📝 SummarXiv

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 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 a theorem of Vop\v{e}nka, Pultr, and Hedrl\'{{\i}}n, which states that any set admits a rigid relational structure.