Topos theory is the part of category theory that studies categories which are toposes. This includes in particular Grothendieck toposes, i.e. categories of sheaves.
There are always two ways to think of topos theory: as being
Discussion in algebraic geometry:
Michael Artin, Alexander Grothendieck, Jean-Louis Verdier (eds.), Théorie des Topos et Cohomologie Etale des Schémas - SGA 4 , LNM 269 Springer Heidelberg 1972.
Monique Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972) (doi:10.1007/978-3-662-59155-0)
A gentle basic introduction is
A quick introduction of the basic facts of Grothendieck topos theory is chapter I, “Background in topos theory” in
Other introductions include
Tom Leinster, An informal introduction to topos theory (2010)
André Joyal, A crash course in topos theory – The big picture, lecture series at Topos à l’IHES, November 2015, Paris
André Joyal, Geometric aspects of topos theory in relation with logical doctrines, talk at New Spaces for Mathematics and Physics, IHP Paris 2015 (video recording)
Peter Johnstone, Topos theory, London Math. Soc. Monographs 10, Acad. Press 1977, xxiii+367 pp. (Available as Dover Reprint, Mineola 2014)
Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic
Peter Johnstone, Sketches of an elephant: a topos theory compendium
Francis Borceux, Handbook of Categorical Algebra 3 - Categories of Sheaves, Cambridge UP 1994 (ISBN:9780521061247)
Introducing even category theory from the scratch while still managing to cover some ground, the following textbook is the royal road to topos theory for people with some background in first-order logic:
Similarly, the following monograph develops topos theory to considerable depth without categorical prerequisites
See also
A survey is in
A nice and concise introduction is available in
Francis Borceux, Some glances at topos theory , lecture notes Como 2018. (pdf, video playlist)
Ieke Moerdijk, Jaap van Oosten, Topos theory Master Class notes (2007) (pdf)
F. William Lawvere, Comments on the development of topos theory, pp.715-734 in Pier (ed.), Development of Mathematics 1950 - 2000 , Birkhäuser Basel 2000. (tac reprint)
Colin McLarty, The Uses and Abuses of the History of Topos Theory , Brit. J. Phil. Sci., 41 (1990) (JSTOR) PDF
A historical analysis of Grothendieck’s 1973 Buffalo lecture series on toposes and their precedents is in
Colin McLarty, Grothendieck’s 1973 topos lectures, Séminaire Lectures grothendieckiennes, 3 May (2018) (YouTube video)
Last revised on October 31, 2020 at 08:27:03. See the history of this page for a list of all contributions to it.