nLab sheaf and topos theory

Idea

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

• about logic

• about geometry.

Related concepts

• topos theory

  • Giraud's theorem

  • fundamental theorem of topos theory

• 2-topos theory

• (∞,1)-topos theory

• higher topos theory

• microlocal sheaf theory

References

Original texts

Discussion in algebraic topology:

• Roger Godement, Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]

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)

Introductions

Brief expositions:

• Michael Barr, Colin McLarty, Charles Wells, Variable set theory, prepared for Scientific American but unpublished (~1985) [pdf, pdf]

• Ieke Moerdijk, Background in topos theory, chapter I in: Classifying Spaces and Classifying Topoi, Lecture Notes in Mathematics 1616, Springer (1995) [doi:10.1007/BFb0094441]

• Luc Illusie, What is… a Topos?, Notices of the AMS 51 9 (2004) [pdf, full volume:pdf]

• Tom Leinster, An informal introduction to topos theory (2010) [arXiv:1012.5647]

• MathProofsable: Toposes - “Nice Places to Do Math” (2017) [video:YT]

• John Baez, Topos Theory in a Nutshell (2021) [web]

• Urs Schreiber: Higher Topos Theory in Physics, Encyclopedia of Mathematical Physics 2nd ed$\;$4 (2025) 62-76 [arXiv:2311.11026, doi:10.1016/B978-0-323-95703-8.00210-X]

Lecture notes:

• Ross Street, A survey of topos theory, lecture notes (1978) [pdf, pdf]

• Oswald Wyler, Lecture Notes on Topoi and Quasitopoi, World Scientific (1991) [doi:10.1142/1047]

• André Joyal: A crash course in topos theory – The big picture, lecture series at Topos à l’IHES, Paris (November 2015) [videos: part 1, 2, 3, 4]

• 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)

• Ieke Moerdijk, Jaap van Oosten, Topos Theory, Master class notes (2007) [pdf]

• Jaap van Oosten, Topos Theory (2018) [pdf, pdf]

• Francis Borceux, Some glances at topos theory, lecture notes, Como (2018) [pdf, pdf, video playlist]

• Pierre Schapira, An Introduction to Categories and Sheaves, lecture notes (2023) [pdf, pdf]

A monograph that aims to be more expository, focusing on presheaf toposes:

• Marie La Palme Reyes, Gonzalo E. Reyes, Houman Zolfaghari, Generic figures and their glueings. A constructive approach to functor categories, Polimetrica (2008) [pdf, ISBN:8876990046]

Monographs

• Peter Johnstone, Topos theory, London Math. Soc. Monographs 10, Acad. Press 1977, xxiii+367 pp. (Available as Dover Reprint, Mineola 2014)

• Michael Barr, Charles Wells, Toposes, Triples, and Theories , Springer Heidelberg 1985. (Available as TAC reprint no.12 2005)

• Colin McLarty, Elementary Categories, Elementary Toposes, Oxford University Press 1992 (ISBN:9780198514732)

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic, Springer (1992) [doi:10.1007/978-1-4612-0927-0]

• Francis Borceux, Handbook of Categorical Algebra 3 - Categories of Sheaves, Cambridge UP (1994) [ISBN:9780521061247, doi:10.1017/CBO9780511525872]

• Peter Johnstone, Sketches of an elephant: a topos theory compendium (2002)

• Alexandru Dimca, Sheaves in Topology, Universitext, Springer (2004) [doi:10.1007/978-3-642-18868-8]

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332 Springer (2006) [doi:10.1007/3-540-27950-4, pdf]

• Garth Warner, Homotopical Topos Theory, EPrint Collection, University of Washington (2012) [hdl:1773/19722, pdf, pdf]

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:

• Robert Goldblatt, Topoi: The Categorial Analysis of Logic, 2nd ed. North-Holland Amsterdam 1984. (Dover reprint New York 2006; free online at Project Euclid.)

See also

• Olivia Caramello, Theories, Sites, Toposes, Oxford UP 2017.

Course notes

A survey is in

• Ross Street, A survey of topos theory (notes for students, 1978) pdf

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)

History

• 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)

• The peripatetic seminar on sheaves and logic 1976-1999