Quotient topos
Contents
1. Definition§
A quotient topos of a topos $\mathcal E$ is a connected geometric morphism $f:\mathcal E\to\mathcal F$, considered up to equivalence under $\mathcal E$. Equivalently, it is the essential image of $f^*$. See Definitions 2.1–2.2 and Remark 2.3 and Definition 3.2.
2. Examples§
- The quotients of the topos of discrete dynamical systems admit an explicit classification.
- Hyperconnected quotients are classified using a Local state classifier when one exists.
3. Properties§
Theorem 3.1 (A proper class of quotients)§★
Let $M_\omega$ be the free monoid on countably infinitely many generators. The topos of right $M_\omega$-sets has a proper class of mutually inequivalent quotient topoi.
See Corollary 6.19. Thus quotient topoi of a Grothendieck topos need not form a set.
4. Related concepts§
5. References§
- Yuhi Kamio and Ryuya Hora, Solution to Lawvere's first problem: A Grothendieck topos that has proper class many quotient topoi, Adv. Math. 487 (2026) 110751.
- Ryuya Hora and Yuhi Kamio, Quotient toposes of discrete dynamical systems, J. Pure Appl. Algebra 228 (2024) 107657, §§3.1 and 5.4.
- Ryuya Hora, Internal Parameterization of Hyperconnected Quotients, Theory and Applications of Categories 42 (2024), no. 11, 263–313, §2 and Theorem 4.1.