nLab sheaf and topos theory



Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory


Category Theory



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


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:


Brief expositions:

Lecture notes:

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


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


  • 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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