poster.tex

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\title{Using the internal language of toposes in algebraic geometry}
\author{Ingo Blechschmidt}
\institute{University of Augsburg}
\date{June 23th, 2014}

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\begin{document}

\begin{frame}[t]\begin{columns}[t]

\begin{column}{0.3\textwidth}
  \begin{alertblock}{Summary}
    With the internal language of toposes, we can
    \begin{itemize}\justifying
    \item express sheaf-theoretic
    concepts in a simple, ele\-ment-ba\-sed language and thus understand them
    in a more conceptual way,
    \item mechanically obtain
    corresponding sheaf-theo\-re\-tic theorems for any (intuitionistic) theorem of
    linear and commutative algebra, and
    \item understand which properties spread from
    points to neighbourhoods.
    \end{itemize}
  \end{alertblock}
  \bigskip

  \begin{block}{What is a topos?}
    A \emph{topos} is a category which has finite limits, is cartesian closed and
    has a subobject classifier. Intuitively, a topos is a category which has
    similar properties to the category of sets.\medskip

    Important examples of toposes are
    the category of sets and
    the category of sheaves on a topological space.
  \end{block}
  \bigskip

  \begin{block}{What is the internal language?}
    The internal language of a topos~$\E$ allows us to
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    construct objects and morphisms of the topos,
    %{\addtocounter{enumi}{1}\usebeamertemplate{enumerate item}}
    formulate statements about them, and
    %{\addtocounter{enumi}{1}\usebeamertemplate{enumerate item}}
    prove such statements
    in a \emph{naive element-based} language.
    The translation of internal statements and proofs into external ones is
    facilitated by an easy mechanical procedure, the \emph{Kripke--Joyal
    semantics}.
    \emph{Special case:} The language of the topos of sets is the usual
    formal mathematical language.

    % Note: The singular is probably better.
    \begin{center}
      \begin{tabular}{ll}
        \toprule
        external point of view & internal point of view \\
        \midrule
        objects of~$\E$ & sets \\
        morphisms of~$\E$ & maps of sets \\
        monomorphisms in~$\E$ & injective maps \\
        epimorphisms in~$\E$ & surjective maps \\
        \bottomrule
      \end{tabular}
    \end{center}
  \end{block}

  \vspace{0.7cm}
  \textbf{GAeL XXII, Trieste, 2014}
\end{column}

\begin{column}{0.3\textwidth}
  \begin{block}{The small Zariski topos}
    Let~$X$ be a scheme. Let~$\Sh(X)$ be the small Zariski topos, i.\,e.\@ the
    topos of set-valued sheaves on~$X$. From the point of view of~$\Sh(X)$,
    the structure sheaf~$\O_X$ looks like an \emph{ordinary ring} (instead of a
    sheaf of rings), and sheaves of~$\O_X$-modules look like \emph{ordinary
    modules} on that ring.\bigskip

    \rotatebox{90}{\tiny Illustration: Carina Willbold}\hspace{-0.25cm}%
    \includegraphics[width=\columnwidth]{images/external-internal}
  \end{block}
  \bigskip

  \begin{block}{Basic example}
    Let~$0 \to \F' \to \F \to \F'' \to 0$ be a short exact sequence of sheaves
    of~$\O_X$-modules. It is well-known that if~$\F'$ and~$\F''$ are of finite type,
    then~$\F$ is as well.\medskip

    A sheaf is of finite type if and only if, internally, it is a
    finitely generated module. Therefore the proposition follows
    \emph{immediately}
    by interpreting the analogous statement of linear algebra
    in the little Zariski topos:
    Let~$0 \to M' \to M \to M'' \to 0$ be a short exact sequence of modules.
    If~$M'$ and~$M''$ are finitely generated, so is~$M$.\medskip

    We can thus recognize notions and statements of scheme theory as notions and
    statements of non-sheafy linear algebra.
    \emph{Caveat:} Non-intuitionistic proofs by contradiction can not be interpreted with the
    internal language.
  \end{block}
\end{column}

\begin{column}{0.3\textwidth}
  \begin{block}{Locally free sheaves}
    Let~$X$ be a reduced scheme. The structure sheaf~$\O_X$ looks like a \emph{field} from the
    internal point of view. Recall that neither the rings of
    local sections nor the stalks are fields.\medskip

    Let~$\F$ be a finite type sheaf of~$\O_X$-modules. 
    Then it is well-known that~$\F$ is locally free on a dense open subset
    of~$X$. (Important hard exercise in Ravi Vakil's notes.) \medskip

    This is an \emph{immediate} application of the following easy lemma of intuitionistic
    linear algebra: Let~$M$ be a finitely generated vector space. Then~$M$ is
    \emph{not not} finite free.
  \end{block}
  \bigskip

  \begin{block}{Rational functions}
    The sheaf~$\K_X$ of rational functions can internally simply be defined as
    the total quotient ring of~$\O_X$.
  \end{block}
  \bigskip

  \begin{block}{Spreading of properties}
    The following metatheorem covers a wide range of cases:
    Let~$\varphi$ be a property which can be formulated without
    using~$\Rightarrow$,~$\neg$,~$\forall$. Then~$\varphi$
    holds at a point if and only if it holds on some open neighbourhood of the
    point.\medskip

    For instance, a sheaf of modules~$\F$ is zero if and only if, from the
    internal perspective, ``$\forall x \in \F{:}\ x = 0$''. Because of
    the~``$\forall$'', a stalk may be zero without the sheaf being zero on a
    neighbourhood.\medskip

    But if~$\F$ is of finite type, the condition can be reformulated using
    generators as ``$x_1 = 0 \wedge \cdots \wedge x_n = 0$''. The metatheorem
    is applicable to this statement and thus a stalk is zero if and only if~$\F$
    is zero on a neighbourhood.
  \end{block}

  \vspace{1cm}
  \begin{alertblock}{Dictionary of external vs. internal notions}
    Expository notes are available at \url{http://tiny.cc/topos} (work in
    progress).
  \end{alertblock}
  \tiny\rmfamily\justifying
  Contents:
  Tensor product of sheaves $=$ internal ordinary tensor product,
  quasicoherent sheaves $=$ internal ordinary modules satisfying an interesting
  condition,
  internal Cartier divisors,
  more metatheorems about spreading of properties,
  pullback along immersions $=$ internal sheafification,
  relative spectrum $=$ internal spectrum,
  scheme dimension $=$ internal Krull dimension of~$\O_X$,
  dense $=$ not not,
  modal operators,
  other toposes,
  group schemes $=$ groups,
  \ldots
\end{column}

\end{columns}\end{frame}

\end{document}