%%%%%%%%%%%%%%%% Section 1, Definitions and basic properties
In this  section we introduce the basic definitions of logarithmic geometry in the sense of \cite{KKato}. Good introductions are given in \cite{KKato} and \cite{Ogus}.

\subsection{Logarithmic structures} % Log Structure

The basic definitions are as follows:

\begin{defn} % defn of monoid
A \emph{monoid} is a commutative semi-group with a unit. A morphism of monoids is required to preserve the unit element. We use $Mon$ to denote the category of Monoids.
\end{defn}

\begin{defn}\label{Def:log-str} % defn of Log str.
Let $X$ be a scheme. A \emph{pre-logarithmic structure} on $X$ is a sheaf of
monoids $\sM$ on the \'etale site $X_{\acute{et}}$ combined with a
morphism of sheaves of monoids: $\alpha
: \sM \longrightarrow \sO_{X}$, called the structure morphism, where
we view $\sO_{X}$ as a monoid under multiplication. A pre-log
structure is called a \emph{log structure} if
$\alpha^{-1}(\sO_{X}^{*})\cong \sO^{*}_{X}$ via $\alpha$. We call the
pair $(X,\sM)$ a \emph{log scheme}. 
\end{defn}

\begin{defn}
The quotient sheaf $\overline{\sM}_{X}=\sM_{X}/\sO_{X}^{*}$ is called the {\em characteristic of the log structure} $\sM_{X}$.
\end{defn}

\begin{defn} % morphism of pre-log str.
Let  $\sM$ and $\sN$ be pre-log structures on $X$. A {\em morphism}
between them is a morphism $\sM \rightarrow \sN$ of sheaves of monoids
which is compatible with the structure morphisms. 
\end{defn}

How should one think of such a beast? There are two extreme cases: 

\begin{enumerate} 
\item If an element $m\in M$ has $\alpha(m)=x \neq 0$, one often thinks
of $m$ as some sort of partial data of a ``branch of the logarithm of
$x$''. Evidently no data is added if $x$ is invertible, but some is
added otherwise. In particular, we will see later that $m$ permits us
to take the logarithmic 
differential $dx/x$ of $x$.

\item If $\alpha(m)=0$ it is often the case that it $m$ comes by
restricting the log structure of an ambient space, and serves as the
``ghost'' of a logarithmic cotangent vector coming from that space. So
the log structure ``remembers'' deformations that are lost when
looking at the underlying scheme.
\end{enumerate}


\subsection{The log structure associated to a pre-log structure} % Log str. assoc. to pre-log str.
We have a natural inclusion $i : $ (log structure on $X$) $\hookrightarrow$ (pre-log structure on $X$) by viewing log structure as a pre-log structure. We now construct a left adjoint.


Let  $\alpha : \sM \rightarrow \sO_{X}$ be a pre-log structure on $X$. We define the  \emph{associated  log structure} $\sM^{a}$ to be the push-out of 
\[\xymatrix{
\alpha^{-1}(\sO_{X}^{*}) \ar[d] \ar[r] & \sM \\
\sO_{X}^{*}
}
\]
in the category of sheaves of monoids on $X_{\acute{et}}$, endowed with $$\sM^{a} \rightarrow \sO_{X}  \ \ (a,b)\mapsto \alpha(a)b \ (a \in \sM, b \in \sO_{X}^{*}).$$

In this way, we obtain a functor $a$ : (pre-log structures on $X$) $\rightarrow$ (log structures on $X$). From the universal property of push-out, any morphism of pre-log structure from a pre-log structure $\sM$ to a log structure on $X$ factor through $\sM^{a}$ uniquely.

 We have the following lemma:

\begin{lem}\cite[1.1.5]{Ogus}
The functor $a$ is left adjoint to $i$.
\end{lem}


\begin{eg}
The category of log structures on $X$ has an initial object, called the trivial log structure, given by the inclusion $\sO_{X}^{*} \hookrightarrow \sO_{X}$. And it also has a final object, given by the identity map $\sO_{X} \rightarrow \sO_{X}$.
\end{eg}

\begin{eg}\label{NClog}
Let $X$ be a regular scheme, $D \subset X$ is a
divisor. We can define a log structure $\sM$ on $X$ associated to the
divisor $D$ as 
$$\sM(U)\ =\ \big\{\,g\in \sO_X(U)\, :\ g|_{U\setminus D} \in \cO_X^*(U\setminus D)\,\big\}\ \subset\ \cO_{X}(U).$$ 

The case where $D$ is a normal
crossings divisor situation is special - we will see later that it is {\em log smooth}.

Note that the concept of normal crossing is
local in the \'etale topology. This is one reason we use
the \'etale topology instead of the Zariski topology. 
\end{eg}

\begin{eg}\label{Afflog}
Let $P$ be a monoid, $R$ a ring, and denote by $R[P]$  the monoid algebra. Denote  $X = \Spec R[P]$. then $X$ has a
canonical log structure associated to the canonical map $P \rightarrow
R[P]$. We denote by $\Spec(P\rightarrow R[P])$ the log
scheme $X$ with this canonical log structure. 
\end{eg}

\subsection{The inverse image} % inverse image of log str.

Let $f:X\rightarrow Y$ be a morphism of schemes. Given a log structure
$\sM_{Y}$ on $Y$, we can obtain a log structure on $X$ called the
invese image of $\sM_{Y}$, to be the log structure associated to the
pre-log structure $f^{-1}(\sM_{Y})\rightarrow
f^{-1}(\sO_{Y})\rightarrow \sO_{X}$. This is usually denoted by
$f^{*}(\sM_{Y})$. Using the inverse image of log structures, we can
give the following definition. 

\begin{defn}\label{Def:log-mor} % morphism of log scheme
A \emph{morphism of log schemes} $(X,\sM_{X})\rightarrow (Y,\sM_{Y})$
consists of a morphism of underlying schemes $f:X\rightarrow Y$, and a
morphism  $f^{\flat}: f^{*}\sM_{Y}\rightarrow \sM_{X}$ of log
structures on X.

We denote by $LSch$ the category of 
log schemes.  
\end{defn}

\begin{eg}
In Example \ref{Afflog}, the log structure on $\Spec(P\rightarrow
R[P])$ can be viewed as the inverse image of the log structure on
$\Spec(P\rightarrow \Z[P])$ via the canonical map
$\Spec(R[P]) \rightarrow \Spec(\Z[P])$. 
\end{eg}

\begin{eg}\label{Logpt} % Log str. from normal crossing divisor and
                        % log pt 
Let $k$ be a field, $Y$=Spec $k[x_{1},\cdots,x_{n}]$,
$D$=$V(x_{1}\cdots x_{r})$. Note that $D$ is a normal crossing divisor
in $Y$. By example \ref{NClog}, we have a log structure $\sM_{Y}$ on
$Y$ associated to the divisor $D$. In fact, $\sM_{Y}$ can be viewed as
a subsheaf of $\sO_{Y}$ generated by $\sO_{Y}^{*}$ and
$\{x_{1},\cdots,x_{r}\}$. 

Consider the inclusion $j: p$ = Spec $k \hookrightarrow Y$ sending the
point to the origin of $Y$. Then $j^{*}\sM_{Y} = k^{*} \oplus \N^{r}$,
and the structure map $j^{*}\sM\longrightarrow \sO_{X}$ is given by
$(a,n_{1},\cdots,n_{r})\mapsto a\cdot 0^{n_{1}+\cdots+n_{r}}$, where
we assume $0^{0} = 1$ and $0^{n}=0$ if $n\neq 0$. Such point with the
log structure above is call a logarithmic point, when $r = 1$ we call
it the standard logarithmic point. 
\end{eg}

\subsection{Charts of log structures} % Charts

\begin{defn}\label{Def:chart} % Charts of log schemes
Let $(X,\sM_{X})$ be a log scheme, and $P$ a monoid. A chart of $\sM_{X}$ is a morphism $P\rightarrow \Gamma(X,\sM_{X})$, such that the log structure associated to the pre-log structure coming from the induced map $P\rightarrow \Gamma(X,\sM_{X})\rightarrow \Gamma(X,\sO_{X})$ is $\sM_{X}$.
\end{defn}

In fact, a chart of $\sM_{X}$ is equivalent to a morphism $$f:(X,\sM_{X}) \rightarrow\Spec(P\rightarrow\Z[P]),$$ such that $f^{\flat}$ is an isomorphism. In general, we have the following:

\begin{lem}\cite[1.1.9]{Ogus} The morphism
$$Hom_{LSch}((X,\sM_{X}),\Spec(P\rightarrow \Z[P]))\ \to \ Hom_{Mon}(P,\Gamma(X,\sM_{X}))$$
associating to $f$ the composition
 $$\xymatrix{P \ar[r]&\Gamma(P_X)\ar[r]^{\Gamma(f^\flat)} & \Gamma(\cM_X)}$$
 is an isomorphism.
\end{lem}

We can also consider the charts of log morphisms.

\begin{defn}\label{Def:chart-mor} %chart of log morphisms
Let $f:(X,\sM_{X})\rightarrow (Y,\sM_{Y})$ be a morphism of log schemes. A chart for $f$ is a triple $(P_{X}\rightarrow\sM_{X},Q_{Y}\rightarrow\sM_{Y},Q\rightarrow P)$ where $P_{X}$ and $Q_{Y}$ are the constant sheaves associated to the monoids $P$ and $Q$, which satisfy the following conditions:
 \begin{enumerate}
   \item $P_{X}\rightarrow\sM_{X}$ and $Q_{Y}\rightarrow\sM_{Y}$ are charts of $\sM_{X}$ and $\sM_{Y}$;
   \item the morphism of monoids $Q\rightarrow P$ makes the following diagram commutative:
\[
\xymatrix{
Q_{X} \ar[r]\ar[d] & P_{X} \ar[d]\\
f^{*}\sM_{Y} \ar[r]& \sM_{X}.
}
\]
 \end{enumerate}
\end{defn}

\subsection{Fine log structures} % Fine log structures
Arbitrary log structures are too wild to manipulate, in rough analogy to arbitrary ringed spaces. Next we will introduce some well-behaved log structures: they are useful for general constructions but a narrower, more geometric  category is desirable.  Given a monoid $P$, we can associate a group 
\begin{equation*}{
P^{gp}:=\{(a,b) |(a,b)\sim(c,d) \ \mbox{if} \ \exists s \in P \ \mbox{such that} \ s+a+d=s+b+c\}.}
\end{equation*}
 Note that any morphism from $P$ to an abelian group factor through $P^{gp}$ uniquely.

\begin{defn}\label{Def:integral} % Integral monoid
$P$ is called {\em integral} if $P\rightarrow P^{gp}$ is injective. It is called \emph{saturated} if it is integral and for any $p\in P^{gp}$, if $n\cdot p \in P$ for some positive integer $n$ then $p\in P$.
\end{defn}

\begin{defn}\label{Def:fine} % Fine log scheme
A log scheme $(X,\sM_{X})$ is said to be {\em fine}, if \'etale locally there is a chart $P\rightarrow \sM$ with $P$ a finitely generated integral monoid. If moreover $P$ is saturated, $(X, \sM_{X})$ is called a {\em fine and saturated} (or {\em fs})  log structure. This is equivalent to saying that for every geometric point $\overline x \to X$ the monoid $\ocM_{\overline x, X}$ is saturated. Finally if $P\simeq \NN^k$ we say that the log strcture is {\em locally free}.
\end{defn}

In the following, we will focus on fine log schemes.