Craig Interpolation for Subgeometric Logics

We show that a vast class of finitary fragments of geometric logic admit a form of Craig interpolation property. In doing so, we provide a new dictionary to import technology from algebraic logic to categorical logic.

💡 Research Summary

The paper “Craig Interpolation for Subgeometric Logics” investigates a broad class of finitary fragments of geometric logic and establishes that they enjoy a form of Craig interpolation. The authors aim to bridge techniques from algebraic logic with categorical logic, providing a unified framework that subsumes many earlier fragment‑specific interpolation results.

The work begins with a historical overview of Craig’s original interpolation theorem (1957) and its extensions by Lyndon, Shütte, and Pitts. Pitts’s categorical proof for intuitionistic and coherent logics serves as a prototype for the authors’ approach. The central goal is to prove interpolation uniformly for any fragment lying between regular and coherent logic, provided the fragment admits an étale map classifier.

To achieve this, the authors introduce the notion of a “doctrine” – a lax‑idempotent pseudo‑monad on the 2‑category Lex of left‑exact categories. A doctrine encodes the semantics of a logical fragment. The key property examined is “preservation of slicing”: a doctrine T preserves slicing if, for any left‑exact category C and any object X in C, the induced functor T(C/X) → T(C)/T(X) is an equivalence. This property is crucial because slicing corresponds syntactically to adding a constant, and its preservation ensures that interpolation can be expressed categorically.

Section 2 shows that slicing in Lex behaves as a colimit rather than a limit, which leads to a “propositional bootstrap” theorem (Theorem 2.2.10): if a doctrine preserves slicing, then its algebraic incarnation alg(T) has interpolation for any co‑comma square. This connects categorical interpolation with the classical algebraic notion of interpolation for posets or lattices.

Section 3 classifies the interpolation property for slicing‑preserving doctrines in terms of an exactness condition. The authors define “t‑conservative maps” between left‑exact categories and prove that they form one half of an orthogonal factorisation system for any finitary doctrine. The main result (Theorem 3.2.3) states that a slicing‑preserving doctrine has interpolation iff t‑conservative maps are closed under co‑comma squares in alg(T). Thus interpolation is reduced to a purely algebraic closure property.

Section 4 introduces the concept of an “étale map classifier” for a fragment H of geometric logic. An étale classifier is a universal object that classifies étale morphisms in the syntactic category of H. The authors prove (Proposition 4.2.5) that if H possesses such a classifier, then the associated doctrine TH automatically preserves slicing. This provides a concrete criterion for checking the slicing condition.

The main theorem (Theorem 5.0.6) combines the previous ingredients: any fragment H between regular and coherent logic that has an étale classifier enjoys Craig interpolation. The proof proceeds by (i) obtaining a slicing‑preserving doctrine from the classifier, (ii) invoking the exactness classification to obtain t‑conservative closure, and (iii) constructing the interpolant via the categorical interpolation condition on lax squares.

An appendix supplies a new proof that the doctrine for coherent logic has interpolation, simplifying Pitts’s original argument by using only classifying toposes rather than Makkai’s filter toposes.

The paper’s contributions are threefold:

1. It provides a uniform categorical framework (doctrines, slicing preservation, t‑conservativity) that captures interpolation for a wide range of logical fragments.
2. It introduces étale classifiers as a semantic bridge linking fragments of geometric logic to slicing‑preserving doctrines.
3. It demonstrates that many previously fragment‑specific interpolation results can be derived as corollaries of the main theorem.

The authors acknowledge limitations: the definition of a “fragment” is non‑syntactic, making it difficult to enumerate all fragments that satisfy the hypotheses; no general method is given for constructing étale classifiers in arbitrary fragments; and the reliance on pseudo‑limits/colimits may require further work to translate results into strict 2‑categorical settings.

Overall, the paper successfully extends Craig interpolation from propositional and coherent settings to a broad class of subgeometric logics, offering new tools for researchers interested in the interplay between algebraic and categorical logic.