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\begin{document}

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\section*{sheaf and topos theory}

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

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

[[!include topos theory - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category Theory}}\label{category_theory}

[[!include category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{introductions}{Introductions}\dotfill \pageref*{introductions} \linebreak
\noindent\hyperlink{textbooks}{Textbooks}\dotfill \pageref*{textbooks} \linebreak
\noindent\hyperlink{course_notes}{Course notes}\dotfill \pageref*{course_notes} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak


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

\emph{Topos theory} is the part of [[category theory]] that studies [[categories]] which are [[topos]]es. This includes in particular [[Grothendieck toposes]], i.e. [[categories of sheaves]].

There are always two ways to think of topos theory: as being

\begin{itemize}%
\item about [[logic]]


\item about [[higher geometry|geometry]].



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

\begin{itemize}%
\item \textbf{topos theory}


\item [[2-topos theory]]


\item [[(∞,1)-topos theory]]


\item [[higher topos theory]]


\item [[microlocal sheaf theory]]



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

\hypertarget{introductions}{}\subsubsection*{{Introductions}}\label{introductions}

A gentle basic introduction is

\begin{itemize}%
\item [[John Baez]], \href{http://math.ucr.edu/home/baez/topos.html}{Topos Theory in a Nutshell}.

\end{itemize}
A quick introduction of the basic facts of [[Grothendieck topos]] theory is chapter I, ``Background in topos theory'' in

\begin{itemize}%
\item [[Ieke Moerdijk]], \emph{Classifying Spaces and Classifying Topoi} Lecture Notes in Mathematics 1616, Springer (1995).

\end{itemize}
Other introductions include

\begin{itemize}%
\item [[Tom Leinster]], \emph{[[Leinster2010|An informal introduction to topos theory]]} (2010)


\item [[André Joyal]], \emph{[[A crash course in topos theory -- The big picture]]}, lecture series at \href{https://indico.math.cnrs.fr/event/747/}{Topos \`a{} l'IHES}, November 2015, Paris


\item [[André Joyal]], \emph{Geometric aspects of topos theory in relation with logical doctrines}, talk at \emph{[[New Spaces for Mathematics and Physics]]}, IHP Paris 2015 (\href{https://www.youtube.com/watch?v=kaZpOEOAUzE}{video recording})



\end{itemize}
\hypertarget{textbooks}{}\subsubsection*{{Textbooks}}\label{textbooks}

The mother of it all though not exactly a textbook

\begin{itemize}%
\item [[Michael Artin|M.Artin]], [[Alexander Grothendieck|A.Grothendieck]], [[J. L. Verdier]] (eds.), \emph{Théorie des Topos et Cohomologie Etale des Schémas - SGA 4} , LNM \textbf{269} Springer Heidelberg 1972.

\end{itemize}
A still very useful reference is the monograph

\begin{itemize}%
\item [[Peter Johnstone]], \emph{Topos theory}, London Math. Soc. Monographs \textbf{10}, Acad. Press 1977, xxiii+367 pp. (Available as Dover Reprint, Mineola 2014)

\end{itemize}
This later grew into the more detailed

\begin{itemize}%
\item [[Peter Johnstone]], [[Elephant|Sketches of an elephant: a topos theory compendium]]

\end{itemize}
Maybe the standard modern textbook on Grothendieck toposes is

\begin{itemize}%
\item [[Saunders MacLane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]}

\end{itemize}
A thorough but clear first introduction to topos theory is

\begin{itemize}%
\item [[Francis Borceux]], \emph{Handbook of Categorical Algebra 3 - Categories of Sheaves} , Cambridge UP 1994.

\end{itemize}
Introducing even category theory from the scratch while still managing to cover some ground, the following textbook is the \emph{royal road to topos theory} for people with some background in [[first-order logic]]:

\begin{itemize}%
\item R. Goldblatt, \emph{Topoi - The Categorical Analysis of Logic} , 2nd ed. North-Holland Amsterdam 1984. (Dover reprint New York 2006; \href{http://projecteuclid.org/euclid.bia/1403013939}{project euclid})

\end{itemize}
Similarly, the following monograph develops topos theory to considerable depth without categorical prerequisites

\begin{itemize}%
\item [[Michael Barr]], [[Charles Wells]], \emph{Toposes, Triples and Theories} , Springer Heidelberg 1985. (Available as \href{http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html}{TAC reprint no.12} 2005)

\end{itemize}
See also

\begin{itemize}%
\item [[Olivia Caramello]], \emph{Theories, Sites, Toposes}, Oxford UP 2017.

\end{itemize}
\hypertarget{course_notes}{}\subsubsection*{{Course notes}}\label{course_notes}

A survey is in

\begin{itemize}%
\item [[Ross Street]], \emph{A survey of topos theory} (notes for students, 1978) \href{http://www.math.mq.edu.au/~street/ToposSurvey.pdf}{pdf}

\end{itemize}
A nice and concise introduction is available in

\begin{itemize}%
\item [[Francis Borceux]], \emph{Some glances at topos theory} , lecture notes Como 2018. (\href{http://tcsc.lakecomoschool.org/files/2018/06/Como2018.pdf}{pdf}, \href{https://www.youtube.com/watch?v=s_fN9euuVAY&list=PLh_3Q6ZRqWs0LBptMGClJ8OArR0fBT6Pp&index=11}{video playlist})


\item [[Ieke Moerdijk]], [[Jaap van Oosten]], \emph{Topos theory} Master Class notes (2007) (\href{http://www.staff.science.uu.nl/~ooste110/syllabi/toposmoeder.pdf}{pdf})



\end{itemize}
\hypertarget{history}{}\subsubsection*{{History}}\label{history}

\begin{itemize}%
\item [[F. William Lawvere]], \emph{Comments on the development of topos theory}, pp.715-734 in Pier (ed.), \emph{Development of Mathematics 1950 - 2000} , Birkh\"a{}user Basel 2000. (\href{http://www.tac.mta.ca/tac/reprints/articles/24/tr24abs.html}{tac reprint})


\item [[Colin McLarty]], \emph{The Uses and Abuses of the History of Topos Theory} , Brit. J. Phil. Sci., 41 (1990) (\href{http://www.jstor.org/stable/687825}{JSTOR}) \href{http://bjps.oxfordjournals.org/content/41/3/351.full.pdf}{PDF}



\end{itemize}
A historical analysis of Grothendieck's 1973 Buffalo lecture series on toposes and their precedents is in

\begin{itemize}%
\item [[Colin McLarty]], \emph{Grothendieck's 1973 topos lectures}, Séminaire Lectures grothendieckiennes, 3 May (2018) (\href{https://www.youtube.com/watch?v=hhWT5V0oaSI}{YouTube video})


\item \emph{\href{http://www.wra1th.plus.com/gcw/math/PSSL/index.html}{The peripatetic seminar on sheaves and logic 1976-1999}}



\end{itemize}
[[!redirects Sheaf and Topos Theory]] [[!redirects sheaf theory]] [[!redirects topos theory]]



\end{document}