fitting
into a commutative square

$\textstyle{\textup{Sm}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)^{\textup{an}}}$$\textstyle{\textup{SmRig}_{S^{\textup{an}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textup{H}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textup{RigH}(S^{\textup{an}}).}$

###### Proof.

This follows from Lemmas 3.11 and 3.12. ∎

######  Remark 3.14 (Change of coefficients).

Let $\lambda\colon{\mathcal{D}}\rightleftarrows{\mathcal{E}}\colon\rho$ be an
adjunction. For any category ${\mathcal{C}}$, since we can compute
equivalences in functor categories objectwise, we get an induced adjunction

$\lambda_{*}\,\colon\,\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{D}})\rightleftarrows\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{E}})\,\colon\,\rho_{*}.$

If ${\mathcal{C}}$ carries a topology, we get an induced adjunction

$\tilde{\lambda}_{*}\,\colon\,\textup{Sh}({\mathcal{C}},{\mathcal{D}})\rightleftarrows\textup{Sh}({\mathcal{C}},{\mathcal{E}})\,\colon\,\tilde{\rho}_{*}$

with $\tilde{\lambda}_{*}\simeq\textup{L}_{\mathcal{E}}\circ\lambda_{*}$ and
$\tilde{\rho}_{*}\simeq\rho_{*}$ (omitting the inclusion functors from
presheaves to sheaves), since we can patch adjunctions together.

If
${\mathcal{C}}\in\\{\textup{Sch}_{S}^{\textup{ft},\operatorname{op}},\textup{Sm}_{S},\textup{Rig}_{S^{\textup{an}}},\textup{SmRig}_{S^{\textup{an}}}\\}$,
then we get an induced adjunction

$\hat{\lambda}_{*}\,\colon\,\textup{Sh}^{\mathbb{A}^{1}}({\mathcal{C}},{\mathcal{D}})\rightleftarrows\textup{Sh}^{\mathbb{A}^{1}}({\mathcal{C}},{\mathcal{E}})\,\colon\,\hat{\rho}_{*}$

on the full subcategories of $\mathbb{A}^{1}$-invariant sheaves. Combining
with Lemma 3.13 we get adjunctions

$\textup{H}(S,\textup{Spc})\rightleftarrows\textup{RigH}(S^{\textup{an}},\textup{Spc})\rightleftarrows\textup{RigH}(S^{\textup{an}},\textup{Pro}^{\omega}(\textup{Spc})).$

In the case, where ${\mathcal{D}}$ and ${\mathcal{E}}$ are commutative algebra
objects in $\textup{Pr}^{L}$ and $\lambda$ is symmetric monoidal, we can give
an alternative description of the adjunction
$\hat{\lambda}\rightleftarrows\hat{\rho}$. We will make this precise in
Section 4.1. The main application of this alternative description will be that
$\hat{\lambda}$ will be a symmetric monoidal functor.

###  3.4 Functoriality

We have defined the motivic homotopy $\textup{RigH}(-,{\mathcal{V}})$ category
with coefficients in any presentable category ${\mathcal{V}}$ (Definition
3.8). However, it suffices to prove most results on functoriality for
coefficients in Spc. This eases computations as we can easily reduce ourselves
to the case of presheaves and localise accordingly. But it is not a drawback
as we will see later in Lemma 4.3 that this implies that also
$\textup{RigH}(-,{\mathcal{V}})$ will satisfy the same functorial properties
as RigH. This will include the localisation sequence proven in Theorem 3.29.

Let $f\colon S\overset{}{\rightarrow}T$ be a morphism of rigid spaces. Then
there is a base change444Here the fibre product denotes the fibre product for
rigid analytic varieties. On affinoids, this corresponds to the completed
tensor product. We drop the completion from our notation. functor

$b\colon\textup{SmRig}_{T}\overset{}{\rightarrow}\textup{SmRig}_{S},X\mapsto
X\times_{T}S\eqqcolon X_{S},$

inducing by precomposition a functor
$f_{*}:=b^{*}\colon\textup{PSh}(\textup{SmRig}_{S})\overset{}{\rightarrow}\textup{PSh}(\textup{SmRig}_{T})$
which restricts to a functor

$f_{*}\colon\textup{RigH}(S)\overset{}{\longrightarrow}\textup{RigH}(T).$

This functor preserves limits and is accessible555On presheaves, this is
clear. Then the left adjoint of $f_{*}$ on RigH is given by
$\textup{L}_{\mathbb{A}^{1}}\textup{L}_{\textup{Nis}}f^{*}$. By abuse of
notation, we will again denote this left adjoint with $f^{*}$., hence admits a
left-adjoint functor

$f^{*}\colon\textup{RigH}(T)\overset{}{\longrightarrow}\textup{RigH}(S).$

The functor $f^{*}$ is a left Kan extension and for $F\in\textup{RigH}(T)$ and
$X\in\textup{SmRig}_{S}$ [Lur09, Prop. 4.3.3.7], one has an equivalence

$(f^{*}F)(X)\simeq\mathop{\textup{colim}}_{\textup{SmRig}_{T}\ni
Y,Y_{S}\overset{}{\rightarrow}X}F(Y).$

In particular, $(f^{*}F)(X\overset{}{\rightarrow}S)\simeq
F(X\overset{}{\rightarrow}S\overset{f}{\rightarrow}T)$ whenever $f$ is smooth.
In this case the functor $f^{*}$ is induced by the restriction along the
postcomposition functor
$\textup{SmRig}_{T}\overset{}{\rightarrow}\textup{SmRig}_{S}$ and as above we
see that $f$ admits a left adjoint

$f_{\sharp}\colon\textup{RigH}(S)\overset{}{\rightarrow}\textup{RigH}(T)$

by formal reasons.

######  Lemma 3.15.

The category $\textup{Sh}_{\textup{Nis}}(\textup{SmRig}_{S})$ is a topological
localisation of $\textup{PSh}(\textup{SmRig}_{S})$. In particular, the
category $\textup{Sh}_{\textup{Nis}}(\textup{SmRig}_{S})$ is an $\infty$-topos
and the localisation
functor$\textup{L}_{\textup{Nis}}\colon\textup{PSh}(\textup{SmRig}_{S})\overset{}{\rightarrow}\textup{Sh}_{\textup{Nis}}(\textup{SmRig}_{S})$
is accessible and left-exact.

###### Proof.

This is a general fact for categories of sheaves for Grothendieck topologies,
see [Lur09, 6.2.2.7, 6.2.1.7]. ∎

######  Lemma 3.16.

The motivic localisation $\textup{L}_{\textup{mot}}$ preserves finite products
and is locally cartesian.

###### Proof.

For $\textup{L}_{\textup{Nis}}$ this follows from Lemma 3.15 and for
$\textup{L}_{\mathbb{A}^{1}}$ this follows from [Hoy17, Prop. 3.4]. ∎

######  Remark 3.17 (Monoidal structure on RigH).

Let $S$ be a rigid space. The category of spaces Spc together with its
cartesian monoidal structure induces a symmetric monoidal structure on
$\textup{PSh}(\textup{SmRig}_{S})$ by the pointwise tensor-product [Lur17,
Rem. 2.1.3.4]. By Lemma 3.16, the localisation $\textup{L}_{\textup{mot}}$ is
compatible with this monoidal structure, inducing a symmetric monoidal
structure on $\textup{RigH}(S)$ [Lur17, Prop. 2.2.1.9]. Note that the
$\otimes$-product on the presheaf level commutes with arbitrary colimits and
thus also on $\textup{RigH}(S)$. If $f\colon S\overset{}{\rightarrow}T$ is a
map of rigid spaces, then $f^{*}$ is a symmetric monoidal functor.666This is
clear as $f^{*}$ on presheaves certainly preserves finite products and as
$\textup{L}_{\mathbb{A}^{1}}\textup{L}_{\textup{Nis}}$ commutes with finite
products, we can reduce to this case. But this is clear, as $f^{*}$ can be
written as a filtered colimit (the fact that this colimit is filtered follows
from [Sta23, 00X3]).

Let us add that $\textup{RigH}(S)$ is obtained via localisation of an
$\infty$-topos at a small set of morphisms. In particular, $\textup{RigH}(S)$
is presentable and even more it is a commutative algebra object in
$\textup{Pr}^{L}$.

######  Proposition 3.18.

Let $f\colon S\rightarrow T$ be a smooth morphism of rigid spaces.

  1. 1.

The pullback $f^{*}\colon\textup{RigH}(T)\rightarrow\textup{RigH}(S)$ admits a
left adjoint $f_{\sharp}$.

  2. 2.

(Smooth projection formula) The canonical map

$f_{\sharp}(f^{*}M\otimes N)\overset{}{\rightarrow}M\otimes f_{\sharp}N$

is an equivalence for all $M\in\textup{RigH}(T)$ and $N\in\textup{RigH}(S)$.

  3. 3.

(Smooth base change) Let $g\colon X\rightarrow T$ be a morphism of rigid
spaces and let

${X_{S}}$${X}$${S}$${T}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{g}$$\scriptstyle{f}$

be a pullback diagram. Then the exchange map
$f^{\prime}_{\sharp}g^{\prime*}\overset{}{\rightarrow}g^{*}f_{\sharp}$ is an
equivalence.

###### Proof.

The first assertion follows from the discussion in the beginning. For (2) and
(3) we may use that $\textup{L}_{\mathbb{A}^{1}}\textup{L}_{\textup{Nis}}$
commutes with finite products and thus, we can check them on the level of
presheaves, where they follow from the construction. ∎

######  Remark 3.19.

Let us note that Proposition 3.18 (3) is equivalent to the right
adjointability of the transposed diagram, i.e.
$f^{\prime}_{\sharp}g^{\prime*}\overset{}{\rightarrow}g^{\prime*}f^{\prime}_{\sharp}$
is an equivalence if and only if the exchange map
$f^{*}g_{*}\overset{}{\rightarrow}g^{\prime}_{*}f^{{}^{\prime}*}$ is an
equivalence.

######  Proposition 3.20.

The following assignments define sheaves for the Nisnevich topology on Rig
with values in $\Pr^{L}$.

  1. 1.

The assignment
$S\mapsto\textup{Sh}_{\textup{Nis}}(\textup{SmRig}_{S}),f\mapsto f^{*}$, and

  2. 2.

The assignment $S\mapsto\textup{RigH}(S),f\mapsto f^{*}.$

###### Proof.

The first part follows from [AGV22, Prop. 2.3.7].

For the second part, let us note that it is enough to show that for any
Nisnevich cover $f\colon U\overset{}{\rightarrow}X$ and any Nisnevich sheaf
$F$ such that $f^{*}F$ is $\mathbb{A}^{1}$-invariant, also $F$ is
$\mathbb{A}^{1}$-invariant. This follows since

$\underline{\operatorname{Hom}}(\mathbb{A}^{1}_{U},f^{*}F)\simeq
f^{*}\underline{\operatorname{Hom}}(\mathbb{A}^{1}_{X},F)$

and since $f^{*}$ is conservative, where $\underline{\operatorname{Hom}}$
denotes the internal $\operatorname{Hom}$ in
$\textup{Sh}_{\textup{Nis}}(\textup{SmRig}_{S})$ (this is completely analogous
to the proof of [AGV22, Thm. 2.3.4]). ∎

In the rest of this subsection, we want to prove that for a closed immersion
$i\colon Z\overset{}{\hookrightarrow}S$ the pushforward $i_{*}$ preserves
weakly contractible colimits. This will follow via direct computation.

Let $S$ be a rigid space. We denote by $\emptyset/S$ the initial object in
$\textup{SmRig}_{S}$. The category of those functors
$F\in\textup{PSh}(\textup{SmRig}_{S})$ such that $F(\emptyset)\simeq\ast$ is
denoted by $\textup{PSh}_{\emptyset}(\textup{SmRig}_{S})$. Equivalently, we
can localise $\textup{PSh}(\textup{SmRig}_{S})$ at the morphism
$\emptyset\overset{}{\rightarrow}\emptyset/S$, where $\emptyset$ is the empty
functor. In particular, The inclusion
$\textup{PSh}(\textup{SmRig}_{S})\overset{}{\rightarrow}\textup{PSh}_{\emptyset}(\textup{SmRig}_{S})$
admits a left adjoint, denoted by $L_{\emptyset}$.

######  Remark 3.21.

By design777A presheaf $F\in\textup{PSh}(\textup{SmRig}_{S})$ is contained in
$\textup{PSh}_{\emptyset}(\textup{SmRig}_{S})$ if and only if its essential
image is weakly contractible., the category
$\textup{PSh}_{\emptyset}(\textup{SmRig}_{S})$ is generated by representables
under weakly contractible colimits (i.e. colimits of shape $I$ where the map
$I\overset{}{\rightarrow}*$ is cofinal).

######  Lemma 3.22.

Let $i\colon Z\hookrightarrow S$ be a closed immersion of rigid spaces. The
functor$i_{\emptyset*}\colon\textup{PSh}_{\emptyset}(\textup{SmRig}_{Z})\rightarrow\textup{PSh}_{\emptyset}(\textup{SmRig}_{S})$
commutes with $\textup{L}_{\textup{Nis}}$ and $\textup{L}_{\textup{mot}}$.

###### Proof.

That $i_{\emptyset*}$ commutes with Nisnevich-sheafification follows with the
same arguments as [Ayo15, Lem. 1.4.19]. For the sake of completion let us
recall the proof.

It is enough to show $\textup{L}_{\textup{Nis}}\circ i_{\emptyset*}\simeq
i_{\emptyset*}\circ\textup{L}_{\textup{Nis}}$. For this let
$F\in\textup{PSh}_{\emptyset}(\textup{SmRig}_{Z})$. Let $U$ be a rigid space
over $S$. We can define

$(i_{\emptyset*}F)^{+}(U)\coloneqq\mathop{\textup{colim}}_{(U_{i})_{i}\in\textup{Cov}_{\textup{Nis}}(U)}\lim_{\Delta}i_{\emptyset*}F(\check{C}(\coprod_{i}U_{i}/U)_{\bullet}),$

where the colimit runs over all Nisnevich coverings
$(U_{i}\overset{}{\rightarrow}U)_{i}$. The sheafification is then given by
applying $(-)^{+}$ infinitely many times (via transfinite induction - cf.
proof of [Lur09, Prop. 6.2.2.7]). In the proof of [AGV22, Lem. 2.2.4] it is
shown that if $U\times_{S}Z$ is non-empty, then any Nisnevich covering of
$U\times_{S}Z$ can be refined by the base change of a Nisnevich covering of
$U$. In particular, the base change map
$\textup{Cov}_{\textup{Nis}}(U)\overset{}{\rightarrow}\textup{Cov}_{\textup{Nis}}(U\times_{S}Z)$
is cofinal. Thus, the definition of $\textup{L}_{\textup{Nis}}$ via the
$(-)^{+}$-construction shows

$\textup{L}_{\textup{Nis}}(i_{\emptyset*}F)(U)\simeq(\textup{L}_{\textup{Nis}}F)(U\times_{S}Z)\simeq
i_{\emptyset*}(\textup{L}_{\textup{Nis}}F)(U).$

For the commutativity with
$\textup{L}^{\\#}_{\mathbb{A}^{1}}\circ\textup{L}_{\textup{Nis}}$ it is enough
to show that for all $U\in\textup{SmRig}_{Z}$, we have that $p\colon
i_{\emptyset*}\mathbb{A}^{1}_{U}\overset{}{\rightarrow}i_{\emptyset*}U$ is an
$\mathbb{A}^{1}_{S}$-equivalence (similar to the proof of [Ayo15, Prop.
1.4.18]). The $0$-section of $\mathbb{A}^{1}_{U}\overset{}{\rightarrow}U$
induces a section $s$ of $p$. We claim that also $s\circ p$ is
$\mathbb{A}^{1}_{S}$-homotopy equivalent the identity on
$i_{\emptyset*}\mathbb{A}^{1}_{U}$. The composition

$\mathbb{A}^{1}_{S}\otimes
i_{\emptyset*}\mathbb{A}^{1}_{U}\overset{}{\rightarrow}i_{\emptyset*}\mathbb{A}^{1}_{Z}\otimes
i_{\emptyset*}\mathbb{A}^{1}_{U}\simeq
i_{\emptyset*}(\mathbb{A}^{1}_{Z}\times_{Z}\mathbb{A}^{1}_{Z}\times_{Z}U)\xrightarrow{m}i_{\emptyset*}(\mathbb{A}^{1}_{Z}\times_{Z}U),$

where $m$ is the multiplication, naturally yields such a homotopy. In
particular, $p$ is $\mathbb{A}^{1}_{S}$-homotopy equivalence. ∎

######  Remark 3.23.

Let us note that $i_{\emptyset*}$ cannot preserve the initial object, when
$U\coloneqq S\setminus Z\neq\emptyset$. This is because
$i_{\emptyset*}(\emptyset/Z)(U)\simeq\emptyset/Z(\emptyset/Z)\simeq\ast\not\simeq\emptyset=\emptyset/S(U)$.
But $i_{\emptyset*}$ preserves weakly contractible colimits, as
$\textup{PSh}_{\emptyset}(\textup{SmRig}_{Z})$ is freely generated under
weakly contractible colimits in $Z$ (and similarly for $S$).

The remark above immediately implies the following.

######  Corollary 3.24.

Let $i\colon Z\overset{}{\hookrightarrow}S$ be a closed immersion of rigid
spaces. Then the functor
$i_{*}\colon\textup{RigH}(Z)\overset{}{\rightarrow}\textup{RigH}(S)$ preserves
weakly contractible colimits.

The only obstruction for $i$ above not preserving colimits is that in
$\textup{RigH}(S)$ the initial and final object may not agree. This is
certainly a "problem" inside Spc, which will go away after stabilising.

###  3.5 Gluing

In this section, we want to prove the existence of a gluing theorem in the
unstable setting which yields a localisation sequence in the stable setting.
Such a theorem has been first proven by Morel-Voevodsky in the algebraic case
[MV99, Thm. 2.21]. In $\mathbb{B}^{1}$-homotopy theory this result is proven
by Ayoub [Ayo15]. We will follow the proof structure presented by Hoyois
[Hoy17, §4].

Notation. In this subsection, we fix a rigid space $S$.

######  Definition 3.25.

Let $i\colon Z\overset{}{\rightarrow}S$ be a closed immersion and let
$U:=S\setminus Z$ be its open complement. Given a pair $(X,t)$ consisting of
$X\in\textup{SmRig}_{S}$ and a partial section $t\colon
Z\overset{}{\rightarrow}X$, i.e. inducing a section $t\colon
Z\overset{}{\rightarrow}X_{Z}:=X\times_{S}Z$, we define the presheaf

$\Phi_{S}(X,t):=(X\sqcup_{X_{U}}U)\times_{i_{*}X_{Z}}S.$

More explicitly, evaluating at $Y\in\textup{SmRig}_{S}$ we have that

$\displaystyle\Phi_{S}(X,t)(Y)=\begin{cases}\operatorname{Hom}_{S}(Y,X)\times_{\operatorname{Hom}_{Z}(Y_{Z},X_{Z})}*&(Y_{Z}\neq\varnothing)\\\
*&(Y_{Z}=\varnothing)\end{cases}$

where $\operatorname{Hom}_{Z}(Y_{Z},X_{Z})$ is pointed at the map
$Y_{Z}\overset{}{\rightarrow}Z\overset{t}{\rightarrow}X_{Z}$.

######  Remark 3.26.

We get a functorial assignment $(X,t)\mapsto\Phi_{S}(X,t)$ and for every
morphism $f\colon T\overset{}{\rightarrow}S$ there is a natural map

$f^{*}\Phi_{S}(X,t)\overset{}{\longrightarrow}\Phi_{T}(X_{T},t_{T})$

which is an isomorphism when $f$ is smooth.

######  Lemma 3.27.

Let $p\colon X^{\prime}\overset{}{\rightarrow}X$ be an étale morphism in
$\textup{SmRig}_{S}$ and let $t\colon Z\overset{}{\rightarrow}X$ and
$t^{\prime}\colon Z\overset{}{\rightarrow}X^{\prime}$ be partial sections
which are compatible, i.e. $t=p\circ t^{\prime}$. Then the induced map

$\Phi_{S}(p)\colon\Phi_{S}(X^{\prime},t^{\prime})\overset{}{\longrightarrow}\Phi_{S}(X,t)$

is a Nisnevich equivalence.

###### Proof.

This is _Etape 2_ in the proof of [Ayo15, Prop. 1.4.21]. ∎

######  Lemma 3.28.

Let $n\geq 1$ and let $\sigma\colon
Z\overset{}{\hookrightarrow}S\overset{}{\rightarrow}\mathbb{A}^{n}_{S}$ be the
restriction of the zero section of the canonical morphism
$\pi\colon\mathbb{A}^{n}_{S}\overset{}{\rightarrow}S$. Then the canonical map
$\Phi_{S}(\mathbb{A}^{n},\sigma)\overset{}{\rightarrow}S$ is an
$\mathbb{A}^{1}$-equivalence.

###### Proof.

A nullhomotopy is given by the map

$\mathbb{A}^{1}\times\Phi_{S}(\mathbb{A}^{n},t)\overset{}{\longrightarrow}\Phi_{S}(\mathbb{A}^{n},t),\quad(a,f)\mapsto
af.$

∎

######  Theorem 3.29 (Gluing).

Let $i\colon Z\overset{}{\hookrightarrow}S$ be a closed immersion of rigid
spaces with open complement $j\colon U\overset{}{\hookrightarrow}S$. Then for
every $F\in\textup{RigH}(S)$ the square

(3.29.5)

is a pushout square (where the maps without labels are the unique ones).

###### Proof.

The category $\textup{RigH}(S)$ is generated by representables under weakly
contractible colimits (cf. Remark 3.21) and all the functors in the square
$(\heartsuit)$ preserve these colimits, either because they are left-adjoint
or by Corollary 3.24. Hence we may assume that $F=\textup{L}_{\textup{mot}}X$
for some $X\in\textup{SmRig}_{S}$ so that it suffices to show that the
canonical map

$X\sqcup_{X_{U}}U\overset{}{\longrightarrow}i_{*}X_{Z}$

in $\textup{PSh}(\textup{SmRig}_{S})$ is motivic equivalence. The latter can
be checked after base change along all maps
$Y\overset{}{\rightarrow}i_{*}X_{Z}$ for $Y\in\textup{SmRig}_{S}$, i.e. it
suffices to show that the maps

$(X\sqcup_{X_{U}}U)\times_{i_{*}X_{Z}}Y\overset{}{\longrightarrow}Y$

are equivalences. We claim that we can reduce to the case where $Y=S$.

Indeed, given $p\colon Y\overset{}{\rightarrow}S$ in $\textup{SmRig}_{S}$ and
denoting by $q\colon Y_{Z}\overset{}{\rightarrow}Z$ and $k\colon
Y_{Z}\overset{}{\hookrightarrow}Y$ base changes of $p$ and $i$, respectively,
we see that the map $(\spadesuit)$ identifies with the map

$p_{\\#}\bigl{(}(p^{*}X\sqcup_{p^{*}X_{U}}p^{*}U)\times_{k_{*}(p^{*}X)_{Y_{Z}}}Y\bigr{)}\overset{}{\rightarrow}p_{\\#}Y$

since we have equivalences

$p^{*}i_{*}X_{Z}\simeq k_{*}q^{*}X_{Z}\simeq k_{*}q^{*}i^{*}X\simeq
k_{*}k^{*}p^{*}X\simeq k_{*}(p^{*}X)_{Y_{Z}}$

where the first one is smooth base change (Proposition 3.18).

By adjunction, the datum of a map $S\overset{}{\rightarrow}i_{*}X_{Z}$
corresponds to a section $t\colon Z\overset{}{\rightarrow}X_{Z}$. Translating
into the notation from Definition 3.25, we have thus reduced to having to show
that the canonical map $\Phi_{S}(X,t)\overset{}{\rightarrow}S$ is a motivic
equivalence for all $(X,t)$ as in loc. cit.

By Nisnevich descent (Proposition 3.20) we may assume that $S$ is affine and
always can replace by an open subspace. Thus we find an open neighbourhood $U$
of $Z$ in $S$, an open neighbourhood $V$ of $t(Z)$ in $X$, and an isomorphism
$V\cong\mathbb{B}^{n}_{U}$ for a suitable $n\geq 1$ such that $t$ identifies
with the partial zero section under this isomorphism [AGV22, Prop. 1.3.16].
Applying Lemma 3.27 to the morphism $\mathbb{B}^{n}\cong
V\overset{}{\hookrightarrow}X$ we obtain a Nisnevich equivalence
$\Phi_{S}(X,t)\overset{\simeq}{\rightarrow}\Phi_{S}(\mathbb{B}^{n},\sigma)$.
Applying the same lemma to the open immersion
$\mathbb{B}^{n}\overset{}{\hookrightarrow}\mathbb{A}^{n}$ we get a Nisnevich
equivalence
$\Phi_{S}(\mathbb{B}^{n},\sigma)\overset{\simeq}{\rightarrow}\Phi_{S}(\mathbb{A}^{n},\sigma)$.
By Lemma 3.28, the presheaf $\Phi_{S}(\mathbb{A}^{n},\sigma)$ is
$\mathbb{A}^{1}$-contractible. ∎

######  Corollary 3.30.

Let $i\colon Z\rightarrow S$ be a closed immersion. Then the functor
$i_{*}\colon\textup{RigH}(Z)\overset{}{\rightarrow}\textup{RigH}(S)$ is fully
faithful.

###### Proof.

This is standard888Proof in other situations can be found in [Ayo15, 1.4.23],
[Hoy17, Cor. 4.19] or [AGV22, Cor. 2.2.2 (1)]. and follows from Theorem 3.29,
smooth base change (cf. Proposition 3.18) and Nisnevich descent. Nevertheless,
let us give a proof for completion.

We have to show that for any $F\in\textup{RigH}(Z)$ the counit
$i^{*}i_{*}F\rightarrow F$ is an equivalence. Using Theorem 3.29, it is enough
to show that $i_{*}$ is conservative. In particular, it is enough to show that
for $f\colon F\rightarrow G$ in $\textup{RigH}(Z)$ such that $i_{*}f$ is an
equivalence, we have that $f$ is an equivalence. By Remark 3.21 it is suffices
to test that $f$ is an equivalence at any $X\in\textup{SmRig}_{Z}$. The proof
of [AGV22, Lem. 2.2.5 (2)] shows that we may assume that there exists an
$S_{X}\in\textup{SmRig}_{S}$ such that $X=S_{X}\times_{S}Z$. In particular,
$f(X)$ is equivalent to $i_{*}f(S_{X})$. ∎

######  Corollary 3.31.

Let us consider a cartesian square

${T_{Z}}$${T}$${Z}$${S}$$\scriptstyle{i^{\prime}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{i}$

where $i$ is a closed immersion. Then the exchange transformation
$f^{*}i_{*}\overset{}{\rightarrow}i^{\prime}_{*}f^{\prime*}$ is an
equivalence.

###### Proof.

This follows from Corollary 3.30 and the smooth base change of Proposition
3.18. ∎

##  4 Stable $\mathbb{A}^{1}$-homotopy theory in rigid geometry

In this section we want to define the stable $\mathbb{A}^{1}$-homotopy
category. To do so, we first want to show that the effective version satisfies
all the functorial properties as RigH. Afterward, we $\otimes$-invert the
pointed projective line. This will yield a stable homotopy functor in the
sense of Ayoub [Ayo07]. In particular, using the results of op. cit., we will
obtain a six functor formalism with respect to algebraic maps.

Notation. In this section, let $S$ be a rigid space and let
${\mathcal{V}}\in\textup{CAlg}(\textup{Pr}^{L,\otimes})$. For results using
the analytification functor, let $R$ be an adic ring with ideal of definition
$I$ and set $B:=\operatorname{Spec}(R)\setminus\textup{V}(I)$ and
$B^{\textup{an}}=\textup{Spf}(R)^{\textup{rig}}$.

###  4.1 Change of coefficients

In Section 3, we constructed the rigid $\mathbb{A}^{1}$-homotopy category via
$\mathbb{A}^{1}$-localisation of Nisnevich sheaves of spaces. In this short
subsection, we want to explain how we can extend our results to
$\mathbb{A}^{1}$-invariant Nisnevich sheaves with ${\mathcal{V}}$-valued
coefficients.

Let us start by giving an alternative description of $\hat{\lambda}_{*}$ and
$\hat{\rho}_{*}$ of Remark 3.14 in the setting, where
${\mathcal{D}},{\mathcal{E}}$ are both contained in
$\textup{CAlg}(\textup{Pr}^{L,\otimes})$ and $\lambda$ is a symmetric monoidal
functor. The functor $\lambda$ induces a map

$\operatorname{Fun}^{R}({\mathcal{C}}^{\operatorname{op}},{\mathcal{D}})\overset{r}{\longrightarrow}\operatorname{Fun}^{R}({\mathcal{C}}^{\operatorname{op}},{\mathcal{E}})$

via postcomposition with $\lambda^{\operatorname{op}}$, where
$\operatorname{Fun}^{R}$ denotes those functors that admit a left adjoint. We
have an equivalence
$\operatorname{Fun}^{R}({\mathcal{C}}^{\operatorname{op}},{\mathcal{D}})\simeq\textup{PSh}({\mathcal{C}})\otimes{\mathcal{D}}$
and analogously for the right hand side; this follows from [Lur17, Prop.
4.8.1.17] and the universal property of the Yoneda embedding [Lur09, Prop
5.1.5.6]. Under this equivalence the functor $r$ corresponds to $\lambda_{*}$.

The localisation functor
$L\colon\textup{PSh}({\mathcal{C}})\overset{}{\rightarrow}\textup{Sh}({\mathcal{C}})$
preserves finite products. Thus, the pointwise monoidal structure on
$\textup{PSh}({\mathcal{C}})$ induces a monoidal structure on
$\textup{Sh}({\mathcal{C}})$. Further, we have
$\textup{Sh}({\mathcal{C}})\otimes{\mathcal{D}}\simeq\textup{Sh}({\mathcal{C}},{\mathcal{D}})$
and similarly for $\textup{Sh}({\mathcal{C}},{\mathcal{E}})$ [Dre18, Prop.
2.4]. This induces a monoidal structure on
$\textup{Sh}({\mathcal{C}},{\mathcal{D}})$ and
$\textup{Sh}({\mathcal{C}},{\mathcal{E}})$. In this case the adjunction

$\tilde{\lambda}_{*}\,\colon\,\textup{Sh}({\mathcal{C}},{\mathcal{D}})\leftrightarrows\textup{Sh}({\mathcal{C}},{\mathcal{E}})\,\colon\,\tilde{\rho}_{*}$

is induced by the unit of the base change functor
$\textup{CAlg}(\textup{Pr}^{L,\otimes})_{/{\mathcal{D}}}\overset{}{\rightarrow}\textup{CAlg}(\textup{Pr}^{L,\otimes})_{/{\mathcal{E}}}$,
cf. [Dre18, §8] for more details.

The proof of [Dre18, Prop. 2.4] also works for the
$\mathbb{A}^{1}$-localisation of $\textup{Sh}({\mathcal{C}})$. In particular,
we see that
$\textup{Sh}^{\mathbb{A}^{1}}({\mathcal{C}},{\mathcal{D}})\simeq\textup{Sh}^{\mathbb{A}^{1}}({\mathcal{C}})\otimes{\mathcal{D}}$
and the adjunction

$\hat{\lambda}_{*}\,\colon\,\textup{Sh}^{\mathbb{A}^{1}}({\mathcal{C}},{\mathcal{D}})\leftrightarrows\textup{Sh}^{\mathbb{A}^{1}}({\mathcal{C}},{\mathcal{E}})\,\colon\,\hat{\rho}_{*}$

is again induced via base change. Applying our discussion above to
${\mathcal{C}}=\textup{SmRig}_{S}$ with the Nisnevich topology, we obtain the
following.

######  Remark 4.1.

We have
$\textup{Sh}_{\textup{Nis}}(\textup{SmRig}_{S})\otimes{\mathcal{V}}\simeq\textup{Sh}_{\textup{Nis}}(\textup{SmRig}_{S},{\mathcal{V}})$
by the above. This is also compatible with $\mathbb{A}^{1}$-localisation,
hence

$\textup{RigH}(S)\otimes{\mathcal{V}}\simeq\textup{RigH}(S,{\mathcal{V}}).$

Furthermore, by Remark 3.17 the category $\textup{RigH}(S)$ is a commutative
algebra object in $\textup{Pr}^{L,\otimes}$. Thus, the same is true for
$\textup{RigH}(X)\otimes{\mathcal{V}}$. In particular, the $\otimes$-product
in $\textup{RigH}(S)\otimes{\mathcal{V}}$ is symmetric and commutes with
arbitrary colimits in both variables.

######  Remark 4.2.

Let us consider the constant functor
$\mathrm{const}\colon\textup{Spc}\overset{}{\rightarrow}\textup{Cond}^{\omega}(\textup{Spc})$.
This is a map of presentable categories by Lemma A.3. As colimits in
$\textup{Cond}^{\omega}(\textup{Spc})$ can be computed pointwise, we see that
$\mathrm{const}$ is colimit preserving. Moreover,
$\textup{Cond}^{\omega}(\textup{Spc})$ is the sheaf category associated to a
Grothendieck topology. In particular, the localisation functor
$\textup{PSh}(\textup{ProFin}^{\omega})\rightarrow\textup{Cond}^{\omega}(\textup{Spc})$
is left exact. Therefore, $\mathrm{const}$ also preserves finite products and
in particular is symmetric monoidal. Thus, the constant functor
$\mathrm{const}$ is a map in $\textup{CAlg}(\textup{Pr}^{L,\otimes})$.
Applying our discussion before to the analytification functor of Remark 3.14,
we obtain a symmetric monoidal colimit preserving functor

$\textup{H}(B,\textup{Spc})\overset{}{\longrightarrow}\textup{RigH}(B^{\textup{an}},\textup{Cond}^{\omega}(\textup{Spc})).$

With the above discussion it is not hard to see that the base change of a
pullback formalism is again a pullback formalism [Dre18, §8]. As our
construction is not directly covered by loc. cit., let us be more precise
and give a statement with proof for completion.

######  Lemma 4.3.

The induced functor

$\textup{RigH}\otimes{\mathcal{V}}\colon\textup{Rig}^{\operatorname{op}}\overset{}{\rightarrow}\textup{CAlg}(\textup{Pr}^{L,\otimes})_{/{\mathcal{V}}},\quad
S\mapsto\textup{RigH}(S)\otimes{\mathcal{V}},\quad f\mapsto
f^{*}\otimes\operatorname{id}_{{\mathcal{V}}}$

has the following properties.

  1. (1)

For any smooth morphism $f$ inside Rig the pullback
$f^{*}\otimes\operatorname{id}_{{\mathcal{V}}}$ admits a left adjoint
$f_{\sharp}\otimes\operatorname{id}_{{\mathcal{V}}}$ satisfying smooth base
change and projection formula (cf. Proposition 3.18).

  2. (2)

For each $S\in\textup{Rig}$ and natural projection
$p\colon\mathbb{A}^{1}_{S}\rightarrow X$ the functor
$p^{*}\otimes\operatorname{id}_{{\mathcal{V}}}$ is fully faithful.

###### Proof.

The proof is similar to [Dre18, Prop. 8.5] but let us give an argument
nonetheless.

If $f$ is smooth, then it admits a left adjoint as this is true for RigH
[Dre18, Lem. 8.4]. The main argument is that
$\textup{RigH}(S)\otimes{\mathcal{V}}\simeq\operatorname{Fun}^{R}(\textup{RigH}(S)^{\operatorname{op}},{\mathcal{V}})$
by [Lur17, Prop 4.8.1.17], where $\operatorname{Fun}^{R}$ denotes those
functor that admit left adjoints. Composing with
$f^{*,\operatorname{op}}\leftrightarrows f_{\sharp}^{\operatorname{op}}$ for
$\textup{RigH}^{\operatorname{op}}$ yields the desired adjunction. This
construction allows us to interpret smooth base change and the projection
formula as composing with the transpose of the exchange transformation resp.
the projection formula in RigH, which is an equivalence by Proposition 3.18.
This shows (1) and analogously also (2). ∎

###  4.2 Effective motives and localisation

Notation. For the rest of this section, we will assume that ${\mathcal{V}}$ is
furthermore stable.

######  Definition 4.4.

We define the effective stable homotopy category with coefficients
${\mathcal{V}}$ by

$\textup{RigSH}^{\textup{eff}}(S,{\mathcal{V}})\coloneqq\textup{RigH}(S)\otimes{\mathcal{V}}.$

We denote the induced functor by
$\textup{RigSH}^{\textup{eff}}_{{\mathcal{V}}}$ and if
${\mathcal{V}}\simeq\operatorname{Sp}$, then we simply write
$\textup{RigSH}^{\textup{eff}}(S)$ and $\textup{RigSH}^{\textup{eff}}$.

######  Remark 4.5.

By Remark 4.1, we see that $\textup{RigSH}^{\textup{eff}}_{\mathcal{V}}$
satisfies Nisnevich descent. Further, by [Lur17, Ex. 4.8.1.23]
$\textup{RigSH}^{\textup{eff}}(S)$ is equivalent to the stabilisation of
$\textup{RigH}(S)$.

######  Notation 4.6.

Let $f\colon T\rightarrow S$ be a morphism of rigid spaces and ${\mathcal{V}}$
a symmetric monoidal stable presentable category. Abusing notation, we will
simply write

$f^{*}\colon\textup{RigSH}^{\textup{eff}}(S,{\mathcal{V}})\overset{}{\rightarrow}\textup{RigSH}^{\textup{eff}}(T,{\mathcal{V}})$

instead of $f^{*}\otimes\operatorname{id}_{{\mathcal{V}}}$ and similarly
$f_{*}$ and $f_{\sharp}$ (if it exists).

######  Remark 4.7.

We have seen in Remark 3.23 that for closed immersions $i\colon
Z\hookrightarrow S$ the pushforward $i_{*}$ on RigH does not preserve colimits
in general. The obstruction is also very clear, in RigH the initial and final
object do not agree. But $i_{*}$ on
$\textup{RigSH}^{\textup{eff}}(S,{\mathcal{V}})$ does certainly preserve the
initial object, as this obstruction vanishes. Therefore $i_{*}$ preserves
colimits.

######  Notation 4.8.

Let $i\colon Z\rightarrow S$ be a closed immersion. Then we denote the right
adjoint of $i_{*}$ by $i^{!}$.

From the gluing in Theorem 3.29, we immediately obtain a localisation sequence

######  Proposition 4.9 (Localisation).

Let $i\colon Z\overset{}{\hookrightarrow}S$ be a closed immersion with open
complement $j\colon U\overset{}{\hookrightarrow}S$. Then for any
$F\in\textup{RigSH}^{\textup{eff}}(S,{\mathcal{V}})$, we have fibre sequences

${j_{\sharp}j^{*}F}$${F}$${i_{*}i^{*}F,}$${i_{*}i^{!}F}$${F}$${j_{*}j^{*}F.}$

###### Proof.

The statement of the lemma is equivalent to the following statement. For any
closed immersion $i\colon Z\hookrightarrow S$ in Rig with open complement
$j\colon U\hookrightarrow S$, the square

${\textup{RigH}(Z)\otimes{\mathcal{V}}}$${\textup{RigH}(S)\otimes{\mathcal{V}}}$${\ast}$${\textup{RigH}(U)\otimes{\mathcal{V}}}$$\scriptstyle{i_{*}\otimes\operatorname{id}_{{\mathcal{V}}}}$$\scriptstyle{j^{*}\otimes\operatorname{id}_{{\mathcal{V}}}}$

is a pullback in $\textup{Pr}^{L}$. In particular, as the functor
$-\otimes{\mathcal{V}}$ preserves colimits, we can use the same arguments as
in the proof of Lemma 4.3 to assume ${\mathcal{V}}\simeq\operatorname{Sp}$. In
view of Remark 4.1, we have
$\textup{RigSH}^{\textup{eff}}(-)\simeq\operatorname{Sp}(\textup{RigH}(-,\textup{Spc}_{*}))$,
where $\textup{Spc}_{*}$ denotes the category of pointed spaces. We have an
equivalence

$\operatorname{Sp}(\textup{RigH}(-,\textup{Spc}_{*}))\simeq\lim(\dots\xrightarrow{\Omega}\textup{RigH}(-,\textup{Spc}_{*}))$

inside $\Pr^{R}$ [Lur17, Prop. 1.4.2.24]. In particular, the second sequence
of the proposition is a fibre sequence if and only if it is so in
$\textup{RigH}(-,\textup{Spc}_{*})$. Since the first sequence is the adjoint
of the second one, we may assume ${\mathcal{V}}\simeq\textup{Spc}_{*}$.

We will show that the first sequence is a cofibre sequence in
$\textup{RigH}(-,\textup{Spc}_{*})$, which proves that the latter one is a
fibre sequence. For this let us note that the forgetful functor
$p\colon\textup{Spc}_{*}\rightarrow\textup{Spc}$ takes pushout squares to
pushout squares and reflects them [Lur09, Prop. 4.4.2.9]. As in the proof of
[Hoy17, Prop. 5.2] let us look at the following two squares

${j_{\sharp}j^{*}p(F)}$${j_{\sharp}j^{*}p(F)\coprod_{U}S}$${p(F)}$${U}$${S}$${i_{*}i^{*}p(F).}$

The outer square is a pushout by Theorem 3.29 (note that
$F\in\textup{RigSH}^{\textup{eff}}(S,{\mathcal{V}})$ comes equipped with a
zero section). The left square is by design a pushout. So the right square is
also a pushout, which is also the image of the first sequence of the
proposition under $p$. ∎

###  4.3 Thom motives and stabilisation

For any vector bundle $p\colon V\rightarrow S$ with zero section $s\colon
S\rightarrow V$, we can define the suspension $\Sigma_{V}\coloneqq
p_{\sharp}s_{*}$ and loop $\Omega_{V}\coloneqq s^{!}p^{*}$. Note that this
yields an adjunction

${\Sigma_{V}\colon\textup{RigSH}^{\textup{eff}}(S,{\mathcal{V}})}$${\textup{RigSH}^{\textup{eff}}(S,{\mathcal{V}})\colon\Omega_{V}.}$

######  Lemma 4.10.

Then the following are equivalent.

  1. (i)

For any smooth morphism $f\colon T\rightarrow S$ of rigid spaces that admits a
section $s$, the Thom transformation $\textup{Th}(f,s)\coloneqq
f_{\sharp}s_{*}$ is an equivalence.

  2. (ii)

For any vector bundle $V$ on $S$ the suspension of the unit $\Sigma_{V}1_{V}$
is $\otimes$-invertible inside
$\textup{RigSH}^{\textup{eff}}(S,{\mathcal{V}})$.

  3. (iii)

The suspension of the unit of the closed unit disc
$\Sigma_{\mathbb{B}^{1}_{S}}1_{\mathbb{B}^{1}_{S}}$ is $\otimes$-invertible.

###### Proof.

Equivalences $(i)\Leftrightarrow(ii)\Leftrightarrow(iii)$ are standard and
follow from the fact that any smooth morphism $f\colon T\rightarrow S$ of
rigid spaces factors locally as an étale morphism
$T\rightarrow\mathbb{B}^{n}_{S}$ followed by the projection [CD19, §2.4]. ∎

######  Remark 4.11.

Let $V\rightarrow S$ be a vector bundle. The localisation sequence in
Proposition 4.9 implies that

$\Sigma_{V}1_{V}\simeq\frac{V}{V\setminus\\{0\\}},$

where the RHS denotes the cofibre of the natural open immersion
$V\setminus\\{0\\}\rightarrow V$ induced by the zero section.

Further, note that we have a commutative diagram with pushout squares

${\mathbb{B}^{1}_{S}\setminus\\{0\\}}$${\mathbb{A}^{1}_{S}\setminus\\{0\\}}$${0}$${\mathbb{B}^{1}_{S}}$${\mathbb{A}^{1}_{S}}$${\frac{\mathbb{A}^{1}_{S}}{\mathbb{A}^{1}_{S}\setminus\\{0\\}}.}$

In particular, we have

$\frac{\mathbb{B}^{1}_{S}}{\mathbb{B}^{1}_{S}\setminus\\{0\\}}\simeq\frac{\mathbb{A}^{1}_{S}}{\mathbb{A}^{1}_{S}\setminus\\{0\\}}$

inside $\textup{RigSH}^{\textup{eff}}(S)$. Thus, condition (iii) in Lemma 4.10
is equivalent to $\Sigma_{\mathbb{A}^{1}_{S}}1_{\mathbb{A}^{1}_{S}}$ being
$\otimes$-invertible.

Finally, let us look at the diagram

${\mathbb{A}^{1}_{S}\setminus\\{0\\}}$${\mathbb{A}^{1}_{S}}$${0}$${\mathbb{A}^{1}_{S}}$${\mathbb{P}^{1}_{S}}$${(\mathbb{P}^{1}_{S},\infty),}$$\scriptstyle{\infty}$$\scriptstyle{0}$

where $(\mathbb{P}^{1},\infty)$ is defined as the pushout of the right square.
All squares in the diagram are a pushout thus we have

$(\mathbb{P}^{1}_{S},\infty)\simeq\frac{\mathbb{A}^{1}_{S}}{\mathbb{A}^{1}_{S}\setminus\\{0\\}}.$

######  Definition 4.12.

We define the category of rigid analytic motivic spectra over $S$

$\textup{RigSH}(S,{\mathcal{V}})\coloneqq\textup{RigSH}^{\textup{eff}}(S,{\mathcal{V}})[(\mathbb{P}^{1}_{S},\infty)^{-1}].$

######  Remark 4.13.

As remarked in [Hoy17, §6.1], the definition of inversion along an element
yields for a morphism of rigid spaces $f\colon S\rightarrow T$ an equivalence

$\textup{RigSH}^{\textup{eff}}(S,{\mathcal{V}})\otimes_{\textup{RigSH}^{\textup{eff}}(S,{\mathcal{V}})}\textup{RigSH}(T,{\mathcal{V}})\simeq\textup{RigSH}(S,{\mathcal{V}})$

via $f^{*}$ as symmetric monoidal categories. In particular, the argumentation
of the proof of Lemma 4.3 shows that $\textup{RigSH}(-,{\mathcal{V}})$ again
satisfies projection formula, smooth base change formula and localisation. We
refer to [Dre18, Prop. 8.5] for a rigorous proof of this statement.
Furthermore, by construction $(\mathbb{P}^{1}_{S},\infty)$ is
$\otimes$-invertible inside $\textup{RigSH}(S,{\mathcal{V}})$.

Lastly, let us remark that by design $\textup{RigSH}(S,{\mathcal{V}})$ is a
commutative ${\mathcal{V}}$-algebra in $\textup{Pr}^{L}$ [Rob15, Def. 2.6] as
$\textup{RigSH}^{\textup{eff}}(S,{\mathcal{V}})\in\textup{CAlg}(\textup{Pr}^{L,\otimes})_{/{\mathcal{V}}}$
by Remark 4.1.

Using this definition, we can conclude that
$X\mapsto\textup{RigSH}(X^{\textup{an}})$, for any $B$-scheme $X$, is a stable
homotopy functor [Ayo07] (see also [Dre18, Prop. 5.11]), i.e. it satisfies the
properties (1)-(6) listed in [Ayo07, §1.4.1].

In particular, [Ayo07, Scholie 1.4.2] implies the existence of a six functor
formalism on the restriction of RigSH to quasi-projective $K$-schemes. Using
the results of Cisinksi-Deglise, we can extend this to all separated finite
type $B$-schemes [CD19, Thm. 2.4.50].

######  Theorem 4.14.

The functor999The functoriality follows similarly as in [Rob14, §9.1].

$\displaystyle\textup{RigSH}_{{\mathcal{V}}}\colon(\textup{Sch}^{\textup{sep,ft}}_{B})^{\operatorname{op}}\rightarrow\textup{CAlg}(\textup{Pr}^{L,\otimes})_{/{\mathcal{V}}},\quad
X\mapsto\textup{RigSH}(X^{\textup{an}},{\mathcal{V}}),\quad f\mapsto f^{*}$

satisfies Nisnevich descent and extends to a full six functor formalism in the
sense of [LZ12]. Further, by passing to homotopy categories this functor
defines a motivic triangulated functor in the sense of [CD19].

###### Proof.

Nisnevich descent follows from Proposition 3.20.

By [Dre18, Prop. 5.11] together with Remark 3.17, Lemma 4.3, Proposition 4.9
and Lemma 4.10 the composition of $\textup{RigSH}_{{\mathcal{V}}}$ with the
homotopy functor yields a stable homotopy $2$-functor in the sense of [Ayo07,
§1.4.1]. Thus, using that any finite type separated $B$-scheme admits a Nagata
compactification [Con07, Thm. 4.1], we can apply [CD19, Thm. 2.4.26] and
[Man22, Prop A.5.10] to obtain a pre-six functor formalism in the sense of
[Man22]. To lift this to a full six functor formalism, we are left to show
that for any proper map of separated finite type $B$-schemes $f\colon
X\rightarrow Y$ the functor $f_{*}$ admits a right adjoint [Man22, Prop.
A.5.10].

Let us assume that ${\mathcal{V}}\simeq\operatorname{Sp}$, then this follows
from [Dre18, Prop. 5.17]. Now let ${\mathcal{W}}$ be a symmetric monoidal
stable presentable category. Since $f_{*}$ is a morphism inside $\Pr^{L}$, so
is the right adjoint $f_{*}\otimes\operatorname{id}_{{\mathcal{W}}}$ of
$f^{*}\otimes\operatorname{id}_{{\mathcal{W}}}$ [Dre18, Lem. 8.4]. In
particular, $f_{*}\otimes\operatorname{id}_{{\mathcal{W}}}$ admits a right
adjoint. ∎

###  4.4 The relation with $\mathbb{P}^{1}$-spectra

In this subsection, we shortly want to relate our definition of the stable
motivic homotopy category with the more classical approach via
$\mathbb{P}^{1}$-spectra. This will be an easy consequence from our
constructions and the analytification functor

$(-)^{\textup{an}}\colon\textup{SH}^{\textup{eff}}(B)\overset{}{\rightarrow}\textup{RigSH}^{\textup{eff}}(B^{\textup{an}}).$

Let us first recall what we mean with $\mathbb{P}^{1}$-spectra following Hovey
[Hov01].A slightly more thorough discussion can be found in [Rob14, §2.3]. Let
${\mathcal{C}}$ be a combinatorial simplicial symmetric monoidal model
category, i.e. a presentable symmetric monoidal category. For any (cofibrant)
object $X\in{\mathcal{C}}$, in fact for any left Quillen endofunctor, one can
define the model category of $X$-spectra in ${\mathcal{C}}$, denoted by
$\operatorname{Sp}^{\mathbb{N}}({\mathcal{C}},X)$. The underlying $1$-category
consists of $X$-spectra $F$, which is a sequence $(F_{0},F_{1},\dots)$ in
${\mathcal{C}}$ together with morphisms $X\otimes F_{n}\rightarrow F_{n+1}$
for all $n\in\mathbb{N}_{0}$. The notion of morphism of $X$-spectra is the
obvious one. We endow $\operatorname{Sp}^{\mathbb{N}}({\mathcal{C}},X)$ with
the stable model structure [Hov01, §3]. Let us remark that in general
$\operatorname{Sp}^{\mathbb{N}}({\mathcal{C}},X)$ is not symmetric monoidal.
For this reason one usually passes to symmetric spectra together with the
stable model structure, denoted by
$\operatorname{Sp}^{\Sigma}({\mathcal{C}},X)$. We do not want to go more into
detail concerning $\operatorname{Sp}^{\Sigma}({\mathcal{C}},X)$ and refer to
[Hov01, §7]. It is important to remark that the associated categories via the
simplicial nerve of $\operatorname{Sp}^{\Sigma}({\mathcal{C}},X)$ and
$\operatorname{Sp}^{\mathbb{N}}({\mathcal{C}},X)$ are not equivalent in
general. However, as discussed in [Hov01, §10], one can see that the only
obstruction is the cyclic permutation $(123)\in\Sigma_{3}$ self map of
$X^{\otimes 3}\coloneqq X\otimes X\otimes X$. To be more precise, Hovey shows
that there is a zig-zag of Quillen equivalences between
$\operatorname{Sp}^{\Sigma}({\mathcal{C}},X)$ and
$\operatorname{Sp}^{\mathbb{N}}({\mathcal{C}},X)$ if the cyclic permutation on
$X\otimes X\otimes X$ is homotopic to the identity. In this case $X$ is called
symmetric. Moreover, Robalo shows that if $X$ is symmetric, we have an
equivalence

${\mathcal{C}}[X^{-1}]\overset{\simeq}{\longrightarrow}\operatorname{Sp}^{\Sigma}({\mathcal{C}},X)$

as presentable symmetric monoidal categories [Rob15, Thm. 2.26].

Now let us come back to our situation and show how we can interpret RigSH as
$\mathbb{P}^{1}$-spectra. For this consider the category
$\textup{RigH}(B^{\textup{an}})^{\wedge}_{*}\coloneqq\textup{RigH}(B^{\textup{an}},\textup{Spc}_{*})$
with the induced $\otimes$-product via Remark 4.1. We also denote by
$\operatorname{\mathbb{G}_{m}}\coloneqq\mathbb{A}^{1}\setminus\\{0\\}$ the
analytic affine line without $0$.

######  Lemma 4.15.

The pointed projective line
$(\mathbb{P}^{1},\infty)\in\textup{RigH}(B^{\textup{an}})^{\wedge}_{*}$ is
symmetric and equivalent to $S^{1}\otimes(\operatorname{\mathbb{G}_{m}},1)$.

In particular, we have

$\textup{RigSH}(B^{\textup{an}})\simeq\operatorname{Sp}^{\mathbb{N}}(\textup{RigH}(B^{\textup{an}})^{\wedge}_{*},(\mathbb{P}^{1},\infty)).$

###### Proof.

Combining our discussion in Section 4.1 and Lemma 3.13, we see that
$(-)^{\textup{an}}$ is a monoidal functor. The cyclic permutation of
$(\mathbb{P}^{1},\infty)^{\otimes 3}\in\textup{H}(B)^{\wedge}_{*}$ is
equivalent to the identity [Voe98, Lem. 4.4]. In particular, the same is true
for $(\mathbb{P}^{1},\infty)^{\textup{an}}$. As the analytification functor
preserves colimits, we see that
$(\mathbb{P}^{1},\infty)^{\textup{an}}\simeq(\mathbb{P}^{1},\infty)\in\textup{RigH}(B^{\textup{an}})^{\wedge}_{*}$.

Moreover, by the same argument we have

$(\mathbb{P}^{1},\infty)\simeq
S^{1}\otimes(\operatorname{\mathbb{G}_{m}},1)\in\textup{RigH}(B^{\textup{an}})^{\wedge}_{*}$

[MV99, Lem. 2.15]. Therefore, we have

$\textup{RigSH}(B^{\textup{an}})\simeq\textup{RigH}(B^{\textup{an}})^{\wedge}_{*}[(\mathbb{P}^{1},\infty)^{-1}],$

as presentable symmetric monoidal categories. ∎

######  Corollary 4.16.

There is an equivalence of presentable symmetric monoidal categories

$\textup{RigSH}(B^{\textup{an}},{\mathcal{V}})\simeq\operatorname{Sp}^{\mathbb{N}}(\textup{RigSH}^{\textup{eff}}(B^{\textup{an}},{\mathcal{V}}),(\mathbb{P}^{1},\infty)).$

###### Proof.

This follows from Lemma 4.15, the monoidality in Section 4.1 and the
discussion in the beginning of this subsection. ∎

##  5 Applications to the K-theory of rigid spaces

Notation. In this entire section, let $(R,R^{+})$ be a Huber pair such that
$R$ is a Tate ring admitting a ring of definition $R_{0}$ which is noetherian
and of finite dimension.101010We need this assumption since pro-cdh descent is
known only for noetherian schemes of finite dimension. Once the latter is
known in a greater generality, our results will be known as well in this
greater generality. We set $S=\operatorname{Spa}(R,R^{+})$ and choose a
pseudo-uniformiser $\pi\in R$ so that $R=R_{0}[\pi^{-1}]$.

###  5.1 Continuous K-theory and analytic K-theory

In this section we recall the definition and basic properties of analytic
K-theory for rigid spaces due to Kerz-Saito-Tamme [KST19a, KST23]. Some
results use the following assumption, cf. [KST19a, §3.2].

Assumption $(\dagger)_{R}$. Every regular and topologically of finite type
$R$-algebra $A$ admits a noetherian ring of definition $A_{0}$ and an
admissible desingularisation, i.e. a proper morphism of schemes $p\colon
X\overset{}{\rightarrow}\operatorname{Spec}(A_{0})$ with $X$ regular such that
the restriction
$p^{-1}(\operatorname{Spec}(A))\overset{}{\rightarrow}\operatorname{Spec}(A)$
is an isomorphism.

#### Pro-categories

Analytic K-theory of rigid spaces is naturally a pro-object in the category of
spectra. Let us quickly recall some facts about pro-categories [Lur18,
§A.8.1]. Let ${\mathcal{C}}$ be a presentable category. Then we can define
$\textup{Pro}({\mathcal{C}})$ as those functors in
$\operatorname{Fun}({\mathcal{C}},\textup{Spc})$, which are accessible and
preserve finite limits. Equivalently, we have
$\textup{Pro}({\mathcal{C}})\simeq\textup{Ind}({\mathcal{C}}^{\operatorname{op}})^{\operatorname{op}}$.
An object $X\in\textup{Pro}({\mathcal{C}})$ can be thought of a limit of a
filtered diagram $\\{X_{i}\\}$, where each $X_{i}$ is contained in the
essential image of the Yoneda embedding
${\mathcal{C}}\rightarrow\textup{Pro}({\mathcal{C}})$. Following [KST19a], we
denote these objects as $\operatorname*{``lim^{\prime\prime}}X_{i}$. We have a
limit functor $\textup{Pro}({\mathcal{C}})\rightarrow{\mathcal{C}}$ that maps
an object $\operatorname*{``lim^{\prime\prime}}X_{i}$ to $\lim
X_{i}\in{\mathcal{C}}$.

Let us remark, that finite limits and colimits in
$\textup{Pro}({\mathcal{C}})$ can be computed pointwise in the following
sense. Let $K$ be a finite simplicial set. The limit functor
$\operatorname{Fun}(K,{\mathcal{C}})\overset{}{\rightarrow}{\mathcal{C}}$
induces a functor
$\textup{Pro}(\operatorname{Fun}(K,{\mathcal{C}}))\rightarrow\textup{Pro}({\mathcal{C}})$,
which we call level-wise limit (similarly for colimits). Furthermore, there
exists a functor
$\textup{Pro}(\operatorname{Fun}(K,{\mathcal{C}}))\rightarrow\operatorname{Fun}(K,\textup{Pro}({\mathcal{C}}))$.
Using this functor we obtain that the levelwise limit of
$F\in\textup{Pro}(\operatorname{Fun}(K,{\mathcal{C}}))$ is the limit of the
induced diagram $K\rightarrow\textup{Pro}({\mathcal{C}})$ (similarly for
colimits) [KST19a, Lem. 2.1].

Let ${\mathcal{C}}=\operatorname{Sp}$. Then $\textup{Pro}(\operatorname{Sp})$
is stable (Lemma 5.2) and we can define homotopy groups on
$\textup{Pro}(\operatorname{Sp})$ pointwise [KST19a]. To be more precise, for
any $n\in\mathbb{Z}$ and any
$\operatorname*{``lim^{\prime\prime}}X_{i}\in\textup{Pro}(\operatorname{Sp})$,
we set

$\pi_{n}(\operatorname*{``lim^{\prime\prime}}X_{i})\coloneqq\operatorname*{``lim^{\prime\prime}}\pi_{n}X_{i}\in\textup{Pro}(\textup{Ab}).$

We will see that $\textup{Pro}(\operatorname{Sp})$ admits a $t$-structure and
the pointwise homotopy groups are precisely the homotopy groups induced by
this $t$-structure (Lemma A.18).

Lastly, let us recall the notion of weak equivalence. In op. cit. it is
shown that there is a functor
$\iota^{*}\colon\textup{Pro}(\operatorname{Sp})\rightarrow\textup{Pro}(\operatorname{Sp}^{+})$,
where $\operatorname{Sp}^{+}$ denotes the category of bounded above spectra.
On objects this functor is given by

$\iota^{*}(\operatorname*{``lim^{\prime\prime}}_{I}X_{i})\simeq\operatorname*{``lim^{\prime\prime}}_{\mathbb{N}\times
I}\tau_{\leq n}X_{i}.$

A morphism of pro-spectra $f\colon X\rightarrow Y$ is a weak equivalence if
$\iota^{*}f$ is an equivalence. Analogously, a fibre sequence in
$\textup{Pro}(\operatorname{Sp})$ is called a weak fibre sequence if it is a
fibre sequence after applying $\iota^{*}$. Note that $f$ is weak equivalence
if and only if $\tau_{\leq n}f$ is an equivalence for some $n\in\mathbb{Z}$
and $\pi_{i}f$ is an isomorphism for all $i\in\mathbb{Z}$ [KST19a, Lem. 2.8].

######  Definition 5.1.

For an category ${\mathcal{C}}$ denote by
$\textup{Pro}^{\omega}({\mathcal{C}})$ the full subcategory of
$\textup{Pro}({\mathcal{C}})$ of objects $X$ that are representable by
diagrams $X\colon I\overset{}{\rightarrow}{\mathcal{C}}$ for a _countable_
cofiltered category $I$.

######  Lemma 5.2.

Let ${\mathcal{C}}$ be a stable category. Then the categories
$\textup{Pro}({\mathcal{C}})$ and $\textup{Pro}^{\omega}({\mathcal{C}})$ both
are stable. In particular, the categories
$\textup{Pro}(\operatorname{Sp}^{+})$ and
$\textup{Pro}^{\omega}(\operatorname{Sp}^{+})$ are stable.

###### Proof.

Since ${\mathcal{C}}$ is stable, the pro-category
$\textup{Pro}({\mathcal{C}})$ is stable as well [KST19a, 2.5]. Since finite
colimits in pro-categories can be computed levelwise, the inclusion
$\textup{Pro}^{\omega}({\mathcal{C}})\overset{}{\hookrightarrow}\textup{Pro}({\mathcal{C}})$
creates finite colimits. Hence $\textup{Pro}^{\omega}({\mathcal{C}})$ has
admits finite colimits and the suspension functor is an equivalence. Thus
$\textup{Pro}^{\omega}({\mathcal{C}})$ is stable [Lur17, 1.4.2.27]. The rest
follows as the category $\operatorname{Sp}^{+}$ is a stable subcategory of
$\operatorname{Sp}$ [Lur17, p. 44]. ∎

In the work of Kerz-Saito-Tamme, the notion of _pro-homotopy invariance_ plays
a major role when working with rings [KST19b], [KST19a, §5.3]. For sheaves,
pro-homotopy invariance is equivalent to $\mathbb{A}^{1}$-invariance.

######  Lemma 5.3.

A sheaf
$F\colon\textup{Rig}_{S}^{\operatorname{op}}\overset{}{\rightarrow}\textup{Pro}({\mathcal{C}})$
is $\mathbb{A}^{1}$-invariant if and only if it is
pro-$\mathbb{B}^{1}$-invariant, i.e. if the canonical map

$F(X)\overset{}{\longrightarrow}\operatorname*{``lim^{\prime\prime}}_{t\mapsto\pi
t}F(X\times\mathbb{B}^{1})$

is an equivalence in $\textup{Pro}({\mathcal{C}})$.

###### Proof.

As in the proof of Lemma 3.3, we compute111111Note that a pro-system
$\operatorname*{``lim^{\prime\prime}}_{i}C_{i}$ is the limit of the constant
pro-systems given by the $C_{i}$ in $\textup{Pro}({\mathcal{C}})$.

$\displaystyle F(X\times\mathbb{A}^{1})\simeq
F(\mathop{\textup{colim}}_{t\mapsto\pi
t}(X\times\mathbb{B}^{1}))\simeq\operatorname*{``lim^{\prime\prime}}_{t\mapsto\pi
t}F(X\times\mathbb{B}^{1}).$

Hence the map $F(X)\overset{}{\rightarrow}F(X\times\mathbb{A}^{1})$ is an
equivalence if and only if the map
$F(X)\overset{}{\rightarrow}\operatorname*{``lim^{\prime\prime}}_{t\mapsto\pi
t}F(X\times\mathbb{B}^{1})$ is an equivalence. ∎

#### Connective analytic K-theory

The idea of (connective) analytic K-theory is to define an analogue of
topological K-theory for rigid spaces. The new idea in [KST19a] is to view
analytic K-theory as an $\mathbb{A}^{1}$-invariant object. This construction
can be enhanced to endow analytic K-theory with the structure of a pro-
spectrum via (2.0.1). Let us be more precise: We define the standard
$n$-simplex with radius $1/\pi^{m}$ as

$\Delta^{n}_{m}\coloneqq\operatorname{Spa}(R\langle\pi^{m}T_{0},\dots,\pi^{m}T_{n}\rangle/(T_{0}+\dots+T_{n}-1),R^{+}\langle\pi^{m}T_{0},\dots,\pi^{m}T_{n}\rangle/(T_{0}+\dots+T_{n}-1)).$

There exists a (non-canonical) $S$-isomorphism between $\Delta^{n}_{m}$ and
$\mathbb{B}^{n}_{1/\pi^{m}}$. Via the standard structure maps we obtain a
cosimplicial rigid space $\Delta^{\bullet}_{m}$. Let $X$ be a rigid space. The
functor
$\Delta^{\operatorname{op}}\overset{}{\rightarrow}\operatorname{Sp}_{\geq 0}$
given by $[n]\mapsto\textup{K}_{\geq 0}(X\times\Delta^{n}_{m})$, where
$\textup{K}_{\geq 0}(-)$ denotes the connective algebraic K-theory spectrum,
defines via geometric realisation a connective spectrum

$\textup{K}_{\geq
0}(X\times\Delta_{m})\coloneqq\mathop{\textup{colim}}_{\Delta^{\operatorname{op}}}\textup{K}_{\geq
0}({\mathcal{O}}(X\times\Delta^{\bullet}_{m})).$

Taking the limit over $m\geq 0$, we can define connective analytic K-theory as
the pro-spectrum

$\textup{k}^{\textup{an}}(X)\coloneqq\operatorname*{``lim^{\prime\prime}}_{m}\textup{K}_{\geq
0}(X\times\Delta_{m}).$

By design, connective analytic $K$-theory is pro-$\mathbb{B}^{1}$-invariant
[KST19a, Prop. 6.3] and hence $\mathbb{A}^{1}$-invariant (Lemma 5.3), i.e.

$\textup{k}^{\textup{an}}(X)\simeq\textup{k}^{\textup{an}}(X\times\mathbb{A}^{1}).$

Applying the same construction to the connected cover $\textup{K}_{\geq 1}$ we
obtain the pro-spectrum $\textup{k}^{\textup{an}}_{\geq 1}(X)$. Note however
that $\textup{k}^{\textup{an}}_{\geq 1}(X)$ is not the connected cover of
$\textup{k}^{\textup{an}}_{\geq 0}(X)$. Nevertheless, we obtain a weak fibre
sequence

$\displaystyle\textup{k}^{\textup{an}}_{\geq
1}(X)\rightarrow\textup{k}^{\textup{an}}(X)\rightarrow\textup{K}_{0}({\mathcal{O}}(X)).$
(5.3.1)

#### Analytic K-theory

Non-connective analytic K-theory is defined by delooping 1-connective analytic
K-theory via the analytic Bass construction [KST19a, §6.3]. Let us denote by
$\mathbb{B}^{-1}$ the adic space $\operatorname{Spa}(R\langle
t^{-1}\rangle,R^{+}\langle t^{-1}\rangle)$. We define
$\Lambda\textup{k}^{\textup{an}}_{\geq 1}$ via the following fibre sequence of
pro-spectra

$\Lambda\textup{k}^{\textup{an}}_{\geq
1}(X)\rightarrow\textup{k}^{\textup{an}}_{\geq
1}(X\times\mathbb{B}^{1})\sqcup_{\textup{k}^{\textup{an}}_{\geq
1}(X)}\textup{k}^{\textup{an}}_{\geq
1}(X\times\mathbb{B}^{-1})\rightarrow\textup{k}^{\textup{an}}_{\geq
1}(X\times\mathbb{B}^{1}\times\mathbb{B}^{-1}).$

Iterating this process, we can define (non-connective) analytic K-theory via
the analytic Bass construction

$\textup{K}^{\textup{an}}(X)\simeq\mathop{\textup{colim}}(\textup{k}^{\textup{an}}_{\geq
1}(X)\xrightarrow{\lambda}\Lambda\textup{k}^{\textup{an}}_{\geq
1}(X)\xrightarrow{\Lambda(\lambda)}\Lambda^{2}\textup{k}^{\textup{an}}_{\geq
1}(X)\rightarrow\dots),$

for more details see section 5.3 below. Though not a purely formal
consequence, it turns out that analytic K-theory satisfies the Bass
Fundamental Theorem, i.e. for any $i\in\mathbb{Z}$ and any affinoid rigid
space $X=\operatorname{Spa}(A,A^{+})$ over $S$ we have an exact sequence

$0\rightarrow\textup{K}^{\textup{an}}_{i}(X)\rightarrow\textup{K}^{\textup{an}}_{i}(X\times\mathbb{B}^{1})\oplus\textup{K}^{\textup{an}}_{i}(X\times\mathbb{B}^{-1})\rightarrow\textup{K}^{\textup{an}}_{i}(X\times\mathbb{B}^{1}\times\mathbb{B}^{-1})\rightarrow\textup{K}^{\textup{an}}_{i-1}(X)\rightarrow
0$

and the map
$\textup{K}^{\textup{an}}_{i}(X\times\mathbb{B}^{1}\times\mathbb{B}^{-1})\rightarrow\textup{K}^{\textup{an}}_{i-1}(X)$
admits a split [KST23, Cor. 2.6].

Again, we define the $i$-th analytic K-group of $X$ as

$\textup{K}^{\textup{an}}_{i}(X):=\pi_{i}\textup{K}^{\textup{an}}(X)\in\textup{Pro}(\textup{Ab}).$

Alternatively, we could apply the Bass construction to
$\textup{k}^{\textup{an}}(X)$ and get a pro-spectrum
$(\textup{k}^{\textup{an}}(X))^{B}$. In general is not clear if both
constructions are equivalent. However, if $X$ is regular and we assume
$(\dagger)$, then
$\pi_{i}(\textup{k}^{\textup{an}}(X))^{B}\cong\textup{K}^{\textup{an}}_{i}(X)$
for all $i\geq 0$. We will recall below from [KST23] that the construction via
$\textup{k}^{\textup{an}}_{\geq 1}$ yields better formal properties.

#### Continuous K-theory

The definition of continuous K-theory is due to Morrow [Mor16] and the theory
has been elaborated by Kerz-Saito-Tamme [KST19a].

Let $A_{0}$ be $\pi$-adic ring for some $\pi\in A_{0}$. Then we set
$\textup{K}^{\textup{cont}}(A_{0})\coloneqq\operatorname*{``lim^{\prime\prime}}\textup{K}(A_{0}/\pi^{n})$
which is an object in $\textup{Pro}^{\omega}(\operatorname{Sp})$. For a Tate
ring $A$ with ring of definition $A_{0}$, we can define
$\textup{K}^{\textup{cont}}(A)$ as the pushout of the diagram , up to weak
equivalence

${\textup{K}(A^{\circ})}$${\textup{K}(A)}$${\textup{K}^{\textup{cont}}(A_{0})}$${\textup{K}^{\textup{cont}}(A).}$

within the category $\textup{Pro}^{\omega}(\operatorname{Sp})$. When passing
to the localisation $\textup{Pro}^{\omega}(\operatorname{Sp}^{+})$, the object
$\textup{K}^{\textup{cont}}(A)$ does not depend on the choice of $A_{0}$
[KST19a, Prop. 5.4].

######  Remark 5.4.

(K-theory of nuclear modules) As a matter of fact, the underlying spectrum
$\underline{\textup{K}}^{\textup{cont}}(A)$, i.e. the image under the limit
functor
$\textup{Pro}^{\omega}(\operatorname{Sp})\overset{}{\rightarrow}\operatorname{Sp}$,
identifies with the K-theory of the dualisable category $\textup{Nuc}(A_{0})$
of nuclear $A_{0}$-modules. As a consequence, we further get an equivalence
$\underline{\textup{K}}^{\textup{cont}}(A)\simeq\textup{K}(\textup{Nuc}(A))$.
For more details, we refer to Andreychev’s thesis [And23, Satz 5.8];
unfortunately, there seems to be no full proof in the literature.

######  Proposition 5.5.

If $X$ is regular and assuming $(\dagger)_{R}$, then
$\textup{K}^{\textup{cont}}$ is $\mathbb{A}^{1}$-invariant, i.e. the canonical
map
$\textup{K}^{\textup{cont}}(X)\overset{\simeq}{\rightarrow}\textup{K}^{\textup{cont}}(X\times\mathbb{A}^{1})$
is an equivalence in $\textup{Pro}(\operatorname{Sp}^{+})$. Moreover, the
connective cover of $\textup{K}^{\textup{cont}}(X)$ agrees with
$\textup{k}^{\textup{an}}(X)$.

###### Proof.

The claims can be checked locally. In this case,
$\textup{K}^{\textup{cont}}(X)\simeq\textup{K}^{\textup{an}}(X)$ [KST19a, Thm.
6.19] and $\textup{K}^{\textup{an}}$ is pro-$\mathbb{B}^{1}$-invariant by
design, hence $\mathbb{A}^{1}$-invariant by Lemma 5.3. ∎

#### Descent for analytic $K$-theory

######  Lemma 5.6 (Nisnevich descent for analytic K-theory).

The presheaf

$\textup{K}^{\textup{an}}\,\colon\,\textup{Rig}_{S}^{\textup{lft},\operatorname{op}}\overset{}{\longrightarrow}\textup{Pro}(\operatorname{Sp}^{+}),\quad\operatorname{Spa}(A,A^{+})\mapsto\textup{K}^{\textup{an}}(A),$

is a Nisnevich sheaf.

###### Proof.

Kerz-Saito-Tamme have shown that analytic K-theory on affinoid adic spaces
over $S$ satisfies Zariski-descent and extends essentially uniquely to a
Zariski sheaf on all adic spaces that are locally of finite type over $S$
[KST23, Thm. 4.1].

For Nisnevich descent, we note that every object in
$\textup{Rig}_{S}^{\textup{lft}}$ is an analytic adic space. Thus we can check
whether a presheaf is a Nisnevich sheaf on elementary Nisnevich squares
[And23, Kor. A.28].

Let $(A,A^{\circ})$ be a Huber pair such that $A$ is Tate. Recall, that we
assume that $A^{\circ}$ is noetherian and satisfies ($\dagger$). Then there is
a fibre sequence [KST23, Thm. 2.10]

$\textup{K}^{\pi^{\infty}}(A)\rightarrow\textup{K}^{\textup{cont}}(A)\rightarrow\textup{K}^{\textup{an}}(A)$

where

$\textup{K}^{\pi^{\infty}}(A)\coloneqq\operatorname*{``lim^{\prime\prime}}_{m}\mathop{\textup{colim}}_{[n]\in\Delta}\textup{K}(A\otimes_{R}R[T_{0},\dots,T_{n}]/(T_{0}+\dots+T_{n}-\pi^{m})).$

Hence one can reduce Nisnevich descent of $\textup{K}^{\textup{an}}$ to
Nisnevich descent of $\textup{K}^{\pi^{\infty}}$ and
$\textup{K}^{\textup{cont}}$. Both $\textup{K}^{\pi^{\infty}}$ and
$\textup{K}^{\textup{cont}}$ are induced by pro-systems of algebraic K-theory.
As the Čech nerve of an open cover is $n$-skeletal, we may use that finite
limits in $\textup{Pro}(\operatorname{Sp}^{+})$ can be computed levelwise and
reduced to Nisnevich descent of algebraic K-theory. ∎

###  5.2 Identifying the representing object of analytic K-theory

Analytic K-theory
$\textup{K}^{\textup{an}}\colon\textup{Rig}_{S}^{\textup{lft},\operatorname{op}}\overset{}{\rightarrow}\textup{Pro}^{\omega}(\operatorname{Sp}^{+})$
is an $\mathbb{A}^{1}$-invariant Nisnevich sheaf, see the preceding
subsection. Since both $\tau_{\geq 1}$ and $\Omega^{\infty}$ are right adjoint
functors, the induced functor

$\textup{Rig}_{S}^{\textup{lft},\operatorname{op}}\overset{}{\rightarrow}\textup{Pro}^{\omega}(\operatorname{Sp}^{+})\overset{}{\longrightarrow}\textup{Pro}^{\omega}(\operatorname{Sp})\overset{\tau_{\geq
1}}{\longrightarrow}\textup{Pro}^{\omega}(\operatorname{Sp}_{\geq
1})\overset{\Omega^{\infty}}{\longrightarrow}\textup{Pro}^{\omega}(\textup{Spc})$

is again an $\mathbb{A}^{1}$-invariant Nisnevich sheaf, hence an object in
$\textup{RigH}(k,\textup{Pro}(\textup{Spc}))$ which we denote by
$\Omega^{\infty}\textup{K}^{\textup{an}}_{\geq 0}$. We would like to identify
this object with more concrete objects. Unfortunately, we only can do this
under the assumption $(\dagger)_{R}$ in the beginning of Section 5.1. In this
case we have for any $R$-algebra $A$ of topologically finite type that
[KST19a, Lem. 7.5]

$\Omega^{\infty}\tau_{\geq
1}\textup{K}^{\textup{an}}(A)\simeq\textup{KV}^{\textup{an}}(A):=\operatorname*{``lim^{\prime\prime}}_{\rho}\,\operatorname{BGL}(A\langle\Delta\rangle_{\rho}).$

This equivalence enables us to identify analytic K-theory similarly as the
classical identification of algebraic K-theory with
$\mathbb{Z}\times\operatorname{BGL}$ in the Morel-Voevodsky category [MV99,
Prop. 3.10].

######  Theorem 5.7 (Representability).

We assume $(\dagger)_{R}$. Then there is a canonical equivalence

$\Omega^{\infty}\textup{K}^{\textup{an}}_{\geq
0}\simeq\textup{L}_{\textup{mot}}(\mathbb{Z}\times\operatorname{BGL})$

in the category $\textup{RigH}(S)$. In particular, for every rigid space $X$
there is a functorial equivalence

$\Omega^{\infty}\textup{K}^{\textup{an}}_{\geq
0}(X)\simeq\underline{\operatorname{Hom}}_{\textup{Cond}^{\omega}(\textup{Spc})}(\textup{L}_{\textup{mot}}X,\textup{L}_{\textup{mot}}(\mathbb{Z}\times\operatorname{BGL}))$

in the category $\textup{Cond}^{\omega}(\textup{Spc})$; here the right-hand
side denotes the enriched Hom-space as a left
$\textup{Cond}^{\omega}(\textup{Spc})$-module, see Appendix B.

###### Proof.

First, we construct a map. For every affinoid algebra $A$ we have a map

$\mathbb{Z}(A)\times\operatorname{BGL}(A)\overset{}{\longrightarrow}\Omega^{\infty}\textup{K}_{\geq
0}(A)\overset{}{\longrightarrow}\Omega^{\infty}\textup{K}^{\textup{an}}_{\geq
0}(A)$

which is functorial in $A$, hence inducing a map
$\mathbb{Z}\times\operatorname{BGL}\overset{}{\rightarrow}\Omega^{\infty}\textup{K}^{\textup{an}}_{\geq
0}$ on the category of presheaves on affinoid rigid spaces. Thus we obtain an
induced map
$\textup{L}_{\textup{mot}}(\mathbb{Z}\times\operatorname{BGL})\overset{}{\rightarrow}\Omega^{\infty}\textup{K}^{\textup{an}}_{\geq
0}$ in $\textup{RigH}(S)$.

Using the fibre sequence (5.3.1) we can separately investigate the induced
maps
$\textup{L}_{\textup{mot}}\operatorname{BGL}\overset{}{\rightarrow}\Omega^{\infty}\textup{K}^{\textup{an}}_{\geq
1}$ on 1-connective covers and
$\textup{L}_{\textup{mot}}\mathbb{Z}\overset{}{\rightarrow}\textup{K}^{\textup{an}}_{0}$
in degree zero. On 1-connective covers, we compute for every affinoid algebra
$A$ as follows.121212Note that $\operatorname{BGL}$ arises from sheafifying
the composition of the presheaf
$\textup{Rig}_{S}^{\operatorname{op}}\overset{}{\rightarrow}\textup{Spc},\,X\mapsto\operatorname{BGL}({\mathcal{O}}_{X}(X)),$
with the inclusion
$\textup{Spc}\overset{}{\hookrightarrow}\textup{Pro}(\textup{Spc})$.

$\displaystyle(\textup{L}_{\mathbb{A}^{1}}\operatorname{BGL})(\operatorname{Spa}(A))$
$\displaystyle\simeq\mathop{\textup{colim}}_{n\in\Delta^{\operatorname{op}}}\,\operatorname{BGL}(A\langle\Delta^{n}\rangle)$
$\displaystyle\simeq\mathop{\textup{colim}}_{n\in\Delta^{\operatorname{op}}}\,\operatorname{Hom}(\mathop{\textup{colim}}_{\rho}\operatorname{Spa}(A\langle\Delta^{n}\rangle_{\rho}),\operatorname{BGL})$
$\displaystyle\simeq\mathop{\textup{colim}}_{n\in\Delta^{\operatorname{op}}}\,\lim_{\rho}\operatorname{Hom}(\operatorname{Spa}(A\langle\Delta^{n}\rangle_{\rho}),\operatorname{BGL}).$
$\displaystyle(\clubsuit)$
$\displaystyle\simeq\lim_{\rho\in\mathbb{N}^{\operatorname{op}}}\,\mathop{\textup{colim}}_{n\in\Delta^{\operatorname{op}}}\,\operatorname{Hom}(\operatorname{Spa}(A\langle\Delta^{n}\rangle_{\rho}),\operatorname{BGL})$
$\displaystyle\simeq\lim_{\rho\in\mathbb{N}^{\operatorname{op}}}\,\operatorname{BGL}(A\langle\Delta\rangle_{\rho})$
$\displaystyle\simeq\textup{KV}^{\textup{an}}(A)\simeq\Omega^{\infty}\tau_{\geq
1}\textup{K}^{\textup{an}}(A)$

where the equivalence $(\clubsuit)$ follows from [KST23, Lemma 2.8]. In degree
zero we have that
$\textup{K}^{\textup{an}}_{0}(A)\cong\textup{K}^{\textup{cont}}_{0}(A)\cong\textup{K}_{0}(A)\cong\mathbb{Z}$
for local rings $A$ which shows the first claim.

It remains to show that

$\Omega^{\infty}\textup{K}^{\textup{an}}_{\geq
0}(X)\simeq\underline{\operatorname{Hom}}_{\textup{Cond}^{\omega}(\textup{Spc})}(\textup{L}_{\textup{mot}}X,\textup{L}_{\textup{mot}}(\mathbb{Z}\times\operatorname{BGL})).$

For this let us note that by construction, we have that
$\textup{L}_{\textup{mot}}$, as a localisation
from$\operatorname{Fun}(\textup{SmRig}_{S}^{\operatorname{op}},\textup{Cond}^{\omega}(\textup{Spc}))\overset{}{\rightarrow}\textup{RigH}(S,\textup{Cond}^{\omega}(\textup{Spc}))$,
is equivalently given by basechange of the motivic localisation functor
$\textup{PSh}(\textup{SmRig}_{S})\overset{}{\rightarrow}\textup{RigH}(S)$. In
particular, it is compatible with the
$\textup{Cond}^{\omega}(\textup{Spc})$-module structure. Thus, we have

$\underline{\operatorname{Hom}}_{\textup{Cond}^{\omega}(\textup{Spc})}(\textup{L}_{\textup{mot}}X,\textup{L}_{\textup{mot}}(\mathbb{Z}\times\operatorname{BGL}))\simeq\underline{\operatorname{Hom}}_{\textup{Cond}^{\omega}(\textup{Spc})}(X,\textup{L}_{\textup{mot}}(\mathbb{Z}\times\operatorname{BGL})),$

where we view $X$ as a functor in
$\operatorname{Fun}(\textup{SmRig}_{S},\textup{Cond}^{\omega}(\textup{Spc}))$
via the composition of the Yoneda functor and the constant functor
$\textup{Spc}\overset{}{\rightarrow}\textup{Cond}^{\omega}(\textup{Spc})$. Our
results above together with the enriched Yoneda lemma finish the proof (cf.
Lemma B.3). ∎

######  Definition 5.8.

For $n\geq d\geq 1\in\mathbb{Z}$ denote by $\textup{Grass}_{d,n}$ be the
Grassmannian $R$-scheme representing submodules
${\mathcal{E}}\subseteq{\mathcal{O}}^{n}$ such that the quotient sheaf
${\mathcal{O}}^{n}/{\mathcal{E}}$ is locally free of rank $n-d$ [GW10, (8.4)].
We write $\textup{Grass}_{d}:=\mathop{\textup{colim}}_{n}\textup{Grass}_{d,n}$
and $\textup{Grass}:=\mathop{\textup{colim}}_{d}\textup{Grass}_{d}$. Denote by
$\textup{Grass}_{d,n}^{\textup{an}}$, $\textup{Grass}_{d}^{\textup{an}}$, and
$\textup{Grass}^{\textup{an}}$ the respective analytifications.

Note that $\textup{Grass}_{d,n}^{\textup{an}}$ coincides with the
representable functor $\textup{Grass}_{d-n}^{\textup{rig}}$ representing
$d$-dimensional subspaces of $\kappa(x)^{n}$ on points
$\operatorname{Spa}(\kappa(x))\overset{}{\rightarrow}S$.

######  Proposition 5.9 (Arndt, Sigloch [Sig16, Prop. 5.34]).

For every $n\geq 1$, there is an equivalence
$\operatorname{BGL}_{n}\simeq\textup{Grass}_{n}^{\textup{rig}}$ in
$\textup{RigH}(S)$. Consequently, there is an equivalence
$\operatorname{BGL}\simeq\textup{Grass}^{\textup{an}}$.

######  Corollary 5.10.

We assume $(\dagger)_{R}$. Then the image of connective algebraic K-theory
under the analytification functor
$\alpha\colon\textup{H}(\operatorname{Spec}(R),\textup{Spc})\overset{}{\rightarrow}\textup{RigH}(S,\textup{Cond}^{\omega}(\textup{Spc}))$
identifies with connective analytic K-theory.

###### Proof.

In $\textup{H}(\operatorname{Spec}(R),\textup{Spc})$, algebraic K-theory
coincides with $\textup{L}_{\textup{mot}}(\mathbb{Z}\times\operatorname{BGL})$
[MV99, Thm. 3.13]. In $\textup{RigH}(S,\textup{Cond}^{\omega}(\textup{Spc}))$,
connective analytic K-theory coincides
with$\textup{L}_{\textup{mot}}(\mathbb{Z}\times\operatorname{BGL})$ (cf.
Theorem 5.7). The identification $\alpha(\mathbb{Z})\simeq\mathbb{Z}$ is
clear. The identification $\alpha(\operatorname{BGL})\simeq\operatorname{BGL}$
follows since $\operatorname{BGL}\simeq\textup{Grass}$ on both sides by [MV99,
Prop. 3.7] and Proposition 5.9. ∎

###  5.3 Analytic Bass delooping

Nonconnective algebraic K-theory is obtained from connective algebraic
K-theory by Bass delooping within the category of spectra [TT90, §6]. The
construction of the delooping uses the cover of $\mathbb{P}^{1}$ by two copies
of $\mathbb{A}^{1}$ whose intersection is $\operatorname{\mathbb{G}_{m}}$.
Such a delooping construction can also be done globally within the category of
presheaves of spectra [Cis13, §2]. For analytic K-theory of rigid spaces, an
analytic analogue of the Bass construction is performed by Kerz-Saito-Tamme
within the category of pro-spectra and using the cover of $\mathbb{P}^{1}$ by
two copies of $\mathbb{B}^{1}$ whose intersection we denote by
$\mathbb{H}_{m}$. Putting 1-connective analytic K-theory into this machine,
one obtains, by definition, nonconnective analytic K-theory, see subsection
5.1. In this subsection, we examine such an analytic Bass delooping within the
category $\textup{PSh}(\textup{Rig}_{S},\textup{Pro}(\operatorname{Sp}^{+}))$
using the alternative cover of $\mathbb{P}^{1}$ by two copies of
$\mathbb{A}^{1}$ whose intersection is $\operatorname{\mathbb{G}_{m}}$, the
analytification of $\operatorname{Spec}(R[t,t^{-1}])$. It turns out that the
$\mathbb{A}^{1}$-delooping is equivalent to the $\mathbb{B}^{1}$-delooping so
that analytic K-theory also satisfies the Bass Fundamental Theorem for
$\mathbb{A}^{1}$ (Corollary 5.16). Our exposition follows closely the
description of the $\mathbb{B}^{1}$-analytic Bass delooping by Kerz-Saito-
Tamme [KST19a, §4.4].

Notation. We fix an embedding
$\mathbb{A}^{1}\overset{}{\hookrightarrow}\mathbb{P}^{1}$ and write
$\mathbb{P}^{1}\setminus\mathbb{A}^{1}=:\\{\infty\\}$ and
$\mathbb{A}^{-1}:=\mathbb{P}^{1}\setminus\\{O\\}$ where $O\in\mathbb{A}^{1}$
denotes the origin. Thus
$\operatorname{\mathbb{G}_{m}}=\mathbb{A}^{1}\cap\mathbb{A}^{-1}$ as subspaces
of $\mathbb{P}^{1}$. Analogously, we write
$\mathbb{B}^{1}:=\operatorname{Spa}(R\langle t\rangle,R^{+}\langle
t\rangle)\overset{}{\hookrightarrow}\mathbb{P}^{1}$ and
$\mathbb{B}^{-1}:=\operatorname{Spa}(R\langle t^{-1}\rangle,R^{+}\langle
t^{-1}\rangle)$, so that
$\mathbb{B}^{1}\cap\mathbb{B}^{-1}=\operatorname{Spa}(R\langle
t,t^{-1}\rangle,R^{+}\langle t,t^{-1}\rangle)=:\mathbb{H}_{\textup{m}}$.
Finally, let ${\mathcal{D}}$ be a stable presentable category; thus it carries
an essentially unique structure of a module over the category of spectra
[Lur17, 4.8.2.18].

######  Definition 5.11.

Let $E\in\textup{PSh}(\textup{Rig}_{S},{\mathcal{D}})$ be a presheaf. We
define a presheaf $\Gamma_{\mathbb{A}^{1}}E$ via

$\Gamma_{\mathbb{A}^{1}}E(X):=E(X\times\mathbb{A}^{1})\sqcup_{E(X)}E(X\times\mathbb{A}^{-1}).$

The commutativity of the square

$\textstyle{\operatorname{\mathbb{G}_{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{A}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{A}^{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{1}}$

yields a map
$\Gamma_{\mathbb{A}^{1}}E(X)\overset{}{\rightarrow}E(X\times\operatorname{\mathbb{G}_{m}})$
so that we can define a presheaf $\Lambda_{\mathbb{A}^{1}}E$ via

$\Lambda_{\mathbb{A}^{1}}E(X):=\textup{fib}\bigl{(}\Gamma_{\mathbb{A}^{1}}E(X)\overset{}{\rightarrow}E(X\times\operatorname{\mathbb{G}_{m}})\bigr{)}.$

Replacing $\mathbb{A}^{1}$, $\mathbb{A}^{-1}$, and
$\operatorname{\mathbb{G}_{m}}$ with $\mathbb{B}^{1}$, $\mathbb{B}^{-1}$, and
$\mathbb{H}_{\textup{m}}$, respectively, we get the presheaves
$\Gamma_{\mathbb{B}^{1}}E$ and $\Lambda_{\mathbb{B}^{1}}E$ as defined in the
affinoid setting by Kerz-Saito-Tamme [KST19a, §4.4].

######  Lemma 5.12.

Given a presheaf $E\in\textup{PSh}(\textup{Rig}_{S},{\mathcal{D}})$, there are
equivalences

$\Lambda_{\mathbb{A}^{1}}E(X)\simeq\Sigma\textup{fib}\bigl{(}E(X\times\mathbb{P}^{1})\overset{}{\rightarrow}E(X)\bigr{)}\simeq\Lambda_{\mathbb{B}^{1}}E(X)$

which are functorial in $X\in\textup{Rig}_{S}$.

###### Proof.

We have a map of cartesian squares

${E(X)}$${E(X\times\mathbb{A}^{1})}$${E(X\times\mathbb{P}^{1})}$${E(X\times\mathbb{A}^{1})}$${\overset{\alpha}{\longrightarrow}}$${E(X\times\mathbb{A}^{-1})}$${\Gamma_{\mathbb{A}^{1}}E(X)}$${E(X\times\mathbb{A}^{-1})}$${E(X\times\operatorname{\mathbb{G}_{m}}).}$

Denoting $\Phi
E(X):=\textup{fib}\bigl{(}E(X\times\mathbb{P}^{1})\overset{}{\rightarrow}E(X)\bigr{)}$,
the fibre of the map $\alpha$ is the cartesian square

$\textstyle{\Phi
E(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{\mathbb{A}^{1}}E(X).}$

This shows the first asserted equivalence. The same argument with
$\Gamma_{\mathbb{B}^{1}}E$ and $\Lambda_{\mathbb{B}^{1}}E$ yields the second
asserted equivalence. ∎

######  Definition 5.13.

Let $E\in\textup{PSh}(\textup{Rig}_{S},{\mathcal{D}})$ carrying a K-module
structure. We fix a map
$t\colon\SS\overset{}{\rightarrow}\Omega\textup{K}(\mathbb{Z}[t,t^{-1}])$
representing the class $t\in\textup{K}_{1}(\mathbb{Z}[t,t^{-1}])$. Then we get
an induced map

$\lambda\colon E(X)\overset{-\cup t}{\longrightarrow}\Omega
E(X\times\operatorname{\mathbb{G}_{m}})\overset{\partial}{\longrightarrow}\Lambda_{\mathbb{A}^{1}}E(X)$

where the first map comes from the cup product on K-theory and the second map
is from the shifted fibre sequence defining $\Lambda_{\mathbb{A}^{1}}E$ above.

######  Definition 5.14.

Let $E\in\textup{PSh}(\textup{Rig}_{S},{\mathcal{D}})$ carrying a K-module
structure. We write $\Lambda
E:=\Lambda_{\mathbb{A}^{1}}E\simeq\Lambda_{\mathbb{B}^{1}}E$ and define the
analytic _analytic Bass construction of_ $E$ as the colimit

$E^{\textup{B}}:=\mathop{\textup{colim}}\bigl{(}E\overset{\lambda}{\longrightarrow}\Lambda
E\overset{\lambda}{\rightarrow}\Lambda^{2}E\overset{\lambda}{\longrightarrow}\ldots\bigr{)}$

within the category $\textup{PSh}(\textup{Rig}_{S},{\mathcal{D}})$.

As a formal consequence from our construction we get the following.

######  Proposition 5.15.

Let
${\mathcal{D}}\in\\{\operatorname{Sp},\textup{Pro}(\operatorname{Sp}),\textup{Cond}^{\omega}(\operatorname{Sp})\\}$.
Let $E\in\textup{PSh}(\textup{Rig}_{S},{\mathcal{D}})$ carrying a K-module
structure and assume that the map $\lambda\colon
E\overset{}{\rightarrow}\Lambda E$ is an equivalence. Then for every
$n\in\mathbb{Z}$ and every $X\in\textup{Rig}_{S}$ we have an exact sequence

$0\overset{}{\longrightarrow}E_{n}(X)\overset{}{\longrightarrow}E_{n}(X\times\mathbb{A}^{1})\oplus
E_{n}(X\times\mathbb{A}^{-1})\overset{\pm}{\longrightarrow}E_{n}(X\times\operatorname{\mathbb{G}_{m}})\overset{\partial}{\longrightarrow}E_{n-1}(X)\overset{}{\longrightarrow}0$

in the category ${\mathcal{D}}^{\heartsuit}$. Furthermore, the map $\partial$
is split by the map

$-\cup
t\,\colon\,E_{n-1}(X)\overset{}{\longrightarrow}E_{n}(X\times\operatorname{\mathbb{G}_{m}})$

induced by the cup product with $t$ from Definition 5.13.

###### Proof.

The same argument as for the $\mathbb{B}^{1}$-delooping applies, see [KST19a,
Prop. 4.13]. ∎

######  Corollary 5.16.

For every $n\in\mathbb{Z}$ and every $X\in\textup{Rig}_{S}$ there is an exact
sequence

$0\overset{}{\longrightarrow}\textup{K}^{\textup{an}}_{n}(X)\overset{}{\longrightarrow}\textup{K}^{\textup{an}}_{n}(X\times\mathbb{A}^{1})\oplus\textup{K}^{\textup{an}}_{n}(X\times\mathbb{A}^{-1})\overset{\pm}{\longrightarrow}\textup{K}^{\textup{an}}_{n}(X\times\operatorname{\mathbb{G}_{m}})\overset{\partial}{\longrightarrow}\textup{K}^{\textup{an}}_{n-1}(X)\overset{}{\rightarrow}0$

of pro-abelian groups where the map $\partial$ has a split.

###### Proof.

This follows from Proposition 5.15 since the map
$\lambda\colon\textup{K}^{\textup{an}}\overset{}{\rightarrow}\Lambda\textup{K}^{\textup{an}}$
is an equivalence [KST23, Cor. 2.6]. ∎

######  Corollary 5.17.

There exists a $\mathbb{P}^{1}$-action on $\mathbb{Z}\times\operatorname{BGL}$
such that the induced $\mathbb{P}^{1}$-spectrum

$\textup{KGL}^{\textup{an}}\coloneqq(\mathbb{Z}\times\operatorname{BGL},\mathbb{Z}\times\operatorname{BGL},\dots)\in\textup{RigSH}(K)$

is a commutative algebra object in $\textup{RigSH}(K)$. Furthermore, under
assumption $(\dagger)$ the $\mathbb{P}^{1}$-spectrum
$\textup{KGL}^{\textup{an}}$ represents (non-connective) analytic $K$-theory.

###### Proof.

Classically, it is known that
$\mathbb{Z}\times\operatorname{BGL}\in\textup{H}(K)$, admits a
$\mathbb{P}^{1}$-action [Voe98, §6.2], [PPR09], [DLØ+07, §3.2]. This action
yields a $\mathbb{P}^{1}$-spectrum
$\textup{KGL}\coloneqq(\mathbb{Z}\times\operatorname{BGL},\mathbb{Z}\times\operatorname{BGL},\dots)$
representing homotopy invariant K-theory. By our discussion in Section 4.4, we
see that the analytification of KGL yields a $\mathbb{P}^{1}$-spectrum
$\textup{KGL}^{\textup{an}}$
inside$\textup{RigSH}(K,\textup{Cond}^{\omega}(\operatorname{Sp}^{+}))$.
Furthermore, by Lemma 5.12 and the construction of homotopy invariant K-theory
ring spectrum via the Bass-construction [Wei13], we see that
$\textup{KGL}^{\textup{an}}$ represents (non-connective) analytic K-theory
under assumption $(\dagger)$ using Theorem 5.7. ∎

###  5.4 Continuous Homotopy K-theory

Under certain finiteness conditions, analytic K-theory is the
$\mathbb{A}^{1}$-localisation of continuous K-theory.

######  Theorem 5.18 ($\mathbb{A}^{1}$-localisation).

Let $A$ be a Tate ring that admits a noetherian and finite dimensional ring of
definition. Then there exists a canonical weak equivalence

$(\textup{L}_{\mathbb{A}^{1}}\textup{K}^{\textup{cont}})(A)\simeq\textup{K}^{\textup{an}}(A).$

###### Proof.

We have the following chain of equivalences:

$\displaystyle(\textup{L}_{\mathbb{A}^{1}}\textup{K}^{\textup{cont}})(A)$
$\displaystyle\simeq\mathop{\textup{colim}}_{n\in\Delta^{\operatorname{op}}}\textup{K}^{\textup{cont}}(\Delta_{A}^{\textup{an},n})$
$\displaystyle(\textup{Lemma}~{}\ref{homotopy-invariantification--lem})$
$\displaystyle\simeq\mathop{\textup{colim}}_{n\in\Delta^{\operatorname{op}}}\operatorname*{``lim^{\prime\prime}}_{j}\textup{K}^{\textup{cont}}(A\langle\Delta_{\pi^{j}}\rangle)$
$\displaystyle(\textup{sheaf condition})$
$\displaystyle\simeq\operatorname*{``lim^{\prime\prime}}_{j}\mathop{\textup{colim}}_{n\in\Delta^{\operatorname{op}}}\textup{K}^{\textup{cont}}(A\langle\Delta_{\pi^{j}}\rangle)$
$\displaystyle(\clubsuit)$
$\displaystyle\simeq\operatorname*{``lim^{\prime\prime}}_{j}\textup{K}^{\textup{cont}}(A\langle\Delta_{\pi^{j}}\rangle)$
$\displaystyle(\textup{definition})$
$\displaystyle\simeq\textup{K}^{\textup{an}}(A)$

For the equivalence $(\clubsuit)$ we use that the geometric realisation
commutes with the pro-limit [KST23, Lem. 2.8] since the spectra in question
are uniformly bounded below by [Dah24, Thm. 8.12]. ∎

Continuous K-theory seems to be the canonical K-theory for rigid analytic
spaces in the view of the Efimov Continuity Theorem [And23, Satz 5.8]. Thus,
in analogy to Weibel’s homotopy K-theory, its $\mathbb{A}^{1}$-localisation is
a natural cohomology theory to study, whence the following.

######  Definition 5.19.

We define _continuous homotopy K-theory_ to be the
$\mathbb{A}^{1}$-localisation of continuous K-theory, i.e.
$\textup{KH}^{\textup{cont}}:=\textup{L}_{\mathbb{A}^{1}}\textup{K}^{\textup{cont}}$.

######  Lemma 5.20.

The functor
$\textup{KH}^{\textup{cont}}\colon\textup{Rig}\overset{}{\rightarrow}\textup{Pro}(\operatorname{Sp}^{+})$
is a Nisnevich sheaf. In particular,
$\textup{KH}^{\textup{cont}}\simeq\textup{L}_{\textup{mot}}\textup{K}^{\textup{cont}}$.

###### Proof.

Let $U\overset{}{\rightarrow}X$ be a Nisnevich cover so that
$X\simeq\mathop{\textup{colim}}_{i\in\Delta^{\operatorname{op}}}U_{i}$ where
$U_{i}$ is the $(i+1)$-fold fibre product of $U$ over $X$. We have the
following equivalences

$\displaystyle\textup{KH}^{\textup{cont}}(X)$
$\displaystyle\simeq\mathop{\textup{colim}}_{n\in\Delta^{\operatorname{op}}}\textup{K}^{\textup{cont}}(X\times\Delta^{\textup{an},n})$
$\displaystyle(\textup{Definition}~{}\ref{defi-continuous-homotopy-k-theory})$
$\displaystyle\simeq\mathop{\textup{colim}}_{n\in\Delta^{\operatorname{op}}}\operatorname*{``lim^{\prime\prime}}_{i}\textup{K}^{\textup{cont}}(U_{i}\times\Delta^{\textup{an},n})$
$\displaystyle(\textup{sheaf condition})$
$\displaystyle\simeq\operatorname*{``lim^{\prime\prime}}_{i}\mathop{\textup{colim}}_{n\in\Delta^{\operatorname{op}}}\textup{K}^{\textup{cont}}(U_{i}\times\Delta^{\textup{an},n})$
$\displaystyle\simeq\operatorname*{``lim^{\prime\prime}}_{i}\textup{KH}^{\textup{cont}}(X)$
$\displaystyle(\textup{Definition}~{}\ref{defi-continuous-homotopy-k-theory})$

which shows the claim. ∎

##  Appendix A Condensed objects

In the main body of this article, we deal with analytic K-theory as defined
and studied by Kerz-Saito-Tamme [KST19a, KST23] which is a functor

$\textup{K}^{\textup{an}}\colon\textup{Rig}\overset{}{\longrightarrow}\textup{Pro}(\operatorname{Sp})$

with values in the category of pro-spectra. Since pro-categories are not that
well-behaved, e.g. computing colimits in them is not easy, we eventually will
postcompose $\textup{K}^{\textup{an}}$ with the canonical functor
$\textup{Pro}(\operatorname{Sp})\overset{}{\rightarrow}\textup{Cond}(\operatorname{Sp})$
to condensed spectra.

Notation. For this entire section, we fix an uncountable strong limit cardinal
$\kappa$.

###  A.1 Condensed objects in a category

Given a category ${\mathcal{C}}$, we work with the categories
$\textup{Cond}^{\omega}({\mathcal{C}})$ of light condensed objects in
${\mathcal{C}}$ and $\textup{Cond}_{\kappa}({\mathcal{C}})$ of
$\kappa$-condensed objects in ${\mathcal{C}}$, as opposed to the large
category
$\textup{Cond}({\mathcal{C}}):=\mathop{\textup{colim}}_{\lambda}\textup{Cond}_{\lambda}({\mathcal{C}})$
of condensed objects (where $\lambda$ runs over all uncountable strong limit
cardinals). One advantage is the following: If the category ${\mathcal{C}}$ is
presentable, then $\textup{Cond}^{\omega}({\mathcal{C}})$ and
$\textup{Cond}_{\kappa}({\mathcal{C}})$ are also presentable (Lemma A.3),
whereas the category $\textup{Cond}({\mathcal{C}})$ is not. Hence we can apply
Adjoint Functor Theorems, for instance. The material about light condensed
objects is extracted from the lectures by Clausen and Scholze on Analytic
Stacks.131313See https://www.youtube.com/playlist?list=PLx5f8IelFRgGmu6gmL-
Kf_Rl_6Mm7juZO.

######  Definition A.1 (Condensed objects).

  1. 1.

A profinite set is called _light_ if it is isomorphic to a countable limit of
finite sets. Denote by $\textup{ProFin}^{\omega}$ the full subcategory of
ProFin that is spanned by light profinite sets. We equip the category
$\textup{ProFin}^{\omega}$ with the topology generated by finite families of
jointly surjective maps.

  2. 2.

Analogously, let $\textup{ProFin}_{<\kappa}$ be the full subcategory of ProFin
that is spanned by $\kappa$-small sets, equipped with the topology generated
by finite families of jointly surjective maps.

Let ${\mathcal{C}}$ be a category that admits finite limits.

  1. 3.

A _light condensed object in_ ${\mathcal{C}}$ is a sheaf $X$ on
$\textup{ProFin}^{\omega}$ with values in ${\mathcal{C}}$. We denote
$\textup{Cond}^{\omega}({\mathcal{C}}):=\textup{Sh}(\textup{ProFin}^{\omega},{\mathcal{C}})$
the category of light condensed objects in ${\mathcal{C}}$.

  2. 4.

A $\kappa$_-small condensed object in_ ${\mathcal{C}}$ is a sheaf on
$\textup{ProFin}_{<\kappa}$ with values in ${\mathcal{C}}$. We denote
$\textup{Cond}_{\kappa}({\mathcal{C}}):=\textup{Sh}(\textup{ProFin}_{<\kappa},{\mathcal{C}})$
the category of $\kappa$-small condensed objects in ${\mathcal{C}}$.

######  Remark A.2.

Explicitly, a presheaf
$X\colon(\textup{ProFin}^{\omega})^{\operatorname{op}}\overset{}{\rightarrow}{\mathcal{C}}$
is a sheaf if and only if it satisfies

  1. 1.

$X(\varnothing)\simeq*$,

  2. 2.

$X(S\sqcup S^{\prime})\overset{\simeq}{\rightarrow}X(S)\times X(S^{\prime})$,
and

  3. 3.

$X(S)\overset{\simeq}{\rightarrow}\lim_{\Delta}X(\check{C}(T\overset{}{\rightarrow}S))$

for all $S,S^{\prime}$ and every surjective map
$T\overset{}{\twoheadrightarrow}S$ in $\textup{ProFin}^{\omega}$.

######  Lemma A.3.

Let ${\mathcal{C}}$ be a presentable category. Then the categories
$\textup{Cond}^{\omega}({\mathcal{C}})$ and
$\textup{Cond}_{\kappa}({\mathcal{C}})$ are presentable.

###### Proof.

Being a category of sheaves on the small site $\textup{ProFin}^{\omega}$, the
category $\textup{Cond}^{\omega}(\textup{Spc})$ is an $\infty$-topos [Lur09,
6.2.2.7]. We have an equivalence
$\textup{Cond}^{\omega}({\mathcal{C}})\simeq\textup{Sh}(\textup{Cond}^{\omega}(\textup{Spc}),{\mathcal{C}})$
[Lur18, 1.3.1.7] and the latter category is presentable as ${\mathcal{C}}$ is
presentable [Lur18, 1.3.1.6]. Same argument for
$\textup{Cond}_{\kappa}({\mathcal{C}})$. ∎

######  Lemma A.4.

Let ${\mathcal{C}}$ be a stable category. Then the categories
$\textup{Cond}^{\omega}({\mathcal{C}})$ and
$\textup{Cond}_{<\kappa}({\mathcal{C}})$ are stable.

###### Proof.

As ${\mathcal{C}}$ is stable, the functor category
$\operatorname{Fun}(\textup{ProFin}^{\omega,\operatorname{op}},{\mathcal{C}})$
is stable [Lur17, 1.1.3.1]. Since the sheafification functor
$\operatorname{Fun}(\textup{ProFin}^{\omega,\operatorname{op}},{\mathcal{C}})\overset{}{\rightarrow}\textup{Cond}^{\omega}({\mathcal{C}})$
is left-exact, the inclusion functor
$\textup{Cond}^{\omega}({\mathcal{C}})\overset{}{\hookrightarrow}\operatorname{Fun}(\textup{ProFin}^{\omega,\operatorname{op}},{\mathcal{C}})$
creates finite limits. Hence $\textup{Cond}^{\omega}({\mathcal{C}})$ admits
finite limits and the suspension functor is an equivalence, thus the category
is stable [Lur17, 1.4.2.27]. Same argument for
$\textup{Cond}_{\kappa}({\mathcal{C}})$. ∎

######  Lemma A.5.

Let ${\mathcal{C}}$ be a category that admits small colimits and finite
limits. Then the categories $\textup{Cond}^{\omega}({\mathcal{C}})$ and
$\textup{Cond}_{\kappa}({\mathcal{C}})$ both admit finite limits. In
particular, they admit the structure of a cartesian symmetric monoidal
category.

###### Proof.

By assumption on ${\mathcal{C}}$, the functor category
$\operatorname{Fun}(\textup{ProFin}^{\omega,\operatorname{op}},{\mathcal{C}})$
admits small colimits and finite limits [Lur09, 5.1.2.3] and this property
passes to its full subcategory $\textup{Cond}^{\omega}({\mathcal{C}})$ of
sheaves. Now the statement about the cartesian symmetric monoidal category
from the existence of finite products [Lur17, 2.4.1.5.(5)]. Same argument for
$\textup{Cond}_{\kappa}({\mathcal{C}})$. ∎

######  Remark A.6 (Extremally disconnected sets).

A profinite set is called an _extremally disconnected set_ if it is a
projective object in the category ProFin. Denote by $\textup{EDS}_{<\kappa}$
the full subcategory of $\textup{ProFin}_{<\kappa}$ spanned by extremally
disconnected sets. By definition of being projective, every cover of profinite
sets $S^{\prime}\overset{}{\rightarrow}S$ with $S$ extremally disconnected has
a split. In particular, the identity map $\operatorname{id}_{S}$ is cofinal
among all covers of $S$. We shall use this property in the proof of Lemma A.15
below.

As a matter of fact, the restriction along the inclusion functor
$\textup{EDS}_{<\kappa}\overset{}{\hookrightarrow}\textup{ProFin}_{<\kappa}$
induces an equivalence from the category
$\textup{Cond}_{\kappa}({\mathcal{C}})$ onto the full
subcategory$\operatorname{Fun}^{\times}(\textup{EDS}_{<\kappa}^{\operatorname{op}},{\mathcal{C}})$
of
$\operatorname{Fun}(\textup{EDS}_{<\kappa}^{\operatorname{op}},{\mathcal{C}})$
which is spanned by those objects that preserve finite products, i.e. sending
finite disjoint unions in $\textup{EDS}_{<\kappa}$ to products in
${\mathcal{C}}$.

######  Remark A.7 (light vs. $\kappa$-small).

Let ${\mathcal{C}}$ be a category that admits limits and colimits. The
restriction along the inclusion functor
$j\colon\textup{ProFin}^{\omega}\overset{}{\hookrightarrow}\textup{ProFin}_{<\kappa}$
induces an adjunction

$j^{*}\colon\operatorname{Fun}(\textup{ProFin}_{<\kappa}^{\operatorname{op}},{\mathcal{C}})\rightleftarrows\operatorname{Fun}(\textup{ProFin}^{\omega,\operatorname{op}},{\mathcal{C}})\colon
R$

where $R$ is a right Kan extension. The functor $j^{*}$ also has a left
adjoint, so preserves limits. Thus it preserves sheaves and restricts to a
functor
$j^{*}\colon\textup{Cond}_{\kappa}({\mathcal{C}})\overset{}{\rightarrow}\textup{Cond}^{\omega}({\mathcal{C}})$.
Now the localisation functor
$\operatorname{Fun}(\textup{ProFin}_{<\kappa}^{\operatorname{op}},{\mathcal{C}})\overset{}{\rightarrow}\textup{Cond}_{\kappa}({\mathcal{C}})\simeq\operatorname{Fun}^{\times}(\textup{EDS}^{\operatorname{op}}_{<\kappa},{\mathcal{C}})$
factors over the restriction functor
$\operatorname{Fun}(\textup{ProFin}_{<\kappa}^{\operatorname{op}},{\mathcal{C}})\overset{}{\rightarrow}\operatorname{Fun}(\textup{EDS}_{<\kappa}^{\operatorname{op}},{\mathcal{C}})$.
Given a light condensed object $X\in\textup{Cond}^{\omega}({\mathcal{C}})$ and
$S_{1},S_{2}\in\textup{EDS}_{<\kappa}$, we have

$\displaystyle RX(S_{1}\sqcup S_{2})$
$\displaystyle\simeq\lim_{\textup{ProFin}^{\omega}\ni
W\overset{}{\rightarrow}S_{1}\sqcup S_{2}}X(W)$
$\displaystyle\simeq\lim_{\textup{ProFin}^{\omega}\ni
W_{1}\overset{}{\rightarrow}S_{1}}X(W_{1})\times\lim_{\textup{ProFin}^{\omega}\ni
W_{2}\overset{}{\rightarrow}S_{2}}X(W_{2})$ $\displaystyle\simeq
RX(S_{1})\times RX(S_{2}).$

Hence the adjunction above restricts to an adjunction

$j^{*}\colon\textup{Cond}_{\kappa}({\mathcal{C}})\rightleftarrows\textup{Cond}^{\omega}({\mathcal{C}})\colon
R.$

Via the explicit formula for the Kan extension $R$ we see that the composition
$j^{*}\circ R$ is equivalent to the identity functor, hence $R$ is fully
faithful.

###  A.2 From pro-objects to condensed objects

Notation. In this subsection, let ${\mathcal{C}}$ be a category that is
accessible, admits colimits and admits finite limits. Denote by
$\textup{Pro}({\mathcal{C}})$ the associated category of pro-objects
${\mathcal{C}}$, cf. the beginning of section 5.1. We have adjunctions

$\textup{const}\colon{\mathcal{C}}\rightleftarrows\textup{Pro}({\mathcal{C}})\colon\textup{lim}\quad\quad\quad\textup{const}\colon{\mathcal{C}}\rightleftarrows\textup{Cond}^{\omega}({\mathcal{C}})\colon\textup{can}$

where the left-adjoints are the respective constant functors which are fully
faithful.

######  Lemma A.8.

Assume additionally that ${\mathcal{C}}$ admits small cofiltered
limits141414In particular, ${\mathcal{C}}$ admits all limits since it has
pullbacks and products (which are cofiltered limits of finite products)
[Lur09, Prop. 4.4.2.6].. Then there is a canonical comparison functor

$\gamma\colon\textup{Pro}({\mathcal{C}})\overset{}{\rightarrow}\textup{Cond}^{\omega}({\mathcal{C}})$

commuting with small limits and fitting into a commutative diagram

---