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

• 2-topos theory

• (∞,1)-topos theory

• higher topos theory

• microlocal sheaf theory

References

Introductions

A gentle basic introduction is

• John Baez, Topos Theory in a Nutshell.

A quick introduction of the basic facts of Grothendieck topos theory is chapter I, “Background in topos theory” in

• Ieke Moerdijk, Classifying Spaces and Classifying Topoi Lecture Notes in Mathematics 1616, Springer (1995).

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)

Textbooks

The mother of it all though not exactly a textbook

• M.Artin, A.Grothendieck, J. L. Verdier (eds.), Théorie des Topos et Cohomologie Etale des Schémas - SGA 4 , LNM 269 Springer Heidelberg 1972.

A still very useful reference is the monograph

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

This later grew into the more detailed

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

Maybe the standard modern textbook on Grothendieck toposes is

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

A thorough but clear first introduction to topos theory is

• Francis Borceux, Handbook of Categorical Algebra 3 - Categories of Sheaves , Cambridge UP 1994.

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:

• R. Goldblatt, Topoi - The Categorical Analysis of Logic , 2nd ed. North-Holland Amsterdam 1984. (Dover reprint New York 2006; project euclid)

Similarly, the following monograph develops topos theory to considerable depth without categorical prerequisites

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

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)

• 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