theory-topos duality

This paper axiomatizes the semantic structure of geometric logic fragments within the 2-category of topoi. To address this, it develops a formal theory of Kan injectivity and pointwise Kan extensions, thereby constructing a unified framework for essential algebraic, disjunctive, regular, coherent, and other geometric logic fragments. It identifies the corresponding lax-idempotent pseudomonad (T^H) on the 2-category of finitely complete categories and proves that its pseudoalgebra 2-category is precisely the 2-categorical analogue of the classifying topos for each fragment. Key contributions include: (i) the first systematic development of Kan extension theory internal to the 2-category of topoi; (ii) a bijective correspondence between geometric logic fragments and lax-idempotent pseudomonads; (iii) several Diaconescu-type representation theorems; and (iv) a conceptual completeness characterization of geometric logic fragments in terms of their classifying topoi.

This work aims to unify the characterization of behavioral equivalences across the linear-time–branching-time spectrum in labeled transition systems. By leveraging topos theory, it interprets behavioral equivalence as localization and, for the first time, establishes a semantic foundation for the process algebraic equivalence spectrum within geometric logic by integrating Grothendieck topologies with an energy-game framework. The main contributions include a geometric closure theorem revealing that the equivalence spectrum forms a bi-Heyting algebra, and the construction of a 30-element closure lattice \( L_{30} \) encompassing 13 classical and 17 novel hybrid equivalences. All results are constructively proved and formally verified in Lean 4/Mathlib.

This work establishes a duality between relations among computational systems—such as bisimulation—and relations among logical predicates, thereby enabling cross-system logical reasoning. By extending Tarski duality and Thomason duality to the relational level for the first time, and integrating tools from category theory, Kripke semantics, and infinitary modal logic, the authors construct a dual framework that systematically links system relations with predicate relations. Building on this foundation, they develop a novel proof system capable of formally relating formulas across distinct systems. The resulting framework provides a robust theoretical basis for program logics and verification of concurrent systems, while significantly broadening the scope of classical duality theory within relational semantics.

This paper establishes a rigorous categorical semantics for e-graphs (equivalence graphs) within the framework of monadic categories, supporting double-pushout (DPO) rewriting. Method: The authors generalize e-graphs to monadic categories by introducing *equivalence hypergraphs* (e-hypergraphs)—a compositional structure whose vertices are algebras over a monad and whose hyperedges encode algebraic operations, thereby internalizing structural equations up to isomorphism. The approach integrates category theory, semilattice-enriched categories, and hypergraph-based combinatorial modeling to yield a sound and complete semantic framework. Contribution/Results: The resulting framework provides an algebraic and monadic foundation for equivalence reasoning in e-graph–based program optimization, and extends the formal applicability of e-graphs to SMT solving and algebraic optimization—enabling principled, categorical treatment of equational rewriting beyond traditional graph-based methods.

This paper addresses the foundational challenge of unifying characterizations of logical equivalence preservation under model composition—such as products, disjoint unions, and sums—in finite model theory. Method: It systematically incorporates Feferman–Vaught–Mostowski (FVM)-style theorems into a game-based comonadic semantics framework, extending coalgebraic semantics for model comparison games to enable parametric modeling of model classes, logical fragments, and composition operations. Contributions: (i) A unified reconstruction of several classical FVM theorems; (ii) the first parametric FVM theorem for arbitrary logics over structural products; (iii) an identification of deep connections between FVM phenomena and monad theory; and (iv) an improvement upon Dawar et al.’s results on C³ logic and cospectrality, thereby providing categorical foundations for modular logical reasoning. The approach yields a principled, compositional account of logical invariance across model constructions, advancing both finite model theory and categorical logic.

This study investigates conceptual completeness for sub-geometric fragments of first-order logic, including coherent logic, regular logic, disjunctive logic, and essentially algebraic logic with falsum. By recasting conceptual completeness as a duality between theories and Grothendieck toposes, and integrating categorical and proof-theoretic methods, the work unifies semantic reconstruction with proof-theoretic perspectives. The main contributions are twofold: first, it establishes that all the aforementioned logical fragments enjoy conceptual completeness and admit conservative embeddings into full geometric logic; second, within this unified framework, it recovers and generalizes Makkai’s semantic reconstruction theorem, thereby revealing a deep correspondence between these logical fragments and their categories of models.

This work formalizes causal reasoning within the framework of toposes, addressing the rigorous verification of interventions, mechanism grafting, and an intuitionistic do-calculus. Leveraging Cubical Agda, it models causal worlds as presheaf categories (1-toposes), defines interventions via characteristic maps of subobject classifiers, and conducts reasoning in the internal intuitionistic language. The main contributions include the first machine-verified core of a topos-theoretic causal model; a correction of the missing Lawvere–Tierney axiom by introducing the double-negation topology; the identification of a novel phenomenon termed the “contextuality barrier”; a proof that interventions and Pearl’s rules are j-stable under arbitrary topologies; and the establishment of an equivalence between counterfactual transportability and j-stability.

This study addresses the complexity in point-based constructions arising from varying morphism definitions in modal logic by establishing a Stone-type duality between an algebraic category equipped with paired modal operators and a category of topological spaces endowed with binary relations. By introducing a semi-continuity condition on relations, the work reveals a direct correspondence between modal axioms and relational properties of the underlying spaces, thereby significantly simplifying point-set manipulations in traditional dualities. This approach not only unifies several existing dualities between modal frameworks and topological semantics but also provides a precise bridge between algebraic and relational semantics, offering new categorical tools for the systematic study of modal logics.

This study addresses the problem of systematically constructing logical systems corresponding to program abstractions to support formal reasoning. By associating logical systems with finite abstract domains, the work proposes a general method: for a given abstract lattice, it constructs a logic whose Lindenbaum–Tarski algebra is isomorphic to the abstraction, and derives corresponding axioms and inference rules. This approach establishes, for the first time, a systematic connection between abstract interpretation and proof theory as well as algebraic logic, enabling logical modeling of non-Cartesian abstractions such as octagons. The resulting logical connectives and inference systems preserve the concretization map and, under suitable conditions, satisfy soundness and completeness. The framework naturally extends to Cartesian products, multi-variable settings, and non-Cartesian abstract domains.

Traditional logical relations struggle to accommodate directed reductions in type theory, impeding normalization proofs in dependent type systems. This work presents the first logical relation model within simplicial homotopy type theory that integrates contravariant computability predicates with directed quotient inductive types. Reduction is internalized as an inequality type, and a comonadic flat modality cleanly separates vertical reductions from horizontal parametricity. Leveraging built-in functoriality and universal properties, the approach supports computability reasoning under directed reduction, enabling a successful proof of directed Boolean normalization. The method extends to systems featuring dependent types and universes, yielding the first formalization of representation independence with proof relevance.