The modal theory of the category of sets
Wojciech Aleksander Wo{\l}oszyn

TL;DR
This paper classifies propositional modal validities in the category of sets based on morphism classes, size, and substitution, using a modal and quantifier elimination approach.
Contribution
It introduces a modality-and-quantifier elimination theorem that enables exact classification of modal validities for various set categories.
Findings
Finite n-element worlds realize Prepartition_n.
Surjection worlds realize Grz.3J_n at the sentential level.
Infinite categories have trivial sentential validities, with function and surjection validities matching Grz.2.
Abstract
We classify the propositional modal validities arising from the category of sets under its natural classes of morphisms. The resulting validities depend on the morphism class, the size of the world, and the permitted substitution instances. Our main technical tool is a modality-and-quantifier elimination theorem for the first-order modal language of equality, reducing formulas to finite Boolean combinations of partition conditions and exact cardinality assertions. This yields exact classifications for the main categories of sets, including the full subcategories of finite sets and of infinite sets. In particular, finite -element worlds in the category of sets with parameters realize ; finite -element worlds in the category of sets and surjections realize at the sentential level; and in the infinite-only subcategories, sentential…
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