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

References

Introductions

Introductions include

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

• 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

A standard textbook is

• Peter Johnstone, Topos theory, London Math. Soc. Monographs 10, Acad. Press 1977, xxiii+367 pp.

This later grew into the more detailed

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

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)

A standard textbook on this case is

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

Introducing even category theory from the scratch while still managing to cover considerable 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)

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)

A survey is in

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

Course notes

• Ieke Moerdijk, Jaap van Oosten, Topos theory Master Class notes (2007) (pdf)

History

• Colin McLarty, The Uses and Abuses of the History of Topos Theory , Brit. J. Phil. Sci., 41 (1990) (JSTOR) PDF

• The peripatetic seminar on sheaves and logic 1976-1999