nLab sheaf and topos theory

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Category Theory

Contents

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

References

Original texts

Discussion in algebraic geometry:

Introductions

A gentle basic introduction, with recommendations for further reading, is

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

Textbooks

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:

See also

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

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

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