Solution to Lawvere's first problem: a Grothendieck topos that has proper class many quotient topoi

This paper solves the first of the open problems in topos theory posted by William Lawvere, concerning the existence of a Grothendieck topos that has proper class many quotient topoi. This paper concretely constructs such Grothendieck topoi, including the presheaf topos on the free monoid generated by countably infinitely many elements PSh(M_ω). Utilizing the combinatorics of the classifying topos of the theory of inhabited objects and with the help of a system of pairing functions, the problem is reduced to a theorem of Vopenka, Pultr, and Hedrlin, which states that any set admits a rigid relational structure.

💡 Research Summary

The paper addresses the first of William Lawvere’s seven open problems in topos theory, namely whether there exists a Grothendieck topos that possesses a “large” collection of quotient topoi—more precisely, a proper class of mutually non‑equivalent quotient topoi (connected geometric morphisms). The authors give an explicit affirmative answer by constructing such a topos and exhibiting a proper class of distinct connected geometric morphisms out of it.

The work begins with a careful clarification of terminology. A quotient topos of a topos (E) is defined as an equivalence class of connected geometric morphisms (f\colon E\to F), where two morphisms are equivalent if there is an equivalence of target topoi making the obvious diagram commute up to natural isomorphism. This eliminates the trivial observation that any topos has a proper class of quotients merely by taking all equivalences of categories.

The technical heart of the paper lies in the interaction between two classifying topoi: the classifying topos (A=