# Rational motivic path spaces and Kim’s relative unipotent section conjecture

Ishai Dan-Cohen and Tomer Schlank

###### Abstract.

We initiate a study of path spaces in the nascent context of _motivic dga’s_ ,
under development in doctoral work by Gabriela Guzman. This enables us to
_reconstruct_ the unipotent fundamental group of a pointed scheme from the
associated augmented motivic dga, and provides us with a factorization of
Kim’s relative unipotent section conjecture into several smaller conjectures
with a homotopical flavor. Based on a conversation with Joseph Ayoub, we prove
that the path spaces of the punctured projective line over a number field are
concentrated in degree zero with respect to Levine’s t-structure for mixed
Tate motives. This constitutes a step in the direction of Kim’s conjecture.

I.D.C. supported by Priority Program 1786 of the Deutsche
Forschungsgemeinschaft: Homotopy theory and algebraic geometry.

T.S. supported by an Alon Fellowship.

##  0\. Introduction

### 0.1.

Let $Z$ be an open integer scheme (open in
$\operatorname{Spec}\mathcal{O}_{K}$, $K$ a number field). For $p\in Z$ a
closed point, we let $Z_{p}$ denote the complete local scheme at $p$. Let $X$
be a smooth curve over $Z$ with étale divisor at infinity. The classical
method of _Chabauty_ for studying the set $X(Z)$ of $Z$-points (at least when
$X$ is proper) constructs a subspace $X(Z_{p})_{\mathrm{Chab}}$ of $X(Z_{p})$
which contains $X(Z)$. This subspace of _Chabauty points_ is defined by
analytic equations. Due to contributions by a range of authors, especially
Coleman, it has turned out to be quite computable via a certain theory of
p-adic integration, producing effective bounds on the number of $Z$-points in
a certain range of cases. Outside of this range however,
$X(Z_{p})_{\mathrm{Chab}}=X(Z_{p})$, so Chabauty’s method breaks down
completely. Moreover, even within this range, $X(Z_{p})_{\mathrm{Chab}}$ is
rarely equal to $X(Z)$.

Better in this respect is Grothendieck’s section conjecture which is expected
to exactly classify the set of $Z$-points via the torsors of étale paths that
connect them to a fixed base-point. Kim’s conjecture, announced in [5] may be
regarded as a variant of the section conjecture which is _unipotent_ on the
one hand, and _relative_ on the other. _Unipotent_ refers to the fact that the
profinite fundamental group of Grothendieck’s conjecture is replaced by its
prounipotent completion. This involves a significant loss of information, and,
correspondingly, to an overabundance of torsors. Kim then restricts attention
to those torsors which are _locally geometric_ , that is, come from $p$-adic
points for some auxiliary prime $p$ of $Z$ (assumed to be of good reduction).
The resulting conjecture is about extending local points to global points,
hence _relative_.

Kim’s conjecture is also closely related to Chabauty’s method. His original
papers on the subject [28, 29] were devoted to the construction of a nested
sequence of analytic subspaces

$\cdots\subset X(Z_{p})_{3}\subset X(Z_{p})_{2}\subset
X(Z_{p})_{1}=X(Z_{p})_{\mathrm{Chab}},$

defined by p-adic _iterated_ integrals, and his foundational theorem there may
be summarized by the inclusion

$X(Z)\subset X(Z_{p})_{\mathrm{Kim}}:=\bigcap_{n}X(Z_{p})_{n}.$

A series of works on the subject by Kim himself, as well as by a number of
other authors [10, 20, 19, 31, 30, 6, 8, 7, 9] have shown that the analytic
spaces $X(Z_{p})_{n}$ (which are finite when not equal to all of $X(Z_{p})$),
are amenable to explicit computation in a range of cases beyond the Chabauty
bound. The conjecture states that the inclusion (*) above is in fact an
equality. Part of our goal here is to explain our belief that this conjecture
should be more tractable than the section conjecture.

### 0.2.

If Kim’s approach to integral points is a classification in terms of torsors
under the unipotent fundamental group, our work provides an a priori finer
classification in terms of torsors under a certain rational loop-space. So
Kim’s homotopy classes of paths derive, in our approach, from _actual_ paths,
in a rational, motivic sense.

Our main theorem is as follows.

######  Theorem 0.2.1 (Factorization of Kim’s cutter).

We assume for technical reasons that $X$ is affine (but we expect this
assumption to be spurious). We then have the following inclusions:

$\displaystyle X(Z)$ $\displaystyle\subset X(Z_{p})_{\mbox{motivically
global}}$ $\displaystyle\subset X(Z_{p})_{\mbox{pathwise motivically global}}$
$\displaystyle\subset X(Z_{p})_{\mbox{pathwise motivically global up to p-adic
\'{e}tale homotopy}}$ $\displaystyle\subset X(Z_{p})_{\mbox{Kim}}.$

Moreover, if $X$ is _mixed Tate_ , then we may replace “p-adic étale homotopy”
with “motivic homotopy”.

###  0.3. Review of Chen’s theorem

In order to explain what the phrases in subscript mean (including the
subscript “Kim”), we first recall Chen’s theorem, following Wojtkowiak’s
account [48], embellished somewhat by a more homotopical language. We start by
recalling a homotopy-theoretic approach to the topological fundamental group.
Let $\mathcal{X}:=X(\mathbb{C})$ and let

$pt\underset{y}{\overset{x}{\rightrightarrows}}\mathcal{X}$

be points. Then there’s a formula from topology:

$x\times_{\mathcal{X}}^{ho}y={{}_{y}P_{x}},$

the space of paths $x\to y$, and

${}_{y}\pi_{x}=\pi_{0}({{}_{y}P_{x}}),$

the space of homotopy classes of paths. In this description, the topological
fundamental group has a natural de Rham analog. Let
$\Omega^{\bullet}(X_{\mathbb{Q}})$ denote the de Rham dga associated to
$X_{\mathbb{Q}}$, and assume $x$, $y$ come from rational points

$\operatorname{Spec}\mathbb{Q}\underset{y}{\overset{x}{\rightrightarrows}}X.$

Then $x$, $y$ induce augmentations

$\mathbb{Q}\underset{y^{*}}{\overset{x^{*}}{\leftleftarrows}}\Omega^{\bullet}(X_{\mathbb{Q}}).$

Let
${}_{y}B_{x}^{\rm{dR}}:=\mathbb{Q}\coprod^{ho}_{\Omega^{\bullet}(X_{\mathbb{Q}})}\mathbb{Q},$
the homotopy pushout being taken in the model category of dga’s. Then
${}_{y}A_{x}^{\rm{dR}}:=H^{0}({{}_{y}B_{x}^{\rm{dR}}})$ has the structure of a
commutative $\mathbb{Q}$-algebra, and there are _path cocomposition maps_

${{}_{z}A_{y}^{\rm{dR}}}\otimes{{}_{y}A_{x}^{\rm{dR}}}\leftarrow{{}_{z}A_{x}^{\rm{dR}}}.$

These give
$\pi_{1}^{\rm{dR}}(X,x):=\operatorname{Spec}{{}_{x}A_{x}^{\rm{dR}}}$ the
structure of a proalgebraic $\mathbb{Q}$-group (which turns out to be
prounipotent, c.f. theorem 6.5), and they give
${{}_{y}\pi_{x}^{\rm{dR}}}:=\operatorname{Spec}{{}_{y}A_{x}^{\rm{dR}}}$ the
structure of a torsor. Chen’s theorem states that the $\mathbb{Q}$-group
$\pi_{1}^{\rm{dR}}(X,x)\otimes\mathbb{C}$ is the complex prounipotent
completion of $\pi_{1}(\mathcal{X},x)$.

#### 0.3.1.

For concreteness, we outline the construction of an explicit map

${}_{y}\pi_{x}\to{{}_{y}\pi_{x}^{\rm{dR}}}(\mathbb{C})$

in terms of iterated integrals (and recall some googlable key-words along the
way). This material will not be used in the sequel. The _Cosimplicial model of
the path space_ is given by

${}_{y}P_{x}^{\bullet}(X)=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.32118pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{pt\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.42761pt\raise 6.65971pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{x}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 31.32118pt\raise
2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.5864pt\raise-7.34026pt\hbox{{}\hbox{\kern 0.0pt\raise
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31.32118pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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70.39061pt\raise 3.01385pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
70.39061pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
70.39061pt\raise-3.01385pt\hbox{\hbox{\kern 0.0pt\raise
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70.39061pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
112.26003pt\raise 3.87495pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
112.26003pt\raise 1.29167pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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with e.g.

$t\in X\hskip 12.91663pt\begin{matrix}\mapsto(x,t)\\\ \mapsto(t,t)\\\
\mapsto(t,y).\end{matrix}$

The _reduced bar construction_ is an explicit simplicial dga
${}_{y}B_{x}^{{\rm{dR}},\bullet}$ with

${}_{y}B_{x}^{{\rm{dR}},n}=\Omega^{\bullet}_{/\mathbb{Q}}(X_{\mathbb{Q}})^{\otimes
n}$

which represents the derived push-out
${}_{y}B_{x}^{\rm{dR}}=\operatorname{tot}{{}_{y}B_{x}^{{\rm{dR}},\bullet}}.$
The Kunneth formula gives rise to an equivalence of simplicial dga’s

$\Omega^{\bullet}_{/\mathbb{Q}}({{}_{y}P_{x}^{\bullet}})\xrightarrow{\sim}{{}_{y}B_{x}^{{\rm{dR}},\bullet}}.$

This gives us an isomorphism

$\operatorname{Spec}H^{0}\big{(}\operatorname{tot}\Omega^{\bullet}_{/\mathbb{Q}}({{}_{y}P_{x}^{\bullet}})\big{)}\xrightarrow{\sim}{{}_{y}\pi_{x}^{\rm{dR}}}.$

We wish to construct a map

${{}_{y}\pi_{x}}\xrightarrow{\Psi}\Big{(}\operatorname{Spec}H^{0}\big{(}\operatorname{tot}\Omega^{\bullet}_{/\mathbb{Q}}({{}_{y}P_{x}^{\bullet}})\big{)}\Big{)}(\mathbb{C})$

Regard the image of a piecewise differentiable path

$\gamma:x\to y$

as a homomorphism

$\Psi\gamma:H^{0}\big{(}\operatorname{tot}\Omega^{\bullet}_{/\mathbb{Q}}({{}_{y}P_{x}^{\bullet}})\big{)}\to\mathbb{C}.$

By the Grothendieck de Rham theorem, an element

$[\omega]\in
H^{0}\operatorname{tot}\Omega^{\bullet}{{}_{y}P^{\bullet}_{x}}\otimes\mathbb{C}$

may be represented by a family $\omega=(\omega_{i})_{i\in\mathbb{N}}$ with
$\omega_{i}\in\Omega^{i}_{\mathrm{smooth}}(X^{i}_{\mathbb{C}}).$ On the other
hand, $\gamma:x\to y$ gives rise to a piecewise differentiable simplex
$\gamma^{i}\in C_{i}^{\mathrm{Sing}}(\mathcal{X}^{i})$ given by

$\textstyle{\mathbb{R}^{i}}$$\textstyle{\\{0\leq t_{1}\leq\dots\leq t_{i}\leq
1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcup}$$\scriptstyle{(\gamma,\dots,\gamma)}$$\textstyle{X^{i}.}$

We set

$\Psi\gamma([\omega])=\sum_{i}\int_{\gamma^{i}}\omega_{i}.$

###  0.4. Motivic reconstruction of the unipotent fundamental group

The point of departure for this project is the following observation. It is
said that “Motives cannot distinguish between points and cycles”. Kim’s theory
shows that the unipotent fundamental group, however, remembers enough about
integral points to provide a classification relative to local points, the
classification being complete whenever his conjecture is true (and there’s a
growing list of cases, c.f. [5]). The unipotent fundamental group, in turn,
may be _reconstructed_ from the data

$\Omega^{\bullet}(X)\underset{y^{*}}{\overset{x^{*}}{\rightrightarrows}}\mathbb{Q},$

_if we include the cup product_. This data should be in some sense motivic. So
we retort that motives _equipped with cup product_ can distinguish points from
cycles.

A category $\mathrm{Mdga}(Z,\mathbb{Q})$ of _motivic dga’s_ , and a
contravariant functor

$C=C^{*}_{\rm{mot}}(\;\cdot\;,\mathbb{Q}):\mathrm{Sm}_{/Z}^{\mathrm{op}}\to\mathrm{Mdga}(Z,\mathbb{Q})$

lifting the cohomological motives functor of Levine [34], Voevodsky [46],
Ayoub [2, 3] is currently under construction in doctoral work by Gabriela
Guzman under Marc Levine. In this setting, integral points $x,y\in X(Z)$ give
rise to augmentations

$C(X)\underset{y^{*}}{\overset{x^{*}}{\rightrightarrows}}\mathbbm{1}.$

We may then form the homotopy pushout

${{}_{y}B_{x}}=\mathbbm{1}\coprod^{ho}_{\operatorname{C}(X)}\mathbbm{1}$

in $\mathrm{Mdga}(Z,\mathbb{Q})$. The resulting objects ${{}_{y}B_{x}}(X)$ are
endowed with _path cocomposition_ morphisms. We refer to ${{}_{y}B_{x}}$ as
the _coordinate ring of rational motivic paths from $x$ to $y$_, or just _path
ring_ for short.111This construction provides a motivic source for an object
whose realizations have appeared in various places in the literature. For
instance, a filtered-$\phi$ realization is constructed in Kim–Hain [32]. A
fairly different construction which gives rise to an object which should play
a similar role appears in Pridham [40, §4.5].

When $X$ is mixed Tate, we may take cohomology with respect to the motivic
t-structure of Levine [33] and compare the result with the unipotent
fundamental group of Deligne-Goncharov [22]

$\pi^{\mathrm{un}}(X,x)=\operatorname{Spec}H^{0}_{t}\big{(}{{}_{x}B_{x}}(X)\big{)},$

a prounipotent group object of the category of mixed Tate motives which
realizes to the prounipotent $\mathbb{Q}$-completion of the Betti fundamental
group $\pi_{1}\big{(}X(\mathbb{C}),x\big{)}$ (depending on the choice of an
embedding $K\subset\mathbb{C}$), to the prounipotent
${\mathbb{Q}_{p}}$-completion of the profinite étale fundamental group
$\hat{\pi}_{1}(X_{\overline{K}},x)$ for every prime $p$ of $Z$, and to the
fundamental group of the category of unipotent connections on $X_{K}$.

###  0.5. Factorization of Kim’s cutter

This slightly different way of constructing the unipotent fundamental group
leads to a natural factorization of “Kim’s cutter” [20]. We define a
_cotorsor_ for $B_{x}$ to be a commutative monoid object $P$ of the
triangulated tensor category of motives $\operatorname{DA}(Z,\mathbb{Q})$ of
Voevodsky, Ayoub [2, 4], and others, together with a morphism of monoid
objects

$B_{x}\otimes^{\mathbb{L}}P\leftarrow P$

which is coassociative, such that
$B_{x}\otimes^{\mathbb{L}}P\xleftarrow{\sim}P\otimes^{\mathbb{L}}P$ and
$H^{0}_{\textrm{\'{e}t}}(P_{\overline{K}},{\mathbb{Q}_{p}})\neq 0.$ In the
mixed Tate setting, we restrict attention to torsors which are, in an
appropriate sense _cellular_.

#### 0.5.1.

We let $\operatorname{DMdga}=\operatorname{DMdga}(Z,\mathbb{Q})$ denote the
homotopy category of $\mathrm{Mdga}(Z,\mathbb{Q})$. To fix ideas let us
restrict attention to the mixed Tate case, and for simplicity, let us assume
that $Z\subset\operatorname{Spec}\mathbb{Z}$. We also fix a base point $x\in
X(Z)$. Let $\pi_{1}^{\operatorname{TM}}(Z)$ denote the fundamental group of
the category of mixed Tate motives. Let $\pi_{1}^{[n]}(X)$ denote the quotient
of $\pi_{1}^{\mathrm{un}}(X)$ by the $n$th step in the descending central
series.

In our construction of the motivic path ring ${{}_{y}B_{x}}$, we may replace
the _arithmetic endpoint_ $y\in X(Z)$ with any _motivic endpoint_

$\xi\in\operatorname{Hom}_{\operatorname{DMdga}}(CX,\mathbbm{1}).$

We are led to a diagram in which the square formed by Kim’s cutter becomes the
union of four smaller squares:

$\textstyle{X(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{X(Z_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}^{p}}$$\textstyle{\operatorname{Hom}_{\operatorname{DMdga}}(\operatorname{C}(X),\mathbbm{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l_{1}}$$\scriptstyle{f_{2}}$$\textstyle{\operatorname{Hom}_{\operatorname{DMdga}}(\operatorname{C}(X_{p}),\mathbbm{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}^{p}}$$\textstyle{\operatorname{Cotors}_{\mathrm{cell}}(B_{x}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{3}}$$\scriptstyle{l_{2}}$$\textstyle{\operatorname{Cotors}(B_{x}(X_{p}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\pi_{1}^{\operatorname{TM}}(Z),\pi_{1}^{\mathrm{un}}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\mathrm{un}}(X)_{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\pi_{1}^{\operatorname{TM}}(Z)_{\mathbb{Q}_{p}},\pi_{1}^{[n]}(X)_{\mathbb{Q}_{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{reg}_{{\mathbb{Q}_{p}}}}$$\textstyle{\pi_{1}^{[n]}(X)_{\mathbb{Q}_{p}}}$

In terms of this diagram, we define

$X(Z_{p})_{\mbox{motivically
global}}:=(f_{1}^{p})^{-1}(\operatorname{Im}l_{1}),$

and similarly for

$X(Z_{p})_{\mbox{pathwise motivically global}},$

and

$X(Z_{p})_{\mbox{pathwise motivically global up to motivic homotopy}}.$

The sets $X(Z_{p})_{n}$ which define $X(Z_{p})_{\mathrm{Kim}}$ are different:
$\operatorname{reg}_{\mathbb{Q}_{p}}$ is a map of
${\mathbb{Q}_{p}}$-varieties, and we take _scheme-theoretic image_ ,
remembering nilpotent structure (in our opinion) when pulling back to
$X(Z_{p})$.

#### 0.5.2.

If we wish to define the map $f_{2}$ in the diagram of segment 0.5.1 by
associating to an augmentation

$\xi:C(X)\to\mathbbm{1}$

the homotopy pushout

$f_{2}(\xi)={{}_{\xi}B_{x}}=\operatorname{hocolim}\left(C(X)\underset{\xi}{\overset{x^{*}}{\rightrightarrows}}\mathbbm{1}\right)$

in $\mathrm{Mdga}$, we need to know that
$H^{0}_{\textrm{\'{e}t}}\big{(}({{}_{\xi}B_{x}})_{\overline{K}},{\mathbb{Q}_{p}}\big{)}\neq
0$. In more intuitive terms, we need to know that every two motivic points are
connected by a p-adic étale path.

Continuing down the left column, if we wish to define the map $f_{3}$ by
associating to a cotorsor $P$ the torsor

$f_{3}(P):=\operatorname{Spec}H^{0}_{t}(P)$

of _motivic homotopy classes of paths_ , we need to know that $H_{t}^{i}(P)=0$
for $i<0$, in other words, that $P$ is concentrated in negative (homological)
degrees. We also need to know that $H^{0}_{t}(P)\neq 0$ (although it will not
follow that every two motivic points are connected by a motivic path, since
there may be no morphisms $\mathbb{Q}(0)\to\operatorname{Spec}H^{0}_{t}(P)$).
A conversation with Joseph Ayoub alerted us to the fact that since $X$ is
affine (this assumption now being essential), we have, moreover,
$H^{i}_{t}(P)=0$ for $i>0$, so $P$ is concentrated in degree $0$. These four
facts follow from our key theorems, the _connectedness_ and _concentration_
theorems.

######  Theorem 0.6 (Connectedness).

Let $Z$ be a Dedekind scheme whose function field $K$ has characteristic zero
(an open integer scheme in our applications). Let $X$ be a smooth scheme over
$Z$ such that

$H^{0}_{\textrm{\'{e}t}}(X_{\overline{K}},{\mathbb{Q}_{p}})={\mathbb{Q}_{p}}.$

Let $\omega,\eta\in\operatorname{Hom}_{\operatorname{DMdga}}(CX,\mathbbm{1})$
be augmentations. We assume $X$ is an affine curve. Then

$H^{0}_{\textrm{\'{e}t}}\big{(}{{}_{\eta}B_{\omega}}(X)_{\overline{K}},{\mathbb{Q}_{p}}\big{)}\neq
0.$

If moreover $Z$ obeys Beilinson-Soulé vanishing (okay for $Z$ an open integer
scheme) and $X$ is mixed Tate, then the same holds for the cohomologies
$H^{i}_{t}$ associated to the mixed Tate t-structure.

######  Theorem 0.7 (Concentration).

Let $Z$ be a Dedekind scheme whose function field $K$ has characteristic zero.
Let $X$ be a smooth affine geometrically connected curve over $Z$, and let
$\omega,\eta\in\operatorname{Hom}_{\operatorname{DMdga}}(CX,\mathbbm{1})$ be
augmentations. Then

$H^{i}_{\textrm{\'{e}t}}\big{(}{{}_{\eta}B_{\omega}}(X)_{\overline{K}},{\mathbb{Q}_{p}}\big{)}=0$

for $i\neq 0$. If moreover $Z$ obeys Beilinson-Soulé vanishing and $X$ is
mixed Tate, then the same holds for the cohomologies $H^{i}_{t}$ associated to
the mixed Tate t-structure.

As a direct corollary of the concentration theorem, we find,

######  Corollary 0.8 (Pathwise motivically global theorem).

Suppose $X$ is a smooth affine curve with étale divisor at infinity over an
open integer scheme $Z$, and let $p\in Z$ be a closed point. Then every point
of $X(Z_{p})$ which is pathwise motivically global up to p-adic étale homotopy
is pathwise motivically global. (In other words, the third inclusion in our
_factorization of Kim’s cutter_ (theorem 0.2.1) is onto.) If $X$ is mixed
Tate, then the same holds with “motivic homotopy” in place of “$p$-adic étale
homotopy”.

### 0.9.

The pattern of symmetry between the global column on the left and the local
column on the right is broken after the third row. There are several reasons
for this. For one, we currently lack a t-structure on the triangulated
category of mixed Tate motives over the local scheme $Z_{p}$. This leads us to
replace nonabelian motivic cohomology with the nonabelian syntomic cohomology

$H^{1}\big{(}\pi_{1}^{\mathrm{MTF\phi}}(Z_{p}),\pi_{1}^{\mathrm{un}}(X)^{F\phi}\big{)}.$

Here $\pi_{1}^{\mathrm{MTF\phi}}(Z_{p})$ denotes the fundamental group of the
category of mixed Tate filtered $\phi$ modules [15] and
$\pi_{1}^{\mathrm{un}}(X)^{\mathrm{F\phi}}$ is, at least a priori, the
filtered $\phi$ realization of the unipotent fundamental group. An appropriate
realization functor defined on the triangulated level gives us a map

$\operatorname{Cotors}(B_{x_{p}})\to
H^{1}\big{(}\pi_{1}^{\mathrm{MTF\phi}}(Z_{p}),\pi_{1}^{\mathrm{un}}(X)^{F\phi}\big{)}.$

The nonabelian syntomic cohomology is then isomorphic to
$\pi_{1}^{\rm{dR}}(X)/F^{0}$, the Tannakian fundamental group of the category
of unipotent vector bundles with integrable connection on $X_{K_{p}}$ modulo
the 0th step in its Hodge filtration. It’s an important feature of Kim’s
construction (and another reason for breaking the symmetry) that the resulting
map

$X(Z_{p})\to\pi_{1}^{\rm{dR}}(X_{K_{p}})/F^{0}$

is given in coordinates by $p$-adic iterated integrals à la Besser–Coleman
[12, 18] which can often be computed algorithmically.

### 0.10.

In particular, for $x,y\in X(Z_{p})$, the Frobenius action on the realization
of the motivic path ring ${}_{y}B_{x}$ should be equal to the one induced by
the Frobenius action on the the category of unipotent connections via the
equivalence of categories of Chiarelotto–Le Stum [16] between the latter
category and the category of unipotent isocrystals on the special fiber. As
far as we can tell, the philosophically correct proof of this comparison
result would proceed by applying theorem 6.5 below to certain topoi associated
to an appropriate rational homotopy theory of $p$-adic mixed Hodge modules.
While the affine stacks of Toën [45] and the methods of Olsson [37] and
Pridham [39] may provide a shortcut, we have elected to cheat by going to the
$p$-adic étale realization (refined somewhat by the recent contribution of
Scholze-Bhatt [13]). The commutativity of the diagram above then follows by
concatenating the triangle of unipotent $p$-adic Hodge theory [37, 29] with
the diagram of segment 7.8 below.

Our approach to the $p$-adic étale realization should be compared with that of
Deligne–Goncharov [22]. While the latter is based on an explicit realization
of the homotopy pushout via the bar construction and on an elaborate
computation in constructible cohomology due to Beilinson, we turn to Morita
theory and Koszul duality to understand the unipotent fundamental groupoid of
an arbitrary topos. Theorem 6.5 (mentioned above) states the result of this
investigation.

### 0.11.

This article represents the beginning of a program. Needless to say, one of
its more ambitious goals is to go in the direction of Kim’s conjecture; we
make no claims as to our level of optimism regarding the prospects of an
actual proof. Short of a proof however, we do expect to find, in each of the
sub-inclusions of theorem 0.2.1, philosophical evidence (as opposed to
numerical evidence) for the conjecture. Indeed, each of the four inclusions
presents itself together with an obvious collection of loose threads at which
to tug. As an example, the conjectured implication “motivically global
$\Rightarrow$ global” leads to the following expectation. Let $X$, $Y$ be
schemes over $Z$, let $p$ be a closed point of $Z$, and let’s assume for
simplicity that $Z\subset\operatorname{Spec}\mathbb{Z}$. Then we may consider
the square

$\textstyle{\operatorname{Hom}(Y,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}(Y_{Z_{p}},X_{Z_{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\operatorname{Hom}_{\operatorname{DMdga}}(CX,CY)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l}$$\textstyle{\operatorname{Hom}_{\operatorname{DMdga}}(CX_{Z_{p}},CY_{Z_{p}}).}$

We should expect that for _many_ schemes $X$, $Y$, we have

$\operatorname{Hom}(Y,X)=f^{-1}(\operatorname{Im}l).$

For instance, since it’s reasonable to expect Kim’s conjecture to extend (via
a suitably relative formulation) to morphisms of anabelian schemes over $Z$,
we should expect the equality (*) to hold at least when $X$, $Y$ are
anabelian. In particular, we should expect an equality whenever $X$, $Y$ are
finite étale over $Z$, or suitable models of hyperbolic curves. Moreover, when
$X$, $Y$ are proper, we may replace $Z$ by $\operatorname{Spec}\mathbb{Q}$. So
if after this grandiose generalization we again specialize to the case
$Z=\operatorname{Spec}\mathbb{Q}$, $X=Y=\operatorname{Spec}k$, $k$ a finite
extension of $\mathbb{Q}$, $p$ a prime at which $k$ is totally split, we’re
led to the expectation that

$\mathrm{Aut}\,_{\operatorname{DMdga}}(CX)=\operatorname{Gal}(k/\mathbb{Q}).$

A proof of this simple statement in motivic rational homotopy theory, which
will appear in a forthcoming article, may be regarded as evidence for a piece
of Kim’s conjecture.

### 0.12.

In this preliminary work, we restrict attention to the _mere set_ of strict
cotorsors in the homotopy category. In doing so, we ignore a geometric /
homotopical structure which is naturally available. Indeed, there’s a moduli
stack of weak cotorsors together with coherent higher homotopies on the model
category or infinity category level, which stands above and behind our set of
cotorsors, much in the same way as the local Selmer variety stands behind the
nonabelian $p$-adic étale cohomology set in Kim’s theory. Another part of our
program will be devoted to a study of functions on the stack of cotorsors and
their pullbacks to $X(Z_{p})$. One possible outcome would give rise to an
interesting class of functions, generalizing the iterated integrals of
Besser–Coleman. The opposite outcome, however — that the resulting functions
are no different than the classical iterated integrals, may enable us to use
the stack of cotorsors to _do_ Chabauty-Kim theory on the triangulated,
motivic level. This could include applications to integral points on varieties
with abelian fundamental group, or to a “Kim-theoretic” proof of Faltings’
theorem via an appropriate lift of Poitou-Tate duality (c.f.
Schlank–Stojanoska [42, 43]), circumventing the $p$-adic regulator isomorphism
conjecture of Bloch–Kato (c.f. Kim [29]).

### Acknowlegements

First and foremost we would like to thank Marc Levine for his interest and his
encouragement, and for many helpful conversations devoted to the topic of this
paper. We would like to thank Federico Binda, Shane Kelly, Adeel Khan, and
Lorenzo Mantovani for many helpful conversations. We would like to thank Brad
Drew for a letter about realization functors. We would like to thank Frédéric
Déglise for helpful correspondence. We would like to thank Minhyong Kim for
his encouragement and for helpful conversations during conferences in Madrid
and at Oxford. We would like to thank Joseph Ayoub for an inspiring
conversation that took place during a conference in Paris. We would like to
thank Martin Gallauer for pointing out a mistake in an earlier draft. Finally,
the first author would like to thank Jochen Heinloth for his hospitality as
his host at the University of Duisburg-Essen, and for his advice regarding
exposition.

##  1\. Review of Guzman’s motivic dga’s

We review results from Guzman [24].

### 1.1.

Let $Z$ be a Dedekind scheme. We let $\mathbb{M}^{\mathrm{eff}}$ denote the
category of complexes of presheaves of $\mathbb{Q}$-vector spaces on the
category of smooth schemes over $Z$, endowed with the
$(\mathbb{A}^{1},{\textrm{\'{e}t}})$-local model structure (see, for instance,
Ayoub [4] for details). We set

$T:=\frac{\mathbb{A}^{1}_{Z}\otimes\mathbb{Q}[0]}{(\mathbb{A}^{1}\setminus\\{0\\})\otimes\mathbb{Q}[0]}$

([4] §2.1), where $\mathbb{Q}[0]$ denotes the constant sheaf associated to
$\mathbb{Q}$ placed in degree $0$; in terms of the object $T$, the Tate object
is given by

$\mathbb{Q}(1):=T[-2].$

We let $\mathbb{M}=\mathbb{M}(Z,\mathbb{Q})$ denote the category of symmetric
$T$-spectra in $\mathbb{M}^{\mathrm{eff}}$ as in Hovey [26, §7], or,
equivalently but more simply, the category of commutative $T$-spectra of Ayoub
[1, §4]. We endow $\mathbb{M}$ with the stable,
$(\mathbb{A}^{1},{\textrm{\'{e}t}})$-local model structure, as in Ayoub [2].
There’s a natural left Quillen functor

$\operatorname{Sus}^{0}_{T}:\mathbb{M}^{\mathrm{eff}}\to\mathbb{M},$

and we set

$\mathbb{Q}(0):=\operatorname{Sus}^{0}_{T}(\mathbb{Q}[0]).$

This makes $\mathbb{M}$ into a monoidal model category with unit object
$\mathbb{Q}(0)$ and tensor product given by the natural (levelwise) tensor
product.

### 1.2.

The Bousfield localizations that intervene in the above construction are made
explicit via descent-theoretic techniques in Cisinski-Déglise [17]. We review
their construction.

#### 1.2.1.

We let $\mathcal{A}$ denote an abelian category. If $E$ is an object and $n$
is an integer, we define

$S^{n}E:=E[n]$

and

$D^{n+1}E=(\cdots\to 0\to E\xrightarrow{Id}E\to 0\to\cdots)$

in cohomological degrees $n-1$, $n$, so that there’s a natural map

$\operatorname{Bdry}(E,n):S^{n}E\to D^{n+1}E.$

#### 1.2.2.

Fix an essentially small collection $\mathcal{G}$ of objects of $\mathcal{A}$
and an essentially small collection $\mathcal{H}$ of complexes in
$\mathcal{A}$. A morphism in the category $\operatorname{Cpx}\mathcal{A}$ of
complexes in $\mathbb{A}$ is a _$\mathcal{G}$ -cofibration_ if it is contained
in the smallest class of maps in $\operatorname{Cpx}\mathcal{A}$ closed under
pushouts, transfinite compositions and retracts, and containing the inclusions
$\operatorname{Bdry}(E,n)$ for $n\in\mathbb{Z}$ and $E\in\mathcal{G}$. A
complex $C$ is _$\mathcal{G}$ -cofibrant_ if

$0\to C$

is a $\mathcal{G}$-cofibration; _$\mathcal{G}$ -local_ if for any $E$ in
$\mathcal{G}$ and $n\in\mathbb{Z}$, the map

$\operatorname{Hom}_{K(\mathcal{A})}(E[n],C)\to\operatorname{Hom}_{D(\mathcal{A})}(E[n],C)$

is bijective; and _$\mathcal{H}$ -flasque_ if for all $n\in\mathbb{Z}$ and
$H\in\mathcal{H}$,

$\operatorname{Hom}_{K(\mathcal{A})}(H,C[n])=0.$

The pair $(\mathcal{G},\mathcal{H})$ forms a _descent structure on
$\mathcal{A}$_ if

DS1:

every element of $\mathcal{H}$ is $\mathcal{G}$-cofibrant and acyclic, and

DS2:

every $\mathcal{H}$-flasque complex is $\mathcal{G}$-local.

A descent structure is said to _flat_ if

FlDS1:

for any $X\in\mathcal{G}$, and any quasi-isomorphism of complexes $f$, the
morphism of complexes $id_{X}\otimes f$ is again a quasi-isomorphism, and

FlDS2:

$\mathcal{G}$ contains the unit object $\mathbbm{1}$ and is essentially closed
under tensor product.

#### 1.2.3.

Let $\mathcal{T}$ be a set of complexes in $\mathcal{A}$. We say that
$\mathcal{T}$ is a _flat localizing family_ if

FlLF1:

the elements of $\mathcal{T}$ are $\mathcal{G}$-cofibrant, and

FlLF2:

for all $E\in\mathcal{G}$ and $T\in\mathcal{T}$, $E\otimes T\in\mathcal{T}$.

#### 1.2.4.

In addition to the data ($\mathcal{A}$, $\mathcal{G}$, $\mathcal{H}$,
$\mathcal{T}$) above, we fix a $\mathcal{G}$-cofibrant complex $S$. We then
have the category $\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})$ of
symmetric $S$-spectra in $\operatorname{Cpx}(\mathcal{A})$. We construct a
model structure on $\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})$ in
four steps, following the usual rubric.

  1. (1)

We start with the _$\mathcal{G}$ -projective model structure_ on
$\operatorname{Cpx}(\mathcal{A})$: the equivalences are defined to be the
quasi-isomorphisms, and the cofibrations are defined to be the
$\mathcal{G}$-cofibrations [17, Theorem 2.5].

  2. (2)

We define the _$\mathcal{T}$ -local $\mathcal{G}$-projective_ model structure
on $\operatorname{Cpx}\mathcal{A}$ in terms of the $\mathcal{G}$-projective
structure as follows. We define a morphism of complexes

$X\to Y$

to be a $\mathcal{T}$-equivalence if for any
$\mathcal{H}\cup\mathcal{T}$-flasque complex $K$, the map

$\operatorname{Hom}_{D(\mathcal{A})}(Y,K)\to\operatorname{Hom}_{D(\mathcal{A})}(X,K)$

is bijective. For the $\mathcal{T}$-local $\mathcal{G}$-projective structure
we take $\mathcal{T}$-equivalences for weak equivalences, and
$\mathcal{G}$-cofibrations for cofibrations. This model structure is equal to
the Bousfield localization of the $\mathcal{G}$-projective model structure by
the class of morphisms

$0\to T[n]$

for $T\in\mathcal{T}$ and $n\in\mathbb{Z}$ [17, Prop 4.3]. We will call the
associated fibrations _$\mathcal{T}$ -local $\mathcal{G}$-projective
fibrations_.

  3. (3)

We define the _$\mathcal{T}$ -local $\mathcal{G}$-projective structure_ on
$\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})$ by taking levelwise
$\mathcal{T}$-equivalences for weak equivalences, and levelwise
$\mathcal{T}$-local $\mathcal{G}$-projective fibrations for fibrations. We
then have the associated homotopy category

$D_{\mathcal{T}}(\mathcal{A},S):=\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})[\\{\mathcal{T}\mbox{-equivalences}\\}^{-1}].$

  4. (4)

Combining Proposition 4.3 with segment 7.8 of [17], we define an $S$-spectrum
$E$ to be an _$\Omega^{\infty}$ -spectrum_ if it is fibrant for the
$\mathcal{T}$-local $\mathcal{G}$-projective structure on
$\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})$ and if, moreover, for
each $n\in\mathbb{N}$, the bonding map

$E_{n}\to\underline{\operatorname{Hom}}_{\operatorname{Cpx}\mathcal{A}}(S,E_{n+1})$

is a $\mathcal{T}$-equivalence. We define a morphism of symmetric $S$-spectra

$A\to B$

to be a _stable $\mathcal{T}$-equivalence_ if for every
$\Omega^{\infty}$-spectrum $E$, the induced map

$\operatorname{Hom}_{D_{\mathcal{T}}(\mathcal{A},S)}(A,E)\leftarrow\operatorname{Hom}_{D_{\mathcal{T}}(\mathcal{A},S)}(B,E)$

is bijective. A morphism of symmetric $S$-spectra is a _stable
$\mathcal{G}$-cofibration_ if it is a cofibration for the $\mathcal{T}$-local
$\mathcal{G}$-projective model structure. We define the _stable
$\mathcal{T}$-local $\mathcal{G}$-projective model structure_ on
$\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})$ by taking the stable
$\mathcal{T}$-equivalences for weak equivalences and the stable
$\mathcal{G}$-cofibrations for cofibrations.

#### 1.2.5.

Recall the _monoid axiom_ on a monoidal model category $M$ of Schwede-Shipley
[44]: every

$\big{(}(\mbox{trivial cofibration})\otimes M\big{)}\mbox{-cell object}$

is a weak equivalence.

######  Theorem 1.2.6 (7.13 and 7.24 of Cisinski-Déglise [17]).

Let $\mathcal{A}$ be an abelian category endowed with a flat descent structure
$(\mathcal{G},\mathcal{H})$, a flat localizing family $\mathcal{T}$, and a
$\mathcal{G}$-cofibrant complex $S$. Then the stable $\mathcal{T}$-local
$\mathcal{G}$-projective structure makes
$\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})$ into a proper, cellular,
symmetric monoidal model category which obeys the monoid axiom. Moreover, an
object of $\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})$ is fibrant if
and only if it is an $\Omega^{\infty}$-spectrum.222We note that the proof of
the monoid axiom is a dévisage which executes itself automatically by
regarding the tensor product as a left Quillen functor from the projective to
the injective model structure, a technique which appears in various places in
the literature.

#### 1.2.7.

We apply the above constructions with $\mathcal{A}$ the category of étale
sheaves of $\mathbb{Q}$-vector spaces on $\operatorname{Sm}/Z$, $\mathcal{G}$
the collection of representable sheaves, $\mathcal{H}$ the family of complexes
of sheaves associated to étale hypercoverings by taking alternating sums of
boundary maps, $\mathcal{T}$ the family of complexes

$\cdots\to 0\to\mathbb{Q}(\mathbb{A}^{1}\times X)\to\mathbb{Q}(X)\to
0\to\cdots$

in degrees $0$, $1$, and $S$ the complex

$\cdots\to 0\to\mathbb{Q}(0)\to\mathbb{Q}(\mathbb{A}^{1}\setminus\\{0\\})\to
0\to\cdots$

in degrees $-1$, $0$. We then have a canonical Quillen equivalence

$\mathbb{M}(Z,\mathbb{Q})=\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})$

with the stable $\mathcal{T}$-local $\mathcal{G}$-fibrant model structure on
the latter.

######  Corollary 1.3 (Guzman).

The motivic model category $\mathbb{M}(Z,\mathbb{Q})$ is proper, cellular,
symmetric monoidal, and obeys the monoid axiom as well as the commutative
monoid axiom of White [47].

###### Proof.

Special case of theorem 1.2.6, except for the commutative monoid axiom. The
latter follows from the (not commutative) monoid axiom since $\mathbb{M}$ is
$\mathbb{Q}$-linear, so that the quotient of a morphism $f$ by a
$\Sigma_{n}$-action is a retract of $f$. ∎

###  1.4. Definition

Following Guzman, we define the _rational motivic model category_
$\mathrm{Mdga}=\mathrm{Mdga}(Z,\mathbb{Q})$ to be the category of commutative
monoids in $\mathbb{M}$. We define a morphism in $\mathrm{Mdga}$ to be a _weak
equivalence_ if the associated morphism in $\mathbb{M}$ is a weak equivalence,
and a _cofibration_ if the associated morphism in $\mathbb{M}$ is a
cofibration.

######  Proposition 1.5 (Guzman [24]).

With these definitions, $\mathrm{Mdga}(Z,\mathbb{Q})$ is a cofibrantly
generated model category.

###### Proof.

This is an application of Theorem 3.2 of [47] in view of corollary 1.3. ∎

######  Lemma 1.6.

Cofibrant objects of $\mathbb{M}$ are flat, that is, if $M$ is a cofibrant
object of $\mathbb{M}$ and $f:A\to B$ is a weak equivalence, then
$id_{M}\otimes f$ is again a weak equivalence.

###### Proof.

Although not stated explicitly by Cinsinski-Déglise, this is essentially a
feature of their construction. In the notation of segment 1.2, we are to show
that for any $\Omega^{\infty}$-spectrum $E$, the induced map

$\operatorname{Hom}_{D_{\mathcal{T}}(\mathcal{A},S)}\Big{(}M\underset{\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})}{\otimes}A,\;E\Big{)}\leftarrow\operatorname{Hom}_{D_{\mathcal{T}}(\mathcal{A},S)}\Big{(}M\underset{\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})}{\otimes}B,\;E\Big{)}$

is bijective. (We’ve decorated the tensor products to emphasize the fact that
they’re _not_ derived.) By adjunction with inner Hom, it’s equivalent to show
that the map

$\operatorname{Hom}_{D_{\mathcal{T}}(\mathcal{A},S)}\big{(}A,\underline{\operatorname{Hom}}_{\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})}(M,E)\big{)}\leftarrow\operatorname{Hom}_{D_{\mathcal{T}}(\mathcal{A},S)}\big{(}B,\underline{\operatorname{Hom}}_{\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})}(M,E)\big{)}$

is bijective. So it’s enough to show that the inner Hom

$\underline{\operatorname{Hom}}_{\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})}(M,E)$

is an $\Omega^{\infty}$-spectrum. Since $M$ is cofibrant for the stable
$\mathcal{T}$-local $\mathcal{G}$-projective structure on
$\operatorname{Spt}_{S}^{\mathfrak{S}}(\mathcal{A})$ and $E$ is fibrant for
the same structure, it follows from the monoidal model category structure on
$\mathbb{M}$ that the inner Hom is fibrant for the same structure, hence an
$\Omega^{\infty}$-spectrum as hoped. ∎

### 1.7.

Recall that that the _pushout product_ $f\oblong g$ of morphisms $f:X\to
X^{\prime}$, $g:Y\to Y^{\prime}$ is by definition the induced morphism

$X\otimes Y^{\prime}\underset{X\otimes Y}{\small\coprod}X^{\prime}\otimes Y\to
X^{\prime}\otimes Y^{\prime}$

######  Lemma 1.8.

The motivic model category $\mathbb{M}$ obeys the _strong commutative monoid
axiom_ of White [47, Definition 3.4]: if $h$ is a cofibration then

$h^{\medcirc n}:=h^{\oblong n}/\Sigma_{n}$

is again a cofibration for all $n\geq 1$; moreover, if $h$ is a trivial
cofibration, then so is $h^{\medcirc n}$.

###### Proof.

$\mathbb{M}$ being monoidal means (in part) that the pushout-product of
cofibrations is a cofibration, and the pushout product of trivial cofibrations
is a trivial cofibration. It follows by induction that if $h$ is a cofibration
(respectively a trivial cofibration), so is $h^{\oblong n}$. Since
$\mathbb{M}$ is $\mathbb{Q}$-linear, $h^{\medcirc n}$ is a retract of
$h^{\oblong n}$, hence again a cofibration (respectively a trivial
cofibration). ∎

### 1.9.

Given morphisms

---
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{A}$$\textstyle{Y}$

let ${{}^{g}f}$ denote the pushout of $f$ along $g$. If $X/\mathcal{M}$ is the
category of morphisms from an object $X$ in a model category $\mathcal{M}$,
then a morphism in $X/\mathcal{M}$ is a _weak equivalence_ if the associated
morphism in $\mathcal{M}$ is a weak equivalence. Recall that a monoidal model
category $\mathcal{M}$ is _h-monoidal_ if for any cofibration

$f:X\to Y,$

and any object $Z$,

hM1:

the co-base-change functor

$(f\otimes id_{Z})_{!}:(X\otimes Z)/\mathcal{M}\to(Y\otimes Z)/\mathcal{M}$

preserves weak equivalences (in other words, for any morphism

$g:X\otimes Z\to A,$

pushout along ${}^{g}(f\otimes id_{Z})$ preserves weak equivalences), and

hM2:

if, moreover, $f$ is a weak equivalence, then so is $f\otimes id_{Z}$.

######  Lemma 1.10.

The motivic model category $\mathbb{M}$ is h-monoidal.

###### Proof.

Combine Proposition 1.13 of Batanin-Berger [11] with Propositions 7.13, 7.19,
and 7.23 of Cinsinski-Déglise [17]. ∎

######  Proposition 1.11.

The category $\mathrm{Mdga}(Z,\mathbb{Q})$ of motivic dga’s is left proper.

###### Proof.

Domains of generating cofibrations are cofibrant. Indeed, before stabilizing,
every object is cofibrant, and it’s straightforward to check that this
property is preserved by stabilization. By lemmas 1.6, 1.8, and 1.10, and
corollary 1.3, Theorem 4.17 of White [47] applies.333Our category is compactly
generated as a mere category and weak equivalences are preserved by filtered
colimits. Therefor $\mathbb{M}$ is a compactly generated model category. ∎

### 1.12.

Using the model structure of proposition 1.5, we may fix once and for all a
fibrant replacement

$\mathbbm{1}\to\mathbbm{1}^{f}$

of the unit object in $\mathrm{Mdga}$. There’s a natural functor

$\operatorname{C_{*}}=\operatorname{C_{*}^{\rm{mot}}}(\;\cdot\;,\mathbb{Q}):\operatorname{Sm}_{Z}\to\operatorname{cc\mathbb{M}}$

to the category of cocommutative comonoids in $\mathbb{M}$ given by

$C_{*}X:=\operatorname{Sus}^{0}_{T}(X\otimes\mathbb{Q}[0])$

with counit

$C_{*}X\to C_{*}Z=\mathbb{Q}(0)$

induced by the structure map of $X$, and comultiplication

$C_{*}X\to C_{*}X\otimes C_{*}X$

induced by the diagonal of $X$. In terms of the object $\mathbbm{1}^{f}$ and
the functor $C_{*}$, we have the following theorem.

######  Theorem 1.13 (Guzman [24]).

The formula

$\operatorname{C}X=\operatorname{C^{*}_{\rm{mot}}}(X,\mathbb{Q}):=\underline{\operatorname{Hom}}_{\mathbb{M}}(C_{*}(X,\mathbb{Q}),\mathbbm{1}^{f})$

defines a functor

$\operatorname{C}=\operatorname{C}_{*}^{\rm{mot}}(\;\cdot\;,\mathbb{Q}):\operatorname{Sm}^{\mathrm{op}}_{Z}\to\mathrm{Mdga}(Z,\mathbb{Q}).$

###### Proof.

For $X\in\operatorname{Sm}_{Z}$, the inner hom above has the structure of a
(strictly!) commutative monoid in $\mathbb{M}$. Moreover, the morphism

$\operatorname{C}(X)\leftarrow\operatorname{C}(Y)$

induced by a morphism of schemes

$X\to Y$

is strictly compatible with the monoid structures. ∎

##  2\. rational motivic path cogroupoid

######  Proposition 2.1.

The natural tensor product

$-\otimes_{\mathbbm{1}}-$

of commutative monoid objects in a monoidal category makes
$\mathrm{Mdga}(Z,\mathbb{Q})$ into a monoidal model category. (The subscript
$\mathbbm{1}$ refers to the unit object $\mathbbm{1}=\mathbb{Q}(0)$ of
$\mathrm{Mdga}$, and plays the role of base-ring.) In particular, the homotopy
category $\operatorname{DMdga}(Z,\mathbb{Q})$ has an induced monoidal
structure. We denote the induced tensor product by

$-\otimes_{\mathbbm{1}}^{\mathbb{L}}-.$

###  2.2. Definition

Fix a cofibrant replacement

$\operatorname{C}(X)^{c}\to\operatorname{C}(X)$

and recall that we’ve fixed a fibrant replacement

$\mathbbm{1}\to\mathbbm{1}^{f}.$

We consider a fixed integral point $x\in X(Z)$. This gives rise to a morphism
of monoid objects

$x^{\lor}:\operatorname{C}(X)^{c}\to\operatorname{C}(X)\xrightarrow{x^{*}}\mathbbm{1}\to\mathbbm{1}^{f}.$

We define the _coordinate ring of rational motivic loops_ , or just _loop
ring_ at $x$, by

$B_{x}:=x^{\lor}\underset{\operatorname{C}(X)^{c}}{\overset{ho}{\amalg}}x^{\lor}.$

More generally, given
$\xi,\eta\in\operatorname{Hom}_{\operatorname{DMdga}}(\operatorname{C}(X),\mathbbm{1})$
we let $\tilde{\xi}$, $\tilde{\eta}$ denote representatives in

$\operatorname{Hom}_{\mathrm{Mdga}}(\operatorname{C}(X)^{c},\mathbbm{1}^{f})$

and we define the _coordinate ring of rational motivic paths_ , or just _path
ring_ from $\xi$ to $\eta$, by

${{}_{\eta}B_{\xi}}:=\tilde{\eta}\underset{\operatorname{C}(X)^{c}}{\overset{ho}{\amalg}}\tilde{\xi}.$

######  Lemma 2.3.

The path ring ${}_{\eta}B_{\xi}$ is independent of the choices of
representatives $\tilde{\eta}$, $\tilde{\xi}$ up to canonical isomorphism in
$\operatorname{DMdga}(Z,\mathbb{Q})$.

###### Proof.

Suppose $\tilde{\eta}^{\prime}:\operatorname{C}(X)^{c}\to\mathbbm{1}^{f}$ is
homotopy equivalent to $\tilde{\eta}$, and let $H:A\times I\to\mathbbm{1}^{f}$
be a homotopy. Then $H$ induces morphisms of diagrams

$\textstyle{\big{(}\mathbbm{1}^{f}\xleftarrow{\tilde{\eta}}\operatorname{C}(X)^{c}\xrightarrow{\tilde{\xi}}\mathbbm{1}^{f}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\big{(}\mathbbm{1}^{f}\xleftarrow{H}\operatorname{C}(X)^{c}\times
I\xrightarrow{\tilde{\xi}}\mathbbm{1}^{f}\big{)}}$$\textstyle{\big{(}\mathbbm{1}^{f}\xleftarrow{\tilde{\eta}^{\prime}}\operatorname{C}(X)^{c}\xrightarrow{\tilde{\xi}}\mathbbm{1}^{f}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

which are object-wise weak equivalences. ∎

### 2.4.

We endow the family

$\left\\{{{}_{\eta}B_{\xi}}\right\\}_{\xi,\eta}$

of objects of $\operatorname{DMdga}(Z,\mathbb{Q})$ with a cogroupoid structure
as follows. The counit of ${}_{\xi}B_{\xi}$ is the morphism

${}_{\xi}B_{\xi}\to\mathbbm{1}$

represented by the obvious maps

${}_{\xi}B_{\xi}\to\mathbbm{1}^{f}\leftarrow\mathbbm{1}$

in $\mathrm{Mdga}(Z,\mathbb{Q})$. For the path cocomposition morphisms, we
write $H:=\operatorname{C}(X)^{c}$ and consider three morphisms

$\zeta,\eta,\xi:H\underset{\to}{\overset{\to}{\to}}\mathbbm{1}^{f}.$

We then have a morphism of diagrams

|
---|---
$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\textstyle{\mathbbm{1}^{f}}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\scriptstyle{\eta}$$\textstyle{\mathbbm{1}^{f}}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H}$$\textstyle{\mathbbm{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbbm{1}^{f}}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\textstyle{H}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{\xi}$$\textstyle{\mathbbm{1}^{f}}$$\textstyle{\mathbbm{1}^{f}}$$\textstyle{\mathbbm{1}^{f}}$

which gives rise to

${{}_{\zeta}B_{\xi}}\to\left(\zeta\coprod_{H}^{ho}\eta\right)\;\coprod^{ho}\;\left(\eta\coprod_{H}^{ho}\xi\right)={{}_{\zeta}B_{\eta}}\bigotimes_{\mathbbm{1}}^{\mathbb{L}}{{}_{\eta}B_{\xi}}.$

We omit the verification of associativity as well as the torsor condition.

##  3\. Cellular motives and cellular motivic dga’s

###  3.1. Cellular motives

We discuss the category

$\operatorname{DA_{cell}}(Z,\mathbb{Q})\subset\operatorname{DA}(Z,\mathbb{Q})$

of _cellular motivic complexes_ , and its relationship to the Tannakian
category of mixed Tate motives constructed by Levine [33]. A similar
discussion may be found in Iwanari [27].

######  Proposition 3.1.1.

Let $M$ be a stable monoidal model category, $T^{c}$ a tensor-triangulated
subcategory of $D:=\operatorname{Ho}M$ with a $t$-structure with heart
$A^{c}$. Let $N^{c}$ be the full subcategory of $M$ whose objects are those of
$T^{c}$. Assume that the objects of $N^{c}$ are compact. Let $N$ be the
closure of $N^{c}$ under small homotopy colimits and desuspensions. Then $N$
is stable monoidal, the t-structure on $T^{c}$ extends to
$T:=\operatorname{Ho}N$, and if $A$ denotes the heart of this t-structure,
then every object of $A$ is a small colimit of objects of $A^{c}$.

######  Remark 3.1.2.

The situation described in the proposition is summarized by the following
diagram,

---
$\textstyle{N^{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T^{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset}$$\textstyle{D}$$\textstyle{A^{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\subset}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$

with input $M$ and $T^{c}$, and output $N$ and $A$.

###### Proof.

We refer to the definition of t-structure given in Lurie [35, Definition
1.2.1.1]. Let $N^{c}_{\geq 0}$ denote the full subcategory of $N^{c}$
corresponding to $T^{c}_{\geq 0}$, and similarly for $N^{c}_{\leq 0}$. Let
$N_{\geq n}$ denote the closure of $N^{c}_{\geq 0}$ under small homotopy
colimits, and similarly for $N_{\leq n}$. Let $T_{\geq 0}$ denote the homotopy
category of $N_{\geq 0}$, and similarly for $T_{\leq 0}$.

We have $\operatorname{Hom}(X,Y)=0$ for $X\in N_{\geq 0}$, $Y\in N_{\leq-1}$.
Indeed,

$\operatorname{Hom}(\operatorname{hocolim}X_{i},\operatorname{hocolim}Y_{j})=\operatorname{holim}\operatorname{Hom}(X_{i},Y_{j})$

by compactness.

An arbitrary object $X=\operatorname{hocolim}X_{i}$ of $T$ fits into an exact
triangle

$X_{\geq 0}\to X\to X_{\leq-1}.$

Indeed, we have triangles like so:

$(X_{i})_{\geq 0}\to X_{i}\to(X_{i})_{\leq-1}.$

Since homotopy colimits commute with finite homotopy limits, hence with exact
triangles, we obtain (*) by taking homotopy colimits.

The remaining properties of a t-structure are clear.

For the statement about the heart, we resort to the language of infinity
categories. The full subcategory $N_{\geq 0,\leq 0}$ of $N$ is discrete, hence
equivalent to the heart; moreover, homotopy colimits in $N_{\geq 0,\leq 0}$
are just ordinary colimits. So it will suffice to show that

$\operatorname{Ind}(N_{\geq 0,\leq 0})=\operatorname{Ind}(N_{\geq
0})\cap\operatorname{Ind}(N_{\leq 0}).$

Suppose that $X$ is in the intersection. Write $X=\operatorname{colim}X_{i}$,
$X_{i}\in N_{\geq 0}$. Consider the cofiber sequence

$\operatorname{colim}\tau_{\geq 1}(X_{i})\to
X\to\operatorname{colim}\tau_{\leq 0}X_{i}.$

The right hand object is in $\operatorname{Ind}(N_{\leq 0})$, as is the middle
object. Hence so is the left hand object. But the left hand object is also in
$N_{\geq 1}$. Hence it is zero. Hence the map on the right is iso. ∎

####  3.1.3. Definition

We apply Proposition 3.1.1 with

$M:=\mathbb{M}(Z,\mathbb{Q})$

and

$T^{c}:=\operatorname{DTM}^{c}(Z,\mathbb{Q})$

the smallest tensor-triangulated subcategory of
$\operatorname{DA}(Z,\mathbb{Q})$ containing the Tate objects. Under the
assumption that $Z$ obeys Beilinson-Soulé vanishing (okay for $Z$ an open
integer scheme), Levine constructs a t-structure on
$\operatorname{DTM}^{c}(Z,\mathbb{Q})$. As in the proposition, we consider the
full subcategory

$\mathbb{M}_{\mathrm{cell}}^{c}\subset\mathbb{M}$

(denoted $N^{c}$ in the proposition) whose objects are in
$\operatorname{DTM}^{c}(Z,\mathbb{Q})$, and we let
$\mathbb{M}_{\mathrm{cell}}$ denote the closure of
$\mathbb{M}_{\mathrm{cell}}^{c}$ under homotopy colimits. We define the
category of _cellular motives_ by

$\operatorname{DA}_{\mathrm{cell}}(Z,\mathbb{Q}):=\operatorname{Ho}\mathbb{M}_{\mathrm{cell}}.$

The proposition then gives us a t-structure on
$\operatorname{DA}_{\mathrm{cell}}(Z,\mathbb{Q})$ extending the one on
$\operatorname{DTM}^{c}(Z,\mathbb{Q})$. Moreover, by the proposition, its
heart $\operatorname{TM}(Z,\mathbb{Q})$ is equal to
$\operatorname{Ind}\operatorname{TM}^{c}(Z,\mathbb{Q})$, the Ind-completion of
the category $\operatorname{TM}^{c}(Z,\mathbb{Q})$ of (compact) mixed Tate
motives.

We summarize the above discussion in the following proposition.

######  Proposition 3.1.4 (Cellular motives).

Assume $Z$ obeys Beilinson-Soulé vanishing. Then Levine’s t-structure on
$\operatorname{DTM}^{c}(Z,\mathbb{Q})$ extends to a t-structure on
$\operatorname{DA}_{\mathrm{cell}}(Z,\mathbb{Q})$. Every object of its heart
$\operatorname{TM}(Z,\mathbb{Q})$ is a small colimit of objects of
$\operatorname{TM}^{c}(Z,\mathbb{Q})$.

###  3.2. Cellular motivic dga’s

#### 3.2.1.

We define the _category of cellular motivic dga’s_

$\mathrm{Mdga}_{\mathrm{cell}}=\mathrm{Mdga}_{\mathrm{cell}}(Z,\mathbb{Q})$

to be the category of commutative monoid objects in
$\mathbb{M}_{\mathrm{cell}}$.

######  Proposition 3.2.2.

The category $\mathrm{Mdga}_{\mathrm{cell}}(Z,\mathbb{Q})$ is closed under
small homotopy colimits.

###### Proof.

Special case of proposition 3.2.3 below. ∎

######  Proposition 3.2.3.

Let $M$ be a monoidophilic model category (monoidal such that
$\operatorname{cMon}(M)$ is a model category with right Quillen forgetful
functor). Let $T$ be a subcategory closed under weak equivalences, small
homotopy colimits and derived tensor products.444The last condition (in the
presence of the first condition) reduces to $T$ closed under (ordinary) tensor
products of cofibrant objects. Then $\operatorname{cMon}(T)$ is closed under
small homotopy colimits.

###### Proof.

It’s enough to check that $\operatorname{cMon}(T)$ is closed under small
homotopy coproducts, realizations and filtered homotopy colimits. (Here
“realization” means homotopy colimit over $\Delta^{{}^{\mathrm{op}}}$.)
Realizations and filtered homotopy colimits are both _sifted_ , hence, by
Corollary 3.2.3.2 of Lurie [35], compatible with the forgetful functor.
Turning to homotopy coproducts in $\operatorname{cMon}(T)$, those are
coproducts of derived tensor products in $T$. ∎

##  4\. Interplay between hopfoids and localization

###  4.1. Definition

A _(set-based) hopfoid_ in a monoidal category $M$ consists of a set $S$, and
a cogroupoid in $\operatorname{cMon}(M)$ with vertices in $S$. In a bit more
detail, a family $\\{_{t}B_{s}\\}_{s,t\in S}$ of objects of $M$, a monoid
structure

$\displaystyle\mathbbm{1}\;\longrightarrow\;{{}_{t}B_{s}},$
$\displaystyle{{}_{t}B_{s}}^{\otimes 2}\;\longrightarrow\;{{}_{t}B_{s}}$

on each ${{}_{t}B_{s}}$, plus counits, cocompositions, and antipodes

$\displaystyle{{}_{s}B_{s}}\to\mathbbm{1}$
$\displaystyle{{}_{u}B_{t}}\otimes{{}_{t}B_{s}}\leftarrow{{}_{u}B_{s}}$
$\displaystyle{{}_{t}B_{s}}\to{{}_{s}B_{t}}$

such that the multiplication maps are unital and associative and the
comultiplication maps are counital, coassociative, and invertible. A
_commutative hopfoid_ is a hopfoid whose multiplication maps are commutative.
As an example, $\operatorname{Spec}$ of a commutative hopfoid in $k$-modules
is a groupoid in $k$-schemes in a set-based sense. A _morphism of hopfoids_
$B\to B^{\prime}$ consists of a family of morphisms

$\\{{{}_{t}B_{s}}\to{{}_{t}B^{\prime}_{s}}\\}_{s,t\in S}$

in $M$ commuting with the structure morphisms of $B$ and $B^{\prime}$. We
write $\operatorname{cHfd}(M)$ for the category of commutative hopfoids in
$M$.

### 4.2.

If $\mathcal{M}$ is a monoidophilic model category (monoidal such that
$\operatorname{cMon}\mathcal{M}$ forms a monoidal model category with right
Quillen forgetful functor) and

$F_{*}:\operatorname{cMon}\mathcal{M}\to\mathcal{M}$

is the forgetful functor, then the right derived functor

$RF_{*}:\operatorname{Ho}\operatorname{cMon}\mathcal{M}\to\operatorname{Ho}\mathcal{M}$

lifts to a functor

$\operatorname{Ho}\operatorname{cMon}\mathcal{M}\to\operatorname{cMon}\operatorname{Ho}\mathcal{M}.$

Given

$\mathbbm{1}\xrightarrow{\epsilon}M,\hskip 21.52771ptM^{\otimes
2}\xrightarrow{\mu}M$

an object of $\operatorname{Ho}\operatorname{cMon}\mathcal{M}$, we define the
associated commutative monoid in $\operatorname{Ho}\mathcal{M}$ as follows.
For the unit we simply take $\operatorname{Ho}(\epsilon)$. For the counit, we
choose a cofibrant replacement

$M^{c}\to M$

and we take the morphism

$M\otimes^{\mathbb{L}}M\to M$

represented by the composit

$M^{c}\otimes M^{c}\to M\otimes M\xrightarrow{\mu}M.$

Verification of the axioms is tedious, but straightforward.

######  Proposition 4.3 (Interplay between Hopfoids and localization).

The functor 4.2($*$) may be upgraded to a functor

$\operatorname{Cogpd}\operatorname{Ho}\operatorname{cMon}\mathcal{M}\;\longrightarrow\;\operatorname{Cogpd}\operatorname{cMon}\operatorname{Ho}\mathcal{M}=\operatorname{cHfd}\operatorname{Ho}\mathcal{M}.$

We omit the details.

##  5\. Connectedness and concentration theorems

### 5.1.

Let $Z$ be an integral scheme with function field $K$ of characteristic zero.
We consider the motivic model category $\mathbb{M}=\mathbb{M}(Z,\mathbb{Q})$,
its homotopy category $\operatorname{DA}=\operatorname{DA}(Z,\mathbb{Q})$, the
category of motivic dga’s $\mathrm{Mdga}=\mathrm{Mdga}(Z,\mathbb{Q})$, its
homotopy category $\operatorname{DMdga}=\operatorname{DMdga}(Z,\mathbb{Q})$,
the category of commutative dga’s over ${\mathbb{Q}_{p}}$:
$\operatorname{dga}=\operatorname{dga}({\mathbb{Q}_{p}})$, its homotopy
category $\operatorname{Ddga}=\operatorname{Ddga}({\mathbb{Q}_{p}})$, as well
as the (unbounded) derived category $D({\mathbb{Q}_{p}})$ of
${\mathbb{Q}_{p}}$-vector spaces.

### 5.2.

There’s a p-adic étale realization functor on the level of model categories
which is covariant, monoidal, colimit preserving and t-exact on the cellular
subcategory $\mathrm{Mdga}_{\mathrm{cell}}(Z,\mathbb{Q})$. We sketch its
construction in the language of infinity categories. We let
$\mathbb{D}A=\mathbb{D}A(Z,\mathbb{Q})$ denote the infinity category obtained
from

$\operatorname{Cpl}\operatorname{\bf{Vect}}(\mathbb{Q}_{Z_{\textrm{\'{e}t}}})\llbracket(\mbox{local
quasi-isomorphisms})^{-1}\rrbracket$

by localization and stabilization as in the model categorical setting. The
language of infinity categories provides a canonical lift

$\mathbb{R}f_{*}{\mathbb{Q}_{p}}_{X_{\overline{K},{\textrm{\'{e}t}}}}\in\mathbb{D}({\mathbb{Q}_{p}})$

of the derived pushforward. The functor

$h_{\overline{K},{\textrm{\'{e}t}}}:\operatorname{Sm}_{/Z}^{{}^{\mathrm{op}}}\to\mathbb{D}({\mathbb{Q}_{p}})$
$X\mapsto\mathbb{R}f_{*}{\mathbb{Q}_{p}}_{X_{\overline{K},{\textrm{\'{e}t}}}}$

satisfies étale descent and $\mathbb{A}^{1}$-invariance, and is
$(\mathbb{P}^{1},\infty)$-stable. It then follows from the universal mapping
property of $\mathbb{D}A$ that there’s a symmetric monoidal functor

$\operatorname{\mathbb{R}e}_{\overline{K},{\textrm{\'{e}t}}}:\mathbb{D}A(Z,\mathbb{Q})\to\mathbb{D}({\mathbb{Q}_{p}})$

which factors $h_{\overline{K},{\textrm{\'{e}t}}}$ up to homotopy (c.f. Robalo
[41, Corollary 2.39]).

### 5.3.

We let $\operatorname{\mathbb{D}Mdga}(Z,\mathbb{Q})$ denote the infinity
category of commutative monoids in $\mathbb{D}A(Z,\mathbb{Q})$ and we let
$\operatorname{\mathbb{D}dga}({\mathbb{Q}_{p}})$ denote the infinity category
of differential graded ${\mathbb{Q}_{p}}$-algebras. According to Hinich [25],

$\operatorname{\mathbb{D}Mdga}=\mathrm{Mdga}\llbracket W^{-1}\rrbracket$

and

$\operatorname{\mathbb{D}dga}=\operatorname{dga}\llbracket W^{-1}\rrbracket$

where, in both cases, $W$ denotes the class of weak equivalences. Cup product
gives $\mathbb{R}f_{*}{\mathbb{Q}_{p}}_{X_{\overline{K},{\textrm{\'{e}t}}}}$
the structure of a differential graded algebra, which we denote by

$\mathbb{C}^{*}_{\textrm{\'{e}t}}(X_{\overline{K}},{\mathbb{Q}_{p}}).$

It follows from the construction of the factorization of
$h_{\overline{K},{\textrm{\'{e}t}}}$ above (§5.2) that the functor
$\operatorname{\mathbb{R}e}_{\overline{K},{\textrm{\'{e}t}}}$ preserves small
(homotopy) colimits. It is then a formal consequence that
$\operatorname{\mathbb{R}e}_{\overline{K},{\textrm{\'{e}t}}}$ can be upgraded
to a functor

$\operatorname{\mathbb{R}e}_{\overline{K},{\textrm{\'{e}t}}}^{a}:\operatorname{\mathbb{D}Mdga}(Z,\mathbb{Q})\to\operatorname{\mathbb{D}dga}({\mathbb{Q}_{p}})$

which factors the functor

$\mathbb{C}^{*}_{\textrm{\'{e}t}}\big{(}(\cdot)_{\overline{K}},{\mathbb{Q}_{p}}\big{)}:\operatorname{Sm}_{/Z}^{{}^{\mathrm{op}}}\to\operatorname{\mathbb{D}dga}({\mathbb{Q}_{p}})$

up to homotopy.

### 5.4.

We note for use in segment 7.9 below that for similar reasons, there’s a
further factorization of the unstructured realization functor
$\operatorname{\mathbb{R}e}_{\overline{K},{\textrm{\'{e}t}}}$ through a
structured realization functor: let
$\mathbb{D}({\mathbb{Q}_{p}}_{Z,{\textrm{\'{e}t}}})$ denote the derived
infinity category of complexes of sheaves of ${\mathbb{Q}_{p}}$ vector spaces;
then there’s a functor

$\operatorname{\mathbb{R}e}_{\textrm{\'{e}t}}:\mathbb{D}A(Z,\mathbb{Q})\to\mathbb{D}({\mathbb{Q}_{p}}_{Z,{\textrm{\'{e}t}}})$

which factors the functor

$X\mapsto\mathbb{R}f_{*}{\mathbb{Q}_{p}}_{X}$

up to homotopy.

This structured realization functor can equally be upgraded to a realization
functor on motivic dga’s. Let
$\operatorname{\mathbb{D}dga}({\mathbb{Q}_{p}}_{Z_{\textrm{\'{e}t}}})$ denote
the infinity category of sheaves of (for us always commutative) differential
graded ${\mathbb{Q}_{p}}_{Z_{\textrm{\'{e}t}}}$-algebras. Then there’s a
functor

$\operatorname{\mathbb{R}e}_{\textrm{\'{e}t}}^{a}:\operatorname{\mathbb{D}Mdga}(Z,\mathbb{Q})\to\operatorname{\mathbb{D}dga}({\mathbb{Q}_{p}}_{Z_{\textrm{\'{e}t}}})$

which factors the functor

$\mathbb{C}^{*}_{\textrm{\'{e}t}}(\;\cdot\;,{\mathbb{Q}_{p}}):\operatorname{Sm}^{{}^{\mathrm{op}}}_{/Z}\to\operatorname{\mathbb{D}dga}({\mathbb{Q}_{p}}_{Z_{\textrm{\'{e}t}}})$

which sends a smooth scheme $X$ to the derived pushforward

$\mathbb{C}^{*}_{\textrm{\'{e}t}}(X,{\mathbb{Q}_{p}}):=\mathbb{R}f_{*}{\mathbb{Q}_{p}}_{X_{\textrm{\'{e}t}}}$

equipped with cup product.

### 5.5.

The forgetful functor

$F_{*}:\mathrm{Mdga}\to\mathbb{M}$

induces a functor

$RF_{*}:\operatorname{DMdga}\to\operatorname{DA}.$

Let $I$ denote the category

$\bullet\leftarrow\bullet\to\bullet.$

For each $i\in\mathbb{Z}$ there’s a commuting diagram like so.

$\textstyle{\operatorname{Ho}(\mathrm{Mdga}^{I})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Re_{\overline{K},{\textrm{\'{e}t}}}^{a,I}}$$\scriptstyle{\operatorname{hocolim}}$$\textstyle{\operatorname{Ho}(\operatorname{dga}^{I})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{hocolim}}$$\textstyle{\operatorname{DMdga}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Re^{a}_{\overline{K},{\textrm{\'{e}t}}}}$$\textstyle{\operatorname{Ddga}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{RF_{*}}$$\textstyle{\operatorname{DA}_{\mathrm{cell}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset}$$\scriptstyle{H_{i}^{t}}$$\textstyle{\operatorname{DA}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Re_{\overline{K},{\textrm{\'{e}t}}}}$$\textstyle{D({\mathbb{Q}_{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H_{i}}$$\textstyle{\operatorname{Ind}\operatorname{TM}^{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Re^{\mathrm{\heartsuit}}_{\overline{K},{\textrm{\'{e}t}}}}$$\textstyle{\operatorname{\bf{Vect}}({\mathbb{Q}_{p}}).}$

Here $\operatorname{\bf{Vect}}{\mathbb{Q}_{p}}$ denotes the category of _all_
vector spaces. The lower-left portion of the diagram is available only under
the assumption that a mixed Tate t-structure exits.

###  5.6. Poof of the connectedness and concentration theorems (0.6 and 0.7)

######  Lemma 5.6.1.

We claim first that if an object $A$ of
$\operatorname{Ind}\operatorname{TM}(Z,\mathbb{Q})$ goes to zero under
$Re^{\mathrm{\heartsuit}}_{\overline{K},{\textrm{\'{e}t}}}$ then $A=0$.

###### Proof.

Write

$A=\operatorname{colim}A_{\bullet}$
$A_{\bullet}:I\to\operatorname{TM}^{c}(Z,\mathbb{Q})$

as a filtered colimit of compact objects. The functor
$Re_{\overline{K},{\textrm{\'{e}t}}}^{\mathrm{\heartsuit}}$ commutes with
colimits since it’s a left adjoint, so $A$ maps to

$\operatorname{colim}Re_{\overline{K},{\textrm{\'{e}t}}}^{\mathrm{\heartsuit,c}}A_{i}.$

The vanishing of a filtered colimit of compact objects means that there exists
a cofinal subdiagram $F:J\to I$ such that for any morphism $f$ in $J$,

$Re_{\overline{K},{\textrm{\'{e}t}}}^{\mathrm{\heartsuit}}(A_{\bullet}(F(f)))=0.$

However, the restriction of
$Re_{\overline{K},{\textrm{\'{e}t}}}^{\mathrm{\heartsuit}}$ to
$\operatorname{TM}^{c}$ is fully faithful, so that in this case we have

$A_{\bullet}(F(f))=0.$

Hence

$A=\operatorname{colim}_{I}A_{\bullet}=\operatorname{colim}_{J}A_{\bullet}=0.\qed$

In view of the lemma, it will be enough to show that
$H_{i}RF_{*}Re^{a}_{\overline{K},{\textrm{\'{e}t}}}({{}_{\eta}B_{\omega}})$ is
nonzero for $i=0$ and is zero for $i\neq 0$.

#### 5.6.2.

We fix cofibrant / fibrant replacements $QX\xrightarrow{c}CX$,
$\mathbbm{1}\xrightarrow{f}P\mathbbm{1}$ in $\mathrm{Mdga}(Z,\mathbb{Q})$, and
suppose $\omega$, $\eta$ are represented by morphisms

$QX\underset{\eta}{\overset{\omega}{\rightrightarrows}}P\mathbbm{1}.$

We switch to working in the infinity category
$\operatorname{\mathbb{D}Mdga}(Z,\mathbb{Q})$ associated to
$\mathrm{Mdga}(Z,\mathbb{Q})$. There we may pick inverses up to homotopy

$QX\xleftarrow{c^{-1}}CX$ $\mathbbm{1}\xleftarrow{f^{-1}}P\mathbbm{1}.$

We let $\omega^{\prime}:=f^{-1}\omega c^{-1}$, $\eta^{\prime}=f^{-1}\eta
c^{-1}$. We then have an equivalence of diagrams

$(QX\underset{\eta}{\overset{\omega}{\rightrightarrows}}P\mathbbm{1})\sim(CX\underset{\eta^{\prime}}{\overset{\omega^{\prime}}{\rightrightarrows}}\mathbbm{1})$

in $\operatorname{\mathbb{D}Mdga}(Z,\mathbb{Q})$555i.e. a morphism in the
diagram category which induces an isomorphism in the homotopy category; see,
for instance, Proposition 1.2.7.3 of Lurie [36] for the formation of the
category of diagrams., and hence an equivalence of pushouts

$P\mathbbm{1}\coprod_{QX}P\mathbbm{1}\sim\mathbbm{1}\coprod_{CX}\mathbbm{1}.$

The functor

$\mathbb{R}e_{\overline{K},{\textrm{\'{e}t}}}^{a}:\operatorname{\mathbb{D}Mdga}(Z,\mathbb{Q})\to\operatorname{\mathbb{D}dga}({\mathbb{Q}_{p}})$

preserves equivalences666c.f. Proposition 1.2.3.1 of Lurie [36] and respects
pushouts and terminal objects. Indeed, the functor

$\mathbb{R}e_{\overline{K},{\textrm{\'{e}t}}}:\mathbb{D}A(Z,\mathbb{Q})\to\mathbb{D}({\mathbb{Q}_{p}})$

from the infinity category associated to the motivic model category to the
derived infinity category of complexes of ${\mathbb{Q}_{p}}$-vector spaces
preserves colimits and is monoidal. Since the forgetful functors

$\operatorname{\mathbb{D}Mdga}(Z,\mathbb{Q})\to\mathbb{D}A(Z,\mathbb{Q})$

and

$\operatorname{\mathbb{D}dga}({\mathbb{Q}_{p}})\to\mathbb{D}({\mathbb{Q}_{p}})$

are conservative and preserve sifted colimits, it follows that the functor of
monoid objects $\mathbb{R}e_{\overline{K},{\textrm{\'{e}t}}}^{a}$ preserves
sifted colimits and finite products, hence all small colimits.

Hence, we have an equivalence

$\mathbb{R}\mathrm{e}_{\overline{K},{\textrm{\'{e}t}}}^{a}(P\mathbbm{1}\coprod_{QX}P\mathbbm{1})={\mathbb{Q}_{p}}\coprod_{C_{\textrm{\'{e}t}}^{*}(X_{\overline{K}})}{\mathbb{Q}_{p}}.$

#### 5.6.3.

We claim that the derived augmentations

$C^{*}_{\textrm{\'{e}t}}(X_{\overline{K}},{\mathbb{Q}_{p}})\underset{\mathbb{R}\mathrm{e}^{a}_{\overline{K},{\textrm{\'{e}t}}}\eta^{\prime}}{\overset{\mathbb{R}\mathrm{e}^{a}_{\overline{K},{\textrm{\'{e}t}}}\omega^{\prime}}{\rightrightarrows}}{\mathbb{Q}_{p}}$

are in the image of the map

$\operatorname{Hom}_{\operatorname{dga}}(C_{\textrm{\'{e}t}}^{*}(X_{\overline{K}}),{\mathbb{Q}_{p}})\to\pi_{0}\operatorname{Hom}_{\operatorname{\mathbb{D}dga}}(C_{\textrm{\'{e}t}}^{*}(X_{\overline{K}}),{\mathbb{Q}_{p}}).$

In fact, we make the stronger claim that the target is a singleton.

#### 5.6.4.

Since $X$ is an affine curve, the dga
$C^{*}_{\textrm{\'{e}t}}=C^{*}_{\textrm{\'{e}t}}(X_{\overline{K}},{\mathbb{Q}_{p}})$
is cohomologically concentrated in degrees $0$ and $1$; recall that we’re also
assuming that the cohomology in degree zero is equal to ${\mathbb{Q}_{p}}$.
This allows us to construct a formal model

${{}^{\prime}C}\to C^{*}_{\textrm{\'{e}t}}.$

We set ${{}^{\prime}C}^{0}={\mathbb{Q}_{p}}$. We let ${{}^{\prime}C}^{1}$ be a
subspace of $C_{\textrm{\'{e}t}}^{1}$ which maps isomorphically to $H^{1}$. We
set

${{}^{\prime}C}={{}^{\prime}C}^{0}[0]\oplus{{}^{\prime}C}^{1}[1]$

with zero differential and product defined by declaring products of elements
of ${{}^{\prime}C}^{1}$ to be zero. Since ${{}^{\prime}C}\subset
C_{\textrm{\'{e}t}}^{*}$ is a quasi-isomorphism, we have a bijection

$\pi_{0}\operatorname{Hom}_{\operatorname{\mathbb{D}dga}({\mathbb{Q}_{p}})}(C_{\textrm{\'{e}t}}^{*},{\mathbb{Q}_{p}})\xrightarrow{\sim}\pi_{0}\operatorname{Hom}_{\operatorname{\mathbb{D}dga}({\mathbb{Q}_{p}})}({{}^{\prime}C},{\mathbb{Q}_{p}}).$

#### 5.6.5.

In order to compute the right hand side, we turn to the explicit cofibrant
replacement of Bousfield–Guggenheim [14, §4.7]. Recall that a cofibrant
replacement of ${{}^{\prime}C}$ is obtained as a colimit of a sequence of
dga’s starting with the initial dga ${\mathbb{Q}_{p}}$:

$\textstyle{{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{1}}$$\textstyle{L_{{}^{\prime}C}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\psi_{1}}$$\textstyle{L_{{}^{\prime}C}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{}^{\prime}C}.}$

A glance at the steps of the construction will show that

$L_{{}^{\prime}C}=\operatorname{colim}_{n}L_{{}^{\prime}C}(n)$

is concentrated in nonnegative cohomological degrees with
$L_{{}^{\prime}C}^{0}={\mathbb{Q}_{p}}$. It will then follow that

$\operatorname{Hom}_{\operatorname{dga}}(L_{{}^{\prime}C},{\mathbb{Q}_{p}})=\\{*\\}$

is a singleton. Since the latter surjects onto the derived hom

$\operatorname{Hom}_{\operatorname{Ddga}}(L_{{}^{\prime}C},{\mathbb{Q}_{p}})=\pi_{0}\operatorname{Hom}_{\operatorname{\mathbb{D}dga}}(L_{{}^{\prime}C},{\mathbb{Q}_{p}}),$

the claim will follow.

#### 5.6.6.

We modify the construction of $L_{{}^{\prime}C}(1)$ given in
Bousfield–Guggenheim in a trivial way: instead of generating the free algebra
on the set of all elements of $Z{{}^{\prime}C}$ as in loc. cit., we choose a
graded basis $\mathcal{B}$ of $ZB$ and set

$L_{{}^{\prime}C}(1):=\bigotimes_{z\in\mathcal{B}}S(|z|).$

Here $|z|$ denotes the dimension of $z$, and $S(n)$ is the free graded-
commutative algebra on one generator $a$ in dimension $n$ with differential
induced by $da=0$.

For the reader’s convenience, we recall from loc. cit. the rest of the
construction. We define $T(n)$ to be the free graded-commutative algebra on
generators $b\in T(n)^{n}$ and $c\in T(n)^{n+1}$ with $db=c$. To construct
$L_{{}^{\prime}C}(n+1)$ we let $R$ be the set of pairs $(w,b)$ such that $w\in
L_{{}^{\prime}C}(n)$, $b\in{{}^{\prime}C}$,

$0\leq|w|=|b|+1,$ $dw=0$

and

$db=\psi_{n}(w),$

and construct $L_{{}^{\prime}C}(n+1)$ by means of a pushout

$\textstyle{\underset{(w,b)\in
R}{\bigotimes}S(|w|)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{{}^{\prime}C}(n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{n+1}}$$\textstyle{\underset{(w,b)\in
R}{\bigotimes}T(|b|)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{{}^{\prime}C}(n+1).}$

Although Bousfield-Guggenheim restrict attention to non-negatively graded
dga’s, it is not hard to check that the same construction works in the model
category of all dga’s. This completes the proof of the claim made in segment
5.6.3.

#### 5.6.7.

The diagram 5.6.3 in $\operatorname{\mathbb{D}dga}({\mathbb{Q}_{p}})$ thus has
a lifting to a diagram

${{}^{\prime}C}\rightrightarrows{\mathbb{Q}_{p}}$

in $\operatorname{dga}({\mathbb{Q}_{p}})$. The limit

${\mathbb{Q}_{p}}\coprod_{C^{*}_{\textrm{\'{e}t}}(X_{\overline{K}})}^{\operatorname{\mathbb{D}dga}}{\mathbb{Q}_{p}}$

of 5.6.3(*) is thus represented by the reduced bar complex, a double complex
which has ${{}^{\prime}C}^{\otimes n}$ on the $n$th line. As instructed by
Ayoub, we mod out by the degeneracies, which are given by all maps

$({\mathbb{Q}_{p}}\oplus{{}^{\prime}C}^{1}[-1])^{\otimes
n-1}\to({\mathbb{Q}_{p}}\oplus{{}^{\prime}C}^{1}[-1])^{\otimes n}$

induced by the inclusion
${\mathbb{Q}_{p}}\to{\mathbb{Q}_{p}}\oplus{{}^{\prime}C}^{1}[-1]$. The result
is ${{}^{\prime}C}^{1}[-1]^{\otimes n}$, so upon passing to the total complex,
we obtain the tensor algebra on ${{}^{\prime}C}^{1}$ in degree $0$, and zero
in all other degrees. This establishes the connectedness and concentration
theorems.

###  5.7. Remark

In the course of the proof, we chose a splitting of the surjection

$C_{\textrm{\'{e}t}}^{1}(X_{\overline{K}},{\mathbb{Q}_{p}})\twoheadrightarrow
H^{1}_{\textrm{\'{e}t}}(X_{\overline{K}},{\mathbb{Q}_{p}}).$

Let us denote its image simply by $C\subset C_{\textrm{\'{e}t}}^{1}$. We noted
that ${\mathbb{Q}_{p}}[0]\oplus C[-1]$ has a quasi-isomorphism to
$C^{*}_{\textrm{\'{e}t}}$ on the one hand, and that the former has only one
augmentation on the other hand. This uniqueness of augmentations is reflected
in the existence of canonical étale paths connecting every two base-points,
depending on the choice of the splitting, a phenomenon which, as far as we
know, has not yet been observed. Perhaps the reason is that the dga
$C_{\textrm{\'{e}t}}^{1}(X_{\overline{K}},{\mathbb{Q}_{p}})$ is a bit
abstract: it is obtained via the general machinery of canonical resolutions
and rectification. Its de Rham cousin $\Omega^{*}$ however, is quite concrete.
The resulting canonical de Rham paths were observed by Deligne [23] in a
special case (under the further assumption that $X$ has a compactification
$\overline{X}$ with $H^{1}(\overline{X},\mathcal{O})=0$).

We sketch a Tannakian construction of such paths which developed in
conversations with Stefan Wewers. In this setting, $C$ represents the choice
of a subspace of $\Omega^{1}(X)$ which maps isomorphically onto
$H^{1}_{\rm{dR}}(X)$. Consider $(E,\nabla)$, a unipotent vector bundle with
integrable connection. Since

$H^{1}(X,\mathcal{O})=0,$

the underlying vector bundle $E$ is trivial. One can check that there’s a
unique $K$-form $V$ of $E$ such that $\nabla$ restricts to a $K$-linear map

$V\to V\otimes C.$

The functor

$\omega_{C}:E\mapsto V$

is a fiber functor. If $x$ is a $K$-point of $X$, then there are canonical
isomorphisms

$V\xrightarrow{\sim}\omega_{x}(E):=E(x),$

which, as $E$ varies, form an isomorphism of fiber functors

$\omega_{C}\xrightarrow{\sim}\omega_{x}.$

###  5.8. Remark

The infinity-categorical steps in our proof can be unravelled into (somewhat
cumbersome) model-categorical ones. If one attempts to work naively with model
categories however, it’s hard to see any direct relation between the left-
derived realization on the level of $I$-diagrams

$Re^{a,I}_{\overline{K},{\textrm{\'{e}t}}}(QX\rightrightarrows
P\mathbbm{1})\in\operatorname{Ho}(\operatorname{dga}^{I})$

(which a priori may require a fibrant replacement of the diagram), with a
diagram of dga’s of the form

$C_{\textrm{\'{e}t}}^{*}(X_{\overline{K}})\rightrightarrows{\mathbb{Q}_{p}}$

to which we can apply Ayoub’s explicit computation via the classical bar
construction.

##  6\. The unipotent étale fundamental group: comparison of external and
internal viewpoints

### 6.1.

We put ourselves in the setting of the introduction ($Z$, $X$, $x$, $p$). We
let $Z_{p}$ denote $\operatorname{Spec}$ of the complete local ring at $p$,
and we let $X_{p}$ and $x_{p}$ denote the base change of $X$ and of $x$ to
$Z_{p}$. We let ét denote the pro-étale topology of Bhatt–Scholze [13]. If $f$
denotes the structure morphism

$f:X_{p}\to Z_{p},$

then $Rf_{*}{\mathbb{Q}_{p}}_{X_{p,{\textrm{\'{e}t}}}}$ has the structure of a
sheaf of a differential graded
${\mathbb{Q}_{p}}_{Z_{\textrm{\'{e}t}}}$-algebras equipped with an
augmentation induced by the section $x_{p}$.

### 6.2.

We consider the model structure on the category
$\operatorname{dga}({\mathbb{Q}_{p}}_{Z_{\textrm{\'{e}t}}})$ of Cisinski-
Déglise [17]. This allows us to take a homotopy colimit in order to form the
_rational p-adic étale loop ring_

$B^{\textrm{\'{e}t}}_{x_{p}}:=\operatorname{hocolim}\left(Rf_{*}{\mathbb{Q}_{p}}_{X_{p,{\textrm{\'{e}t}}}}\underset{x_{p}^{*}}{\overset{x_{p}^{*}}{\rightrightarrows}}{\mathbb{Q}_{p}}_{Z_{p,{\textrm{\'{e}t}}}}\right).$

More generally, as in the motivic setting, if

$Rf_{*}{\mathbb{Q}_{p}}_{X_{p,{\textrm{\'{e}t}}}}\underset{\eta}{\overset{\xi}{\rightrightarrows}}{\mathbb{Q}_{p}}_{Z_{p,{\textrm{\'{e}t}}}}$

are any two augmentations, the we may form the _rational p-adic étale path
ring_

${{}_{\eta}B^{\textrm{\'{e}t}}_{\xi}}:=\operatorname{hocolim}\left(Rf_{*}{\mathbb{Q}_{p}}_{X_{p,{\textrm{\'{e}t}}}}\underset{\eta}{\overset{\xi}{\rightrightarrows}}{\mathbb{Q}_{p}}_{Z_{p,{\textrm{\'{e}t}}}}\right).$

With these definitions the family
$\\{{{}_{\eta}B^{\textrm{\'{e}t}}_{\xi}}\\}_{\xi,\eta}$ forms a cogroupoid in
$\operatorname{Ddga}({\mathbb{Q}_{p}}_{Z_{p,{\textrm{\'{e}t}}}})$. Each
${\mathbb{Q}_{p}}_{Z_{p,{\textrm{\'{e}t}}}}$-algebra
$H^{0}({{}_{\eta}B^{\textrm{\'{e}t}}_{\xi}})$ is locally constant, so the
family $\\{H^{0}({{}_{\eta}B^{\textrm{\'{e}t}}_{\xi}})\\}_{\xi,\eta}$ may
equally be regarded as a groupoid in Galois representations.

### 6.3.

This section concerns the commutativity of the diagram below

$\textstyle{X(Z_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{\operatorname{Hom}_{\operatorname{Ddga}({\mathbb{Q}_{p}}_{Z_{\textrm{\'{e}t}}})}(Rf_{*}{\mathbb{Q}_{p}}_{X_{\textrm{\'{e}t}}},{\mathbb{Q}_{p}}_{Z_{\textrm{\'{e}t}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Cotors}B_{x_{p}}^{\textrm{\'{e}t}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(G_{K_{p}},\operatorname{Spec}H^{0}B^{\textrm{\'{e}t}}_{x_{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(G_{K_{p}},\mathrm{Aut}\,^{\otimes}\omega_{x_{p}}^{\mathrm{un}})}$

with

$\mu(y)=\operatorname{Isom}^{\otimes}(\omega_{x_{p}}^{\mathrm{un}},\omega_{y}^{\mathrm{un}}),$

see theorem 6.6 below. We begin in segment 6.4 in a general setting.

###  6.4. The unipotent fundamental group of a topos

If $X$ is a topos and $x,y$ are points, then we can associate to $x$ and $y$ a
diagram

$\mathbb{R}\Gamma(\mathbb{Q}_{X})\underset{y^{*}}{\overset{x^{*}}{\rightrightarrows}}\mathbb{Q}$

in the model (or infinity) category of (always commutative) dga’s. (The model
category structure follows, for instance, by combining Cisinski–Déglise [17]
with White [47] as we did for $\mathrm{Mdga}$.) We then let ${}_{y}B_{x}$ be
the homotopy colimit; its cohomology $H^{0}(_{y}B_{x})$ is a
$\mathbb{Q}$-algebra. Let $\mathrm{Un}(\mathbb{Q}_{X})$ denote the category of
unipotent local systems, and let $\omega_{x}^{\mathrm{un}}$,
$\omega_{y}^{\mathrm{un}}$ denote the fiber functors

$\mathrm{Un}(\mathbb{Q}_{X})\underset{}{\overset{}{\rightrightarrows}}\operatorname{\bf{Vect}}\mathbb{Q}$

associated to $x$ and to $y$. Then there’s an isomorphism

$\operatorname{Spec}H^{0}({{}_{y}B_{x}})=\operatorname{Isom}^{\otimes}(\omega_{x}^{\mathrm{un}},\omega_{y}^{\mathrm{un}}).$

Our goal here is to prove this statement, amplified by a certain base-changing
property in a relative setting.

#### 6.4.1.

For $X$ a topos and $K$ a field, we write $\operatorname{Cpl}(K_{X})$ for the
model category of complexes of $K_{X}$-modules, and $D(K_{X})$ for its
homotopy category, the (unbounded) derived category. We write
$\operatorname{dga}(K_{X})$ for the model category of commutative differential
graded $K_{X}$-algebras, and $\operatorname{Ddga}(K_{X})$ for its homotopy
category.

#### 6.4.2.

If $f:X\to Z$ is a morphism of topoi, then there’s a commutative differential
graded $K_{Z}$-algebra $C(X)=C_{Z}(X,K)$ which represents $Rf_{*}K_{Z}$ which
is contravariant in topoi over $Z$. In one of two possible constructions, we
note that $f_{*}$, being right adjoint to a monoidal functor, is lax-monoidal,
hence sends algebras to algebras. This means, for instance, that for
$E,F\in\operatorname{Cpl}(K_{X})$, there’s a morphism

$f_{*}E\otimes f_{*}F\to f_{*}(E\otimes F).$

Fixing a fibrant-replacement functor on $\operatorname{dga}(K_{X})$, we let
$P_{X}$ denote the fibrant replacement of the unit object $K_{X}$. We then set

$C(X):=f_{*}P_{X}.$

If $m$ denotes the multiplication map

$P_{X}\otimes P_{X}\to P_{X},$

then the product structure on $C(X)$, for instance, is given by the composite

$f_{*}P_{X}\otimes f_{*}P_{X}\to f_{*}(P_{X}\otimes
P_{X})\xrightarrow{f_{*}m}P_{X}.$

The dga $C(X)$ may also be interpreted as the endomorphism ring of the unit
object $K_{X}$, a construction which is a bit awkward on the model categorical
level, but automatic in the infinity-categorical setting.

#### 6.4.3.

Assume given in addition to the morphism $f:X\to Z$ of topoi, a pair of
sections $\xi,\eta:X\leftleftarrows Z$. There is then a diagram

$C(X)\underset{\eta}{\overset{\xi}{\rightrightarrows}}K_{Z}$

in $\operatorname{dga}(K_{Z})$. We let ${}_{\eta}B_{\xi}$ denote its homotopy
colimit. When $\eta=\xi$ we write simply $B_{\xi}$. As the sections $\xi$,
$\eta$ vary, the objects ${}_{\eta}B_{\xi}$ form a commutative Hopfoid (4.1)
in $D(K_{Z})$. We may then take cohomology with respect to the canonical
t-structure to obtain a commutative Hopfoid
$\\{H^{0}({{}_{\eta}B_{\xi}})\\}_{\xi,\eta}$ in the category

$\operatorname{\bf{Vect}}K_{Z}$

of $K_{Z}$-modules.

#### 6.4.4.

If $Z$ is a topos and $K$ a field, we let

$D_{\mathrm{cell}}(K_{Z})\subset D(K_{Z})$

denote the the smallest tensor-triangulated subcategory closed under small
direct sums and containing the constant sheaf $K_{Z}$. Its heart, which we
denote by $\operatorname{Cell}(K_{Z})$ is equal to the Ind-completion of the
category $\operatorname{Uni}(K_{Z})$ of unipotent local systems.

######  Theorem 6.5 (Unipotent fundamental group of pointed topos).

Let $K$ be a field containing $\mathbb{Q}$. Consider a diagram of topoi

$\textstyle{X_{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{z}}$$\scriptstyle{\zeta}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\operatorname{Set}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{z}$$\scriptstyle{x}$$\scriptstyle{y}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\scriptstyle{\eta}$

in which all upwards arrows are sections of the respective downwards arrows,
and all three squares commute. Assume the base-change property

$z^{*}C(X)=C(X_{z}),$

as well as the following connectivity property: every locally constant sheaf
on $X_{z}$ which vanishes at $x$ vanishes globally. Let
$\omega_{x}^{\mathrm{un}}$, $\omega_{y}^{\mathrm{un}}$ denote the fiber
functors

$\operatorname{Uni}(K_{X_{z}})\underset{}{\overset{}{\rightrightarrows}}\operatorname{\bf{Vect}}^{c}(K)$

induced by $x$, $y$. (Here $\operatorname{\bf{Vect}}^{c}$ denotes the category
of finite dimensional vector spaces.) Then we have an isomorphism

$\operatorname{Spec}z^{*}H^{0}({{}_{\eta}B_{\xi}})=\operatorname{Isom}^{\otimes}(\omega_{x}^{\mathrm{un}},\omega_{y}^{\mathrm{un}})$

which respects the groupoid structure as the sections $\xi$, $\eta$, $x$, $y$
vary.

We begin by recalling two parts of Tannaka duality.

#### 6.5.1.

Suppose $T$ is a Tannakian category over a field $k$ and $\omega$ is a
$k$-rational fiber functor. Then the fundamental group

$\pi_{1}(X,\omega):=\mathrm{Aut}\,^{\otimes}\omega$

is _characterized_ as the unique proalgebraic affine $k$-group whose
representation category fits into a commutative triangle

---
$\textstyle{\operatorname{\bf{Rep}}^{c}_{k}\big{(}\pi_{1}(T,\omega)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{\operatorname{\bf{Vect}}^{c}k.}$

(The superscript $c$ refers to _compact objects_.)

#### 6.5.2.

If $\eta$ is another fiber functor, then the Tannakian path torsor

${}_{\eta}P_{\omega}(X):=\operatorname{Isom}^{\otimes}(\omega,\eta)$

is characterized as the unique $\pi_{1}(T,\omega)$–$\pi_{1}(T,\eta)$–bitorsor
such that pushout along ${}_{\eta}P_{\omega}$ fits into a commuting triangle

---
$\textstyle{\operatorname{\bf{Rep}}^{c}_{k}\big{(}\pi_{1}(T,\omega)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T.}$$\textstyle{\operatorname{\bf{Rep}}^{c}_{k}\big{(}\pi_{1}(T,\eta)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

#### 6.5.3.

If $A$ is a dga, we let $\operatorname{DMod}A$ denote the derived category of
$A$-modules. If $Y$ is a topos (regarded as topos over $\operatorname{Set}$),
then Morita theory, as formulated by Lurie [35, §7.1.2.1–7] provides an
equivalence of triangulated categories

$D_{\mathrm{cell}}(K_{Y})=\operatorname{DMod}C(Y,K).$

Moreover, this equivalence is functorial in the following way. If

$f:Y^{\prime}\to Y$

is a morphism of topoi, then the induced morphism of dga’s

$CY^{\prime}\leftarrow CY$

gives rise, via the bar construction, to a morphism $f^{*}$ of triangulated
categories, and the square

$\textstyle{\operatorname{DMod}CY^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{DMod}CY\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{D_{\mathrm{cell}}K_{Y^{\prime}}}$$\textstyle{D_{\mathrm{cell}}K_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

commutes.

#### 6.5.4.

If $B$ is a differential graded coalgebra, we let
$\operatorname{D^{co}Comod}B$ denote the coderived category of comodules of
Positselski [38, §4.4], a certain variant of the derived category, which comes
equipped with a natural t-structure whose heart is the same as the heart of
the derived category of comodules. If $Y$ is a topos, and
$y:\operatorname{Set}\to Y$ is a point, then the Koszul duality of Positselski
[38, §6.3] provides an equivalence of triangulated categories

$\operatorname{DMod}CY=\operatorname{D^{co}Comod}B_{y}.$

Moreover, this equivalence is functorial in the following way. If

$f:Y^{\prime}\to Y$

is a morphism of topoi and $y^{\prime}$ is a point of $Y$ lying above $y$,
then the a natural morphism of comonoids in dga’s

$f_{*}:B_{y^{\prime}}\to B_{y}$

gives rise, by forgetting along $f_{*}$, to a morphism $f^{*}$ of triangulated
categories, and the square

$\textstyle{\operatorname{D^{co}Comod}B_{y^{\prime}}}$$\textstyle{\operatorname{D^{co}Comod}B_{y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{\operatorname{DMod}CY^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{DMod}CY\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

commutes.

#### 6.5.5.

Applying segments 6.5.3 and 6.5.4 with $Y=X_{z}$, and
$Y^{\prime}=\operatorname{Set}$, we obtain a commuting triangle of
triangulated categories

---
$\textstyle{D_{\mathrm{cell}}K_{X_{z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x^{*}}$$\textstyle{\operatorname{D^{co}Comod}B_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x^{*}}$$\textstyle{D(K).}$

On both sides, the natural t-structure is the one induced by the natural
t-structure on $D(K)$. In the case of $\operatorname{D^{co}Comod}B_{x}$, this
is clear; in the case of $D_{\mathrm{cell}}K_{X_{z}}$, this follows from the
connectivity property included in our assumptions, in view of the local-
constancy of cohomology sheaves in the cellular category. It follows that the
equivalence of triangulated categories is compatible with t-structures, and
that the induced equivalence of hearts is compatible with the fiber functors.

#### 6.5.6.

Denoting the heart of the natural t-structure by $\heartsuit$, we have

$\operatorname{Comod}H^{0}z^{*}B_{\xi}=\heartsuit\operatorname{Comod}z^{*}B_{\xi}$

because of flatness of modules over a field. We also have

$z^{*}B_{\xi}=B_{x}$

because of the base-changing property assumed in the statement. Combining
these observations with the Morita theory and Koszul duality discussed above,
we obtain a sequence of equivalences of neutralized Ind-Tannakian categories,
which we may summarize as follows:

$\displaystyle\operatorname{Comod}z^{*}H^{0}B_{\xi}$
$\displaystyle=\operatorname{Comod}H^{0}z^{*}B_{\xi}$
$\displaystyle\overset{\mbox{flatness}}{=}\heartsuit\operatorname{D^{co}Comod}z^{*}B_{\xi}$
$\displaystyle\overset{\mbox{base
change}}{=}\heartsuit\operatorname{D^{co}Comod}B_{x}$
$\displaystyle\overset{\mbox{Koszul
duality}}{=}\heartsuit\operatorname{DMod}\big{(}C(X_{z})\big{)}$
$\displaystyle\overset{\mbox{Morita theory}}{=}\heartsuit
D_{\mathrm{cell}}(K_{X_{z}})$ $\displaystyle=\operatorname{Cell}(K_{X_{z}}).$

The isomorphism of groups (6.5(*) for $\eta=\xi$ and $x=y$) now follows from
our clause 6.5.1 of Tannaka duality.

#### 6.5.7.

For the isomorphisms of path torsors, we claim that the triangle

---
$\textstyle{\operatorname{D^{co}Comod}z^{*}B_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\otimes(z^{*}{{}_{\eta}B_{\xi}})}$$\textstyle{D_{\mathrm{cell}}(K_{X_{z}})}$$\textstyle{\operatorname{D^{co}Comod}z^{*}B_{y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$

commutes. Indeed, this amounts to the property of the bar construction

$B(M,A,x)\otimes_{B(x,A,x)}B(x,A,y)=B(M,A,y)$

applied to $A=CX_{z}$ and $M$ a cellular complex of $K_{X_{z}}$-modules. We
may then take $H^{0}$’s on the left and heart on the right, and apply part
6.5.2 of Tannaka duality to obtain the hoped-for isomorphism. This completes
the proof of theorem 6.5.

Our application is a direct consequence of theorem 6.5:

######  Theorem 6.6 (Comparison of étale-internal and étale-external
constructions).

. Let $Z$ be a Dedekind scheme with function field $K$,

$f:X\to Z$

a smooth scheme over $Z$, $z=\operatorname{Spec}\overline{K}$ a generic
geometric point of $Z$, $\xi,\eta$ sections of $f$ inducing sections $x,y$ of
the pullback $f_{z}$ of $f$ to $z$ as in the the diagram of theorem 6.5. We
let ét denote the pro-étale topology of Bhatt-Scholze [13]. Let

$\mathrm{Uni}({\mathbb{Q}_{p}}_{{X_{z}}_{\textrm{\'{e}t}}})\underset{\omega_{y}^{\mathrm{un}}}{\overset{\omega_{x}^{\mathrm{un}}}{\rightrightarrows}}\operatorname{\bf{Vect}}^{c}({\mathbb{Q}_{p}})$

denote the fiber functors on the category of unipotent local systems on
$X_{z}$ induced by $x$ and $y$. Then there’s a $G_{K}$-equivariant isomorphism

$\operatorname{Isom}^{\otimes}(\omega_{x}^{\mathrm{un}},\omega_{y}^{\mathrm{un}})=\operatorname{Spec}z^{*}H^{0}\big{(}\operatorname{hocolim}_{\operatorname{dga}({\mathbb{Q}_{p}}_{Z_{\textrm{\'{e}t}}})}(Rf_{*}{{\mathbb{Q}_{p}}}_{X_{\textrm{\'{e}t}}}\underset{\eta^{*}}{\overset{\xi^{*}}{\rightrightarrows}}{{\mathbb{Q}_{p}}}_{Z_{{\textrm{\'{e}t}}}})\big{)}.$

###### Proof.

The isomorphism furnished by theorem 6.5 is $G_{K}$-equivariant by
functoriality. ∎

We’ll denote both sides of equation 6.6(*) by
${{}_{y}\pi_{x}^{\textrm{\'{e}t}}}(X_{\overline{K}})$, or by
$\pi_{1}^{\textrm{\'{e}t}}(X_{\overline{K}},x)$ when $x=y$.777Let us emphasize
that these are prounipotent completions of the classical profinite fundamental
group and path torsors.

##  7\. h-cotorsors and Factorization of Kim’s cutter

### 7.1.

We consider a smooth affine curve $X$ with étale divisor at infinity over an
open integer scheme $Z\subset\operatorname{Spec}\mathcal{O}_{K}$, and a
$Z$-point $x\in X(Z)$. We then have the loop ring $B_{x}$ of segment 2.2.
According to §2.4 and 4.3 , $B_{x}$ has the structure of a commutative Hopf
algebra object of the monoidal category
$\operatorname{DA}=\operatorname{DA}(Z,\mathbb{Q})$.

### 7.2.

To define the notion of _cotorsor_ we wish to consider, we regard $B_{x}$ as a
cogroup object of the category of commutative monoid objects in
$\operatorname{DA}$. We define a _cotorsor_ to be an object $P$ of
$\operatorname{cMon}\operatorname{DA}$ equipped with a coaction map

$B_{x}\otimes^{\mathbb{L}}_{\mathbbm{1}}P\leftarrow P$

such that

$B_{x}\otimes^{\mathbb{L}}_{\mathbbm{1}}P\xleftarrow{\sim}P\otimes^{\mathbb{L}}P$

and

$H^{0}_{\textrm{\'{e}t}}(P_{\overline{K}},{\mathbb{Q}_{p}})\neq 0.$

Here the symbol $\otimes^{\mathbb{L}}_{\mathbbm{1}}$ refers to the monoidal
structure on $\operatorname{cMon}\operatorname{DA}$ induced by the monoidal
structure $\otimes^{\mathbb{L}}$ on $\operatorname{DA}$. A cotorsor is
_cellular_ if the associated object of $\operatorname{DA}$ is cellular. We let
$\operatorname{Cotors}_{\mathrm{cell}}(B_{x})$ denote the set of cellular
cotorsors. We define

$f_{2}:\operatorname{Hom}_{\operatorname{DMdga}}(\operatorname{C}(X),\mathbbm{1})\to\operatorname{Cotors}_{{}_{\mathrm{cell}}}(B_{x})$

by

$f_{2}(\omega):={{}_{\omega}B_{x}}.$

###  7.3. Lemma

Let $D$ be a triangulated tensor category with a t-structure $t$, and suppose
$B$ is a cogroup object of $\operatorname{cMon}D$. Fix $i\neq 0$ and assume
$H^{i}_{t}(B)=0$. If $P$ is a cotorsor under $B$, then $H^{i}_{t}(P)=0$.

###### Proof.

By functoriality of the Kunneth decomposition, we have an isomorphism

$0=H_{t}^{i}(B)\otimes H_{t}^{0}(P)\xleftarrow{\sim}H_{t}^{i}(P)\otimes
H_{t}^{0}(P).$

Since $H_{t}^{0}(P)\neq 0$, it follows that $H_{t}^{i}(P)=0$ as claimed. ∎

### 7.4.

If $X$ is mixed Tate, then we may take cohomology with respect to the
t-structure of section 3. In view of concentration 0.7, the pro- mixed Tate
motive $H^{0}_{t}(B_{x})$ has the structure of a commutative Hopf algebra. We
set

$\pi_{1}^{\mathrm{un}}(X,x):=\operatorname{Spec}H^{0}_{t}(B_{x}).$ 888We do
not carry out the task of comparing our unipotent fundamental group with the
one constructed by Deligne–Goncharov [22]. We note however that our group has
all the needed properties (comparison with Tannakian étale and de Rham
fundamental groups, relationship with motivic Ext groups) to carry out
“motivic Chabauty–Kim theory”, as in Dan-Cohen–Wewers [21] and Dan-Cohen [19].

### 7.5.

In view of lemma 7.3, the concentration theorem (0.7), the cellular motives
theorem (3.1.4), and interplay between Hopfoids and localization (4.3), if $P$
is a cellular cotorsor under $B_{x}$, then $H^{0}_{t}(P)$ is a cotorsor under
$H^{0}_{t}(B_{x})$ in
$\operatorname{cMon}\operatorname{Ind}\operatorname{TM}(Z,\mathbb{Q})$. Thus,
in terms of the canonical fiber functor $\omega_{\mathrm{can}}$, we find that
$\operatorname{Spec}\omega_{\mathrm{can}}H^{0}_{t}(P)$ is a
$\pi_{1}^{\operatorname{TM}}(Z,\mathbb{Q})$-equivariant
$\pi_{1}^{\mathrm{un}}(X,x)$-torsor.

Actually, if $\omega$ is any $\mathbb{Q}$-rational fiber functor on
$\operatorname{TM}(Z,\mathbb{Q})$, then the pointed set

$H^{1}\big{(}\operatorname{TM}(Z,\mathbb{Q}),\pi_{1}^{\mathrm{un}}(X,x)\big{)}:=H^{1}\Big{(}\pi_{1}\big{(}\operatorname{TM}(Z,\mathbb{Q}),\omega\big{)},\omega\big{(}\pi_{1}^{\mathrm{un}}(X,x)\big{)}\Big{)}$

is independent of the choice of $\omega$ (up to canonical bijection). Indeed,
its elements are in bijection with cotorsors in
$\operatorname{cMon}\operatorname{Ind}\operatorname{TM}(Z,\mathbb{Q})$.
Following Deligne [23], we let $\operatorname{Spec}$ of a commutative monoid
object denote itself regarded as an object of the opposite category. We define

$f_{3}:\operatorname{Cotors}_{\mathrm{cell}}(B_{x})\to
H^{1}\big{(}\operatorname{TM}(Z,\mathbb{Q}),\pi_{1}^{\mathrm{un}}(X,x)\big{)}$

by

$f_{3}(P):=\operatorname{Spec}H^{0}_{t}(P).$

### 7.6.

For $\mathfrak{p}$ a prime of $Z$, we let $K_{\mathfrak{p}}$ denote the
complete local field of $Z$ at $\mathfrak{p}$. Let
$\omega^{\mathrm{un}}_{x_{\mathfrak{p}}}$ denote the fiber functor on the
category of unipotent $p$-adic local systems on
$X_{\overline{K}_{\mathfrak{p}}}$ induced by $x$. We define

$\mu_{3}^{\mathfrak{p}}:X(Z_{\mathfrak{p}})\to
H^{1}\big{(}G_{K_{\mathfrak{p}}},\mathrm{Aut}\,^{\otimes}(\omega_{x_{\mathfrak{p}}}^{\mathrm{un}})\big{)}$

by

$\mu_{3}^{\mathfrak{p}}(y)=\operatorname{Isom}^{\otimes}(\omega_{x_{\mathfrak{p}}}^{\mathrm{un}},\omega_{y}^{\mathrm{un}}).$

### 7.7.

If $P$ is a cotorsor for $B_{x}(X_{\mathfrak{p}})$, then

$H^{0}(P_{\overline{K}_{\mathfrak{p}}},{\mathbb{Q}_{p}}):=H^{0}(\mathrm{Re}_{\overline{K}_{\mathfrak{p}},{\textrm{\'{e}t}}}P)$

has the structure of a commutative algebra and a cotorsor in the category of
$p$-adic $G_{K_{\mathfrak{p}}}$-representations. By the comparison theorem
(6.6), its spectrum is thus a $G_{K_{\mathfrak{p}}}$-equivariant
$\pi_{1}^{\textrm{\'{e}t}}(X_{K_{\mathfrak{p}}},x)$-torsor. We define

$f^{\mathfrak{p}}_{3}:\operatorname{Cotors}_{\mathrm{cell}}\big{(}B_{x}(X_{\mathfrak{p}})\big{)}\to
H^{1}\big{(}G_{K_{\mathfrak{p}}},\pi_{1}^{\textrm{\'{e}t}}(X_{K_{\mathfrak{p}}},x)\big{)}$

by

$f^{\mathfrak{p}}_{3}(P)=\operatorname{Spec}H^{0}\big{(}B_{x}(X_{\mathfrak{p}})_{\overline{K}},{\mathbb{Q}_{p}})\big{)}.$

### 7.8.

We let $X_{\mathfrak{p}}$ denote the base-change of $X$ to $Z_{\mathfrak{p}}$.
For $X$ mixed Tate, and $p$ a prime of $\mathbb{Z}$ lying below $Z$, we obtain
a diagram like so,

$\textstyle{X(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{\prod_{\mathfrak{p}|p}X(Z_{\mathfrak{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}^{p}}$$\scriptstyle{\mu_{1}}$$\scriptstyle{\mu_{2}}$$\scriptstyle{\mu_{3}}$$\scriptstyle{\mu^{[n]}}$$\textstyle{\operatorname{Hom}_{\operatorname{DMdga}}(\operatorname{C}(X),\mathbbm{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l_{1}}$$\scriptstyle{f_{2}}$$\textstyle{\prod_{\mathfrak{p}|p}\operatorname{Hom}_{\operatorname{DMdga}}(\operatorname{C}(X_{\mathfrak{p}}),\mathbbm{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}^{p}}$$\textstyle{\operatorname{Cotors}_{\mathrm{cell}}(B_{x}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{3}}$$\scriptstyle{l_{2}}$$\textstyle{\prod_{\mathfrak{p}|p}\operatorname{Cotors}\big{(}B_{x}(X_{\mathfrak{p}})\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{3}^{p}}$$\textstyle{H^{1}(\pi_{1}^{\operatorname{TM}}(Z),\pi_{1}^{\mathrm{un}}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l_{3}}$$\scriptstyle{f_{4}}$$\textstyle{\prod_{\mathfrak{p}|p}H^{1}\big{(}G_{K_{\mathfrak{p}}},\pi_{1}^{\textrm{\'{e}t}}(X_{\overline{K}_{\mathfrak{p}}},x)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{4}^{p}}$$\textstyle{H^{1}(\pi_{1}^{\operatorname{TM}}(Z),\pi_{1}^{\mathrm{un}}(X)^{[n]})_{{\mathbb{Q}_{p}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l^{[n]}}$$\textstyle{\prod_{\mathfrak{p}|p}H^{1}\big{(}G_{K_{\mathfrak{p}}},\pi_{1}^{\textrm{\'{e}t}}(X_{\overline{K}_{\mathfrak{p}}},x)^{[n]}\big{)},}$

in which $\mu_{3}$ sends a point $y\in X(Z_{\mathfrak{p}})$ to the Tannakian
path torsor of unipotent étale paths from $x$ to $y$.

### 7.9.

We claim that the diagram of segment 7.8 is commutative. The only nontrivial
commutativities concern the small square bounded above by $l_{2}$, and the
small triangle bounded on the left by $\mu_{2}$. We start with the former.
Recall that there’s a monoidal $p$-adic étale realization functor

$\mathrm{Re}_{\textrm{\'{e}t}}:\operatorname{DA}(Z_{p},{\mathbb{Q}_{p}})\to
D({{\mathbb{Q}_{p}}}_{{Z_{p}}_{\textrm{\'{e}t}}})$

(5.2). The composite functor

$\operatorname{DA}_{\mathrm{cell}}(Z,{\mathbb{Q}_{p}})\subset\operatorname{DA}(Z,{\mathbb{Q}_{p}})\to\operatorname{DA}(Z_{p},{\mathbb{Q}_{p}})\to
D({{\mathbb{Q}_{p}}}_{{Z_{p}}_{\textrm{\'{e}t}}})$

is compatible with t-structures. So there’s a commuting pentagon

$\textstyle{\operatorname{Cotors}_{\mathrm{cell}}B_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Cotors}_{\mathrm{cell}}B_{x_{Z_{p}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Cotors}B_{x_{K_{p}}}^{\textrm{\'{e}t}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}\big{(}\operatorname{TM}(Z,\mathbb{Q}),\pi_{1}^{\mathrm{un}}(X,x)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}\big{(}G_{K_{p}},\pi_{1}^{\textrm{\'{e}t}}(X,x)\big{)},}$

whence the hoped-for square.

We turn to the small triangle bounded on the left by $\mu_{2}$. In view of the
commutativity established in section 6, it will suffice to construct a
commuting square

$\textstyle{\operatorname{Hom}_{\operatorname{DMdga}(Z_{\mathfrak{p}},\mathbb{Q})}(CX_{\mathfrak{p}},\mathbbm{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\operatorname{Ddga}({\mathbb{Q}_{p}}_{Z_{\mathfrak{p},{\textrm{\'{e}t}}}})}(C_{\textrm{\'{e}t}}X_{\mathfrak{p}},{\mathbb{Q}_{p}}_{Z_{\mathfrak{p},{\textrm{\'{e}t}}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Cotors}B_{x}(X_{\mathfrak{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Cotors}B^{\textrm{\'{e}t}}_{x}(X_{\mathfrak{p}}).}$

This commutativity is the same, mutatis mutandis, as the equality established
in segment 5.6.2 with the structured realization functor of segment 5.4 in
place of the unstructured realization functor.

### 7.10.

If $X$ is not mixed Tate, we resort to a less motivic version of the above
diagram with the nonabelian absolute Galois cohomology
$H^{1}\big{(}G_{K},\pi_{1}^{\textrm{\'{e}t}}(X_{\overline{K}},x)\big{)}$ (as
well as its $[n]$-variant) in place of the nonabelian absolute motivic
cohomologies appearing in the lower left corner.

This completes the proof theorem 0.2.1 of the introduction in a more general
setting. We restate the theorem in this setting in terms of diagram 7.8 above.

We denote scheme theoretic image by $\overline{\operatorname{Im}}$.

######  Theorem 7.11 (Factorization of Kim’s cutter).

Suppose $Z$ is an open integer scheme and $X$ a smooth affine curve over $Z$
with étale divisor at infinity. Then in terms of the diagram of segment 7.8,
we have a sequence of inclusions

$\textstyle{X(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcap}$$\textstyle{\left(\prod_{\mathfrak{p}|p}X(Z_{\mathfrak{p}})\right)_{\mbox{motivically
global}}=\mu_{1}^{-1}(\operatorname{Im}l_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcap}$$\textstyle{\left(\prod_{\mathfrak{p}|p}X(Z_{\mathfrak{p}})\right)_{\mbox{pathwise
motivically
global}}=\mu_{2}^{-1}(\operatorname{Im}l_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcap}$$\textstyle{\left(\prod_{\mathfrak{p}|p}X(Z_{\mathfrak{p}})\right)_{\overset{\mbox{pathwise
motivically global}}{\mbox{up to $p$-adic \'{e}tale
homotopy}}}=\mu_{3}^{-1}(\operatorname{Im}l_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcap}$$\textstyle{\left(\prod_{\mathfrak{p}|p}X(Z_{\mathfrak{p}})\right)_{\mbox{Kim}}=\bigcap_{n}(\mu^{[n]})^{-1}(\overline{\operatorname{Im}}\;l^{[n]}).}$

If $X$ is mixed Tate, we may replace “$p$-adic étale homotopy” by “motivic
homotopy”.

### 7.12.

We note that although the crystalline condition on Galois-representations
plays a key role in the computability of the locus $X(Z_{p})_{\mbox{Kim}}$, it
is not needed for the definition.

### 7.13.

Finally, we note that the “pathwise motivically global theorem” (0.8) follows
from the concentration theorem (0.7) in view of lemma 7.3.

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ID: Department of Mathematics, Ben-Gurion University of the Negev.
<EMAIL_ADDRESS>TS: Einstein Institute of Mathematics, Hebrew University of
Jerusalem<EMAIL_ADDRESS>