UWO Reading Seminar on Topos Theory, Fall 2025

Organisers: Theofanis Chatzidiamantis-Christoforidis [tchatzid 'at' uwo.ca], Thomas Thorbjørnsen [tthorbjr 'at' uwo.ca].

Meetings: Mondays, 13:30-15:30, room MC108.

Schedule

• [15.09.2025] Talk 1: Introduction.
  Speaker: Theofanis Chatzidiamantis-Christoforidis.
  References: [6, sections 5.1-5.4][13, part 1, sections 1-3][15, sections I.1-I.6].


• [22.09.2025] Talk 2: Heyting Algebras and Logic in a Topos.
  Speaker: Zack Dooley.
  References: [6, sections 6.1-6.2][15, sections I.7-I.9][17, chapter 13].


• [29.09.2025] Talk 3: Grothendieck Topoi, part I.
  Speaker: Mac Martin.
  References: [6, sections 3.1-3.2][15, sections III.1, III.2, III.4-III.8][12, sections 16.1-16.3 and 17.1-17.4].


• [06.10.2025] Talk 4: Grothendieck Topoi, part II.
  Speaker: Alex Zwart.
  References: [Stacks, chapters 34.1-34.4][19, chapter 2].


• [15.10.2025] Talk 5: Lawvere-Tierney Topologies. Date and time change: Wednesday October 15, 15:30.
  Speaker: Elias Vandenberg.
  References: [6, chapter 9][10, sections A2.1, A4.4-A4.5][15, chapter 5].


• [20.10.2025] Talk 6: Colimits and Slices of Topoi.
  Speaker: Adrien Grenier.
  Reference: [6, sections 5.5, 5.7-5.8]


• [27.10.2025] Talk 7: The Mitchell-Bénabou Language
  Speaker: Dan Christensen.


• [03.11.2025] Talk 8: Forcing Semantics I.
  Speaker: Theofanis Chatzidiamantis-Christoforidis.


• [10.11.2025] Talk 9: Forcing Semantics II.
  Speaker: Zack Dooley.


• [17.11.2025] Talk 10: Classifying Topoi.
  Speaker: Benni Ngo.


• [24.11.2025] Talk 11: Set-theoretic Forcing and ETCS.
  Speaker: Theofanis Chatzidiamantis-Christoforidis.


• [01.12.2025] Talk 12: Forcing in a Topos.
  Speaker: Thomas Thorbjørnsen.

Useful References

1. Anel, M., Biedermann, G., Finster, E., & Joyal, A. (2022). Left-exact localizations of \(\infty\)-topoi I: Higher sheaves. Advances in Mathematics, 400.
2. Anel, M., Biedermann, G., Finster, E., & Joyal, A. (2023). Left-exact Localizations of \(\infty\)-Topoi III: The Acyclic Product. arXiv: 2308.15573.
3. Anel, M., Biedermann, G., Finster, E., & Joyal, A. (2024). Left-exact localizations of \(\infty\)-topoi II: Grothendieck topologies. Journal of Pure and Applied Algebra, 228(3).
4. Anel, M., & Joyal, A. (2021). Topo-logie. In M. Anel & G. Catren (Eds.), New Spaces in Mathematics: Formal and Conceptual Reflections (pp. 155–257). Cambridge University Press.
5. Bell, J. L. (2005). Set Theory: Boolean-Valued Models and Independence Proofs. Oxford University Press.
6. Borceux, F. (1994). Handbook of categorical algebra. 3: Categories of sheaves (Vol. 52, p. xviii+522). Cambridge University Press, Cambridge.
7. Caramello, O. (2018). Theories, sites, toposes (p. xii+368). Oxford University Press, Oxford.
8. Chow, T. Y. (2008). A beginner’s guide to forcing. arXiv: 0712.1320.
9. Jech, T. (2003). Set theory (millennium, p. xiv+769). Springer-Verlag, Berlin.
10. Johnstone, P. T. (2002a). Sketches of an elephant: a topos theory compendium. Vol. 1 (Vol. 43, p. xxii+468+71). The Clarendon Press, Oxford University Press, New York.
11. Johnstone, P. T. (2002b). Sketches of an elephant: a topos theory compendium. Vol. 2 (Vol. 44, pp. i–xxii, 469–1089 and I1–I71). The Clarendon Press, Oxford University Press, Oxford.
12. Kashiwara, M., & Schapira, P. (2006). Categories and sheaves (Vol. 332, p. x+497). Springer-Verlag, Berlin.
13. Lambek, J., & Scott, P. J. (1986). Introduction to higher order categorical logic (Vol. 7, p. x+293). Cambridge University Press, Cambridge.
14. Lurie, J. (2009). Higher topos theory (Vol. 170, p. xviii+925). Princeton University Press, Princeton, NJ.
15. Mac Lane, S., & Moerdijk, I. (1994). Sheaves in geometry and logic (p. xii+629). Springer-Verlag, New York.
16. Makkai, M., & Reyes, G. E. (1977). First order categorical logic: Vol. Vol. 611 (p. viii+301). Springer-Verlag, Berlin-New York.
17. McLarty, C. (1992). Elementary Categories, Elementary Toposes. Oxford University Press.
18. Moerdijk, I. (1995). Classifying spaces and classifying topoi (Vol. 1616, p. vi+94). Springer-Verlag, Berlin.
19. Olsson, M. (2016). Algebraic spaces and stacks (Vol. 62, p. xi+298). American Mathematical Society, Providence, RI.
20. Rasekh, N. (2021). Filter quotients and non-presentable \((\infty,1)\)-toposes. Journal of Pure and Applied Algebra, 225(12).
21. Rasekh, N. (2022). A Theory of Elementary Higher Toposes. arXiv: 1805.03805.
22. Rezk, C. (2019). Lectures on Higher Topos Theory. Available at https://rezk.web.illinois.edu/leeds-lectures-2019.pdf.
23. Riehl, E. (2024). On the \(\infty\)-topos semantics of homotopy type theory. Bulletin of the London Mathematical Society, 56(2), 461–517.
24. Shulman, M. (2019). All \((\infty,1)\)-toposes have strict univalent universes. arXiv: 1904.07004.