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

TL;DR

This paper constructs a Grothendieck topos with a proper class of quotient topoi, solving Lawvere's first open problem in topos theory by leveraging combinatorics and a theorem on rigid relational structures.

Contribution

It provides a concrete construction of a Grothendieck topos with a proper class of quotient topoi, addressing a longstanding open problem in topos theory.

Findings

Constructed a Grothendieck topos with class-many quotient topoi

Reduced the problem to a theorem on rigid relational structures

Utilized combinatorics of the classifying topos of inhabited objects

Abstract

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_{\omega}). 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.