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\title{The Art of Sketching (1): Sketch of the Real Number Line}
\author{Pierre Ageron}
\date{Research Report 2000--15}

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\newtheorem{proposition}{Proposition}

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\maketitle

\begin{abstract}
The series \emph{The Art of Sketching} aims to enrich the traditional collection of examples of sketches that have been constructed so far and to illustrate recent theoretical results about sketches.
In this first installment, we describe a sketch whose category of models is equivalent to the completed real line $\overline{\mathbb{R}}=[-\infty,+\infty]$; then we indicate how to modify it to obtain categories of equivalent models, respectively, to $\mathbb{R}\cup\{-\infty\}$, $\mathbb{R}\cup\{+\infty\}$, $\mathbb{R}$, and $\mathbb{R}\setminus\mathbb{Q}$.
\end{abstract}

\section{Cuts and Sketches}

The goal is to sketch the completed real line $\overline{\mathbb{R}}=[-\infty,+\infty]$, viewed (in the category-theoretic sense) as the set of Dedekind cuts that has $\mathbb{Q}$ as support and is equipped with a map from $1$.
Recall that a cut is a pair $(A',A'')$ of parts of $\mathbb{Q}$ such that
\[
A'\cap A''=\varnothing,
\qquad
A'\cup A''=\mathbb{Q},
\]
and every element of $A'$ is less than every element of $A''$.

To represent $-\infty$ or $+\infty$, one does not impose that $A'$ and $A''$ are nonempty.
To avoid having representations for irrational numbers (the item itself and its complement in $A'$ or $A''$), we restrict to left cuts, verifying for all $p<r<q$ in $\mathbb{Q}$:
\[
\text{if }p,q\in A'\text{ then }r\in A',
\qquad
\text{and if }p,q\in A''\text{ then }r\in A''.
\]

Cuts are in bijection with decreasing maps $M$ from $\mathbb{Q}$ to $\{0,1\}$:
\[
q\in A' \Longleftrightarrow M(q)=1,
\qquad
q\in A'' \Longleftrightarrow M(q)=0.
\]
Left cuts correspond to those maps satisfying, for all $q<q'$,
\[
M(q)=\inf_{q<r<q'} M(r).
\]

To represent elements of $\overline{\mathbb{R}}$ as set-valued functors, we note that the category $\{0,1\}$ is equivalent to the full subcategory of \textbf{Set} whose objects are sub-singletons.
Hence a model in \textbf{Set} can be seen as a cone.
This yields the following projective sketch $E_{\overline{\mathbb{R}}}$.

\begin{proposition}
The category of models of the following sketch is equivalent to the ordered set $\overline{\mathbb{R}}=[-\infty,+\infty]$:
\begin{itemize}[leftmargin=1.5em]
\item support: the ordered set $(\mathbb{Q},\ge)$;
\item for each $q\in\mathbb{Q}$, a distinguished discrete-indexing projective cone at $q$;
\item for each $q\in\mathbb{Q}$, a distinguished projective cone expressing the interval condition above.
\end{itemize}
\end{proposition}

\noindent
\textbf{Proof.}
A model $M$ of $E_{\overline{\mathbb{R}}}$ in \textbf{Set} identifies with a map from $\mathbb{Q}$ to $\{0,1\}$, hence with an element of $\overline{\mathbb{R}}$.
Among these maps, exactly those satisfying the second family of cones are retained, namely
\[
\forall q\in\mathbb{Q},\quad M(q)=1 \Rightarrow M(q')=1\text{ for all }q'<q,
\]
which is equivalent to the inequality condition on associated elements.
Therefore
\[
\mathrm{Mod}(E_{\overline{\mathbb{R}}})\simeq\overline{\mathbb{R}}.
\]

\section{Some Non-Algebraic Variants}

Let $\Gamma$ be the ordered set $\mathbb{R}\cup\{-\infty\}$ (that is, $\mathbb{R}_0\cup\{+\infty\}$ in the notation of the original text).
The preceding analysis can be read as excluding from $\overline{\mathbb{R}}$ the constant functor $M(q)=1$ for all $q$, equivalently requiring
\[
\prod_{q\in\mathbb{Q}} M(q)=\varnothing.
\]
To express this, one enlarges the support of the sketch by adding cones corresponding to
\begin{itemize}[leftmargin=1.5em]
\item the empty-indexing inductive cone (interpreted as $+\infty$),
\item a non-discrete-indexing projective cone (for $q\in\mathbb{Q}$),
\end{itemize}
and equivalently enforces
\[
\lim_{q\in\mathbb{Q}} M(q)=1.
\]

Similarly, to sketch $\mathbb{R}\cup\{-\infty\}$ one modifies the support in the opposite direction and distinguishes the corresponding supplementary cones.
Taking support $(\widehat{\mathbb{Q}},\ge)$ and distinguishing all projective and inductive cones from both constructions gives a sketch of the real line itself, denoted $E_{\mathbb{R}}$, with
\[
\mathrm{Mod}(E_{\mathbb{R}})\simeq\mathbb{R}.
\]

Finally, by replacing one of the discrete projective cones by an appropriate non-discrete one, we obtain a sketch $E_{\mathbb{R}\setminus\mathbb{Q}}$ whose category of models is equivalent to the ordered set $\mathbb{R}\setminus\mathbb{Q}$.
Indeed, a model corresponds to an irrational iff, for every rational $q$,
\[
M(q)=\lim_{q'<q} M(q').
\]

Thus one sketches the doubleton functor (inclusion) $\mathbb{R}\setminus\mathbb{Q}\to\mathbb{R}$.
(And similarly, as left as an exercise in the original text, one can sketch the doubleton functor $\mathbb{Q}\to\mathbb{R}$.)

\section{Remarks}

The sketches above provide interesting illustrations of several recent classification theorems for sketches and categories accessible in nature.
From the sketch $E_{[-\infty,+\infty]}$ we note in particular:
\begin{enumerate}[label=(\arabic*),leftmargin=1.6em]
\item all distinguished projective cones are countably indexed and nonempty;
\item all distinguished inductive cones are of empty indexing.
\end{enumerate}

A natural question is which properties of the associated category $[-\infty,+\infty]$ are reflected by (1) and (2) together with the results cited in \cite{Ageron95,Ageron99}.
The following statement is verified in the source text.

\begin{proposition}
The categories of models of sketches satisfying (1) and (2) are exactly (up to equivalence) the accessible categories with directed colimits such that:
\begin{itemize}[leftmargin=1.5em]
\item[(a)] they have $\aleph_1$-filtered limits and contain a set of $\aleph_1$-presentable objects whose directed colimit is each object;
\item[(b)] they have strict initial object;
\item[(c)] they have nonempty projective limits.
\end{itemize}
\end{proposition}

In particular, the ordered set $[-\infty,+\infty]$, seen as a category, satisfies these conditions.
All of its objects are $\aleph_1$-presentable, and $-\infty$ is initial but not strict.

For $E_{[-\infty,+\infty]}$, one can formulate a more precise characterization by separating two classes of model categories:
\begin{enumerate}[label=(\arabic*),leftmargin=1.6em]
\item all distinguished projective cones are of countable indexing and nonempty;
\item all distinguished inductive cones are of empty indexing.
\end{enumerate}

This aligns with known theorems on categories accessible with finite projective limits and the role of conditional projective limits.
A modified sketch can also be built in which all cones are as above except one non-discrete inductive cone indexed by $(\mathbb{Q},\ge)$.
This yields an equivalent sketch that represents real numbers no longer as cuts in $\mathbb{Q}$, but as cuts in $\mathbb{R}$.

\section*{Bibliography}
\begin{thebibliography}{9}
\bibitem{AdamekKoubekVelebil}
J. Ad\'amek, V. Koubek, J. Velebil,
\emph{Duality between finitary varieties and algebraic theories}, prepublication, 13 p.

\bibitem{Ageron92}
P. Ageron,
\emph{The logic of structures}, Journal of Pure and Applied Algebra 79 (1992), 15--34.

\bibitem{Ageron95}
P. Ageron,
\emph{Cat\'egories accessibles \`a limites projectives non vides et cat\'egories accessibles \`a limites projectives finies},
Diagrammes 34 (1995), 1--10.

\bibitem{Ageron96}
P. Ageron,
\emph{Cat\'egories accessibles \`a produits fibr\'es}, Diagrammes 36 (1996), 1--11.

\bibitem{Ageron97}
P. Ageron,
\emph{Limites projectives conditionnelles dans les cat\'egories accessibles}, Diagrammes 38 (1997), 3--18.

\bibitem{Ageron99}
P. Ageron,
\emph{Limites inductives point par point dans les cat\'egories accessibles}, Diagrammes, to appear.

\bibitem{BorceuxQuinteiroRosicky}
F. Borceux, C. Quinteiro, J. Rosick\'y,
\emph{A theory of enriched sketches}, Theory and Applications of Categories 4 (1998), 47--72.
\end{thebibliography}

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