Topos Theory

Time and Place:

Wednesday 16:15-18:00 in the MPI Seminar Room


David Carchedi, Max Planck Institute for Mathematics, office: 314


Course structure:

There will be homework problems each week. Each class session shall start with a lecture, a short break, and then a session dedicated to the previous week's homework problems. The homework problems will appear on this site.


Topos theory has many different guises. On one hand, a Grothendieck topos is a generalization (in fact categorification) of a topological space, a viewpoint which underpinned Grothendieck's own intuition on topoi, and aided his proof of one of the Weil conjectures. On the other hand, every topos can be thought of as a mathematical universe itself in which one can do mathematics. In fact, there is a duality between Grothendieck topoi and certain first-order theories of logic, called geometric theories. The interplay of these many different sides of topos theory is what makes the subject so rich.

As a first introduction to this subject, this course will take a geometric emphasis, spending more time developing the role of topoi as generalized spaces, and as universes of generalized spaces.

The course is now over. Here are the complete lecture notes:


Students wishing to take this course should be familiar with some basic concepts from category theory, e.g. natural transformations, limits and colimits, adjunctions etc. Familiarity with 2-categories/bicategories is helpful, but not at all necessary.


This course will not be officially following any book, but here is a list of resources:

The following book is an excellent introduction to topos theory:

  • Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic
  • It covers many (but not all) topics of this course in a greater detail than possible in these lectures (and also many other interesting topics). It is a very instructive read.

    A much larger and even more exhaustive source is:

  • Peter Johnstone, Sketches of an elephant: a topos theory compendium
  • This reference is not recommended to learn the subject, as it is too vast, but it is quite useful to use as a reference to look up definitions.

    We also highly recommend:

  • Francis Borceux, Handbook of Categorical Algebra: Volume 3, Sheaf Theory
  • for a very formally categorical introduction to both topos theory and the theory of locales.

    Finally, here is a link to slides from a course on topos theory (with a slightly different emphasis than this course) given by Olivia Caramello:

  • Topos Theory (Cambridge)