A solution to the first Lawvere's problem A Grothendieck topos that has a proper class many quotient topoi

Abstract

This paper solves the first problem of the open problems in topos theory posted by William Lawvere, which asks the existence of a Grothendieck topos that has a proper class many quotient topoi. This paper concretely constructs such Grothendieck topoi, including the presheaf topos of the free monoid generated by countably infinite elements $\mathbf{PSh}(M_\omega)$. Utilizing the combinatorics of the classifying topos of the theory of inhabited objects and considering pairing functions, the problem is reduced to making rigid relational structures. This is accomplished by using Kunen's theorem on elementary embeddings in set theory.