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\begin{document}

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\section*{locally decidable topos}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


[[!redirects QD topos]] [[!redirects locally decidable object]]

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Loosely speaking a \textbf{locally decidable topos} is a topos that is locally Boolean. Whereas in a [[Boolean topos]] every object is decidable, in a locally decidable topos every object is a quotient of a [[decidable object]]. This results in a reasonable approximation to the concept of `being almost Boolean'.

In Lawvere's approach to cohesion locally decidable toposes are one of the principal classes of \textbf{petit toposes} (Lawvere 1991). In the 1991 paper Lawvere also hypothesizes the discrete base topos $\mathcal{S}$ that he contrasts with the cohesive topos of spaces, to be locally decidable.\footnote{Lawvere uses the term `QD'-topos ( \emph{quotient of decidable}) for `locally decidable topos' in this paper. He also uses `SUD'-object (from \emph{separable-unramified-decidable} to indicate equivalent concepts in algebra-topology-logic) for what we call a locally decidable object. In later papers Lawvere uses also the terms `locally separable'or `adequately separable'. The idea to take a locally decidable base topos is probably suggested by [[Galois theory]] where `locally decidable' corresponds roughly to `product of separable field extensions' for an algebra over a field $k$. This also connects the notion `[[decidable object|decidable]]' via Grothendieck's Galois theory to the topological notion `unramified'.} 

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

An object $X$ in a topos $\mathcal{E}$ is called \emph{locally decidable} (or, a \emph{quotient of a decidable object}) iff there is an epimorphism $Y\twoheadrightarrow X$ such that $Y$ is a [[decidable object]]. $\mathcal{E}$ is called \emph{locally decidable} iff every object $X$ is locally decidable.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The [[classifying topos]] $Set[\mathbf{D}]=[FinSet_{mono},Set]$ for the [[theory of decidable objects]] $\mathbf{D}$ is locally decidable. Here $FinSet_{mono}$ is the category of finite sets and monomorphisms, whose opposite category is the carrier of the site for the so called (Myhill-) \emph{[[Schanuel topos]]}.


\item Every [[localic topos]] is locally decidable.


\item For every [[Grothendieck topos]] $\mathcal{E}$ the full subcategory $\mathcal{E}_{QD}$ of locally decidable objects is a locally decidable topos.



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item \textbf{Proposition}. A topos $\mathcal{E}$ is locally decidable iff there is a [[localic topos|localic]] geometric morphism to a Boolean topos, or iff $\mathcal{E}$ has a site $(\mathcal{C}, J)$ with all morphisms in $\mathcal{C}$ epic.


\item Locally decidable (presheaf) toposes are `co-\'e{}tendues' in the sense that for a small $\mathcal{C}$ the functor category $[\mathcal{C},Set]$ is locally decidable precisely if $[\mathcal{C}^{op},Set]$ is an [[étendue]]. Also the all-epic-site property is dual to the all-monic-site property of \'e{}tendues. Both concepts are subsumed under the notion of having a (sub canonical) site representation with no (non-trivial) [[idempotents]] (McLarty 2006, Lawvere 2007,2008).


\item A locally decidable topos has a [[localic topos|localic geometric morphism]] to the [[Schanuel topos]] (cf. Johnstone 2002, p.794).



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Boolean topos]]
\item [[petit topos]]
\item [[étendue]]
\item [[decidable object]]
\item [[theory of decidable objects]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Peter Freyd]], \emph{All topoi are localic - or, why permutation models prevail} , JPAA \textbf{46} (1987) pp.49-58.


\item [[Simon Henry]], \emph{On toposes generated by finite cardinal objects} , arxiv:1505.04987 (2015). (\href{http://arxiv.org/pdf/1505.04987v1.pdf}{pdf})


\item [[Peter Johnstone]], \emph{Quotients of decidable objects in a topos} , Math. Proc. Camb. Phil. Soc. \textbf{93} (1983) pp.409-419.


\item [[Peter Johnstone]], \emph{How general is a generalized space?} , London Math. Soc. LNS \textbf{93} (1985) pp.77-111.


\item [[Peter Johnstone]], \emph{Sketches of an Elephant II}, Oxford UP 2002. (sec. C5.4 pp.792-803)


\item [[F. William Lawvere]], \emph{Qualitative Distinctions between some Toposes of Generalized Graphs} , Cont. Math. \textbf{92} (1989) pp.261-299.


\item [[F. William Lawvere]], \emph{Some Thoughts on the Future of Category Theory} , pp.1-13 in Springer LNM vol. 1488 (1991).


\item [[F. William Lawvere]], \emph{Axiomatic Cohesion} , TAC \textbf{19} no. 3 (2007) pp.41-49. (\href{http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf}{pdf})


\item [[F. William Lawvere]], \emph{Cohesive Toposes: Combinatorial and Infinitesimal Cases}, Como Ms. 2008. (\href{http://comocategoryarchive.com/Archive/temporary_new_material/FWLawvere-Cohesive-Toposes-Como-January-2008.pdf}{pdf})


\item [[Colin McLarty]], \emph{Every Grothendieck Topos has a One-Way Site} , TAC \textbf{16} no. 5 (2006) pp.123-126. (\href{http://www.tac.mta.ca/tac/volumes/16/5/16-05.pdf}{pdf})



\end{itemize}


\end{document}