Conceptual completeness for subgeometric logics

Abstract: We explore the notion of conceptual completeness for a fragment of geometric logic in the framework developed by the first and third author. Unlike its traditional interpretation as a reconstruction of syntax from semantics, in this paper we characterise conceptual completeness of a fixed fragment in terms of a duality between theories and topoi. We then show that conceptually complete fragments are conservatively embedded in full geometric logic, thus casting conceptual completeness in a new proof-theoretic light. We give a new proof of conceptual completeness for coherent logic, and we also show that regular, disjunctive, and essentially algebraic logic with falsum are conceptually complete. Finally, we show that our notion is equivalent to a traditional reconstruction result under the assumption of completeness with respect to set-based models: in the coherent case, we thus recover Makkai's original reconstruction theorem via ultracategories.

• The paper establishes a new categorical framework that reframes conceptual completeness as the 2-fully-faithfulness of classifying topos functors.
• It leverages modular 2-categorical methods, Kan injectivity, and dual adjunctions linking syntactic categories with topoi, generalizing Makkai’s theorem.
• The approach provides a unified proof for various logical fragments, such as coherent, regular, and finitary disjunctive logics, with implications for topos-theoretic semantics.

Conceptual Completeness for Subgeometric Logics

Introduction and Motivation

The paper "Conceptual completeness for subgeometric logics" (2607.02250) proposes a new categorical and proof-theoretic framework to analyze conceptual completeness phenomena for broad, non-geometric fragments of first-order logic. The work is driven by a desire to generalize and clarify the deep relationships between syntax and semantics beyond the classical setting of geometric or coherent logic, such as the ones characterized by Makkai’s strong conceptual completeness theorem. Rather than reconstructing syntax solely from the category of models with additional "ultrastructural" data, the authors present conceptual completeness as a duality between certain classes of theories (given by fragments of geometric logic) and categories of topoi, relying on weak Kan injectivity and modular 2-categorical tools. In doing so, the analysis shifts from purely set-based semantics to intrinsic, topos-theoretic interpretations, with an emphasis on abstract categorical dualities and the modular study of logical fragments.

Framework: Subgeometric Logics and Kan Injectivity

The foundation of this analysis is the perspective that a "logic," or more precisely a fragment $H$ of geometric logic, is determined by a set of semantic prescriptions: specifically, a class of geometric morphisms between topoi. A topos is considered "formally in $H$" if it satisfies weak right Kan injectivity with respect to all morphisms in $H$. This notion of semantic prescription, originally due to [dilibertiLogicConcepts2category2025], unifies a range of traditional logics (coherent, regular, essentially algebraic, etc.) and supports a categorical treatment of logical fragments as Kan injectivity classes in the 2-category of Grothendieck topoi.

For each such logic $H$, three functorial constructions are central:

• The 2-category $\mathrm{WRInj}(H)$ of topoi and morphisms formally in $H$.
• The syntactic category $\mathrm{Syn}^H(X)$ associated to a topos $X$ in $H$.
• The monad of $H$-syntax $H$0, capturing the freely generated syntactic structure appropriate for the logic fragment.
• The classifying topos $H$1 assigned to a $H$2-algebra $H$3.

By reorganizing traditional model theory and categorical logic through these constructions, the paper enables a modular investigation of the conceptual completeness property for a wide hierarchy of subgeometric logics.

Main Results: Categorical Duality and Conceptual Completeness

A central achievement is the reframing of conceptual completeness as the 2-fully-faithfulness (i.e., reflection) of the classifying topos functor $H$4 from the 2-category of $H$5-algebras to $H$6. This is seen as a dual adjunction

$H$7

which is a reflection precisely when every $H$8-algebra is realized as the syntactic category of its classifying topos.

This definition is shown to be modular and amenable to a uniform study of conceptual completeness across a broad class of logics, notably:

• Coherent logic: The paper provides a new proof of conceptual completeness for pretopoi, recovering Makkai's syntactic-semantics duality in terms of ultracategories and Kan injectivity.
• Regular logic: The analogous result is obtained for regular (Barr-exact) categories and their associated topoi.
• Essentially algebraic logic with falsum: The result holds via analysis of lex categories with initial objects, using dense embeddings.
• Finitary disjunctive logic: Conceptual completeness is shown for lextensive categories via a novel treatment of pure embeddings and the associated sheaf topoi.

The technical machinery avoids ad hoc case distinctions, enabling arguments that are canonical and categorical in nature, and allowing explicit computation of the relevant syntactic and classifying categories for these fragments.

Proof-Theoretic and Semantic Implications

A significant claim is that conceptually complete fragments $H$9 are not only semantically well-behaved, but also enjoy a strong conservativity property with respect to the proof theory of full geometric logic. Specifically, they are conservatively embedded in geometric logic: the corresponding adjunction $H$0 is idempotent, and the induced interpretation from the fragment's syntactic category to its full geometric closure is conservative.

Further, the analysis shows that for logics complete with respect to set-based models (i.e., classifying topoi have enough points), the novel syntactic notion of conceptual completeness is, in fact, equivalent to the traditional, semantic model-theoretic version à la Makkai—recovering Makkai's original conceptual completeness as a special case for coherent logic.

Relationship to Virtual Ultracategories and Geometric Completions

In the final sections, the authors establish connections with recent work on virtual ultracategories [saadiaExtendingConceptualCompleteness2025, hamadGeneralisedUltracategoriesConceptual2025, vangoolToposesEnoughPoints2026], which provide a highly abstract framework for model theory, generalizing the role of ultraproducts and ultrafilters.

The key insight is that the class of Kan-injective topoi for a logically-specified class of morphisms corresponds under a duality to categories of models equipped with prescribed semantic structure (e.g., ultracategories in the case of coherent logic). The classifying topos construction is thus shown to encapsulate both semantic and syntactic completeness, and the diagrammatic relationships between the various adjunctions and dualities further clarify the Rold of topos-theoretic semantics as a bridge between syntax and semantics in abstract logic.

Notably, the distinction is made between syntactic aspects (definability from Kan injectivity and topos classifiers) and semantic completeness with respect to set-based models, with the latter being strictly necessary only when one wishes to reconstruct the classical, model-theoretic conceptual completeness phenomenon.

Perspectives and Future Developments

The framework developed in the paper paves the way for several potential theoretical advances:

• Extension to larger/infinite fragments (e.g., $H$1-coherent, infinitary, or other non-traditionally algebraic fragments), leveraging the modularity of Kan injectivity.
• A general study of exotic or unclassified logics as new Kan injectivity classes, possibly leading to unanticipated completeness phenomena.
• Further study of the interplay between semantic prescriptions and categorical presentations of logic, connecting more deeply with notions of accessible and presentable categories, and the role of topoi as syntax-semantics intermediaries.
• The possibility of identifying logics not enjoying conceptual completeness, and clarifying the limits of the approach.

Conclusion

This work broadens the abstract understanding of conceptual completeness by moving beyond the classical framework of geometric logic and set-based semantics. By tying conceptual completeness to the fully faithfulness of classifying topos functors for subgeometric logics, and employing right Kan injectivity in the 2-category of topoi, the paper offers a categorical, modular, and syntactic perspective on the dualities between theories and categories of models. This approach both encompasses and generalizes Makkai’s celebrated theorems, provides new proofs for a range of logical fragments, and sets the stage for future developments in abstract categorical logic and model theory. The fusion of 2-categorical methods, topos theory, and semantic prescription yields a significant conceptual advancement in the formal study of logic.

Reference:

"Conceptual completeness for subgeometric logics" (2607.02250)