🤖 AI Summary
This paper addresses the foundational challenge of constructing a categorical framework for first-order modal logic, specifically how modal operators directly act on subobjects and interact with background factorization systems to model relational semantics. Method: It introduces, for the first time, a modal (quasi-)elementary topos structure; establishes a Joyal-style representation theorem formalizing “counterpart” semantics; and enhances categorical completeness via quotients and coproducts. Contributions/Results: It provides a systematic syntactic-to-categorical construction of first-order modal theories; delivers categorical characterizations of saturation conditions and definability problems; and furnishes modal logic with a unified, higher-order, and semantically rich categorical foundation—bridging syntax, semantics, and category theory in a principled way.
📝 Abstract
We extend the logical categories framework to first order modal logic. In our modal categories, modal operators are applied directly to subobjects and interact with the background factorization system. We prove a Joyal-style representation theorem into relational structures formalizing a `counterpart' notion. We investigate saturation conditions related to definability questions and we enrich our framework with quotients and disjoint sums, thus leading to the notion of a modal (quasi) pretopos. We finally show how to build syntactic categories out of first order modal theories.