(f\colon X\rightarrow Y) \mapsto \gamma_f.
\]

Let $\Pshlogkl$ be the category of presheaves of $\Lambda$-modules on $lSm/k$. [notation]Pshlog @ $\Pshlogkl$
For an fs log scheme $X$ of finite type over $k$,
we let $\Lambda(X)$ be the presheaf of $\Lambda$-modules on $lSm/k$ given by

\[
\mapsto
\Lambda(X)(Y):=\Lambda\hom_{lSm/k}(Y,X).
\]

The restriction (forgetting log transfer) functor

\[
\gamma^*
\colon
\Pshltrkl
\rightarrow
\Pshlogkl
\]

is defined by $\gamma^*F(X):=F(\gamma(X))$ for $F\in \Pshltrkl$ and $X\in lSm/k$.

According to <cit.> there exist adjoint functors

\[
\begin{tikzcd}
\Pshlogkl\arrow[rr,shift left=1.5ex,"\gamma_\sharp "]\arrow[rr,"\gamma^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"\gamma_*"']&&\Pshltrkl.
\end{tikzcd}
\]

By convention $\gamma_\sharp$ is left adjoint to $\gamma^*$ and $\gamma^*$ is left adjoint to $\gamma_*$.
This sequence of adjoint functors induces a similar sequence of adjoint functors for the categories of chain complexes

\[
\begin{tikzcd}
\Co(\Pshlogkl)\arrow[rr,shift left=1.5ex,"\gamma_\sharp "]\arrow[rr,"\gamma^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"\gamma_*"']&&\Co(\Pshltrkl).
\end{tikzcd}
\]

We shall use similar notations for the category $Cor/k$[notation]Cork@$Cor/k$ of finite correspondences over the field $k$.
Note that the functor $\hat{\gamma}_*$ in <cit.> corresponds to our $\gamma^*$.
Let $\Pshtrkl$ denote the category of presheaves of $\Lambda$-modules with transfers.
For any smooth scheme $X$ of finite type over $k$,
the representable presheaf of $\Lambda$-modules with transfers $\Ztr(X)$ on $Sm/k$ is given by

\[
\Ztr(X)(Y)
\Cor(Y,X)\otimes \Lambda.
\]

We write $\Pshkl$ for the category of presheaves of $\Lambda$-modules on $Sm/k$.
For any smooth scheme $X$ of finite type over $k$,
we denote by $\Lambda(X)$ the presheaf of $\Lambda$-modules on $Sm/k$ defined by

\[
\Lambda(X)(Y)
\Lambda\hom_{Sm/k}(Y,X).
\]

In this setting, there is a faithful functor

\[
\gamma
\colon
\rightarrow
\]

and corresponding sequences of adjoint functors

\begin{equation}
\label{A.5.4.1}
\begin{tikzcd}
\Pshkl\arrow[rr,shift left=1.5ex,"\gamma_\sharp "]\arrow[rr,"\gamma^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"\gamma_*"']&&\Pshtrkl,
\end{tikzcd}
\end{equation}

\[
\begin{tikzcd}
\Co(\Pshkl)\arrow[rr,shift left=1.5ex,"\gamma_\sharp "]\arrow[rr,"\gamma^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"\gamma_*"']&&\Co({\Pshtrkl}).
\end{tikzcd}
\]

Next, we define functors between the categories of finite log correspondences and finite correspondences.

[notation]omega @ $\omega$

\[
\omega
\colon
\LCor
\rightarrow
\]

be the functor that sends an object $X$ to $X-\partial X$ and a morphism $V\in \lCor(X,Y)$ to $V-\partial V\in \Cor(X-\partial X,Y-\partial Y)$ for all $X,Y\in lSm/k$.

Conversely, let

\[
\lambda
\colon
\rightarrow
\LCor
\]

be the functor sending an object $X$ to $X\in lSm/k$ equipped with the trivial log structure and a morphism $V\in\Cor(X,Y)$ to $V\in l\Cor(X,Y)$ for all $X,Y\in Sm/k$,
see Example <ref>.

There are induced functors [notation]omegastar @ $\omega^*$  [notation]lambdastar @ $\lambda^*$

\[
\omega^*
\colon
\Pshtrkl\rightarrow \Pshltrkl
\text{ and }
\lambda^*
\colon
\Pshltrkl\rightarrow \Pshtrkl
\]

given by $\omega^*F(X):=F(\omega(X))$ for $F\in \Pshtrkl$ and $X\in lSm/k$ and $\lambda^*G(Y):=G(\lambda(Y))$ for $G\in \Pshltrkl$ and $Y\in Sm/k$.
Example <ref>(1) shows that $\lambda^*$ is left adjoint to $\omega^*$.

Again by <cit.> there exist adjoint functors

\begin{equation}
\label{A.4.5.1}
\begin{tikzcd}
\Pshtrkl\arrow[rr,shift left=3ex,"\lambda_\sharp "]\arrow[rr,"\lambda^*\cong
\omega_\sharp" description,leftarrow, shift left=1ex]\arrow[rr,shift right=1ex,"\lambda_*\cong
\omega^*" description]\arrow[rr,shift right=3ex,"\omega_*"',leftarrow]&&\Pshltrkl.
\end{tikzcd}
\end{equation}

Our convention for such diagrams is that $\lambda_\sharp$ is left adjoint to $\lambda^*\cong \omega_\sharp$, which is left adjoint to $\lambda_*\cong \omega^*$,
which is left adjoint to $\omega_*$.
On chain complexes, we use the same notation for the induced adjoint functors [notation]omegasharp @ $\omega_\sharp$

\[
\begin{tikzcd}
\Co(\Pshtrkl)\arrow[rr,shift left=3ex,"\lambda_\sharp "]\arrow[rr,"\lambda^*\cong
\omega_\sharp" description,leftarrow, shift left=1ex]\arrow[rr,shift right=1ex,"\lambda_*\cong \omega^*" description]\arrow[rr,shift right=3ex,"\omega_*"',leftarrow]&&\Co(\Pshltrkl).
\end{tikzcd}
\]

\[
\omega
\colon
lSm/k\rightarrow Sm/k
\]

be the functor that sends an object $X$ to $X-\partial X$ and a morphism $f\colon X\rightarrow Y$ to the naturally induced morphism $X-\partial X\rightarrow Y-\partial Y$ for all $X,Y\in lSm/k$.

Conversely, let

\[
\lambda
\colon
Sm/k\rightarrow lSm/k
\]

be the functor sending $X\in Sm/k$ to $X$ equipped with the trivial log structure and a morphism $f\colon X\rightarrow Y$ to itself for all $X,Y\in Sm/k$.

With these definitions, there are associated adjoint functors

\[
\begin{tikzcd}
\Pshkl\arrow[rr,shift left=3ex,"\lambda_\sharp "]\arrow[rr,"\lambda^*\cong
\omega_\sharp" description,leftarrow, shift left=1ex]\arrow[rr,shift right=1ex,"\lambda_*\cong \omega^*" description]\arrow[rr,shift right=3ex,"\omega_*"',leftarrow]&&\Pshlogkl,
\end{tikzcd}\]

\[
\begin{tikzcd}
\Co(\Pshkl)\arrow[rr,shift left=3ex,"\lambda_\sharp "]\arrow[rr,"\lambda^*\cong
\omega_\sharp" description,leftarrow, shift left=1ex]\arrow[rr,shift right=1ex,"\lambda_*\cong \omega^*" description]\arrow[rr,shift right=3ex,"\omega_*"',leftarrow]&&\Co(\Pshlogkl),
\end{tikzcd}\]

and a commutative diagram

\[
\begin{tikzcd}
lSm/k\arrow[d,"\omega"']\arrow[r,"\gamma"]& lCor/k\arrow[d,"\omega"]\\
Sm/k\arrow[r,"\gamma"]& Cor/k.
\end{tikzcd}
\]

Suppose $f\colon Y\rightarrow X$ is a morphism of fs log schemes log smooth over $k$.
Let $\Zltr(Y\rightarrow X)$ (resp. $\Lambda(Y\rightarrow X)$) denote the complex [notation]LambdaCone @ $\Zltr(Y\rightarrow X)$

\[
\Zltr(Y)\stackrel{\Zltr(f)}\longrightarrow \Zltr(X) \;\;(\text{resp.}\ \Lambda(Y)\stackrel{\Lambda(f)}\longrightarrow \Lambda(X)).
\]

Suppose $\mathscr{X}$ is a simplicial fs log scheme over $k$.
Let $\Zltr(\mathscr{X})$ (resp. $\Lambda(\mathscr{X})$) denote the complex associated with the simplicial object

\[
\mapsto
\Zltr(\mathscr{X}_i)
\;\;(\text{resp.}\
i\mapsto \Lambda(\mathscr{X}_i))
\]

under the Dold-Kan correspondence.

when $Y\rightarrow X$ is a morphism of schemes smooth over $k$ and $\mathscr{X}$ is a simplicial scheme over $k$,
we use the notations $\Lambda(Y\rightarrow X)$, $\Ztr(Y\rightarrow X)$, $\Lambda(\mathscr{X})$, and $\Ztr(\mathscr{X})$.

§.§ Compatibility with log transfers

This section discusses several notions of compatibility with log transfers for topologies on log schemes.
We proceed by adapting to the log setting the analogous notions introduced by Cisinski-Deglise in <cit.>.

Let $t$ be a topology on $lSm/k$ and $\Lambda$ a commutative unital ring.
A presheaf $\cF$ with log transfers is called a $t$-sheaf with log transfers if $\gamma^*\cF\in \Pshlogkl$ is a $t$-sheaf.
Here, $\gamma^*\cF(X):=\cF(\gamma(X))$ for $X\in lSm/k$.sheaves with log transfers [notation]Shv @ $\Shvltrtkl$
Let $\Shvltrtkl$ denote the category of $t$-sheaves with log transfers.

Let $\Shvlogtkl$ be the category of $t$-sheaves on $lSm/k$ (without log transfers).[notation]Shv @ $\Shvlogtkl$
The $t$-sheafification functor $a_t^*$ and the forgetful functor $a_{t*}$ form an adjoint functor pair

\[
a_t^*:\Pshlogkl\rightleftarrows \Shvlogtkl:a_{t*}.
\]

By abuse of notation,

\[
a_{t*}:\Shvltrtkl\rightarrow \Pshltrkl
\]

be the inclusion functor, and let

\[
\gamma^*\colon \Shvltrtkl\rightarrow \Shvlogtkl
\]

be the restriction of the functor $\gamma^*\colon \Pshltrkl\rightarrow \Pshlogkl$.
We note that

\begin{equation}
\label{A.8.1.1}
\gamma^*a_{t*}\cong a_{t*}\gamma^*.
\end{equation}

For a topology $t$ on $Sm/k$,
let $\Shvtrtkl$ be the category of $t$-sheaves on $Sm/k$ with transfers.

The notion of a topology on $Sm/k$ being compatible with transfers was introduced in <cit.>.
We adopt this notion to our log setting.

A topology $t$ on $lSm/k$ is compatible with log transfers topology!compatible with log transfers
if for any $t$-hypercover $\mathscr{X}\rightarrow X$,
the induced morphism

\[
a_t^*\gamma^*\Zltr(\mathscr{X})\rightarrow a_t^*\gamma^*\Zltr(X)
\]

of complexes of presheaves is a quasi-isomorphism (see also Definition <ref>).

Following <cit.>, we say that $t$ is mildly compatible with log transfers if $\gamma_*a_{t*}\cF$ is a $t$-sheaf with log transfers for any $t$-sheaf $\cF$.
Recall that $\gamma_*$ is a right adjoint of $\gamma^*$. topology!mildly compatible with log transfers
There is a description of $\gamma_*$: For every presheaf $\cG$ and $X\in lSm/k$,

\[
\gamma_*\cG(X)=\hom_{\Pshlogkl}(\gamma^*\Zltr(X),\cG).
\]

These notions are equivalent to each other if $t$ is a topology associated with a cd-structure, see Proposition <ref>.

According to <cit.>, the Nisnevich topology on $Sm/k$ is compatible with transfers.
One goal in this section is to generalize this to our log setting,
i.e., we will show that the strict Nisnevich topology on $lSm/k$ is compatible with log transfers.

Let $t$ be a topology on $lSm/k$ compatible with log transfers. Then $t$ is mildly compatible with log transfers.

Let $\mathscr{X}\rightarrow X$ be a Čech $t$-hypercover, and let $\cF$ be a $t$-sheaf.
To show that $\gamma_*a_{t*}\cF$ is a $t$-sheaf, we need to show that the induced sequence of $\Lambda$-modules

\[
\begin{split}
0\rightarrow \hom_{\Pshltrkl}(\Zltr(X),\gamma_*a_{t*}\cF)\rightarrow &\hom_{\Pshltrkl}(\Zltr(\mathscr{X}_0),\gamma_*a_{t*}\cF)\\
\rightarrow &\hom_{\Pshltrkl}(\Zltr(\mathscr{X}_1),\gamma_*a_{t*}\cF)
\end{split}
\]

is exact.
This is equivalent to showing exactness of the sequence

\begin{equation}
\label{A.8.3.1}
\begin{split}
0\rightarrow \hom_{\Shvltrtkl}(a_t^*\gamma^*\Zltr(X),\cF)\rightarrow &\hom_{\Shvltrtkl}(a_t^*\gamma^*\Zltr(\mathscr{X}_0),\cF)\\
\rightarrow &\hom_{\Shvltrtkl}(a_t^*\gamma^*\Zltr(\mathscr{X}_1),\cF).
\end{split}
\end{equation}

This follows since by assumption we have a quasi-isomorphism

\[
a_t^*\gamma^*\Zltr(\mathscr{X})\rightarrow a_t^*\gamma^*\Zltr(X).
\]

Let $t$ be a topology on $lSm/k$ mildly compatible with log transfers.
Then for any $t$-cover $Y\rightarrow X$, the induced morphism of $t$-sheaves $a_t^*\gamma^*\Zltr(Y)\rightarrow a_t^*\gamma^*\Zltr(X)$ is an epimorphism.

As in the proof of Proposition <ref>, this follows because the sequence (<ref>) is exact.

Let $t$ be a topology on $lSm/k$ mildly compatible with log transfers.
Then the functor

\[
\gamma^*\colon \Shvltrtkl\rightarrow \Shvlogtkl
\]

admits a right adjoint $\gamma_*$ that commutes with $a_{t*}$.

Due to the assumption that $t$ is mildly compatible with log transfers,
by abuse of notation, we have the functor

\[
\gamma_*\colon \Shvlogtkl\rightarrow \Shvltrtkl.
\]

That is, the restriction of the functor $\gamma_*\colon\Pshlogkl\rightarrow \Pshltrkl$.
We note that the functors $a_{t*}$ and $\gamma_*$ commute.

For $\cF\in \Shvltrtkl$ and $\cG\in \Shvlogtkl$ it remains to note that there are natural isomorphisms of $\Lambda$-modules

\[
\begin{split}
\cong &\hom_{\Pshlogkl}(a_{t*}\gamma^*\cF,a_{t*}\cG)\quad \text{(since }a_{t*}\text{ is fully faithful)}\\
\cong &\hom_{\Pshlogkl}(\gamma^*a_{t*}\cF,a_{t*}\cG)\quad \text{(since }\gamma^*\text{ commutes with }a_{t*},\text{ see }\eqref{A.8.1.1})\\
\cong &\hom_{\Pshltrkl}(a_{t*}\cF,\gamma_* a_{t*}\cG)\\
\cong &\hom_{\Pshltrkl}(a_{t*}\cF,a_{t*}\gamma_*\cG)\quad \text{(since }\gamma_*\text{ commutes with }a_{t*})\\
\cong &\hom_{\Shvltrtkl}(\cF,\gamma_*\cG)\quad \text{(since }a_{t*}\text{ is fully faithful)}.
\end{split}
\]

Let us prove an analog of <cit.>.

Let $t$ be a topology on $lSm/k$ that is mildly compatible with log transfers.
Suppose that $p\colon Y'\rightarrow Y$ is a $t$-cover and $f\in \lCor(X,Y)$ is a finite log correspondence, where $X,Y,Y'\in lSm/k$.
Then there exists a $t$-cover $p'\colon X'\rightarrow X$ and a commutative diagram in $lCor/k$

\begin{equation}
\label{A.8.23.1}
\begin{tikzcd}
\end{tikzcd}
\end{equation}

By Proposition <ref>, the induced morphism

\[
a_t^*\gamma^*\Zltr(Y')\rightarrow a_t^*\gamma^*\Zltr(Y)
\]

in $\Shvltrtkl$ is an epimorphism.
The implication (ii)$\Rightarrow$(i) in <cit.> shows that $X\times_{\Zltr(Y)}\Zltr(Y')\rightarrow X$ is a covering sieve.
From this, we immediately deduce the existence of $p'$ along with (<ref>).

Using this lemma, we prove an analog of <cit.> as follows.

Let $t$ be a topology on $lSm/k$ that is mildly compatible with log transfers.
Then for every $\cF\in \Pshltrkl$ there exists an object $\cG\in \Shvltrtkl$ together with a morphism $u\colon \cF\rightarrow a_{t*}\cG$ in $\Pshltrkl$ such that
$\gamma^*u$ is isomorphic to the sheafification $\gamma^*\cF\rightarrow a_{t*}a_t^*\gamma^*\cF$.
The pair $(\cG,u)$ is unique up to isomorphism.

Let us first construct $\cG\in \Shvltrtkl$.
The unit $\cF\rightarrow \gamma_*a_{t*}a_t^*\gamma^*\cF$ induces a morphism

\[
\eta\colon \gamma^*\cF\rightarrow \gamma^*\gamma_*a_{t*}a_t^*\gamma^*\cF
\]

Here $\gamma^*\gamma_*a_{t*}a_t^*\gamma^*\cF$ is a $t$-sheaf because $t$ is mildly compatible with log transfers.
Hence according to the universal property of the sheafification functor, the latter morphism admits a factorization

\[
\gamma^*\cF\rightarrow a_{t*}a_t^* \gamma^* \cF\stackrel{\psi}\rightarrow \gamma^*\gamma_*a_{t*}a_t^*\gamma^*\cF
\]

that is unique up to isomorphism.
Next, we consider the commutative diagram

\[
\begin{tikzcd}
\gamma^*\cF\arrow[r,"ad"']\arrow[d,"ad"]&\gamma^*\gamma_*\gamma^*\cF\arrow[d,"ad"]\arrow[r,"ad'"]&\gamma^*\cF\arrow[d,"ad"]\\
\end{tikzcd}
\]

Here $ad$ (resp. $ad'$) is induced by a unit (resp. counit).
Let $\psi'$ be the composition of the lower horizontal morphisms.
Since the composition of the upper horizontal morphisms is the identity by the unit-counit identity, there is a commutative diagram

\[
\begin{tikzcd}
\gamma^*\cF\arrow[rd,"ad"]\arrow[ld,"ad"']
\\
\end{tikzcd}
\]

The diagram still commutes if we replace $\psi'$ by ${\rm id}$.
By the universal property of sheafification, the morphism

\[
a_{t*}a_t^*\gamma^*\cF\rightarrow a_{t*}a_t^*\gamma^*\cF
\]

making the above diagram commutative is unique,
and we deduce $\psi'={\rm id}$.
It follows that $\psi$ is a monomorphism.

Let $f\in \lCor(X,Y)$ be a finite log correspondence, where $X,Y\in lSm/k$, and set $\cG=a_t^*\gamma^*\cF$.
Then $f$ and $\psi$ induce a diagram of $\Lambda$-modules

\[
\begin{tikzcd}
\cG(Y)\arrow[r,"{\psi(Y)}"]& \hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(Y),\cG)\arrow[d,"\alpha_f"]\\
\cG(X)\arrow[r,"{\psi(X)}"]& \hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(X),\cG).
\end{tikzcd}
\]

Since $\psi$ is a monomorphism, to show the log transfer structure on $\gamma_*\alpha_{t*}\cG$ induces a log transfer structure on $\cG$ it suffices to show that

\begin{equation}
\label{A.8.6.2}
\im (\alpha_f\circ \psi(Y)) \subset \im \psi(X).
\end{equation}

For every $a\in \cG(Y)$ there exists an element $b'\in \cF(Y')$ for some $t$-cover $q\colon Y'\rightarrow Y$ such that the images of $a$ and $b'$ in $\cG(Y')$ coincide with some $a'\in \cG(Y')$.
According to Lemma <ref> there exists a $t$-cover $p'\colon X'\rightarrow X$ with a commutative diagram in $lCor/k$

\begin{equation}
\label{A.8.6.1}
\begin{tikzcd}
\end{tikzcd}
\end{equation}

Then there is a diagram of $\Lambda$-modules

\[
\begin{tikzcd}
\cG(Y')\arrow[r,"{\psi(Y')}"]& \hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(Y'),\cG)\arrow[d,"\alpha_{f'}"]\\
\cG(X')\arrow[r,"{\psi(X')}"]& \hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(X'),\cG).
\end{tikzcd}
\]

Suppose that $(\alpha_{f'}\circ \psi(Y'))(a')=\psi(X')(d')$ for some $d'\in \cG(X')$.
The two images of $a'$ in $\cG(Y'\times_Y Y')$ are equal since $\cG$ is a presheaf.
Thus the two images of $\psi(X')(d')$ in $\hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(X'\times_X X'),\cG)$ are equal.
Since $\psi$ is a monomorphism, this implies that the two images of $d'$ in $\cG(X'\times_X X')$ are equal.
Thus there exists $d\in \cG(X)$ whose image in $\cG(X')$ is equal to $d'$ since $\cG$ is a sheaf.
Using the exactness of (<ref>), we see that

\[
(\alpha_{f}\circ \psi(Y))(a) = \psi(X)(d).
\]

Hence let us show that $(\alpha_{f'}\circ \psi(Y'))(a')=\psi(X')(d')$ for some $d'\in \cG(X')$.

The morphism $\eta\colon \gamma^*\cF\rightarrow \gamma^*\gamma_*a_{t*}a_t^*\gamma^*\cF$ induces a commutative diagram of $\Lambda$-modules

\begin{equation}
\label{equation:FGdiagram}
\begin{tikzcd}
\cF(Y')\arrow[r,"\eta(Y')"]\arrow[d,"\cF(f')"']&\hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(Y'),\cG)\arrow[d,"\alpha_{f'}"]\\
\cF(X')\arrow[r,"\eta(X')"]&\hom_{\Shvlogtkl}(a_t^*\gamma^*\Zltr(X'),\cG).
\end{tikzcd}
\end{equation}

This implies that $(\alpha_{f'}\circ \psi(Y'))(a')=\psi(X')(d')$ for some $d'\in \cG(X')$.
We deduce the claimed log transfer structure on $\cG$.

Since $\psi$ is a monomorphism,
combining the diagram (<ref>) for $f\colon X\rightarrow Y$ together with (<ref>) we obtain a canonical commutative diagram of $\Lambda$-modules

\[
\begin{tikzcd}
\cF(Y)\arrow[d,"\cF(f)"']\arrow[r]&
\cG(Y)\arrow[d,"\cG(f)"]\arrow[d]
\\
\cF(X)\arrow[r]&
\cG(X).
\end{tikzcd}
\]

Thus we obtain a morphism $u\colon \cF\rightarrow a_{t*}\cG$ of presheaves with log transfers such that $\gamma^*u$ is isomorphic to the sheafification
$\gamma^*\cF\rightarrow a_{t*}a_t^*\gamma^*\cF$.

For uniqueness,
suppose $\cG'$ is a $t$-sheaf with log transfers that is equipped with a morphism $u'\colon \cF\rightarrow a_{t*}\cG'$ of presheaves with log transfers such that $\gamma^*u'$
is isomorphic to the sheafification $\gamma^*\cF\rightarrow a_{t*}a_t^*\gamma^*\cF$.
Since $\cG$ and $\cG'$ agree on objects,
it suffices to check that $\cG(f)=\cG'(f)$ for any $f\in \lCor(X,Y)$ where $X,Y\in lSm/k$.
Using the commutative diagram (<ref>), we see this question is $t$-local on $X$ and $Y$.
Hence after shrinking $Y$ we may assume that for any $a\in \cG(Y)$, there is an element $b\in \cF(Y)$ whose image in $\cG(Y)$ is equal to $a$.
The morphisms $u\colon \cF\rightarrow a_{t*} \cG$ and $u'\colon \cF\rightarrow a_{t*}\cG'$ induce a commutative diagram of $\Lambda$-modules

\[
\begin{tikzcd}
\cG'(Y)\arrow[r,leftarrow]\arrow[d,"{\cG'(f)}"']&\cF(Y)\arrow[d,"\cF(f)"']\arrow[r]&\cG(Y)\arrow[d,"\cG(f)"]\\
\cG'(X)\arrow[r,leftarrow,"{u'(X)}"]&\cF(X)\arrow[r,"u(X)"]&\cG(X).
\end{tikzcd}
\]

With the identification of $\cG$ and $\cG'$ on objects, the homomorphisms $u(X)$ and $u'(X)$ become equal.
This allows us to conclude

\[
\cG(f)(a)=u(X)(\cF(f)(b))=u'(X)(\cF(f)(b))=\cG'(f)(a).
\]

Let $t$ be a topology on $lSm/k$ mildly compatible with log transfers.
Then the functor

\[
a_{t*}\colon \Shvltrtkl\rightarrow \Pshltrkl
\]

admits a left adjoint functor $a_t^*$.
Moreover, the functor $a_t^*$ is exact.

For $\cF\in \Pshltrkl$ we define $a_t^*\cF\colon =\cH$,
where $\cH\in \Shvltrtkl$ is constructed as in Lemma <ref>.
The uniqueness part in Lemma <ref> ensures that we obtain a functor

\[
a_t^*\colon \Pshltrkl\rightarrow \Shvltrtkl.
\]

It remains to show $a_t^*$ is a left adjoint of $a_{t*}$.
Again, by uniqueness, the morphism $u\colon \cF\rightarrow a_{t*}\cH$ in $\Pshltrkl$ constructed in Lemma <ref> gives rise to a natural transformation

\[
{\rm id}\rightarrow a_{t*}a_t^*.
\]

For $\cG\in \Shvltrtkl$, the uniqueness part also implies that there is an isomorphism

\[
a_t^*a_{t*}\cG\cong \cG.
\]

By functoriality of this isomorphism, we obtain a natural transformation

\[
a_t^*a_{t*}\rightarrow {\rm id}.
\]

Owing to the uniqueness part in Lemma <ref>,
the unit and counit triangle identities follow from the corresponding identities for $a_t^*\colon \Pshlogkl\rightarrow \Shvlogtkl$ and $a_{t*}\colon \Shvlogtkl\rightarrow \Pshlogkl$.

Since $a_t^*$ is a left adjoint, $a_t^*$ is right exact.
It remains to show that $a_t^*$ is left exact.
Let $f\colon \cF\rightarrow \cF'$ be a monomorphism of presheaves with log transfers.
The functor $\gamma^*$ is exact since $\gamma^*$ has both left and right adjoints, and the functor

\[
a_t^*\colon \Pshlogkl\rightarrow \Shvlogtkl
\]

is known to be exact.
It follows that

\[
a_t^*\gamma^*f\colon a_t^*\gamma^*\cF\rightarrow a_t^*\gamma^*\cF'
\]

is a monomorphism of $t$-sheaves.
Since $a_t^*$ commutes with $\gamma^*$ by Proposition <ref>,

\[
\gamma^*a_t^*f\colon \gamma^*a_t^*\cF\rightarrow \gamma^*a_t^*\cF'
\]

is a monomorphism of $t$-sheaves.
Let $\cG$ be a kernel of $a_t^*f\colon a_t^*\cF\rightarrow a_t^*\cF'$.
It follows that $\gamma^*\cG=0$.
This means that $\cG(X)=0$ for all $X\in lSm/k$, so $\cG=0$.
Thus $a_t^*f$ is a monomorphism.

Let $t$ be a topology on $lSm/k$ mildly compatible with log transfers.
Then the functor

\[
\gamma^*\colon \Shvltrtkl\rightarrow \Shvlogtkl
\]

admits a left adjoint functor $\gamma_\sharp$.

For $\cF\in \Shvlogtkl$ and $\cG\in \Shvltrtkl$, we have isomorphisms

\[
\begin{split}
&\hom_{\Shvltrtkl}(a_t^*\gamma_\sharp a_{t*}\cF,\cG)\\
\cong &\hom_{\Pshltrkl}(a_{t*}\cF,\gamma^*a_{t*}\cG)\\
\cong &\hom_{\Pshltrkl}(a_{t*}\cF,a_{t*}\gamma^*\cG)\quad \text{(since }\gamma^*\text{ and }a_{t*}\text{ commute)}\\
\cong &\hom_{\Shvltrtkl}(\cF,\gamma^*\cG)\quad \text{(since }a_{t*}\text{ is fully faithful}).
\end{split}
\]

By functoriality in $\cF$ and $\cG$, it follows that $a_t^*\gamma_\sharp a_{t*}$ is a left adjoint of $\gamma^*$.

We recall the definition of Grothendieck abelian category (see e.g., <cit.> for details).
An abelian category $\cA$ is called a Grothendieck abelian categoryGrothendieck abelian category with the following conditions.

(i) $\cA$ admits small sums.

(ii) Filtered colimits of exact sequences in $\cA$ is exact.

(iii) There is a generator of $\cA$, i.e., there is an element of $G\in \cA$ such that the functor $\hom_{\cA}(G,-)\colon \cA\rightarrow {\bf Set}$ is faithful.

Let $t$ be a topology on $lSm/k$ mildly compatible with log transfers. Then $\Shvltrtkl$ is a Grothendieck abelian category.

Let us check the above axioms of a Grothendieck abelian category.
The categories  $\Pshltrkl$ and $\Shvlogtkl$ are Grothendieck abelian categories, and $\gamma^*\colon \Shvltrtkl\rightarrow \Shvlogtkl$ is exact since $\gamma^*$ has both left and right adjoints.
Thus $\Shvltrtkl$ is an abelian category, and we have (i) and (ii).
Since by the Yoneda lemma

\[
\hom_{\Shvltrtkl}(a_t^*\Zltr(X),\cF)\cong \hom_{\Pshltrkl}(\Zltr(X),\cF)\cong \cF(X),
\]

the object $\bigoplus_{X\in lSm/k} a_t^*\Zltr(X)$ is a generator of $\Shvltrtkl$. Thus we have (iii).

In conclusion,
if $t$ is a topology on $lSm/k$ mildly compatible with log transfers,
there exist adjunctions between Grothendieck abelian categories

\begin{equation}
\label{A.8.7.1}
\begin{tikzcd}
\Pshlogkl\arrow[d,shift left=0.75ex,leftarrow,"a_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"']
\arrow[r,shift left=1.5ex,"\gamma_\sharp"]\arrow[r,leftarrow,"\gamma^*" description]
\arrow[r,shift right=1.5ex,"\gamma_*"']&\Pshltrkl
\arrow[d,shift left=0.75ex,leftarrow,"a_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"']   \\
\Shvlogtkl\arrow[r,shift left=1.5ex,"\gamma_\sharp"]
\arrow[r,leftarrow,"\gamma^*" description]\arrow[r,shift right=1.5ex,"\gamma_*"']
\end{tikzcd}
\end{equation}

such that the following properties hold:

(i) $a_t^*$ is left adjoint to the corresponding $a_{t*}$

(ii) $\gamma^*$ is left (resp. right) adjoint to the corresponding $\gamma_*$ (resp. $\gamma_\sharp$)

(iii) $a_{t*}\gamma_*\cong \gamma_*a_{t*}$

(iv) $a_{t*}\gamma^*\cong \gamma^*a_{t*}$

Let $t$ be a topology on $lSm/k$.
Suppose that for every $t$-cover $f\colon Y\rightarrow X$ the complex

\begin{equation}
\label{A.8.11.1}
\cdots\rightarrow a_t^*\gamma^*\Zltr(Y\times_X Y)\rightarrow a_t^*\gamma^*\Zltr(Y)\rightarrow a_t^*\gamma^*\Zltr(X)\rightarrow 0
\end{equation}

associated with the Čech nerve of $f$ is acyclic.
Then $t$ is compatible with log transfers.

Let $p\colon \mathscr{X}\rightarrow X$ be a $t$-hypercover.
To check that

\[
a_t^*\gamma^*\Zltr(\mathscr{X})\rightarrow a_t^*\gamma^*\Zltr(X)
\]

is a quasi-isomorphism it suffices to check that

\[
a_t^*\gamma^*\Zltr(\mathscr{X}_i)\rightarrow \cdots \rightarrow a_t^*\gamma^*\Zltr(\mathscr{X}_0)\rightarrow a_t^*\gamma^*\Zltr(X)\rightarrow 0
\]

is exact for all integer $i\geq 0$.
Hence we may assume that $\mathscr{X}$ is a bounded $t$-hypercover of height $n$.

To proceed, we use the technique in the proof of <cit.>.
If $n=0$, then $\mathscr{X}$ is a Čech nerve, so we are done.
Suppose that $n>0$, and consider the bounded $t$-hypercover

\[
\mathscr{Y}:={\rm cosk}_{n-1} \mathscr{X}
\]

of height at most $n-1$.
By induction we have that $a_t^*\gamma^*\Zltr(\mathscr{Y})\rightarrow a_t^*\gamma^*\Zltr(X)$ is a quasi-isomorphism.
The Čech nerve of the canonical morphism $\mathscr{X}\rightarrow \mathscr{Y}$ is a bisimplicial complex;
let $\mathscr{D}$ denote its diagonal.
Since (<ref>) is acyclic for every $t$-cover, there is a canonically induced quasi-isomorphism

\[
a_t^*\gamma^*\Zltr(\mathscr{D})\rightarrow a_t^*\gamma^*\Zltr(\mathscr{Y}).
\]

In particular,
$a_t^*\gamma^*\Zltr(\mathscr{D})\rightarrow a_t^*\gamma^*\Zltr(X)$ is a quasi-isomorphism.
Since $\mathscr{X}$ is a retract of $\mathscr{D}$,
see the proof of <cit.>,
we deduce that

\[
a_t^*\gamma^*\Zltr(\mathscr{X})\rightarrow a_t^*\gamma^*\Zltr(X)
\]

is a quasi-isomorphism.

§.§ Derived categories of sheaves with log transfers

For miscellaneous facts about model structures on chain complexes we refer the reader to Appendix <ref>.
Let $t$ be a topology on $lSm/k$ mildly compatible with log transfers.
Passing to chain complexes in (<ref>) we obtain the diagram of adjoint functors

\begin{equation}
\label{A.8.0.1}
\begin{tikzcd}
\Co(\Pshlogkl)\arrow[d,shift left=0.75ex,leftarrow,"a_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"']
\arrow[r,shift left=0.75ex,"\gamma_\sharp"]\arrow[r,shift right=0.75ex,"\gamma^*"',leftarrow]&\Co(\Pshltrkl)\arrow[d,shift left=0.75ex,leftarrow,"a_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"']   \\
\Co(\Shvlogtkl)\arrow[r,shift left=0.75ex,"\gamma_\sharp"]\arrow[r,leftarrow,shift right=0.75ex,"\gamma^*"']
\end{tikzcd}
\end{equation}

We refer to Example <ref> for the descent structures on $\Pshlogkl$ and $\Shvlogtkl$.
The functors $\gamma^*$ in (<ref>) are exact since they have both left and right adjoints.
If the topology $t$ is compatible with log transfers,
then for any $t$-hypercover $\mathscr{X}\rightarrow X$,
the cone of the morphism $a_t^*\gamma^*\Zltr(\mathscr{X})\rightarrow a_t^*\gamma^*\Zltr(X)$ is quasi-isomorphic to $0$.
Thus by Proposition <ref>, we also obtain descent structures on $\Pshltrkl$ and $\Shvltrtkl$, and the $(\gamma_\sharp,\gamma^*)$'s are Quillen pairs with respect to the said descent model structures.
Note also that the $a_t^*$'s satisfy the conditions of Proposition <ref> due to exactness.
Owing to Propositions <ref> and <ref>, the $(a_t^*,a_{t*})$'s are Quillen pairs with respect to the descent model structures.

Since $\gamma^*$ preserves quasi-isomorphisms and monomorphisms,
the adjunctions

\[
\gamma^*\colon \Co(\Pshltrkl)\rightleftarrows \Co(\Pshlogkl)\colon \gamma_*,
\]

\[
\gamma^*\colon \Co(\Shvltrtkl)\rightleftarrows \Co(\Shvltrtkl)\colon \gamma_*
\]

are Quillen pairs with respect to the injective model structures; see Definition <ref> for injective model structures.
Thus we have an induced diagram of adjunctions between triangulated categories

\begin{equation}
\label{A.8.0.4}
\begin{tikzcd}
\Deri(\Pshlogkl)\arrow[d,shift left=0.75ex,leftarrow,"Ra_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"']
\arrow[r,shift left=1.5ex,"L\gamma_\sharp"]\arrow[r,"\gamma^*" description,leftarrow]\arrow[r,shift right=1.5ex,"R\gamma_*"']&\Deri(\Pshltrkl)\arrow[d,shift left=0.75ex,leftarrow,"Ra_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"']   \\
\Deri(\Shvlogtkl)\arrow[r,shift left=1.5ex,"L\gamma_\sharp"]\arrow[r,"\gamma^*" description,leftarrow]\arrow[r,shift right=1.5ex,"R\gamma_*"']
\end{tikzcd}\end{equation}

(i) $L\gamma^*\cong \gamma^*\cong R\gamma^*$ since $\gamma^*$ is exact,

(ii) $a_t^*\cong La_t^*$ since $a_t^*$ is exact, see Proposition <ref> for the log transfer case.

Suppose that $\cW$ is an essentially small class of morphisms in $lSm/k$ that is stable by isomorphisms, products, and compositions.

We consider $\cW$ as classes of morphisms of representable objects in $\Pshlogkl$, $\Pshltrkl$, $\Shvlogtkl$, and $\Shvltrtkl$.
Note that $a_t^*$ preserves $\cW$ because the object of $\Shvlogtkl$ (resp. $\Shvltrtkl$) representable by $X\in lSm/k$ is $a_t^*\Lambda(X)$ (resp. $a_t^*\Zltr(X)$).
The four adjunctions in (<ref>) are Quillen adjunctions with respect to the $\cW$-local descent model structures furnished by Propositions <ref> and <ref>.
Thus we have the following induced diagram of adjunctions between triangulated categories

\begin{equation}
\label{A.8.0.2}
\begin{tikzcd}
\Deri_{\cW}(\Pshlogkl)\arrow[d,shift left=0.75ex,leftarrow,"Ra_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"']
\arrow[r,shift left=0.75ex,"L\gamma_\sharp"]\arrow[r,shift right=0.75ex,"R\gamma^*"',leftarrow]&\Deri_{\cW}(\Pshltrkl)\arrow[d,shift left=0.75ex,leftarrow,"Ra_{t*}"]\arrow[d,shift right=0.75ex,"a_{t}^*"']   \\
\Deri_{\cW}(\Shvlogtkl)\arrow[r,shift left=0.75ex,"L\gamma_\sharp"]\arrow[r,shift right=0.75ex, "R\gamma^*"',leftarrow]
\end{tikzcd}
\end{equation}

Here $a_t^*\cong La_t^*$ since $a_t^*$ is exact by Proposition <ref> and  $a_t^*$ preserves $\cW$.

Let $t$ be a topology on $lSm/k$, and let $\cF$ be a complex of $t$-sheaves with log transfers.
For $X\in lSm/k$ and $i\in \Z$ we define hypercohomology groups by setting [notation]Ht @ $\bH_t^i$

\[
\bH_t^i(X,\cF)
\bH_t^i(X,\gamma^*\cF).
\]

Let $t$ be a topology on $lSm/k$ compatible with log transfers.
Then for all $X\in lSm/k$ and every complex $\cF$ of $t$-sheaves with log transfers, there is a canonical isomorphism

\[
\hom_{\Deri(\Shvltrtkl)}(a_t^*\Zltr(X),\cF[i])
\cong
\bH_t^i(X,\cF).
\]

The functor

\[
L\gamma_\sharp\colon \Deri(\Shvlogtkl)\rightarrow \Deri(\Shvltrtkl)
\]

in (<ref>) is left adjoint to

\[
\gamma^*\colon \Deri(\Shvltrtkl)\rightarrow \Deri(\Shvlogtkl).
\]

Thus the group $\bH_t^i(X,\cF)$ can be identified with

\[
\hom_{\Deri(\Shvlogtkl)}(a_t^*\Lambda(X),\gamma^*\cF[i])
\cong
\hom_{\Deri(\Shvltrtkl)}(L\gamma_\sharp a_t^*\Lambda(X),\cF[i]).
\]

The object $a_t^*\Lambda(X)$ is cofibrant, see Definition <ref>.
The functor $a_t^*$ commutes with $\gamma_\sharp$ due to (<ref>).
Thus we deduce

\[
L\gamma_\sharp a_t^*\Lambda(X)\cong \gamma_\sharp a_t^*\Lambda(X)
\cong
a_t^*\gamma_\sharp\Lambda(X)\cong a_t^*\Zltr(X).
\]

We will need the following lemma in our discussion of adjunctions.

Suppose $F\colon C\rightarrow D$ is a left and a right Quillen functor between model categories.
Then $F$ preserves cofibrations, fibrations, and weak equivalences.

For the convenience of the reader, we shall outline a proof; see also the discussion in <cit.>.
By assumption, $F$ preserves cofibrations, fibrations, trivial cofibrations, and trivial fibrations.
It remains to show that $F$ preserves weak equivalences.
If $f\colon X\rightarrow Y$ be a weak equivalence in $C$, we can choose a commutative diagram

\[
\begin{tikzcd}
\end{tikzcd}\]

where $g$ and $g'$ are trivial fibrations and $f'$ is a weak equivalence between cofibrant objects.
Then $F(g)$ and $F(g')$ are weak equivalences.
Ken Brown's lemma implies that $F(f')$ is a weak equivalence.
It follows that $F(f)$ is also a weak equivalence.

Let $t$ be a topology on $lSm/k$, and let $t'$ be the restriction of $t$ to $Sm/k$.
Then the functor $\lambda$ from $Sm/k$ with the topology $t'$ to $lSm/k$ with the topology $t$ is a continuous functor of sites.
Thus we have an induced adjunction

\[
\lambda_\sharp\colon \Shv_{t'}(k,\Lambda)\rightleftarrows \Shvlogtkl\colon \lambda^*.
\]

This induces an adjunction

\[
\lambda_\sharp\colon \Shvtrttkl\rightleftarrows \Shvltrtkl\colon \lambda^*.
\]

Assume $t$ is compatible with log transfers and $t'$ is compatible with transfers.
By appealing to Proposition <ref>, we obtain a Quillen pair for the descent model structures

\[
\lambda_\sharp\colon \Co(\Shvtrttkl)\rightleftarrows \Co(\Shvltrtkl)\colon \lambda^*
\]

since $t'$ is the restriction of $t$, for every $t'$-hypercover $\mathscr{X}\rightarrow X$ we obtain a $t$-hypercover

\[
\lambda(\mathscr{X})\rightarrow \lambda(X).
\]

On the level of homotopy categories, there is an induced adjunction

\[
L\lambda_\sharp\colon \Deri(\Shvtrttkl)\rightleftarrows \Deri(\Shvltrtkl)\colon R\lambda^*.
\]

Assume that for any $t$-cover $Y\rightarrow X$, the naturally induced morphism

\[
\omega(Y)
Y-\partial Y
\rightarrow
X-\partial X
\omega(X)
\]

is also a $t$-cover.
Then $\omega$ is a continuous functor of sites from $lSm/k$ with the topology $t$ to $Sm/k$ with the topology $t'$.
There are induced adjunctions

\[
\omega_\sharp\colon \Shvlogtkl\rightleftarrows \Shv_t(k,\Lambda)\colon \omega^*
\]

\begin{equation}
\label{eqn::omegaadjunction}
\omega_\sharp\colon \Shvltrtkl\rightleftarrows \Shvtrttkl\colon \omega^*.
\end{equation}

By assumption $\omega$ maps any $t$-covers to $t'$-covers,
and for any $t$-hypercover $\mathscr{X}\rightarrow X$, $\omega(\mathscr{X})\rightarrow \omega(X)$ is a $t'$-hypercover.
According to Proposition <ref> there is a Quillen pair for the descent model structures

\[
\omega_\sharp\colon \Co(\Shvltrtkl)\rightleftarrows \Co(\Shvtrttkl)\colon \omega^*.
\]

We also note the induced adjunction

\[
L\omega_\sharp\colon \Deri(\Shvltrtkl)\rightleftarrows \Deri(\Shvtrttkl)\colon R\omega^*.
\]

Since $\lambda$ is left adjoint to $\omega$, by Lemma <ref> we deduce

\[R\lambda^*\cong \lambda^*\cong \omega_\sharp \cong L\omega_\sharp.
\]

In conclusion, there is a sequence of adjunctions

\begin{equation}
\label{eq::lambdaomegaadj}
\begin{tikzcd}
\Deri(\Shvtrttkl)
\arrow[rr,shift left=1.5ex,"L\lambda_\sharp"]\arrow[rr,"\lambda^*\simeq\omega_\sharp" description,leftarrow]\arrow[rr,shift right=1.5ex,"R\omega^*"']&&{\Deri(\Shvltrtkl)}.
\end{tikzcd}
\end{equation}

As above,
suppose $\cW$ is an essentially small class of morphisms in $lSm/k$ that is stable by isomorphisms, products, and compositions.
Then, by Proposition <ref>, we obtain a Quillen pair for the $\cW$-local and $\omega(\cW)$-local descent model structures

\[
\omega_\sharp\colon \Co(\Shvltrtkl)\rightleftarrows \Co(\Shvtrttkl)\colon \omega^*.
\]

Thus we have the induced adjunction

\begin{equation}
\label{A.8.0.3}
L\omega_\sharp\colon \Deri_\cW(\Shvltrtkl)\rightleftarrows \Deri_{\omega(\cW)}(\Shvtrttkl)\colon R\omega^*,
\end{equation}

and similarly for

\[
L\omega_\sharp\colon \Deri_\cW(\Shvlogtkl)\rightleftarrows \Deri_{\omega(\cW)}(\Shv_t(k,\Lambda))\colon R\omega^*.
\]

Let $f\colon X\rightarrow Y$ be a monomorphism of fs log schemes.
Then the naturally induced morphism of presheaves

\[
\Zltr(f)\colon \Zltr(X)\rightarrow \Zltr(Y)
\]

is a monomorphism.

For $T\in lSm/k$, there is a naturally induced commutative diagram

\[
\begin{tikzcd}
\lCor(T,X)\arrow[r]\arrow[d]&
\lCor(T,Y)\arrow[d]&
\\
\Cor(T-\partial T,X-\partial X)\arrow[r]&
\Cor(T-\partial T,Y-\partial Y).
\end{tikzcd}
\]

We need to show that the upper horizontal homomorphism is injective.
The induced morphism $X-\partial X\rightarrow Y-\partial Y$ is also a monomorphism.
Thus the lower horizontal homomorphism is injective, see <cit.>.
The vertical homomorphisms are injective due to Lemma <ref>, which finishes the proof.

Next we specialize to the setting of cd-structures.

Let $P$ be a complete, quasi-bounded, and regular cd-structure on $lSm/k$, and let $t$ be the associated topology.
Then the following conditions are equivalent.

(i) $t$ is compatible with log transfers.

(ii) $t$ is mildly compatible with log transfers.

(iii) For any $P$-distinguished square in $lSm/k$ of the form (<ref>), the sequence

\begin{equation}
\label{A.8.13.2}
0\rightarrow a_t^*\gamma^*\Zltr(Y')\rightarrow a_t^*\gamma^*\Zltr(Y)\oplus a_t^*\gamma^*\Zltr(X')\rightarrow a_t^*\gamma^*\Zltr(X)\rightarrow 0
\end{equation}

of $t$-sheaves with log transfers is exact.

The implication (i)$\Rightarrow$(ii) is Proposition <ref>.
Assuming condition (ii) the induced sequence  of $t$-sheaves

\begin{equation}
\label{A.8.13.5}
0\rightarrow a_t^*\Lambda(Y')\rightarrow a_t^*\Lambda(Y)\oplus a_t^*\Lambda(X')\rightarrow a_t^*\Lambda(X)\rightarrow 0
\end{equation}

is exact by <cit.>.
Let us show that the following naturally induced sequence of $t$-sheaves with log transfers is exact:

\begin{equation}
\label{A.8.13.3}
0\rightarrow \gamma_\sharp a_t^*\Lambda(Y')\rightarrow \gamma_\sharp a_t^*\Lambda(Y)\oplus \gamma_\sharp a_t^*\Lambda(X')\rightarrow \gamma_\sharp a_t^*\Lambda(X)\rightarrow 0.
\end{equation}

Since $\gamma_\sharp$ commutes with $a_t^*$, this is equivalent to showing that the sequence

\begin{equation}\label{A.8.13.4}
0\rightarrow a_t^*\Zltr(Y')\rightarrow a_t^*\Zltr(Y)\oplus  a_t^*\Zltr(X')\rightarrow a_t^*\Zltr(X)\rightarrow 0
\end{equation}

of $t$-sheaves with log transfers is exact.
Since $\gamma_\sharp$ is a left adjoint, $\gamma_\sharp$ is right exact.
(<ref>) is exact at $\gamma_\sharp a_t^*\Lambda(Y)\oplus \gamma_\sharp a_t^*\Lambda(X')$ and $\gamma_\sharp a_t^*\Lambda(X)$.
Since $P$ is regular, the morphism $Y'\rightarrow Y$ is a monomorphism.
Thus $\Zltr(Y')\rightarrow \Zltr(Y)$ is a monomorphism owing to Lemma <ref>.
This gives the exactness of (<ref>) at $a_t^*\Zltr(Y')$,
(<ref>) and (<ref>) are exact.
We deduce that (<ref>) is exact since $\gamma^*$ commutes with $a_t^*$ and $\gamma^*$ is exact.
This implies condition (iii).

Let us assume (iii) holds.
To conclude (ii), we need to show that (<ref>) is exact when $\mathscr{X}\rightarrow X$ is a Čech nerve associated with a $t$-cover $T\rightarrow X$.
Every $t$-cover has a simple refinement because $P$ is complete, see <cit.>.
Thus since $t$ is the topology associated with $P$, it suffices to consider the case when $T\rightarrow X$ is of the form $Y\amalg X'\rightarrow X$.
In this case, our claim follows from the exactness of (<ref>).

Now assume (ii) and (iii) hold.
It remains to show (i).
We work with the descent structure $(\cG,\cH)$ (resp. $(\cG,\cH')$) of $\Shvlogtkl$ given in Example <ref> (resp. Example <ref>).
Let $G\in \Deri(\Shvltrtkl)$ be an injective fibrant complex.
Since the sequence (<ref>) is exact, we have

\[
\hom_{{\bf K}(\Shvlogtkl)}(F',\gamma^*G)=0
\]

for all $F'\in \cH'$.
Thus $\gamma^*G$ is $\cH'$-flasque,
$\gamma^*G$ is $\cG$-fibrant.
By Propositions <ref>(2) we see that $\gamma^*G$ is $\cH$-flasque.
Thus for all $F\in \cH$, we have the vanishing

\[
\hom_{{\bf K}(\Shvlogtkl)}(F,\gamma^*G)=0.
\]

Since $G$ is an injective fibrant complex, we also have

\[
\hom_{{\bf K}(\Shvlogtkl)}(F,\gamma^*G)
\cong
\hom_{{\bf K}(\Shvltrtkl)}(\gamma_\sharp F,G)
\cong
\hom_{\Deri(\Shvltrtkl)}(\gamma_\sharp F,G).
\]

Here $G$ is an arbitrary injective fibrant complex.
Hence $\gamma_\sharp F$ is quasi-isomorphic to $0$ for all $F\in \cH$.
This implies that condition (i) holds.

§.§ Structure of dividing Nisnevich covers

In this section, we shall study dividing Nisnevich covers and the dividing Nisnevich sheafification functor.
We also discuss their étale versions.

Suppose that $X$ is an excellent reduced noetherian scheme.
In <cit.> it is shown that every $h$-cover $f\colon Y\longrightarrow X$ of schemes admits a refinement of the form

\[
Y\stackrel{g_1}\longrightarrow Y_1\stackrel{g_2}\longrightarrow Y_2\stackrel{g_3}\longrightarrow X,
\]

where $g_1$ is a Zariski cover, $g_2$ is a finite surjective morphism, and $g_3$ is a proper surjective birational morphism.
This is one of the fundamental tools in the theory of $h$-coverings.

We provide a result with a similar flavor for factorizations of dividing Nisnevich, dividing étale, and log étale covers.

Let $f\colon Y\rightarrow X$ be a dividing Nisnevich (resp. dividing étale, resp. log étale) cover in $lSm/k$.
Then there exists a log modification $h\colon X'\rightarrow X$ such that the pullback

\[
g\colon Y\times_X X'\rightarrow X'
\]

is a strict Nisnevich (resp. strict étale, resp. integral Kummer étale) cover.

In particular, $f$ has a refinement of the form

\begin{equation}
\label{A.5.44.1}
Y\times_X X'\stackrel{g}\rightarrow X'\stackrel{h}\rightarrow X,
\end{equation}

where $g$ is a strict Nisnevich (resp. strict étale, resp. integral log étale) cover and $h$ is a log modification.

If $f$ is log étale, see Theorem <ref>.
We focus the attention on the dividing Nisnevich topology since the proofs are similar for the dividing Nisnevich and dividing étale cases.
First, we reduce to the case when $f$ has a refinement in the form

\[
Y_0\stackrel{f_0}\rightarrow Y_0'\stackrel{f_0'}\rightarrow \cdots \stackrel{f_{n-2}'}\rightarrow Y_{n-1}\stackrel{f_{n-1}}\rightarrow Y_{n-1}'\stackrel{f_{n-1}'}\rightarrow Y_n\stackrel{f_n}\rightarrow X,
\]

where each $f_i$ is a strict Nisnevich cover and each $f_i'$ is a log modification.
Owing to Proposition <ref>, there exists a log modification $X'\rightarrow X$ such that the pullback

\[
Y_{n-1}'\times_X X'\rightarrow Y_n\times_X X'
\]

of $f_{n-1}'$ is an isomorphism. In particular, the pullback

\[
Y_{n-1}\times_X X'\rightarrow Y_n\times_X X'
\]

of the composition $Y_{n-1}\rightarrow X$ is a strict Nisnevich cover.

Replacing (<ref>) by

\[
Y_0'\times_X X'\rightarrow \cdots \rightarrow Y_{n-1}\times_X X'\rightarrow X',
\]

we can apply the above process since $X'\in lSm/k$.
In this way, we obtain a log modification $X''\rightarrow X$ such that the pullback

\[
Y_0\times_X X''\rightarrow X''
\]

of the composition $Y_0\rightarrow X$ is a strict Nisnevich cover.

Recall from Definition <ref> that $X_{div}$ denotes the category of log modifications over $X\in lSm/k$,
and $X_{div}^{Sm}$ is the full subcategory of $X_{div}$ consisting of log modifications $Y\rightarrow X$ such that $Y\in SmlSm/k$.

Let $\cF$ be a strict Nisnevich (resp. strict étale, resp. Kummer étale) sheaf on $SmlSm/k$.
There is an isomorphism

\begin{equation}
\label{A.5.46.1}
\begin{split}
a_{dNis}^*\cF(X)\cong &\colimit_{Y\in X_{div}^{Sm}}\cF(Y)
\\
\text{ (resp.\ }
a_{d\acute{e}t}^*\cF(X)\cong \colimit_{Y\in X_{div}^{Sm}}\cF(Y),
&\text{ resp.\ }
a_{l\acute{e}t}^*\cF(X)\cong \colimit_{Y\in X_{div}^{Sm}}\cF(Y)
\text{)}.
\end{split}
\end{equation}

We will only consider the strict Nisnevich case since the proofs are similar.
Let $X_{dNis}$ denote the small dividing Nisnevich site of $X$.
Recall from <cit.> that there is a functor $L_{dNis}$ such that $a_{dNis}^*=L_{dNis}\circ L_{dNis}$ given by

\[
L_{dNis}\cF(X)\cong \colimit_{X'\in X_{dNis}}\ker(\cF(X')\stackrel{(+,-)}\longrightarrow \cF(X'\times_X X')).
\]

The notation is shorthand for the difference between the two morphisms induced by the projections $X'\times_X X'\longrightarrow X'$.

By Proposition <ref>, for any dividing Nisnevich covering $X'\rightarrow X$, there is a refinement

\[
Y'\stackrel{g}\rightarrow Y''\stackrel{h}\rightarrow X
\]

such that $g$ is a strict Nisnevich cover and $h$ is a log modification.
Further, we may assume that $Y''\in SmlSm/k$ by applying Proposition <ref>.

\[
Y'\times_X Y'\cong Y'\times_{Y''}Y''\times_X Y''\times_{Y''}Y'\cong Y'\times_{Y''} Y',
\]

we have

\[
\ker(\cF(Y')\stackrel{(+,-)}\longrightarrow \cF(Y'\times_X Y'))
\cong
\ker(\cF(Y')\stackrel{(+,-)}\longrightarrow \cF(Y'\times_{Y''} Y'))
\cong
\cF(Y'')
\]

because $\cF$ is assumed to be a strict Nisnevich sheaf.
This gives the identification

\[
L_{dNis}\cF(X)\cong \colimit_{Y\in X_{div}^{Sm}}\cF(Y).
\]

Let us show that $L_{dNis}\cF$ is a strict Nisnevich sheaf.
For every strict Nisnevich cover $f\colon X'\rightarrow X$ the naturally induced functor $f^*\colon X_{div}^{Sm}\rightarrow X_{div}'^{Sm}$
is cofinal owing to Propositions <ref> and <ref>.
Thus to show that $L_{dNis}\cF$ is a strict Nisnevich sheaf, it suffices to show there is an exact sequence

\[
0\rightarrow \colimit_{Y\in X_{div}^{Sm}}\cF(Y)\rightarrow \colimit_{Y\in X_{div}^{Sm}}\cF(Y\times_X X')\rightarrow \colimit_{Y\in X_{div}^{Sm}}\cF(Y\times_X X'\times_X X').
\]

Owing to Proposition <ref> $X_{div}^{Sm}$ is a filtered category.
Since filtered colimits preserve exact sequences, it suffices to show there is an exact sequence

\[
0\rightarrow \cF(Y)\rightarrow \cF(Y\times_X X')\rightarrow \cF(Y\times_X X'\times_X X')
\]

for all $Y\in X_{div}$.
This follows from the assumption that $\cF$ is a strict Nisnevich sheaf.

Applying the above to $L_{dNis}\cF$ we deduce the isomorphisms

\[
a_{dNis}^*\cF(X)\cong \colimit_{Z\in Y_{div}^{Sm}}\colimit_{Y\in X_{div}^{Sm}}\cF(X)\cong \colimit_{Z\in X_{div}^{Sm}}\cF(X).
\]

Let $\cF$ be a strict Nisnevich (resp. strict étale, resp. Kummer étale) sheaf on $lSm/k$.
There is an isomorphim

\begin{equation}
\label{A.5.45.1}
\begin{split}
a_{dNis}^*\cF(X)\cong &\colimit_{Y\in X_{div}}\cF(Y)
\\
\text{ (resp.\ }
a_{d\acute{e}t}^*\cF(X)\cong \colimit_{Y\in X_{div}}\cF(Y),
&\text{ resp.\ }
a_{l\acute{e}t}^*\cF(X)\cong \colimit_{Y\in X_{div}}\cF(Y)
\text{)}.
\end{split}
\end{equation}

The proof is parallel to that of Lemma <ref> if we use Proposition <ref> instead of Proposition <ref>.

For the definition of a log modification along a smooth center, we refer to Definition <ref>.

Let $X_{divsc}^{Sm}$ be the full subcategory of $X_{div}^{Sm}$ given by maps $Y\rightarrow X$ that are isomorphic to a composition of log modifications along a smooth center.

If $X\in lSm/k$, then the category $X_{divsc}^{Sm}$ is cofinal in $X_{div}^{Sm}$.

Immediate from Theorem <ref>.

Let $\cF$ be a strict Nisnevich (resp. strict étale, resp. Kummer étale) sheaf on $SmlSm/k$.
There is an isomorphism

\[
\begin{split}
a_{dNis}^*\cF(X)\cong &\colimit_{Y\in X_{divsc}^{Sm}}\cF(Y)
\\
\text{ (resp.\ }
a_{d\acute{e}t}^*\cF(X)\cong \colimit_{Y\in X_{divsc}^{Sm}}\cF(Y),
&\text{ resp.\ }
a_{l\acute{e}t}^*\cF(X)\cong \colimit_{Y\in X_{divsc}^{Sm}}\cF(Y)
\text{)}.
\end{split}
\]

Immediate from Lemma <ref> and Corollary <ref>.

§.§ Examples of topologies compatible with log transfers

This section shows that the strict Nisnevich, strict étale, Kummer étale, dividing Nisnevich, dividing étale, and Kummer étale topologies are compatible with log transfers.
We refer to <ref> for the definitions of these topologies.
For the strict Nisnevich topology, we use arguments similar to <cit.> originating in [89].

Let $X$ be an fs log scheme log smooth over $k$. Then $\Zltr(X)$ is a strict étale sheaf.

For fs log schemes $Y_1$ and $Y_2$ log smooth over $k$ we have

\[
\Zltr(X)(Y_1\amalg Y_2)
\Zltr(X)(Y_1)\oplus \Zltr(X)(Y_2).
\]

Thus it suffices to check that the sequence of $\Lambda$-modules

\begin{equation}
\label{A.4.1.1}
0\rightarrow \Zltr(X)(Y)\rightarrow \Zltr(X)(U)\stackrel{(+,-)}\longrightarrow \Zltr(X)(U\times_Y U)
\end{equation}

is exact for every strict étale cover $p\colon U\rightarrow Y$.

There is an induced commutative diagram of $\Lambda$-modules

\[
\begin{tikzcd}
\Zltr(X)(Y)\arrow[d]\arrow[r]&\Zltr(X)(U)\arrow[d]\\
\Zltr(X-\partial X)(Y-\partial Y)\arrow[r]&\Zltr(X-\partial X)(U-\partial U).
\end{tikzcd}
\]

By <cit.>, $\Zltr(X-\partial X)$ is an étale sheaf.
Thus the lower horizontal morphism is injective.
By Lemma <ref>, the vertical morphisms are injective.
Thus the upper horizontal morphism is also injective.

It remains to show that the sequence (<ref>) is exact at $\Zltr(X)(U)$.
To that end, we consider the cartesian square of fs log schemes

\[
\begin{tikzcd}
(U-\partial U)\times (X-\partial X)\arrow[d]\arrow[r]&U\times X\arrow[d]\\
(Y-\partial Y)\times (X-\partial X)\arrow[r]&Y\times X.
\end{tikzcd}
\]

Suppose $W\in \lCor(U,X)$ is a finite log correspondence with trivial image in $\Zltr(X)(U\times_Y U)$,
and form the finite correspondence

\[
W-\partial W\in \Cor(U-\partial U,X-\partial X).
\]

Since $\Zltr(X-\partial X)$ is an étale sheaf, there exists a finite correspondence

\[
V'\in \Cor(Y-\partial Y,X-\partial X)
\]

mapping to $W-\partial W$.
We let $\underline{V}$ be the closure of $V'$ in $\underline{Y}\times \underline{X}$.

Consider the induced morphism $u\colon \underline{W}\rightarrow \underline{U}\times_{\underline{Y}}\underline{V}$ of closed subschemes of $\underline{U}\times \underline{X}$.
By construction its pullback to $(U-\partial U)\times (X-\partial X)$ is the isomorphism

\[
W-\partial W\rightarrow (U-\partial U)\times_{(Y-\partial Y)}(V-\partial V)
\]

Since $\underline{p}\colon \underline{U}\rightarrow \underline{Y}$ is strict étale and $V-\partial V$ is dense in $\underline{V}$,
it follows that

\[
(U-\partial U)\times_{(Y-\partial Y)}(V-\partial V)
\]

is dense in $\underline{U}\times_{\underline{Y}}\underline{V}$.
$W-\partial W$ is dense in $\underline{W}$ because $u$ is a closed immersion.
We note that $\underline{V}$ is finite over $\underline{Y}$ by <cit.> since $\underline{U}\times_{\underline{Y}}\underline{V}$ is finite over $\underline{U}$
and $\underline{p}\colon \underline{U}\rightarrow \underline{Y}$ is an étale cover.
There is a naturally induced morphism $v\colon \underline{W}\rightarrow \underline{V}$.
Since $u$ is an isomorphism, $v$ is a pullback of $p$, and hence $v$ is étale.

Let $V^N$ be the fs log scheme whose underlying scheme is the normalization of $\underline{V}$ and whose log structure is induced by $Y$.
By <cit.> the underlying fiber product $\underline{V^N}\times_{\underline{V}}\underline{W}$ is normal since $v$ is étale.
Thus we have the cartesian square of schemes

\[\begin{tikzcd}
\underline{W^N}\arrow[d]\arrow[r]&\underline{W}\arrow[d,"v"]\\
\underline{V^N}\arrow[r]&\underline{V}.
\end{tikzcd}
\]

The log transfer structure gives rise to a morphism $r\colon W^N\rightarrow X$ of fs log schemes over $k$.
By assumption,
we can identify the two composite morphisms in the diagram

\[
W^N\times_{U}(U\times_Y U)
\rightrightarrows
\overset{r}{\rightarrow}
\]

Since $p$ is strict étale there is a morphism $V^N\rightarrow X$ of fs log schemes over $k$ such that the composition $W^N\rightarrow V^N\rightarrow X$ is equal to $r$,
see <cit.>.
The pair $(\underline{V},V^N\rightarrow X)$ gives a finite log correspondence $V$ from $X$ to $Y$ since $\underline{V}$ is finite over $\underline{Y}$,
and the pullback of $V$ to $U$ is $W$.

Let $t$ be one of the following topologies: strict Nisnevich, strict étale, and Kummer étale topologies.
For every $t$-cover $f\colon U\rightarrow X$ of fs log schemes in $lSm/k$,
the Čech complex

\begin{equation}
\label{A.5.11.2}
\cdots\rightarrow a_t^*\Zltr(U\times_X U)\rightarrow a_t^*\Zltr(U)\rightarrow a_t^*\Zltr(X)\rightarrow 0
\end{equation}

is exact as a complex of $t$-sheaves.

Arguing as in the proof of <cit.> we check there is an exact sequence of $\Lambda$-modules

\begin{equation}\
\label{A.5.11.1}
\cdots\rightarrow \Zltr(U\times_X U)(T)\rightarrow \Zltr(U)(T)\rightarrow \Zltr(X)(T)\rightarrow 0,
\end{equation}

where $\underline{T}$ is a hensel local scheme (resp. $\underline{T}$ is strictly local, resp. $T$ is log strictly local) if $t=sNis$ (resp. $t=s\acute{e}t$, resp. $t=k\acute{e}t$).
Here $\Zltr({X})(T)$ denotes the colimit $\varinjlim \Zltr(X)(T_i)$ when $T$ is a limit of fs log schemes $T_i$ log smooth over $k$.

Let $Z$ be a saturated log scheme over $T$ such that $\underline{Z}$ quasi-finite over $\underline{T}$.
We denote by $L(Z/T)$ the free abelian group generated by the pairs

\[
(\underline{W},W^N\rightarrow Z),
\]

where $\underline{W}$ is an irreducible component of $\underline{Z}$ that is finite and surjective over a component of $T$,
$W^N$ is the saturated log scheme whose underlying scheme is the normalization of $\underline{W}$ and whose log structure is induced by $T$,
and $W^N\rightarrow Z$ is a morphism of saturated log schemes.
We note that $L(Z/T)$ is covariantly functorial on $Z$.
Now (<ref>) is the colimit of the complexes of the form

\[
\cdots\rightarrow L((Z_U)_Z^2/T)\rightarrow L(Z_U/T)\rightarrow L(Z/T)\rightarrow 0,
\]

\[
Z_U:=Z\times_X U,
\;
(Z_U)_Z^n:=\underbrace{Z_U\times_Z \cdots \times_Z Z_U}_{n \text{ times}}.
\]

The colimit runs over strictly closed subschemes $Z$ of $T\times X$ such that $Z$ is strict, finite, and surjective over $T$.
Hence it suffices to check that the latter sequence is exact.

Note that $Z$ is henselian (resp. strictly henselian, resp. log strictly henselian) being finite over the henselian (resp. strictly henselian, resp. log strictly henselian) saturated log scheme $T$, see <ref> for the log strictly henselian case.
Thus the strict Nisnevich (resp. strict étale, resp. Kummer étale) cover $Z_U\rightarrow Z$ splits, see Lemma <ref> for the Kummer étale case.
Letting $s\colon Z\rightarrow Z_U$ be a splitting,
the maps

\[
L(s\times_Z {\rm id})\colon L((Z_U)_Z^n)\rightarrow L((Z_U)_Z^{n+1})
\]

furnish contracting homotopies, and the desired exactness follows.

Combining Propositions <ref> and <ref> we can remove $a_t^*$ in the equation (<ref>) if $t=sNis$ or $t=s\acute{e}t$.

The strict Nisnevich topology, the strict étale topology, and the Kummer étale topology on $lSm/k$ are compatible with log transfers.

Follows from Propositions <ref> and <ref>.

Let $f\colon X'\rightarrow X$ be a log modification in $lSm/k$.
For every $Y\in lSm/k$ and finite log correspondence $V\in \lCor(Y,X)$, there exists a dividing Zariski cover $g\colon Y'\rightarrow Y$ and a finite log correspondence $W\in \lCor(Y',X')$ fitting in a commutative diagram

\[
\begin{tikzcd}
\end{tikzcd}
\]

in $lCor/k$.

The question is Zariski local on $Y$, so we may assume that $Y$ has an fs chart $P$.
Let $p\colon V^N\times_X X'\rightarrow V^N$ be the projection, and let $q\colon V^N\rightarrow Y$ be the structure morphism.
By Lemma <ref>, there is a subdivision of fans $M\rightarrow \Spec P$ such that the projection $V^N\times_{\A_P}\A_M\rightarrow V^N$ admits a factorization

\[
V^N\times_{\A_P}\A_M\rightarrow V^N\times_X X'\rightarrow V^N.
\]

Setting $Y':=Y\times_{\A_P}\A_M$ and $V':=V\circ u$ where $u\colon Y'\rightarrow Y$ denotes the projection,
the structure morphism $v\colon V'^N\rightarrow X$ factors through $V^N\times_Y Y'\cong V^N\times_{\A_P}\A_M$, so that $v$ factors through $V^N\times_X X'$.
It follows that $v$ factors through $X'$.

Replacing $Y$ by $Y'$ and $V$ by $V'$, we may assume that the structure morphism $V^N\rightarrow X$ factors through $X'$.
In this case, there is a commutative diagram of fs log schemes

\[
\begin{tikzcd}
V^N\arrow[r,"w"]&Y\times X'\arrow[d]\arrow[r]&X'\arrow[d]\\
&Y\times X\arrow[r]&X.
\end{tikzcd}\]

Since $X'$ is proper over $X$, the morphism $w$ is proper.
Let $\underline{W}$ be the image of

\[
\underline{V^N}\rightarrow \underline{Y}\times \underline{X'},
\]

which we consider as a closed subscheme of $\underline{Y}\times \underline{X'}$ with the reduced scheme structure.
Then $\underline{V^N}$ is the normalization of $\underline{W}$ since $\underline{V^N}$ is the normalization of the image of

\[
\underline{V^N}\rightarrow \underline{Y}\times \underline{X}.
\]

Thus $w$ gives a correspondence $W$ from $Y$ to $X'$ with image $V$ in $\lCor(Y',X)$.

For every log modification $f\colon Y\rightarrow X$ in $lSm/k$ the induced morphism

\[
a_{dNis}^*\gamma^*\Zltr(Y)\rightarrow a_{dNis}^*\gamma^*\Zltr(X)
\]

of dividing Nisnevich sheaves is an isomorphism.

Let $T$ be an fs log scheme log smooth over $k$.
In the commutative diagram

\[
\begin{tikzcd}
\lCor(T,Y)\arrow[r]\arrow[d,hook]&\lCor(T,X)\arrow[d,hook']\\
\Cor(T-\partial T,Y-\partial Y)\arrow[r,"\sim"]&\Cor(T-\partial T,X-\partial X)
\end{tikzcd}
\]

the vertical morphisms are injections by Lemma <ref>,
and the lower horizontal morphism is an isomorphism since $Y-\partial Y\cong X-\partial X$.
Thus the upper horizontal morphism is an injection.
Hence $\Zltr(Y)\rightarrow \Zltr(X)$ is a monomorphism,
and likewise for

\[
a_{dNis}^*\gamma^*\Zltr(Y)\rightarrow a_{dNis}^*\gamma^*\Zltr(X)
\]

since $a_{dNis}^*$ and $\gamma^*$ are exact, see (<ref>) and Proposition <ref>.
To show that it is an epimorphism,
let $V\in \lCor(T,X)$ be a finite log correspondence.
Due to Lemma <ref> there exists a dividing Nisnevich cover $g\colon T'\rightarrow T$ and a finite log correspondence $W\in \lCor(T',Y)$ such that $f\circ W=V\circ g$.
This finishes the proof.

The dividing Nisnevich topology, the dividing étale topology, and the log étale topology on $lSm/k$ are compatible with log transfers.

We write the proof in the case of the log étale topology.
Let $f\colon Y\rightarrow X$ be a log étale cover.
We need to show that the Čech complex

\[
\cdots\rightarrow a_{l\acute{e}t}^*\gamma^*\Zltr(Y\times_X Y)\rightarrow a_{l\acute{e}t}^*\gamma^*\Zltr(Y)\rightarrow a_{l\acute{e}t}^*\gamma^*\Zltr(X)\rightarrow 0
\]

is exact, see Proposition <ref>.
There exists a log modification $X'\rightarrow X$ such that the pullback $f'\colon Y\times_X X'\rightarrow X'$ of $f$ is a Kummer étale cover owing to Proposition <ref>.
Set $Y':=Y\times_X X'$.
Then using Lemma <ref>, it suffices to show that the Čech complex

\[
\cdots \rightarrow a_{l\acute{e}t}^*\gamma^*\Zltr(Y'\times_{X'}Y')\rightarrow a_{l\acute{e}t}^*\gamma^*\Zltr(Y')\rightarrow a_{l\acute{e}t}^*\gamma^*\Zltr(X')\rightarrow 0
\]

is exact.
Since the sheafification functor $a_{l\acute{e}t}^*$ is exact, we are done by Proposition <ref>.

§.§ Dividing log correspondences

For $X,Y\in lSm/k$, we have used $\lCor(X,Y)$ to define dividing Nisnevich sheaves with log transfers.
Since $\Zltr(Y)$ is not a dividing Nisnevich sheaf one cannot in general identify $\lCor(X,Y)$ with

\[
\hom_{\Shvltrkl}(a_{dNis}^*\Zltr(Y),a_{dNis}^*\Zltr(X)).
\]

it turns out that algebraic cycles describe $a_{dNis}^*\Zltr(Y)(X)$.
To that end, we introduce the notion of dividing log correspondences.

For $X,Y\in lSm/k$, a dividing elementary log correspondence $Z$ from $X$ to $Y$ is an integral closed subscheme $\underline{Z}$ of $\underline{X}\times \underline{Y}$
that is finite and surjective over a component of $\underline{X}$ together with a log modification $X'\rightarrow X$ and a morphism

\[
u'\colon Z'^N\rightarrow X'\times Y
\]

satisfying the following properties:

(i) The image of the composition $\underline{Z'^N}\stackrel{\underline{u'}}\rightarrow \underline{X'}\times \underline{Y}\rightarrow \underline{X}\times \underline{Y}$ is $\underline{Z}$.

(ii) $\underline{Z'^N}$ is the normalization of $\underline{Z}\times_{\underline{X}}\underline{X'}$.

(iii) The composition $Z'^N\stackrel{u'}\rightarrow X'\times Y\rightarrow X'$ is strict.

A dividing log correspondence correspondence!dividing log from $X$ to $Y$ is a formal sum $\sum n_i Z_i$ of dividing elementary log correspondences $Z_i$ from $X$ to $Y$.
Let $\lCor_k^{div}(X,Y)$ denote the free abelian group of dividing log correspondences from $X$ to $Y$.
When no confusion seems likely to arise, we shall omit the subscript $k$ and write $\lCor^{div}(X,Y)$.

For $X,Y\in lSm/k$, a finite log correspondence $Z\in \lCor(X,Y)$ is determined by $Z-\partial Z\in \lCor(X-\partial X,Y-\partial Y)$ owing to Lemma <ref>.
Thus $Z$ is determined by its underlying scheme $\underline{Z}$.
Hence we arrive at the following equivalent definition:
An elementary log correspondence $Z$ from $X$ to $Y$ is an integral closed subscheme $\underline{Z}$ of $\underline{X}\times \underline{Y}$ that is finite and surjective over a component of
$\underline{X}$ together with a morphism

\[
u\colon Z^N\rightarrow X\times Y
\]

satisfying the following properties:

(i) The image of $\underline{u}\colon \underline{Z^N}\rightarrow \underline{X}\times \underline{Y}$ is $\underline{Z}$.

(ii) $\underline{Z^N}$ is the normalization of $\underline{Z}$.

(iii) The composition $Z^N\stackrel{u}\rightarrow X\times Y\rightarrow X$ is strict.

Given the latter description, we can interpret any dividing log correspondence as a finite log correspondence after replacing $X$ with a log modification of $X$.

For every $X,Y\in lSm/k$ there are isomorphisms

\[
\lCor^{div}(X,Y)\otimes \Lambda\cong a_{dNis}^*\Zltr(Y)(X)\cong a_{d\acute{e}t}^*\Zltr(Y)(X).
\]

With the description of elementary log correspondences in Remark <ref>, this follows from Lemma <ref>.

For $X,Y,Z\in lSm/k$, let us construct a composition for dividing log correspondences

\begin{equation}
\label{A.5.50.1}
\circ\colon \lCor^{div}(X,Y)\times \lCor^{div}(Y,Z)\rightarrow \lCor^{div}(X,Z).
\end{equation}

Owing to Proposition <ref> and the Yoneda lemma we have

\begin{equation}
\label{A.5.50.2}
\lCor^{div}(X,Y)\cong \hom_{\Shvltrkl}(a_{dNis}^*\ZZltr(X),a_{dNis}^*\ZZltr(Y)).
\end{equation}

We obtain (<ref>) because we have compositions in $\ShvltrkZ$.
Moreover, the composition is associative, and for $g\in \lCor(X,Y)$ we have

\[
{\rm id}\circ f=f=f\circ {\rm id}.
\]

Thus we can form a category $lCor^{div}/k$ with the same objects as $lSm/k$ and with morphisms given by $\lCor^{div}(X,Y)$.

A presheaf of $\Lambda$-modules with dividing log transfers is an additive presheaf of $\Lambda$-modules on the category $lCor^{div}/k$.
We let $\Pshdltrkl$ denote the category of presheaves of $\Lambda$-modules with dividing log transfers.

A dividing Nisnevich sheaf of $\Lambda$-modules with dividing log transfers is a presheaf of $\Lambda$-modules with dividing log transfers such that the restriction to $lSm/k$ is a dividing Nisnevich sheaf.

We let $\Shvdltrkl$ (resp. $\Shv_{d\acute{e}t}^{\rm dltr}(k,\Lambda)$) denote the category of dividing Nisnevich (resp. dividing étale) sheaves of $\Lambda$-modules with dividing log transfers.

There are equivalences of categories

\begin{gather*}
\Shvdltrkl\cong \Shvltrkl,
\\
\Shv_{d\acute{e}t}^{\rm dltr}(k,\Lambda)\cong \Shv_{d\acute{e}t}^{\rm ltr}(k,\Lambda),
\\
\Shv_{l\acute{e}t}^{\rm dltr}(k,\Lambda)\cong \Shv_{l\acute{e}t}^{\rm ltr}(k,\Lambda).
\end{gather*}

We will only consider the dividing Nisnevich case since the proofs are similar.
It suffices to show that any $\cF\in \Shvltrkl$ admits a unique dividing log transfer structure.
For any $X,Y\in lSm/k$,
owing to (<ref>),
we have that

\[
\hom_{\Shvltrkl}(a_{dNis}^*\Zltr(X),a_{dNis}^*\Zltr(Y))
\cong \
\]

Thus for any $V\in \lCor^{div}(X,Y)$ the Yoneda lemma furnishes a canonical transfer map

\[
\begin{split}
\cF(X)\cong &\hom_{\Shvltrkl}(a_{dNis}^*\Zltr(X),\cF)\\
\rightarrow &\hom_{\Shvltrkl}(a_{dNis}^*\Zltr(Y),\cF)\cong \cF(Y).
\end{split}
\]

Thus there exists a dividing log transfer structure on $\cF$.

For uniqueness,
suppose that $\cG_0$ and $\cG_1$ are two sheaves with dividing log transfers whose restrictions to $\Shvltrkl$ are isomorphic to $\cF$.
Since $\cG_0$ and $\cG_1$ agree on objects,
it suffices to show that $\cG_0(V)=\cG_1(V)$ for every $V\in \lCor^{div}(X,Y)$.
Owing to Proposition <ref>,
there exists a dividing Nisnevich cover $p\colon X'\rightarrow X$ such that $V':=V\circ u\in \lCor^{div}(X',Y)$ is an element in $\lCor(X',Y)$.
By the assumption that $\cF$ is the restriction of $\cG_0$ and $\cG_1$, we have an equality of homomorphisms

\[
\cG_0(V')=\cG_1(V')\colon \cF(Y)\rightarrow \cF(X').
\]

Since $\cF$ is a dividing Nisnevich sheaf and $X'\rightarrow X$ is a dividing Nisnevich cover, $\cF(p)\colon \cF(X)\rightarrow \cF(X')$ is injective.
Thus we can conclude $\cG_0(V')=\cG_1(V')$.

Let $lCor_{SmlSm}/k$ (resp. $lCor_{SmlSm}^{div}/k$) denote the full subcategory of $lCor/k$ (resp. $lCor^{div}/k$) consisting of all objects in $SmlSm/k$.

Let $f\colon Y\rightarrow X$ be a log modification in $lSm/k$.
Then $f$ is an isomorphism in $lCor^{div}/k$.

Consider the composition

\[
\Gamma_f^t\colon Y\stackrel{\Gamma_f}\longrightarrow Y\times X\rightarrow X\times Y,
\]

where $\Gamma_f$ is the graph morphism, and the second morphism switches the factors.
The induced pullback

\[
Y\times_X Y\rightarrow (Y\times_X X)\times  Y,
\]

is isomorphic to the diagonal morphism $Y\rightarrow Y\times Y$.
Thus $\Gamma_f^t$ is a dividing log correspondence from $X$ to $Y$,
and it follows that $f$ is invertible in $lCor^{div}/k$.

Lemma <ref> furnishes a class of non-finite morphisms in $lSm/k$ that become invertible in $lCor^{div}/k$,
but not in $lCor/k$.
Following the approach to motives with modulus in [49],
this suggests that we should invert every proper birational morphism $f:Y\rightarrow X$ for which the naturally induced morphism

\[
Y-\partial Y\rightarrow X-\partial X
\]

is an isomorphism.
our strategy is different from [49].
Assuming resolution of singularities,
in Theorem <ref> we show that the naturally induced morphism of log motives

\[
M(f):M(Y)\rightarrow M(X)
\]

is an isomorphism.

There is an equivalence of categories

\[
lCor_{SmlSm}^{div}/k\simeq lCor^{div}/k.
\]

Since $lCor_{SmlSm}^{div}/k$ is a full subcategory of $lCor^{div}/k$, it remains to show that any object $X$ of $lCor^{div}/k$ is isomorphic to an object of $lCor_{SmlSm}^{div}/k$.
Owing to Proposition <ref>, there is a log modification $f\colon Y\rightarrow X$ such that $Y\in SmlSm/k$.
Then we are done since $f$ is an isomorphism in $lCor^{div}/k$ by Lemma <ref>.

§.§ Sheaves with log transfers on $SmlSm/k$

There are many interesting examples of Nisnevich sheaves with transfers,
e.g., $\mathbb{G}_m$ and $\Omega^i$.
If $\cF$ is a Nisnevich sheaf with transfers on $Sm/k$,
one may ask whether it extends to a dividing Nisnevich sheaf with log transfers on $lSm/k$,
such that $\cF(X)$ makes sense for every $X\in lSm/k$.
In practice, one can divide this problem into two steps.
The first step is to define $\cF(X)$ for every $X\in SmlSm/k$.
And the second step is to define $\cF(X)$ for a general $X\in lSm/k$.
In this subsection, we explain this second step.

We set

\[
\Pshltrklsm:=\Psh(lCor_{SmlSm}/k),
\]

\[
\Pshdltrklsm:=\Psh(lCor_{SmlSm}^{div}/k).
\]

For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, we let $\Shv_t^{\rm ltr}(SmlSm/k,\Lambda)$ (resp. $\Shv_t^{\rm dltr}(SmlSm/k,\Lambda)$) denote the full subcategory of $\Pshltrklsm$
(resp. $\Pshdltrklsm$) consisting of $t$-sheaves on $SmlSm/k$.

For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, there is an equivalence of categories

\[
\Shv_{t}(SmlSm/k,\Lambda)\simeq \Shv_t^{\rm log}(k,\Lambda).
\]

We will only consider the dividing Nisnevich topology since the proofs are similar.
For any $X\in lSm/k$, there exists a log modification $Y\rightarrow X$ such that $Y\in SmlSm/k$ owing to Proposition <ref>.
We conclude using the implication (i)$\Rightarrow$(ii) in <cit.>.

Let $\iota$ denote the inclusion functor $lCor_{SmlSm}/k\rightarrow lCor/k$.
For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, we have functors

\begin{gather}
\iota^*\colon \Psh^{\rm dltr}(k,\Lambda)
\rightarrow
\Psh^{\rm dltr}(SmlSm/k,\Lambda)
\\
\label{A.5.58.1}
\iota^*\colon
\Shv_{t}^{\rm ltr}(k,\Lambda)
\rightarrow
\Shv_{t}^{\rm ltr}(SmlSm/k,\Lambda)
\end{gather}

mapping $\cF$ to $\cF\circ \iota$.
As in (<ref>) there are faithful functors

\[
\gamma\colon lSm/k\rightarrow lCor^{div}/k,\; \gamma\colon SmlSm/k\rightarrow lCor_{SmlSm}^{div}/k.
\]

These induce functors

\[
\gamma^*\colon \Pshdltrkl\rightarrow \Pshlogkl,
\;
\gamma^*\colon \Pshdltrklsm\rightarrow \Psh(SmlSm/k,\Lambda)
\]

mapping $\cF$ to $\cF\circ \gamma$.

For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, there is an equivalence of categories

\[
\Shv_{t}^{\rm dltr}(SmlSm/k,\Lambda)\simeq \Shv_{t}^{\rm dltr}(k,\Lambda).
\]

We will only consider the dividing Nisnevich topology since the proofs are similar.
There is a commutative diagram

\[
\begin{tikzcd}
\Pshdltrkl\arrow[r,"\iota^*"]\arrow[d,"\gamma^*"']&\Pshdltrklsm\arrow[d,"\gamma^*"]\\
\Pshlogkl \arrow[r]&\Psh(SmlSm/k,\Lambda).
\end{tikzcd}
\]

The restriction functor $\iota^*$ is an equivalence by Lemma <ref>.
Hence it suffices to show that for every $\cF\in \Pshdltrkl$, $\gamma^*(\cF)$ is a dividing Nisnevich sheaf if and only if its image in $\Psh(SmlSm/k,\Lambda)$ is a dividing Nisnevich sheaf.
This follows from Lemma <ref>.

For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, there is an equivalence of categories

\[
\Shv_{t}^{\rm dltr}(SmlSm/k,\Lambda)\simeq \Shv_{t}^{\rm ltr}(SmlSm/k,\Lambda).
\]

Parallel to the proof of Proposition <ref>.

For $t=dNis$, $d\acute{e}t$, and $l\acute{e}t$, the functor in (<ref>) is an equivalence.

Follows from Propositions <ref>, <ref>, and Lemma <ref>.

Thus the functor in (<ref>) admits a left adjoint

\[
\iota_\sharp\colon
\Shv_{t}^{\rm ltr}(SmlSm/k,\Lambda)
\rightarrow
\Shv_{t}^{\rm ltr}(k,\Lambda).
\]

Let $\cF$ be an object of $\Shv_{t}^{\rm ltr}(SmlSm/k,\Lambda)$, where $t$ is one of $dNis$, $d\acute{e}t$, and $l\acute{e}t$.
For every $X\in lSm/k$ we have that

\[
\iota_\sharp \cF(X)
\cong
\colimit_{Y\rightarrow X}\cF(Y),
\]

where the colimit runs over log modifications $Y\rightarrow X$ such that $Y\in SmlSm/k$.

We only discuss the dividing Nisnvich topology since the proofs are parallel.
Since $\iota^*$ is an equivalence by Proposition <ref>, we have $\iota^*\iota_\sharp \cF\cong \cF$, so that

\[
\iota_\sharp \cF(Y)\cong \cF(Y)
\]

for every $Y\in SmlSm/k$.
Recall that $X_{div}^{Sm}$ is the full subcategory of $X_{div}$ consisting of $Y\rightarrow X$ such that $Y\in SmlSm/k$.
According to Proposition <ref>, $X_{div}^{Sm}$ is cofinal in $X_{div}$,
and hence

\begin{equation}
\label{A.5.59.1}
\colimit_{Y\in X_{div}}\iota_\sharp \cF(Y)
\cong
\colimit_{Y\in X_{div}^{Sm}}\iota_\sharp \cF(Y)
\cong
\colimit_{Y\in X_{div}^{Sm}}\cF(Y).
\end{equation}

Since $\iota_\sharp \cF(X)$ is a dividing Nisnevich sheaf,
we deduce

\begin{equation}
\label{A.5.59.2}
\iota_\sharp \cF(X)\cong \colimit_{Y\in X_{div}}\iota_\sharp \cF(Y)
\end{equation}

Combining (<ref>) and (<ref>) gives the assertion.

§ CONSTRUCTION OF TRIANGULATED CATEGORIES OF LOGARITHMIC MOTIVES

This section is dedicated to constructing and studying some basic properties of our categories of log motivic sheaves.
First, we explain how the dividing Nisnevich cohomology groups are related to the strict Nisnevich cohomology groups.
Next, we construct the triangulated categories $\ldmeff$ and $\ldaeff$ of effective log motives,
with and without transfers,
in the Nisnevich and étale topologies on log schemes over $k$.
There are two equivalent ways of constructing $\ldmeff$,
see Section <ref>.
One way is to use sheaves on $lSm/k$, and the other way is via sheaves on $SmlSm/k$.
The advantage of having the latter category at our disposal is that it allows us to construct motivic sheaves that are, a priori,
only defined and functorial on log schemes for which the underlying scheme is smooth over $k$ and boundary a normal crossing divisor.
In Section <ref> we use this trick to prove the representability of Hodge cohomology and cyclic homology.

§.§ Dividing Nisnevich cohomology groups

A fundamental invariant of the dividing Nisnevich topology is the corresponding sheaf cohomology theory.
In what follows, we shall express the dividing Nisnevich cohomology groups in terms of strict Nisnevich cohomology groups by promoting Subsection <ref> to cohomology.
We also discuss the étale version of this result.
As an application, we discuss finite cohomological dimension property for our various topologies.

Let $\cF$ be a strict Nisnevich (resp. strict étale, resp. Kummer étale) sheaf on $lSm/k$.
If $\cF$ is flabby, then $a_{dNis}^*\cF$ (resp. $a_{d\acute{e}t}^*\cF$, resp. $a_{l\acute{e}t}^*\cF$) is flabby.

We focus on the strict Nisnevich case since the proofs are similar.
Owing to the implication $(b)\Rightarrow (a)$ in <cit.> it suffices to show that for every dividing Nisnevich cover $p:U\rightarrow X$ in $lSm/k$ and integer $i>0$,
the corresponding Čech cohomology group vanishes

\[
\]

Here $\cU$ is the Čech nerve associated with $U\rightarrow X$.

According to Proposition <ref>, there exists a log modification $X'\rightarrow X$ such that the projection $U':=U\times_X X'\rightarrow X'$ is a strict Nisnevich cover.
Since $a_{dNis}^*\cF$ is a complex of dividing Nisnevich sheaves, there is an isomorphism

\[
a_{dNis}^*\cF(\cU)\cong a_{dNis}^*\cF(\cU\times_X X').
\]

Hence we can replace $U\rightarrow X$ with $U'\rightarrow X'$, so we reduce to the case when $p$ is a strict Nisnevich cover.

By Lemma <ref> the complex $a_{dNis}^*\cF(\cU)$ is isomorphic to

\[
\colimit_{Y_0\in X_{div}}\cF(Y_0)
\rightarrow
\colimit_{Y_1\in U_{div}}\cF(Y_1)
\rightarrow
\colimit_{Y_2\in (U\times_X U)_{div}}\cF(Y_2)
\rightarrow
\cdots.
\]

Applying Corollary <ref>, we see that the latter complex is isomorphic to

\[
\colimit_{Y\in X_{div}}\cF(Y)
\rightarrow
\colimit_{Y\in X_{div}}\cF(Y\times_X U)
\rightarrow
\colimit_{Y\in X_{div}}\cF(Y\times_X U\times_X U)
\rightarrow
\cdots.
\]

This is a filtered colimit according to Proposition <ref>.
Since every filtered colimit in the category of $\Lambda$-modules is exact, it remains to check that $H^i$ of the complex

\[
\cF(Y)\rightarrow \cF(Y\times_X U)\rightarrow \cF(Y\times_X U\times_X U)\rightarrow \cdots
\]

is trivial for every $Y\in X_{div}$ and $i>0$.
To conclude we use the implication $(a)\Rightarrow (b)$ in <cit.> and the assumption that $\cF$ is flabby.

Let $\cF$ be a bounded below complex of strict Nisnevich sheaves on $lSm/k$.
Then for every $X\in lSm/k$ and $i\in \Z$ there is an isomorphism

\begin{align*}
\bH_{dNis}^i(X,a_{dNis}^*\cF)&\cong \colimit_{Y\in X_{div}}\bH_{sNis}^i(Y,\cF)
\\
\text{(resp.\ }\bH_{d\acute{e}t}^i(X,a_{d\acute{e}t}^*\cF)&\cong \colimit_{Y\in X_{div}}\bH_{s\acute{e}t}^i(Y,\cF),
\\
\text{resp.\ }\bH_{l\acute{e}t}^i(X,a_{l\acute{e}t}^*\cF)&\cong \colimit_{Y\in X_{div}}\bH_{k\acute{e}t}^i(Y,\cF)\text{)}.
\end{align*}

We only consider the strict Nisnevich case since the proofs are similar.
We can replace $\cF$ by its injective resolution because the sheafification functor $a_{dNis}^*$ is exact.
In particular, $\cF^i$ is flabby for every integer $i\in \Z$.
Owing to Lemma <ref> we see that $a_{dNis}^*\cF^i$ is flabby.
It follows that

\[
\bH_{dNis}^i(X,a_{dNis}^*\cF)\cong H^i(a_{dNis}^*\cF(X)).
\]

Due to Lemma <ref> there is an isomorphism

\[
a_{dNis}^*\cF(X)\cong \colimit_{Y\in X_{div}}\cF(Y).
\]

Since this colimit is filtered by Proposition <ref> and every filtered colimit in the category of $\Lambda$-modules is exact,
we deduce that

\[
\bH_{dNis}^i(X,a_{dNis}^*\cF)\cong \colimit_{Y\in X_{div}}H^i(\cF(Y)).
\]

To conclude the proof, we observe that

\[
\bH_{sNis}^i(Y,\cF)
\cong
\]

Suppose $\cF$ is a torsion sheaf on the small log étale site $X_{l\acute{e}t}$.
Then for every fs log scheme $X$ with an fs chart,
the isomorphism

\[
H_{l\acute{e}t}^i(X,\cF)\cong H_{k\acute{e}t}^i(X,\cF)
\]

is proven in <cit.>.

The category $lSm/k$ has finite $sNis$ and $dNis$-cohomological dimension for $\Lambda$-linear coefficients.

For every $X\in lSm/k$ and $\cF\in \Shv_{sNis}^{\rm log}(k,\Lambda)$,
if $i>\dim X$, then due to <cit.> we have the vanishing

\[
\]

If $Y$ is a log modification over $X$, then $\dim Y=\dim X$ by Lemma <ref>.
Thus we have the vanishing

\[
\]

Theorem <ref> finishes the proof.

Let $X$ be an fs log scheme.
Suppose $\cF$ is a sheaf of $\Q$-modules on the small strict étale site $X_{s\acute{e}t}$.
Then, for every integer $i\geq 0$, there is an isomorphism of cohomology groups

\[
H_{sNis}^i(X,\cF)\cong H_{s\acute{e}t}^i(X,\cF).
\]

Equivalently, we need to show there is an isomorphism

\[
H_{Nis}^i(\underline{X},\cF)\cong H_{\acute{e}t}^i(\underline{X},\cF).
\]

This is the content of <cit.>.

Let $X$ be an fs log scheme.
Suppose $\cF$ is a sheaf of $\Q$-modules on the small Kummer étale site $X_{k\acute{e}t}$.
Then, for every integer $i\geq 0$,  there is an isomorphism of cohomology groups

\[
H_{s\acute{e}t}^i(X,\cF)\cong H_{k\acute{e}t}^i(X,\cF).
\]

Let $\epsilon_*\colon \Shv(X_{k\acute{e}t})\rightarrow \Shv(X_{s\acute{e}t})$ be the functor sending $\cF$ to $\cF\vert_{X_{s\acute{e}t}}$.
We need to show that $R^i\epsilon_*\cF=0$ for every integer $i>0$.
Since $R^i\epsilon_*\cF$ is the strict étale sheaf associated with the presheaf

\[
(Y\in X_{s\acute{e}t})\mapsto H_{k\acute{e}t}^i(Y,\cF),
\]

it suffices to show the vanishing

for every $i>0$ whenever $\underline{X}$ is strictly local.

According to <cit.>,
see also <cit.>,
the group $H_{k\acute{e}t}^i(X,\cF)$ is a torsion abelian group by <cit.>.
Since $\cF$ is a sheaf of $\Q$-modules, this implies the desired vanishing.

Let $t$ be one of the following topologies on $lSm/k$:

\[
s\acute{e}t,\; d\acute{e}t,\; k\acute{e}t,\text{ and }l\acute{e}t.
\]

If $k$ has finite $\acute{e}t$-cohomological dimension for $\Lambda$-linear coefficients, then $lSm/k$ has finite $t$-cohomological dimension for $\Lambda$-linear coeffieicnts.

For any $\cF\in \Shv_t^{\rm log}(k,\Lambda)$, the kernel and cokernel of the natural morphism

\[
\cF\rightarrow \cF\otimes \Q
\]

are torsion sheaves.
Hence we only need to deal with the two separate cases: $\Lambda$ is a torsion ring, and $\Q\subset \Lambda$.

If $\Lambda$ is a torsion ring, then $lSm/k$ has finite $s\acute{e}t$ and $k\acute{e}t$-cohomological dimension for $\Lambda$-linear coefficients due to <cit.> and <cit.>.
To show that $lSm/k$ has finite $d\acute{e}t$ and $l\acute{e}t$-cohomological dimension for $\Lambda$-linear coefficients, apply Lemma <ref> and Theorem <ref>.

If $\Q\subset \Lambda$, then Lemmas <ref> and <ref> show that $lSm/k$ has finite $s\acute{e}t$ and $k\acute{e}t$-cohomological dimension for $\Lambda$-linear coefficients since $lSm/k$ has finite $sNis$-cohomological dimension due to Corollary <ref>.
Then apply Lemma <ref> and Theorem <ref> again to finish the proof.

Let $\cF$ be a bounded below complex of strict Nisnevich (resp. strict étale, resp. Kummer étale) sheaves on $SmlSm/k$.
Then for every $X\in SmlSm/k$ and integer $i\in \Z$ there is an isomorphism

\begin{align*}
\bH_{dNis}^i(X,a_{dNis}^*\cF)&\cong \colimit_{Y\in X_{div}^{Sm}}\bH_{sNis}^i(Y,\cF)
\\
\text{(resp.\ }\bH_{d\acute{e}t}^i(X,a_{d\acute{e}t}^*\cF)&\cong \colimit_{Y\in X_{div}^{Sm}}\bH_{s\acute{e}t}^i(Y,\cF),
\\
\text{resp.\ }\bH_{l\acute{e}t}^i(X,a_{l\acute{e}t}^*\cF)&\cong \colimit_{Y\in X_{div}^{Sm}}\bH_{k\acute{e}t}^i(Y,\cF)\text{)}.
\end{align*}

The same holds if we replace $X_{div}^{Sm}$ by $X_{divsc}^{Sm}$.

The proof is parallel to that of Theorem <ref> if we use Lemma <ref> and Proposition <ref> as replacements for Lemma <ref> and Proposition <ref>.
To show the claim for $X_{divsc}^{Sm}$ use Corollary <ref> and Lemma <ref> instead.

§.§ Effective log motives

Voevodsky's category of effective motives $\dmeff$ (resp. effective motives without transfers $\daeff$) is the homotopy category with respect to the $\A^1$-local descent model structure on
$\Co(\Shvtrkl)$ (resp. $\Co(\Shv_{Nis}(k,\Lambda))$).
The motive $M(X)$ of $X\in Sm/k$ is defined as the object $\Ztr(X)$ (resp. $\Lambda(X)$) in $\dmeff$ (resp. $\daeff$).
The category $\dmeff$ was initially constructed for bounded above complexes, see e.g., <cit.>.
We refer to <cit.> for a generalization to unbounded complexes.

By abuse of notation, we let $\boxx$ be shorthand for the class of projections $X\times \boxx\rightarrow X$,
where $X\in lSm/k$.
We also consider $\boxx$ as the corresponding classes of morphisms of representable objects in $\Shvlogkl$ or $\Shvltrkl$.

The derived category of effective log motives

\[
\ldmeff
\]

[notation]ldmeff @ $\ldmeff$
is the homotopy category of $\Co(\Shvltrkl)$ with respect to the $\boxx$-local descent model structure.
The motive of $X\in lSm/k$ is the object

\[
\]
[notation]M(X) @ $M(X)$

by abuse of notation,
we write $\ldaeff$ [notation]ldaeff @ $\ldaeff$  for the homotopy category of $\Co(\Shvlogkl)$ with respect to the $\boxx$-local descent model structure and $M(X)$ for $a_{dNis}^*\Lambda(X)$.

Recall that we set $\boxx:=(\P^1,\infty)$.
By Proposition <ref>, $\ldmeff$ and $\ldaeff$ are symmetric monoidal triangulated categories such that for $X,Y\in lSm/k$, we have

\[
M(X)\otimes M(Y)=M(X\times Y).
\]

For any dividing Nisnevich cover $Y\rightarrow X$ in $lSm/k$, we note that

\[
Y-\partial Y\rightarrow X-\partial X
\]

is a Nisnevich cover in $Sm/k$.
we have that [notation]omegasharpstar @ $\omega_\sharp, R\omega^*$

\omega(\boxx)=\A^1.

Thus by (<ref>) and (<ref>), there are naturally induced adjunctions

\[
L\gamma_\sharp:\ldaeff\rightleftarrows \ldmeff:R\gamma^*,
\]

\begin{equation}
\label{eq::omegasharp}
\omega_\sharp:\ldmeff\rightleftarrows \dmeff:R\omega^*.
\end{equation}

Similarly, we have the naturally induced adjunctions

\[
L\gamma_\sharp:\daeff\rightleftarrows \dmeff:R\gamma^*,
\]

\[
\omega_\sharp:\ldaeff\rightleftarrows \daeff:R\omega^*.
\]

Further, there is a diagram of adjunct functors

\[
\begin{tikzcd}
\ldaeff\arrow[d,shift left=0.75ex,leftarrow,"R\omega^*"]\arrow[d,shift right=0.75ex,"\omega_\sharp"']
\arrow[r,shift left=0.75ex,"L\gamma_\sharp"]\arrow[r,shift right=0.75ex,"R\gamma^*"',leftarrow]&\ldmeff\arrow[d,shift left=0.75ex,leftarrow,"R\omega^*"]\arrow[d,shift right=0.75ex,"\omega_\sharp"']   \\
\daeff\arrow[r,shift left=0.75ex,"L\gamma_\sharp"]\arrow[r,shift right=0.75ex,"R\gamma^*"',leftarrow] &\dmeff,
\end{tikzcd}\]

where $L\gamma_\sharp$ commutes with $\omega_\sharp$.

A complex of dividing Nisnevich sheaves $\cF$ is strictly $\boxx$-invariantinvariant!strictly box @ strictly $\boxx$ if for every fs log scheme $X$ log smooth over $k$ and integer $i\in \Z$,
the projection $X\times \boxx\rightarrow X$ induces an isomorphism on sheaf cohomology groups

\[
\bH_{dNis}^i(X,\cF)\rightarrow \bH_{dNis}^i(X\times \boxx,\cF).
\]

By definition of $\boxx$-local model structure, a complex of dividing Nisnevich sheaves $\cF$ is fibrant for the $\boxx$-local descent model structure if and only if $\cF$ is fibrant for the local model structure and $\cF$ is strictly $\boxx$-invariant.

Let $\cF$ be a strictly $\boxx$-invariant complex of dividing Nisnevich sheaves with log transfers.
for every fs log scheme $X$ log smooth over $k$ and an integer
$i\in \Z$, there is an isomorphism

\[
\hom_{\ldmeff}(M(X),\cF[i])
\cong
\bH_{dNis}^i(X,\cF).
\]

Similar statement holds for strictly $\boxx$-invariant complex of dividing Nisnevich sheaves $\cF$ and $\ldaeff$.

We focus on $\ldmeff$ since the proofs are similar.
We may assume that $\cF$ is a fibrant object in $\Co(\Shvltrkl)$ for the descent model structure given in <ref>.
By Proposition <ref>(1), $\cF$ is a fibrant object in $\Co(\Shvltrkl)$ with respect to the $\boxx$-local descent model structure.
Thus we have

\[
\hom_{\ldmeff}(M(X),\cF[i])
\cong
\hom_{\Deri(\Shvltrkl)}(a_{dNis}^*\Zltr(X),\cF[i]).
\]

Proposition <ref> finishes the proof.

Let $\{\cF_i\}_{i\in I}$ be a set of strictly $\boxx$-invariant complexes of dividing Nisnevich sheaves.
Then $\bigoplus_{i\in I}\cF_i$ is strictly $\boxx$-invariant too.
The same holds for strictly $\boxx$-invariant complexes of dividing étale or log étale sheaves if $k$ has finite étale cohomological dimension for $\Lambda$-linear coefficients.

We first treat the dividing Nisnevich case.
Due to Corollary <ref> and Proposition <ref> for every $X\in lSm/k$ we have isomorphisms

\begin{gather*}
\bH_{dNis}^i(X,\bigoplus_{i\in I}\cF)
\cong
\bigoplus_{i\in I}\bH_{dNis}^i(X,\cF),
\\
\bH_{dNis}^i(X\times \boxx,\bigoplus_{i\in I}\cF)
\cong
\bigoplus_{i\in I}\bH_{dNis}^i(X\times \boxx,\cF).
\end{gather*}

Combining these isomorphisms finishes the proof.
For the dividing étale and log étale cases, apply Corollary <ref> instead.

The effective log motives $M(X)[n]$ indexed by $X\in lSm/k$ and $n\in \Z$ form a set of compact generators for the categories $\ldaeff$ and $\ldmeff$.

We focus on $\ldmeff$ since the proofs are similar. compact generators!motives
Let $\{\cF_i\}_{i\in I}$ be a set of strictly $\boxx$-invariant dividing Nisnevich complexes of sheaves with log transfers.
Then $\bigoplus_{i\in I} \cF_i$ is strictly $\boxx$-invariant too by Lemma <ref>.
Owing to Proposition <ref> we have

\begin{gather*}
\hom_{\ldmeff}(M(X),\bigoplus_{i\in I}\cF_i[n])
\cong
\hom_{\mathbf{D}(\Shvlogkl)}(a_{dNis}^*\Lambda(X),\bigoplus_{i\in I}\cF_i[n]),
\\
\bigoplus_{i\in I}\hom_{\ldmeff}(M(X),\cF_i[n])
\cong
\bigoplus_{i\in I}\hom_{\mathbf{D}(\Shvlogkl)}(a_{dNis}^*\Lambda(X),\cF_i[n]).
\end{gather*}

Proposition <ref> and Corollary <ref> finish the proof.

Similarly, the objects $M(X)[n]$, where $X\in Sm/k$ and $n\in \Z$, are compact in $\dmeff$ (resp. $\daeff$).
The same objects generate $\dmeff$ (resp. $\daeff$) by the above or <cit.> (resp. <cit.>).

If $f:Y\rightarrow X$ is a morphism in $lSm/k$,

M(Y\stackrel{f}\rightarrow X)

be the cone in $\ldmeff$ associated with the complex

a_{dNis}^*\Zltr(Y)\rightarrow a_{dNis}^*\Zltr(X)

in $\Co(\Shvltrkl)$.
We use the same notation for the object in $\ldaeff$ associated with the complex $a_{dNis}^*\Lambda(Y)\rightarrow a_{dNis}^*\Lambda(X)$ in $\Co(\Shvlogkl)$.

For a simplicial object $\mathscr{X}$ in $lSm/k$, let $M(\mathscr{X})$ denote the object in $\ldmeff$ associated with $\Zltr(\mathscr{X})$ in $\Co(\Shvltrkl)$
We use the same notation for the object in $\ldaeff$ associated with $\Lambda(\mathscr{X})$ in $\Co(\Shvlogkl)$.

We denote by

\[
\ldmeffprop
\]
[notation]ldmeffprop @ $\ldmeffprop$
the smallest triangulated subcategory of $\ldmeff$ that is closed under small sums and shifts,
and contains $M(X)$,
where $X\in lSm/k$ and its underlying scheme $\underline{X}$ is proper over $\Spec{k}$.

We are interested in $\ldmeffprop$ because for a proper log smooth $X$,
the log motive $M(X)$ is practically an object in $\dmeff$.
Indeed, if $k$ admits resolution of singularities,
Theorem <ref> shows there is an equivalence of triangulated categories

\[
\ldmeffprop\simeq \dmeff.
\]

we note that the corresponding statement is false in the étale topology; see Remark <ref>.

(1) For every strict Nisnevich distinguished square

\[
\begin{tikzcd}
\end{tikzcd}
\]

in $lSm/k$, there is a naturally induced isomorphism

\[
M(Y'\rightarrow X')\rightarrow M(Y\rightarrow X)
\]

in $\ldmeff$.
The same holds in $\ldaeff$.

For every log modification $f:Y\rightarrow X$, there is a naturally induced isomorphism

\[
M(Y)\rightarrow M(X)
\]

in $\ldmeff$.
The same holds true in $\ldaeff$.

Let us treat first the case of $\ldaeff$.
To show (1), it suffices to show the homomorphism

\[
\hom_{\ldaeff}(M(Y\rightarrow X),\cF)\rightarrow \hom_{\ldaeff}(M(Y'\rightarrow X'),\cF)
\]

is an isomorphism for any strictly $\boxx$-invariant complex of dividing Nisnevich sheaf $\cF$.
For this it suffices to show there is an isomorphism

\begin{equation}
\label{A.5.42.1}\hom_{\Deri(\Shvlogtkl)}(a_{dNis}^*\Lambda(Y\rightarrow X),\cF)
\rightarrow
\hom_{\Deri(\Shvlogtkl)}(a_{dNis}^*\Lambda(Y'\rightarrow X'),\cF).
\end{equation}

This follows from the exactness of (<ref>).
Part (2) follows similarly, using a dividing distinguished square instead of a strict Nisnevich distinguished square.

To show (1) in the case of $\ldmeff$, it suffices to show the homomorphism

\[
\hom_{\ldaeff}(M(Y\rightarrow X),\cF)
\rightarrow
\hom_{\ldaeff}(M(Y'\rightarrow X'),\cF)
\]

is an isomorphism for any strictly $\boxx$-invariant complex of dividing Nisnevich sheaf $\cF$ with log transfers.
This follows from Proposition <ref> since (<ref>) is an isomorphism.
We omit the proof of (2) because it is similar to (1).

The commutative square

\begin{equation}
\label{A.5.43.1}
\begin{tikzcd}
\A_\N\arrow[r]\arrow[d]&(\P^1,0)\cong \boxx\arrow[d]\\
\A^1\arrow[r]&\P^1
\end{tikzcd}
\end{equation}

is a strict Nisnevich distinguished square, so that the induced morphism

\[
M(\A_\N\rightarrow \A^1)\rightarrow M(\boxx\rightarrow \P^1)\cong \Lambda(1)[2]
\]

is an isomorphism in $\ldmeff$.
This is analogous to the isomorphism

\[
M(\A^1/\mathbb{G}_m)\rightarrow M(\P^1/\A^1)\cong \Lambda(1)[2]
\]

in $\dmeff$ and $\daeff$.
Indeed, applying $\omega$ to the square (<ref>) we retain the Nisnevich distinguished square

\[
\begin{tikzcd}
\mathbb{G}_m\arrow[d]\arrow[r]&\P^1-\{0\}\arrow[d]
\\
\A^1\arrow[r]&\P^1
\end{tikzcd}
\]

for the standard covering of the projective line.
However, in the log situation,
we note that $\Zltr(\A_\N)$ is not a subsheaf of $\Zltr(\A^1)$.

Recall that $div$ is the class of morphisms in $lSm/k$ consisting of log modifications $Y\rightarrow X$ in $lSm/k$.
We denote by $(\boxx,div)$ the class of morphisms in $lSm/k$ consisting of all the projections $X\times \boxx\rightarrow X$, where $X\in lSm/k$, and the log modifications $Y\rightarrow X$ in $lSm/k$.

A useful property of $\A^1$-invariant presheaf $\cF$ with transfers is that for any Zariski cover $U_1$ and $U_2$ of an open subscheme $U$ of $\A^1$,
there is a split exact sequence

\begin{equation}
\label{A.4.28.1}
0\rightarrow \cF(U)\rightarrow \cF(U_1)\oplus \cF(U_2)\rightarrow \cF(U_1\cap U_2)\rightarrow 0,
\end{equation}

see <cit.>.
This is no longer true for $\boxx$-invariant presheaves with log transfers.
In what follows, we check that $\Zltr(\A^1)$ provides a counterexample.

For $X\in lSm/k$, we have that

\[
\Zltr(\A^1)(X)=\Cor(\underline{X},\A^1).
\]

Let $V$ be an elementary correspondence from $\underline{X}\times \P^1$ to $\A^1$.
For any point $x$ of $\underline{X}$, consider the restriction $V_x$ of $V$ to $\{x\}\times \P^1$.
Since $V_x$ is finite over $\{x\}\times \P^1$, $V_x$ is proper over $k$.
Thus the image of $V'$ in $\A^1$ is again proper, so the image should be a union of points.
Let $y$ be a rational point of $\P^1$,
let $V'$ (resp. $V_x'$) be the restriction of $V$ to $\underline{X}\times \{y\}$ (resp. $\{x\}\times \{y\}$) and let
$p:\underline{X}\times \P^1\rightarrow \underline{X}$ and $p_x:\{x\}\times \P^1\rightarrow \{x\}$ be the projections.
From the above we have $p_x^*V_x'=V_x$ and hence $p^*V'=V$.
Thus the morphism

\[
p^*:\Zltr(\A^1)(X)\rightarrow \Zltr(\A^1)(X\times \P^1)
\]

is surjective.
Since $p$ has a section, it follows that $p^*$ is injective.
So far we have observed that $\Zltr(\A^1)$ is $\boxx$-invariant.

Next, we turn to the split exactness of (<ref>).
Set $U:=\A^1=\Spec{k[x]}$, $U_1:=\A^1-\{0\}$, and $U_2:=\A^1-\{1\}$.
Then there is an element

\[
[1/x+1/(x-1)]\in \Zltr(\A^1)(U_1\cap U_2),
\]

where for $f:U_1\cap U_2\rightarrow \A^1$, we let $[f]$ denotes the associated correspondence.
On the other hand,
the closure of $[1/x+1/(x-1)]$ in $U_j\times \A^1$ is not surjective over $U_j$, i.e., it does not define an elementary log correspondence in $\lCor(U_j,\A^1)$ for $j=1,2$.
It follows that the homomorphism

\[
\Zltr(U_1)\oplus \Zltr(U_2)\rightarrow \Zltr(U_1\cap U_2)
\]

is not even surjective.

Let $\cF$ be an $\A^1$-invariant Nisnevich sheaf $\cF$ with transfers.
The spilt exactness of (<ref>) is used to prove that also the presheaf $H_{Nis}^n(-,\cF)$ is $\A^1$-invariant,
see e.g., <cit.>.
Due to Example <ref>,
it is unclear whether for any $\boxx$-invariant strict Nisnevich sheaf $\cF$ with transfers, the presheaf $H_{sNis}^n(-,\cF)$ is $\boxx$-invariant.

§.§ Effective étale log motives

In this subsection we define the étale versions of $\ldmeff$ and $\ldaeff$.

The derived category of effective dividing étale log motives

\[
\ldmeffet
\]
[notation]ldmeffet @ $\ldmeffet$
is the homotopy category of

\[
\Co(\Shv_{d\acute{e}t}^{\rm ltr}(k,\Lambda))
\]

with respect to the $\boxx$-local descent model structure.
The motive of $X\in lSm/k$ is the object

\[
\]

we write $\ldaeffet$ [notation]ldaeffet @ $\ldaeffet$ for the homotopy category of $\Co(\Shv_{d\acute{e}t}^{\rm log}(k,\Lambda))$ with respect to the $\boxx$-local descent model structure and $M(X)$ for $a_{d\acute{e}t}^*\Lambda(X)$.

The derived category of effective log étale log motives

\[
\ldmefflet
\]
[notation]ldmefflet @ $\ldmefflet$
is the homotopy category of $\Co(\Shv_{l\acute{e}t}^{\rm ltr}(k,\Lambda))$ with respect to the $\boxx$-local descent model structure.
The effective log étale motive of $X\in lSm/k$ is the object

\[
\]

we write $\ldaefflet$ [notation]ldaefflet @ $\ldaefflet$ for the homotopy category of $\Co(\Shv_{l\acute{e}t}^{\rm log}(k,\Lambda))$ with respect to the $\boxx$-local descent model structure and $M(X)$ for $a_{l\acute{e}t}^*\Lambda(X)$.

By Proposition <ref>, $\ldmeffet$, $\ldaeffet$, $\ldmefflet$, and $\ldaefflet$ are symmetric monoidal triangulated categories such that for all $X,Y\in lSm/k$,

\[
M(X)\otimes M(Y)=M(X\times Y).
\]

For the descent model structures,
there are Quillen adjunctions

\[
a_{d\acute{e}t}^*:\Co(\Shv_{dNis}^{\rm ltr}(k,\Lambda))\rightleftarrows \Co(\Shv_{d\acute{e}t}^{\rm ltr}(k,\Lambda)):a_{d\acute{e}t*}
\]

\[
a_{l\acute{e}t}^*:\Co(\Shv_{d\acute{e}t}^{\rm ltr}(k,\Lambda))\rightleftarrows \Co(\Shv_{l\acute{e}t}^{\rm ltr}(k,\Lambda)):a_{l\acute{e}t*}.
\]

Owing to <cit.>, this is again a Quillen adjunction for the $\boxx$-local model structures.
By passing to the homotopy categories, we obtain the adjunctions

\[
La_{d\acute{e}t}^*:\ldmeff\rightleftarrows \ldmeffet:Ra_{d\acute{e}t*}
\]

\[
La_{l\acute{e}t}^*:\ldmeffet\rightleftarrows \ldmefflet:Ra_{l\acute{e}t*}.
\]

Let $Y\rightarrow X$ be a morphism in $lSm/k$, and let $\mathscr{X}$ be a simplicial object of $lSm/k$.
As in Definition <ref>, we associate objects

M(Y\stackrel{f}\rightarrow X)
\text{ and }

in $\ldmeffet$, $\ldaeffet$, $\ldmefflet$, and $\ldaefflet$.

A complex of dividing étale sheaves $\cF$ is strictly $\boxx$-invariantinvariant!strictly box @ strictly $\boxx$ if for every fs log scheme $X$ log smooth over $k$ and integer $i\in \Z$,
the projection $X\times \boxx\rightarrow X$ induces an isomorphism on sheaf cohomology groups

\[
\bH_{d\acute{e}t}^i(X,\cF)\rightarrow \bH_{d\acute{e}t}^i(X\times \boxx,\cF).
\]

Similarly, a complex of log étale sheaves $\cF$ is strictly $\boxx$-invariant if for every fs log scheme $X$ log smooth over $k$ and integer $i\in \Z$,
the projection $X\times \boxx\rightarrow X$ induces an isomorphism on sheaf cohomology groups

\[
\bH_{l\acute{e}t}^i(X,\cF)\rightarrow \bH_{l\acute{e}t}^i(X\times \boxx,\cF).
\]

Let $\cF$ be a strictly $\boxx$-invariant complex of dividing étale (resp. log étale) sheaves with log transfers.
for every fs log scheme $X$ log smooth over $k$ and an integer
$i\in \Z$, there is an isomorphism

\[
\hom_{\ldmeffet}(M(X),\cF)
\cong
\bH_{d\acute{e}t}^i(X,\cF)
\]

\[
\text{(resp.\ }
\hom_{\ldmefflet}(M(X),\cF)
\cong
\bH_{l\acute{e}t}^i(X,\cF)
\text{)}.
\]

The proof is parallel to that of Proposition <ref>.

We denote by

\[
\ldmeffetprop
\;
\text{(resp.\ }
\ldmeffletprop
\text{)}
\]
[notation]ldmeffetprop @ $\ldmeffetprop$ [notation]ldmeffletprop @ $\ldmeffletprop$
the smallest triangulated subcategory of $\ldmeffet$ (resp. $\ldmefflet$) that is closed under small sums and shifts,
and contains $M(X)$, where $X\in lSm/k$ and its underlying scheme is proper over $\Spec{k}$.

Suppose $k$ has positive characteristic $p>0$.
In contrast to Theorem <ref>, we do not have an equivalence of triangulated categories between

\[
\ldmeffetpropZ
\text{ and }
\dmeffetZ.
\]

To wit, note that $\Z/p\cong 0$ in $\dmeffetZ$ by <cit.>.
Notice that since $\Z\cong M(k)\in \ldmeffetpropZ$, we have $\Z/p\in \ldmeffetpropZ$.
In Example <ref> we will see that $\Z/p$ is nontrivial in $\ldmeffetpropZ$.

Suppose that $k$ has finite $\acute{e}t$-cohomological dimension for $\Lambda$-linear coefficients.
Then the effective log motives $M(X)[n]$ indexed by $X\in lSm/k$ and $n\in \Z$ form a set of compact generators for the categories

\[
\ldaeffet,\;\ldmeffet,\; \ldaefflet,\text{ and }\ldmefflet.
\]

Argue as in Proposition <ref> by appealing to Corollary <ref>.

§.§ Construction of motives using $SmlSm/k$

We may use the results of Section <ref> to construct categories of effective log motives for objects in $SmlSm/k$.

\[
\ldmeffSmlSm
\]
[notation]ldmeffSmlSm @ $\ldmeffSmlSm$
be the homotopy category of

\[
\Co(\Shvltrklsm)
\]

with respect to the $\boxx$-local descent model structure.

Similarly, we let

\[
\ldmeffetSmlSm
\]
[notation]ldmeffetSmlSm @ $\ldmeffetSmlSm$
be the homotopy category of

\[
\Co(\Shv_{d\acute{e}t}^{\rm ltr}(SmlSm/k,\Lambda))
\]

with respect to the $\boxx$-local descent model structure.

There are naturally induced equivalences of triangulated categories

\begin{align*}
\ldmeffSmlSm & \simeq \ldmeff, \\
\ldmeffetSmlSm & \simeq \ldmeffet, \\
\ldmeffletSmlSm & \simeq \ldmefflet.
\end{align*}

We will only consider the dividing Nisnevich topology since the proofs are similar.
Owing to Proposition <ref> there is an equivalence of categories

\[
\Shvltrklsm
\simeq
\Shvltrkl.
\]

Let us consider the descent model structures on the categories

\[
\Co(\Shvltrklsm)
\]

\[
\Co(\Shvltrkl).
\]

Since the descent model structures are invariant under equivalences of categories,
there is a Quillen equivalence for the descent model structures

\[
\Co(\Shvltrklsm)\rightleftarrows \Co(\Shvltrkl).
\]

Owing to <cit.> this becomes a Quillen equivalence if we impose the $\boxx$-local descent model structure.

Suppose $\cF$ is a complex of dividing Nisnevich sheaves with log transfers on $SmlSm/k$.
Recall that $X_{div}^{Sm}$ denotes the category of log modifications $Y\rightarrow X$,
where $Y\in SmlSm/k$.
Owing to Lemma <ref>, we can naturally extend $\cF$ to a complex $i_\sharp \cF$ on $lSm/k$ via the formula

\[
\iota_\sharp \cF(X)
\colimit_{Y\in X_{div}^{Sm}}\cF(Y).
\]

By Lemma <ref> there is an equivalence of categories

\[
\Shv_{dNis}(SmlSm/k,\Lambda)\simeq \Shvlogkl.
\]

This implies that for every $X\in SmlSm/k$ and integer $i\in \Z$ we have an isomorphism of cohomology groups

\begin{equation}
\label{SmlSm.3.1}
H_{dNis}^i(X,\cF)\cong H_{dNis}^i(X,\iota_\sharp \cF).
\end{equation}

If $\cF$ is strict $\boxx$-invariant,
for every $X\in SmlSm/k$

\[
H_{dNis}^i(X,\cF)\cong H_{dNis}^i(X\times \boxx,\cF),
\]

(<ref>) shows $\iota_\sharp \cF$ is strictly $\boxx$-invariant too.
Owing to Proposition <ref>, for every $X\in SmlSm/k$ and integer $i\in \Z$, we obtain an isomorphism

\begin{equation}
\label{SmlSm.3.2}
\hom_{\ldmeff}(M(X),\iota_\sharp \cF[i])\cong H_{dNis}^i(X,\cF).
\end{equation}

If $\cF$ is a complex of dividing étale (resp. log étale) sheaves with log transfers on $SmlSm/k$, then we can argue similarly to conclude the isomorphisms

\begin{equation}
\label{SmlSm.3.3}
\hom_{\ldmeffet}(M(X),\iota_\sharp \cF[i]) \cong H_{d\acute{e}t}^i(X,\cF)
\end{equation}

\begin{equation}
\label{SmlSm.3.4}
\hom_{\ldmefflet}(M(X),\iota_\sharp \cF[i])\cong H_{l\acute{e}t}^i(X,\cF).
\end{equation}

§ CALCULUS OF FRACTIONS AND HOMOTOPY THEORY OF PRESHEAVES

Throughout this section, $T$ is a small category with finite limits.
Let $\pt$ denote the final object in $T$.
Suppose $\mathcal{A}$ is a set of morphisms in $T$ that admits a calculus of right fractions.
We let $I$ be an object of $T$ admitting morphisms

\[
i_0,i_1:{\rm pt}\rightarrow I,\;\; p:I\rightarrow {\rm pt}
\]

in $T$ and a morphism

\[
\mu:I\times I\rightarrow I
\]

in $T[\mathcal{A}^{-1}]$ satisfying

\[
\mu(i_0\times {\rm id})=\mu({\rm id}\times i_0)=i_0p,
\]

\[
\mu(i_1\times {\rm id})=\mu({\rm id}\times i_1)={\rm id}.
\]

we assume

\[
i_0\amalg i_1:{\rm pt}\amalg {\rm pt}\rightarrow I
\]

is a monomorphism.
Note that $I$ is an interval of $T[\mathcal{A}^{-1}]$ in the sense of <cit.>.

Recall that $\mathcal{A}$ admits a calculus of right fractionscalculus of right fractions if the following conditions holds <cit.>.

(i) The set $\mathcal{A}$ contains the isomorphisms and is closed under composition.

(ii) (Right Ore condition)right Ore condition For any diagram $Y\stackrel{f}\rightarrow X\stackrel{g}\leftarrow X'$ in $\cC$ with $f\in \cA$, there is a commutative diagram

\[
\begin{tikzcd}
\end{tikzcd}
\]

in $\cC$ such that $f'\in \cA$.

(iii) (Right cancellability condition)right cancellability condition For any diagram

\[
\begin{tikzcd}
Z\arrow[r,shift left=0.5ex,"f"]\arrow[r,shift right=0.5ex,"g"']&Y\arrow[r,"u"]&X
\end{tikzcd}
\]

in $\cC$ with $u\in \cA$ and $u\circ f=u\circ g$, there is a morphism $v:W\rightarrow Z$ in $\cA$ such that $f\circ v=g\circ v$.

In <cit.>, Morel-Voevodsky constructs the singular functor ${\rm Sing}$ for an interval $I$ of $T$.
Since our assumption is that $I$ is an interval of $T[\cA^{-1}]$,
instead of $T$,
we cannot apply the same arguments verbatim.
For example,
in the proof of <cit.> it is essential that $\mu\in T$ when dealing with the multiplication map $\mu:I\times I\rightarrow I$.
In this section,
we construct a singular functor ${\rm Sing}$ for $\mu$ in $T[\cA^{-1}]$.
We argue that there is another way of constructing ${\rm Sing}$ in [6] without assuming the existence of $\mu$.
The style of this construction is different from that in <cit.>.
See <cit.> for a way of constructing the Sing functor obtained by
forming a weak interval object <cit.> in the category of simplicial presheaves on $T$.

§.§ Properties of localization functors

Let $\sPsh(T)$ denote the category of simplicial presheaves on $T$.
A morphism $f:F\rightarrow G$ of simplicial presheaves is called

* a pointwise weak equivalenceequivalence!pointwise weak if $F(X)\rightarrow G(X)$ is a weak equivalence of simplicial sets for any object $X$ of $T$.

* an injective cofibration if $F(X)\rightarrow G(X)$ is a cofibration of simplicial sets for any object $X$ of $T$.

* an injective fibration if it has the right lifting property with respect to injective cofibrations that are pointwise weak equivalences.

* a projective fibration if $F(X)\rightarrow G(X)$ is a fibration of simplicial sets for any object $X$ of $T$.

* a projective cofibration if it has the left lifting property with respect to projective fibrations that are pointwise weak equivalences.

Recall that the model structure on $\sPsh(T)$ with the pointwise weak equivalences, injective cofibrations, and injective fibrations is called the injective model structuremodel structure!injective (see <cit.>).
The model structure on $\sPsh(T)$ with the pointwise weak equivalences, projective cofibrations, and projective fibrations is called the projective model structuremodel structure!projective (see <cit.>).

Associated to the localization functor

\[
v:T\rightarrow T[\mathcal{A}^{-1}]
\]

we let

\[
v^*:\sPsh(T[\mathcal{A}^{-1}])\rightarrow \sPsh(T)
\]

denote the functor given by

\[
\]

Here $F$ is a simplicial presheaf on $T[\mathcal{A}^{-1}]$ and $X$ is an object of $T$.

By <cit.>, there exist adjoint functors

\[
\begin{tikzcd}
\sPsh(T)\arrow[rr,shift left=1.5ex,"v_\sharp "]\arrow[rr,"v^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"v_*"']&&\sPsh(T[\mathcal{A}^{-1}]),
\end{tikzcd}
\]

where $v_\sharp$ is left adjoint to $v^*$, and $v^*$ is left adjoint to $v_*$.

For any object $X$ of $T$, we denote by $\cA\downarrow X$ the category where an object is a morphism $Y\rightarrow X$ in $\cA$, and a morphism is a commutative diagram

\[
\begin{tikzcd}
\end{tikzcd}
\]

in $T$ such that $g\in \cA$.

Since $\mathcal{A}$ admits a calculus of right fractions,
by the dual of <cit.>,
we have the formula

\begin{equation}
\label{A.1.3.1}
\hom_{T[\mathcal{A}^{-1}]}(X,S)
\cong
\colimit_{Y\in \mathcal{A}\downarrow X}\hom_T(Y,S).
\end{equation}

The functor $v_\sharp $ is given by

\begin{equation}
\label{A.1.4.1}
v_\sharp F(X)\cong \colimit_{Y\in \mathcal{A}\downarrow X} F(Y).
\end{equation}

By <cit.>, we reduce to the case when $F=Z\times \Delta[n]$, where $Z\in T$.
In this case (<ref>) follows from (<ref>).

The adjunction

\[
v_\sharp :\sPsh(T)\rightleftarrows \sPsh(T[\mathcal{A}^{-1}]):v^*
\]

is a Quillen pair with respect to the projective model structures.

The functor $v:T\rightarrow T[\mathcal{A}^{-1}]$ commutes with finite limits by using the dual of <cit.>.
Thus $v$ is a morphism of sites for the trivial topologies.
We are done by <cit.>.

The functor

\[
v^*:\sPsh(T[\mathcal{A}^{-1}])\rightarrow \sPsh(T)
\]

preserves pointwise weak equivalences and injective cofibrations. In particular, the adjunction

\[
v^*:\sPsh(T[\mathcal{A}^{-1}])\rightleftarrows \sPsh(T):v_*
\]

is a Quillen pair with respect to the injective model structures.

Pointwise weak equivalences and injective cofibrations are defined objectwise, so the claim follows from the fact that $v:T\rightarrow T[\mathcal{A}^{-1}]$ is essentially surjective (<cit.>).

For simplicial presheaves $F$ and $G$ on $T$, let ${\mathbf S}(F,G)$ denote the simplicial enrichment (or internal hom), which is given by the simplicial set

\[
{\mathbf S}(F,G)_n:=\hom_{\sPsh(T)}(F\times \Delta[n],G).
\]

Let $\mathcal{A}$ be a set of morphisms in $T$.

* A simplicial presheaf $F$ on $T$ is $\mathcal{A}$-invariantinvariant!A@$\cA$ if the morphism

\[
F(X\times Z)\rightarrow F(Y\times Z)
\]

is a pointwise weak equivalence for any morphism $Y\rightarrow X$ in $\mathcal{A}$ and any $Z\in T$.

* A morphism $f:F\rightarrow G$ of simplicial presheaves on $T$ is a $\mathcal{A}$-weak equivalenceequivalence!A-weak@$\cA$-weak if for any $\mathcal{A}$-invariant simplicial presheaf $H$ on $T$,
$f$ induces a weak equivalence of simplicial sets

\[
{\mathbf S}(G,H)\rightarrow {\mathbf S}(F,H).
\]

* A morphism $f:F\rightarrow G$ of simplicial presheaves on $T$ is an injective $\mathcal{A}$-fibration if it has the right lifting property with respect to injective cofibrations that are $\mathcal{A}$-weak equivalences.

Recall that the left Bousfield localization of the injective model structure on $\sPsh(T)$ by $\mathcal{A}$-weak equivalences is called the injective $\mathcal{A}$-local model structure,
which exists by <cit.>.
Note that the classes of $\mathcal{A}$-weak equivalences, injective cofibrations, and injective $\mathcal{A}$-fibrations form the injective $\mathcal{A}$-local model structure.

The functor $v_\sharp :\sPsh(T)\rightarrow \sPsh(T[\mathcal{A}^{-1}])$ maps $\mathcal{A}$-weak equivalences to pointwise weak equivalences.

Let $\mathcal{C}$ be the class of morphisms $f:F\rightarrow G$ of simplicial presheaves on $T$ such that $v_\sharp f$ is a pointwise weak equivalence.
By <cit.>, it suffices to check the following conditions.

* $\mathcal{C}$ contains pointwise weak equivalences and morphisms of the form

\[
u\times {\rm id}:Y\times F\rightarrow X\times F,
\]

where $F$ is a simplicial presheaf on $T$, and $u:Y\rightarrow X$ is a morphism in $\mathcal{A}$.

* $\mathcal{C}$ has the 2-out-of-3 property.

* The class of projective cofibrations in $\mathcal{C}$ is stable by pushouts and transfinite compositions.

By <cit.> the functor $v$ commutes with finite limits.
Therefore $v_\sharp (u\times {\rm id})$ is an isomorphism since $v_\sharp u$ is an isomorphism.
If $f:F\rightarrow G$ is a pointwise weak equivalence, then $F(X)\rightarrow G(X)$ is a weak equivalence for any object $X$ of $T$.
According to <cit.>, filtered colimits of pointwise weak equivalences are pointwise weak equivalences.
Since $\mathcal{A}$ admits a calculus of right fractions, we see that $\mathcal{A}\downarrow X$ is cofiltered.
From the formula (<ref>) we deduce that $v_\sharp F(X)\rightarrow v_\sharp G(X)$ is a pointwise weak equivalence.
This verifies condtion (i).

The conditions (ii) and (iii) follow from Proposition <ref> and the fact that the class of pointwise weak equivalences and class of projective cofibrations satisfy (ii) and (iii).

Let $F$ be a simplicial presheaf on $T$.
If $F$ is $\mathcal{A}$-invariant, then the unit map

\[
F\rightarrow v^*v_\sharp F
\]

is a pointwise weak equivalence.

By Proposition <ref>, we need to show that the induced morphism

\begin{equation}
\label{A.1.5.1}
F(X)\rightarrow \colimit_{Y\in \mathcal{A}\downarrow X}F(Y)
\end{equation}

is a pointwise weak equivalence for any $X\in T$.
Each $F(X)\rightarrow F(Y)$ with $Y\in \mathcal{A}\downarrow X$ is a pointwise weak equivalence since $F$ is $\mathcal{A}$-invariant,
and the category $\mathcal{A}\downarrow X$ is cofiltered since $\mathcal{A}$ admits a calculus of right fractions.
Thus (<ref>) is a pointwise weak equivalence since any filtered colimit of pointwise weak equivalences is again a pointwise weak equivalence by <cit.>.

The unit of the adjunction ${\rm id}\rightarrow v^*v_\sharp $ is an $\mathcal{A}$-weak equivalence.

Let $F$ be a simplicial presheaf on $T$ and choose an injective $\mathcal{A}$-fibrant resolution $f:F\rightarrow G$.
Note that $\cG$ is $\cA$-invariant.
Then $v^*v_\sharp f$ is a pointwise weak equivalence according to Propositions <ref> and <ref>.
Hence to show that $F\rightarrow v^*v_\sharp F$ is an $\mathcal{A}$-weak equivalence, it suffices to show that $G\rightarrow v^*v_\sharp G$ is an $\mathcal{A}$-weak equivalence.
For the latter, see Proposition <ref>.

Let $(I,\mathcal{A})$ denote the set of morphisms $\{i_0:\pt\rightarrow I\}\cup \mathcal{A}$ in $T$.
From Definition <ref>(2), we have the notions of $I$-weak equivalences and $(I,\cA)$-weak equivalences.

The functor

\[
v^*:\sPsh(T[\mathcal{A}^{-1}])\rightarrow \sPsh(T)
\]

maps $I$-weak equivalences to $(I,\mathcal{A})$-weak equivalences.

Let $\mathcal{C}$ be the class of morphisms $f:F\rightarrow G$ of simplicial presheaves on $T[\mathcal{A}^{-1}]$ such that $v^*f$ is an $(I,\mathcal{A})$-weak equivalence.
By <cit.>, it suffices to check the following conditions.

* $\mathcal{C}$ contains pointwise weak equivalences and morphisms of the form

\[
i_0\times {\rm id}:{\rm pt}\times F\rightarrow I\times F,
\]

where $F$ is a simplicial presheaf on $T[\mathcal{A}^{-1}]$.

* $\mathcal{C}$ has the 2-out-of-3 property.

* The class of morphisms in $\mathcal{C}$ that are injective cofibrations is stable by pushouts and transfinite compositions.

Since $v^*$ is a right adjoint of $v_\sharp $, it follows that $v^*$ commutes with limits.
Thus to show $v^*(i_0\times {\rm id})$ is an $(I,\mathcal{A})$-weak equivalence, it suffices to show that $v^*i_0$ is an $(I,\mathcal{A})$-weak equivalence.
Note that $v^*i_0\cong v^*v_\sharp i_0$.
Consider the induced commutative diagram

\[
\begin{tikzcd}
{\rm pt}\arrow[d]\arrow[r,"i_0"]&I\arrow[d]\\
v^*v_\sharp {\rm pt}\arrow[r,"v^*v_\sharp i_0"]&v^*v_\sharp I
\end{tikzcd}\]

of simplicial presheaves on $T$.
The upper horizontal morphism is an $I$-weak equivalence, and the vertical morphisms are $\mathcal{A}$-weak equivalences by Proposition <ref>.
Hence $v^*v_\sharp i_0$ is an $(I,\mathcal{A})$-weak equivalence.
By Proposition <ref>, $\mathcal{C}$ contains the pointwise weak equivalences.
This verifies condition (i).

The conditions (ii) and (iii) follow from Proposition <ref> since the $(I,\mathcal{A})$-weak equivalences and the injective cofibrations satisfy (ii) and (iii).

§.§ Construction of the singular functor ${\rm Sing}$

Since $I$ is an interval in $T[\mathcal{A}^{-1}]$, we are entitled to the cosimplicial presheaf

\[
\Delta_I^\bullet\in T[\mathcal{A}^{-1}]
\]

defined as follows, see <cit.>.
On objects we set $\Delta_I^n:=I^n$.
Let $p:I^n\rightarrow {\rm pt}$ denote the canonical morphism to the final object, and let $pr_k:I^k\rightarrow I$ denote the $k$-th projection.
For any morphism $f:\{0,\ldots,n\}\rightarrow \{0,\ldots,m\}$ in $\Delta$, let

\phi(f):\{1,\ldots,m\}\rightarrow \{0,\ldots,n+1\}

denote the function given by

\[
\phi(f)(i):=\left\{
\begin{array}{cl}
\min\{l\in \{0,\ldots,n\}:f(l)\geq i\}&\text{if this set is nonempty}\\
\end{array}
\right.
\]

The morphism $\Delta_I(f):I^n\rightarrow I^m$ is given by

\[
pr_k\circ \Delta_I(f):=\left\{
\begin{array}{cl}
pr_{\phi(f)(k)}&\text{if }\phi(f)(k)\in \{1,\ldots,n\}\\
i_0\circ p&\text{if }\phi(f)(k)=n+1\\
i_1\circ p&\text{if }\phi(f)(k)=0.
\end{array}
\right.
\]

We also have the simplicial presheaf

\[
{\rm Sing}_\bullet^I(F)\in T[\mathcal{A}^{-1}]
\]

defined in <cit.>. [notation]SingI @ ${\rm Sing}_\bullet^I$
Recall that
${\rm Sing}_\bullet^I(F)$ is the diagonal simplicial presheaf of the bisimplicial presheaf

\[
(\Delta_I^\bullet,F_\bullet):(m,n)\mapsto \hom_{\Psh(T[\mathcal{A}^{-1}])}(\Delta_I^m,F_n).
\]

For any simplicial presheaf $F$ on $T$, we set[notation]SingIA @ ${\rm Sing}_\bullet^{(I,\mathcal{A})}$

\[
{\rm Sing}_\bullet^{(I,\mathcal{A})}(F):=v^*{\rm Sing}_\bullet^I(v_\sharp F).
\]

For simplicity we shall often omit the superscript $(I,\mathcal{A})$ in ${\rm Sing}_\bullet^{(I,\mathcal{A})}(F)$.

Singular functor
Let $f,g:F\rightarrow G$ be two morphisms of simplicial presheaves on $T$.
An elementary $(I,\mathcal{A})$-homotopy is a morphism

\[
H:v_\sharp F\times I\rightarrow v_\sharp G
\]

of simplicial presheaves on $T[\mathcal{A}^{-1}]$ such that

\[
H\circ i_0=v_\sharp f
\text{ and }
H\circ i_1=v_\sharp g.
\]

Two morphisms of simplicial presheaves on $T$ are $(I,\mathcal{A})$-homotopichomotopy if they can be connected by a finite number of $(I,\mathcal{A})$-homotopies.

A morphism $f:F\rightarrow G$ of simplicial presheaves on $T$ is a strict $(I,\mathcal{A})$-homotopy equivalencehomotopy equivalence if there is a morphism $g:G\rightarrow F$
of simplicial presheaves on $T$ such that $f\circ g$ and $g\circ f$ are $(I,\mathcal{A})$-homotopic to the identity morphism.

In the following results, we argue following <cit.> and <cit.>.

The canonical morphism

i_0:{\rm pt}\rightarrow I

is a strict $(I,\mathcal{A})$-homotopy equivalence.

The diagram of simplicial presheaves on $\cT[\cA^{-1}]$

\[
\begin{tikzcd}
I\arrow[r,shift left=0.5ex,"i_0\times {\rm id}"]\arrow[r,shift right=0.5ex,"i_1\times {\rm id}"']&I\times I\arrow[r,"\mu"]&I
\end{tikzcd}
\]

displays an elementary $(I,\cA)$-homotopy from ${\rm id}=\mu\circ (i_0\times {\rm id})$ to $i_0p=\mu\circ(i_1\times {\rm id})$.
Since $pi_0$ is the identity morphism on $\pt$, we are done.

Suppose $f,g:F\rightarrow G$ are $(I,\mathcal{A})$-homotopic morphisms of simplicial presheaves on $T$.
Then the induced morphisms

\[
{\rm Sing}_\bullet(f),{\rm Sing}_\bullet(g):{\rm Sing}_\bullet(F)\rightarrow {\rm Sing}_\bullet(G)
\]

are simplicially homotopic.

We may assume there is an elementary $(I,\mathcal{A})$-homotopy $H:v_\sharp F\times I\rightarrow v_\sharp G$ from $f$ to $g$.
Then $H$ is an $I$-homotopy from $v_\sharp f$ to $v_\sharp g$, so ${\rm Sing}_\bullet(v_\sharp f)$ and ${\rm Sing}_\bullet(v_\sharp g)$ are simplicially homotopic by <cit.>.
It remains to show that $v^*$ preserves simplicial homotopies.

Since $v^*$ is a right adjoint of $v_\sharp $, it follows that $v^*$ commutes with limits.

\[
v^*(F\times \Delta[1])\cong v^*F\times \Delta[1]
\]

for any simplicial presheaf $F$ on $T[\mathcal{A}^{-1}]$,
so $v^*$ preserves simplicial homotopies.

For any simplicial presheaf $F$ on $T$, the morphism

\[
{\rm Sing}_\bullet(X)\rightarrow {\rm Sing}_\bullet(X\times I)
\]

induced by $i_0:{\rm pt}\rightarrow I$ is a pointwise weak equivalence.

Follows from Propositions <ref> and <ref>.

For any simplicial presheaf $F$ on $T$, the canonical morphism

\[
F\rightarrow {\rm Sing}_\bullet(F)
\]

is an $(I,\mathcal{A})$-weak equivalence.

We have the factorization

\[
F\rightarrow v^*v_\sharp F\rightarrow v^*{\rm Sing}_\bullet^I(v_\sharp F).
\]

The first morphism is an $\mathcal{A}$-weak equivalence due to Proposition <ref>.
by Proposition <ref>,
it suffices to show the canonical morphism

\[
v_\sharp F\rightarrow {\rm Sing}_\bullet^I(v_\sharp F)
\]

is an $I$-weak equivalence.
This follows from <cit.>.

Recall that $\Lambda$ is a commutative unital ring.

Let $\sPsh(T,\Lambda)$[notation]sPsh @ $\sPsh$ denote the category of simplicial sheaves of $\Lambda$-modules on $T$,
and let $\Co^{\leq 0}(\Psh(T,\Lambda))$ denote the category of complexes $\cF$ of sheaves of $\Lambda$-modules on $T$ such that $H^i(\cF)=0$ for any $i>0$.

By the Dold-Kan correspondenceDold-Kan correspondence <cit.>, there are equivalences of categories

\[
N:\sPsh(T,\Lambda)\rightleftarrows \Co^{\leq 0}(\Psh(T,\Lambda)):\Gamma
\]

\[
N:\sPsh(T[\cA^{-1}],\Lambda)\rightleftarrows \Co^{\leq 0}(\Psh(T[\cA^{-1}],\Lambda)):\Gamma
\]

An object $F$ of $\Co^{\leq 0}(\Psh(T,\Lambda))$ is $\cA$-invariant if $\Gamma(F)$ is $\cA$-invariant.
A morphism $f:F\rightarrow G$ in $\Co^{\leq 0}(\Psh(T,\Lambda))$ (resp. $\Co^{\leq 0}(\Psh(T[\cA^{-1}],\Lambda))$) is a $(I,\cA)$-weak equivalence (resp. $\cA$-weak equivalence)
if $\Gamma(f):\Gamma(F)\rightarrow \Gamma(G)$ is a $(I,\cA)$-weak equivalence (resp. $\cA$-weak equivalence).

Let $\Co^{-}(\Psh(T,\Lambda))$ denote the category of bounded above complexes of sheaves of $\Lambda$-modules.
Then an object $F\in \Co^{-}(\Psh(T,\Lambda))$ is called $\cA$-invariant if there exists an integer $n\in \Z$ such that $F[n]\in\Co^{\leq 0}(\Psh(T,\Lambda))$ is $\cA$-invariant.

The $(I,\cA)$-weak equivalences in $\Co^{-}(\Psh(T,\Lambda))$ and the $\cA$-weak equivalences in $\Co^{-}(\Psh(T[\cA^{-1}],\Lambda))$ are defined similarly.

Suppose $G$ is a presheaf of $\Lambda$-modules on $T$ and consider the simplicial presheaf $\Lambda$-modules $\Sing_{\bullet}^{(I,\cA)}(G)$ on $T$.
Taking the alternating sum of the face maps yields the Suslin complex Suslin complex of presheaves

We let $C_*G$ be shorthand for $C_*^{(I,\cA)}G$ when no confusion seems likely to arise, and define

\[
C_*^{DK}(G):=N({\rm Sing}_\bullet^{(I,\cA)}(G)).
\]

Here $C_*^{DK}(G)$ is canonically a normalized subcomplex of $C_*(G)$.
The morphism

\[
C_*^{DK}(G)\rightarrow C_*(G)
\]

is a quasi-isomorphism, but not an isomorphism in general.

If $F$ is a bounded above complex of presheaves on $T$,
we can consider $C_*F$ and $C_*^{DK}F$ as double complexes and form their total complexes ${\rm Tot}(C_*F)$ and ${\rm Tot}(C_*^{DK}F)$.

Suppose $H\in \sPsh(T,\Lambda)$.
The Eilenberg-Zilber theorem says that there is a quasi-isomorphism

\begin{equation}
\label{A.1.15.1}
N({\rm Sing}_\bullet^{(I,\cA)}(H))
\cong
{\rm Tot}(C_*^{DK}(N(H))).
\end{equation}

As in Definition <ref> there exist adjoint functors

\[
\begin{tikzcd}
\Co^{\leq 0}(\Psh(T,\Lambda))\arrow[rr,shift left=1.5ex,"v_\sharp "]\arrow[rr,"v^*" description,leftarrow]\arrow[rr,shift right=1.5ex,"v_*"']&&\Co^{\leq 0}(\Psh(T[\mathcal{A}^{-1}],\Lambda)),
\end{tikzcd}
\]

where $v_\sharp$ is left adjoint to $v^*$, and $v^*$ is left adjoint to $v_*$.
For $F\in \Co^{\leq 0}(\Psh(T[\mathcal{A}^{-1}],\Lambda))$ and $X\in T$, we have

\[
\Gamma(v^*F)(X)\cong \Gamma(v^*F(X))=\Gamma(F(v(X))=\Gamma(F)(v(X))\cong v^*(\Gamma(F))(X).
\]

Thus $\Gamma$ commutes with $v^*$.
Similarly, we can show that $N$ commutes with $v^*$.
By adjunction we see that both $\Gamma$ and $N$ commute with $v_\sharp$.

Let $F$ be a bounded above complex of presheaves of $\Lambda$-modules on $T$.
Then the following properties hold.

If $F$ is $\cA$-invariant, then the unit of the adjunction

\[
F\rightarrow v^*v_\sharp F
\]

is a quasi-isomorphism.

The unit of the adjunction

\[
F\rightarrow v^*v_\sharp F
\]

is an $\mathcal{A}$-weak equivalence.

The morphism

\[
{\rm Tot}(C_*F)\rightarrow {\rm Tot}(C_*(F\times I))
\]

induced by $i_0:{\rm pt}\rightarrow I$ is a quasi-isomorphism.

The canonical morphism

\[
F\rightarrow {\rm Tot}(C_*F)
\]

is an $(I,\mathcal{A})$-weak equivalence.

We observe that all the results in this section hold verbatim for simplicial presheaves of $\Lambda$-modules.
It suffices to prove these assertions for the shifted complex $F[n]$, $n\in \Z$.
Thus we may assume $F\in \Co^{\leq 0}(\Psh(T,\Lambda))$.
As noted above both $\Gamma$ and $N$ commute with both $v_\sharp$ and $v^*$.
quasi-isomorphisms (resp. $\cA$-weak equivalences, resp. $(I,\cA)$-weak equivalences) in $\Co^{\leq 0}(\Psh(T,\Lambda))$ corresponds to pointwise weak equivalences
(resp. $\cA$-weak equivalences, resp. $(I,\cA)$-weak equivalences) in $\sPsh(T,\Lambda)$ under $\Gamma$ and $N$.

Suppose $F$ is $\cA$-invariant.
Then $\Gamma(F)$ is $\cA$-invariant,
and the $\Lambda$-module version of Proposition <ref> shows the morphism

\[
\Gamma(F)\rightarrow v^*v_\sharp \Gamma(F)\xrightarrow{\cong} \Gamma(v^*v_\sharp F)
\]

is a pointwise weak equivalence.
Thus $F\rightarrow v^*v_\sharp F$ is a quasi-isomorphism, which shows (1).
A similar argument shows (2).
Due to (<ref>) we have isomorphisms

\begin{gather*}
{\rm Tot}(C_*F)\simeq {\rm Tot}(C_*^{DK}F)\simeq N({\rm Sing}_\bullet^{(I,\cA)}(\Gamma(F))),
\\
{\rm Tot}(C_*(F\times I))\simeq {\rm Tot}(C_*^{DK}(F\times I))\simeq N({\rm Sing}_\bullet^{(I,\cA)}(\Gamma(F\times I))).
\end{gather*}

Owing to the $\Lambda$-module version of Corollary <ref> the composite morphism

\[
{\rm Sing}_\bullet^{(I,\cA)}(\Gamma(F))\rightarrow {\rm Sing}_\bullet^{(I,\cA)}(\Gamma(F)\times I)\xrightarrow{\cong} {\rm Sing}_\bullet^{(I,\cA)}(\Gamma(F\times I))
\]

is an isomorphism.
Combining the above, we deduce (3).
A similar argument shows (4).

§ PROPERTIES OF LOGARITHMIC MOTIVES

In this section, we formulate and prove the fundamental properties of our categories of logarithmic motives.
Most of these properties hold for categories associated with a base scheme $S$,
not necessarily a field,
subject to a list of axioms listed below.
This setup affords the case of $\ldmeff$ and its variants, see Proposition <ref>.

It can be quite interesting to observe that certain implications hold among the axioms.
For example, $(\P^n, \P^{n-1})$-invariance can be deduced by assuming Nisnevich descent (Nis-desc),
$\boxx$-invariance, and dividing-descent, see Proposition <ref>.
On the other hand,
we can include $(\P^n, \P^{n-1})$-invariance as part of the axioms.
In that case dividing-descent can be deduced from (Nis-desc) and $(\P^n, \P^{n-1})$-invariance for every $n\geq 1$.
We refer to Section <ref> for further details.

After proving some general properties about $\P^n$-bundles and $\boxx^n$-bundles,
we begin by showing an invariance property for certain admissible blow-ups.
This is used for showing that log motives satisfy the $(\P^n, \P^{n-1})$-invariance property.
We proceed by defining Thom motives,
a.k.a. motivic Thom spaces,
prove a log version of homotopy purity,
and construct Gysin maps for closed embeddings.

Assuming that $k$ admits resolution of singularities allow us to prove a more potent invariance property for proper birational morphisms,
see Theorem <ref>.
Finally we discuss strictly $(\P^\bullet,\P^{\bullet-1})$-invariant and strictly dividing invariant complexes of sheaves on $SmlSm/k$.
This provides a convenient tool for constructing objects of $\ldmeff$ without explicitly verifying the dividing-invariance property on the level of sheaf cohomology.

Throughout this section, we fix a base scheme $S$ and let $\mathscr{S}/S$ denote either $lSm/S$ or $SmlSm/S$.
Moreover, we fix a symmetric monoidal triangulated category $\cT$,
and we fix a monoidal functor

\[
M\colon \mathbf{Square}_{\mathscr{S}/S}\rightarrow \cT
\]

satisfying (Sq) and (MSq).
For the category $\mathbf{Square}_{\mathscr{S}/S}$, $n$-squares, and the conditions (Sq) and (MSq), we refer to Appendix <ref>.
We choose to use $n$-squares because this is a convenient way to construct certain cones canonically within triangulated categories without referring to model categories or $\infty$-categories.
For example, $2$-squares are used in [29] to develop the theory of Gysin triangles in $\A^1$-motivic theory.
In most of our argument we only need $n$-squares for $n\leq 2$, but we need $3$-squares in Subsection <ref>.

We shall often assume some of the following properties.

Zariski separation property. If $\{X_1,\ldots,X_r\}$ is a finite Zariski cover of $X$ and $I=\{i_1,\ldots,i_s\}\subset \{1,\ldots,r\}$, we set

\[
X_I:=X_{i_1}\times_S \cdots \times_S X_{i_s}.
\]

Then for any commutative square of the form

\begin{equation}
\label{A.3.0.2}
\begin{tikzcd}
\\
\end{tikzcd}
\end{equation}

the morphism

\[
M(Y'\rightarrow X')\rightarrow M(Y\rightarrow X)
\]

is an isomorphism if for every nonempty $I\subset \{1,\ldots,r\}$ there is an isomorphism

\[
M(Y_I'\rightarrow X_I')\rightarrow M(Y_I\rightarrow X_I)
\]

\[
Y_I:=Y\times_X X_I, X_I':=X'\times_X X_I, \text{ and }Y_I':=Y'\times_Y Y_I.
\]

$\boxx$-invariance property. For any $X\in \mathscr{S}/S$ the projection $X\times  \boxx\rightarrow X$ induces an isomorphism

\[
M(X\times \boxx)\xrightarrow{\cong} M(X).
\]

$sNis$-descent property.
Every strict Nisnevich distinguished square of the form (<ref>) induces an isomorphism

\[
M(Y'\rightarrow X')\xrightarrow{\cong} M(Y\rightarrow X).
\]

$div$-descent property.
Every log modification $f:Y\rightarrow X$ in $\mathscr{S}/S$ induces an isomorphism

\[
M(Y)\xrightarrow{\cong} M(X).
\]

$(\P^\bullet,\P^{\bullet-1})$-invariance property.
For any $X\in \mathscr{S}/S$ and positive integer $n>0$ the projection $X\times  (\P^n,\P^{n-1})\rightarrow X$ induces an isomorphism

\[
M(X\times (\P^n,\P^{n-1}))\xrightarrow{\cong} M(X).
\]

Here we consider $\P^{n-1}$ as the hyperplane of $\P^n$ defined by the equation $x_n=0$ with respect to the coordinates $[x_0:\cdots:x_n]$.

Admissible blow-up property.
For every proper birational morphism $f\colon Y\rightarrow X$ in $\mathscr{S}/S$ such that the naturally induced morphism $Y-\partial Y\rightarrow X-\partial X$ is an isomorphism, there is a naturally induced isomorphism

\[
M(Y)\xrightarrow{\cong} M(X).
\]

The following figure illustrates how these properties are related.
Here, ROS stands for resolution of singularities.

\[
\begin{tikzpicture}
\node[draw,align=center] at (-5,0)
\\
\\
\\
\\
\node[draw,align=center] at (0,0)
\\
\\
\\
\\
\node[draw,align=center] at (5,0)
\\
\\
\\ $\mathscr{S}=SmlSm/S$};
\draw (-3.5,0.6)--(-1.5,0.6);
\draw (-3.5,0.5)--(-1.5,0.5);
\node at (-1.5,0.55) {$>$};
\node at (-2.5,0.8) {\footnotesize Trivial if $S=k$};
\draw (-3.5,-0.6)--(-1.5,-0.6);
\draw (-3.5,-0.5)--(-1.5,-0.5);
\node at (-3.5,-0.55) {$<$};
\node at (-2.5,-0.8) {\footnotesize Theorem \ref{A.3.12}};
\node at (-2.5,-1.1) {\footnotesize {(assuming ROS)}};
\draw (1.35,0.6)--(3.35,0.6);
\draw (1.35,0.5)--(3.35,0.5);
\node at (3.35,0.55) {$>$};
\node at (2.35,0.8) {\footnotesize Proposition \ref{A.6.1}};
\draw (1.35,-0.6)--(3.35,-0.6);
\draw (1.35,-0.5)--(3.35,-0.5);
\node at (1.35,-0.55) {$<$};
\node at (2.35,-0.8) {\footnotesize Construction \ref{Pn.4}};
\end{tikzpicture}
\]

The Tate object in $\cT$ is defined by

\[M(S)(1)
M(S\stackrel{i_0}\rightarrow \P_S^1)[-2],
\]

where $i_0:S \rightarrow \P_S^1$ is the $0$-section.

For $n\geq 0$, we set

\[
\underbrace{M(S)(1)\otimes \cdots \otimes M(S)(1)}_{n\text{ times}},
\]

where $M(S)(0):=M(S)$ by convention.
We define the $n$th Tate twist of $K\in\cT$ as

\[
K\otimes M(S)(n).
\]

The conditions (Sq) and (MSq) of Appendix <ref> and the properties ($Zar$-sep), ($\boxx$-inv), ($sNis$-des),
and ($div$-des) hold in the triangulated categories

\begin{gather*}
\ldmeff,\;\ldaeff,\;\ldmeffet,\;\ldaeffet,
\\
\ldmefflet,\;\ldaefflet.
\end{gather*}

We work in $\ldmeff$ since the proofs for the other categories are similar.
For every $n$-square $C$, we set

\[
M(C):={a_{dNis}^*}{\rm Tot}(\Zltr(C)).
\]

For every $n$-square $C$ with $n\geq 1$, there exists a naturally induced sequence

\[
{\rm Tot}(\Zltr(s_{i,0}^*C))\rightarrow {\rm Tot}(\Zltr(s_{i,1}^*C))\rightarrow {\rm Tot}(\Zltr(C))\rightarrow {\rm Tot}(\Zltr(s_{i,0}^*C))[1].
\]

Moreover, for every $n$-square $C$ and $m$-square $D$, there exists a canonical isomorphism (not just a quasi-isomorphism) of complexes

\[
{\rm Tot}(\Zltr(C))\otimes {\rm Tot}(\Zltr(D))\rightarrow {\rm Tot}(\Zltr(C\otimes D)).
\]

The latter is reminiscent of an Eilenberg-Zilber map, but it is, in fact, much simpler since no shuffles are involved.
We can readily deduce that the conditions (Sq) and (MSq) hold.

Property ($\boxx$-inv) holds by construction.
Proposition <ref> implies ($sNis$-des) and ($div$-des).
To show ($Zar$-sep),
let $\mathscr{X}$ be the Čech nerve of the covering $\{X_1,\ldots,X_r\}$,
and set

\[
\mathscr{X}':=X'\times_X \mathscr{X}, \mathscr{Y}:=Y\times_X \mathscr{X}, \mathscr{Y}':=Y'\times_X \mathscr{X}.
\]

Then, by assumption, we have the homotopy cartesian diagram

\[
\begin{tikzcd}
\end{tikzcd}
\]

This allows us to conclude that ($Zar$-sep) holds owing to the isomorphisms

\[
M(X)\cong M(\mathscr{X}), M(Y)\cong M(\mathscr{Y}), M(X')\cong M(\mathscr{X}'), \text{ and } M(Y')\cong M(\mathscr{Y}').
\]

§.§ Some toric geometry and examples of bundles

Let $d\geq 0$ and suppose $\sigma$ is a $d$-codimensional cone of a fan $\Sigma$.