Topos theory

An elementary topos can be regarded as a category that is “enough like the category of sets” to interpret intuitionistic higher-order logic. In this sense, topos theory is a branch of categorical logic. But that is only one way to think about what a topos is.

Literature

Surveys

  • Baez, 2021: Topos theory in a nutshell
  • Baez, 2020: Lecture notes on topos theory (Azimuth 1 ,2 ,3 ,4 ,5 ,6 ,7 ,8 )
  • Blechschmidt, 2022: Exploring mathematical objects from custom-tailored mathematical universes (doi, arxiv)
  • Kostecki, 2011: An introduction to topos theory (pdf)

Books on topos theory, in roughly increasing order of difficulty:

  • Lawvere & Schanuel, 2009: Conceptual mathematics, 2nd ed. (doi)
    • Elementary introduction to the topos of sets
  • Reyes, Reyes, Zolfaghari, 2004: Generic figures and their glueings: A constructive approach to functor categories (online , pdf)
    • A nice book on “combinatorial toposes”, such as the topos of graphs
    • Main examples also appear in videos by MathProofsable
    • Sec 9.1 has interesting interpretation of the two negation operations in categories of presheaves: Reyes et al, 1994: The non-boolean logic of natural language negation (doi)
  • Goldblatt, 1984: Topoi: The categorical analysis of logic (online )
    • Level similar to Reyes et al, but focus on logic, not geometry
  • Mac Lane & Moerdijk, 1992: Sheaves in geometry and logic: A first introduction to topos theory (doi, nLab )
  • Johnstone, 2002: Sketches of an Elephant: A topos theory compendium (nLab )
    • The bible of topos theory
    • In three volumes, two of which are published (~1200 pages)
    • Sequel to Johnstone, 1977: Topos theory (nLab )