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



A gentle basic introduction 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


The mother of it all though not exactly a textbook

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

Maybe the standard modern textbook on Grothendieck toposes is

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

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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