Finitary Fragments of Geometric Logic

• Finitary fragments of geometric logic restrict disjunctions to finite sets, ensuring well-behaved categorical semantics and sharp interpolation results.
• The associated doctrines preserve filtered pseudocolimits and slice maps through étale-map classifiers, reinforcing connections to algebraic and categorical logic.
• These fragments underpin practical applications in model theory and topos theory, offering concrete insights into interpolation, definability, and subgeometric logics.

Finitary fragments of geometric logic constitute a class of first-order logical systems with strictly controlled syntactic resources. While geometric logic permits formulas constructed by finite conjunctions, arbitrary (potentially infinitary) disjunctions, and existential quantifications, its finitary fragments restrict the formation of disjunctions to finite sets only. This locality in formula-building imbues the associated logics with well-behaved categorical semantics, characterized by doctrines preserving filtered pseudocolimits, and enables sharp model-theoretic and proof-theoretic properties such as Craig interpolation in a broad spectrum of settings. Connections to algebraic logic, categorical logic, and topos theory play a central structural role in characterizing and proving these properties (Liberti et al., 16 Jan 2026).

1. Formal Structure of Geometric Logic Fragments

Geometric logic operates over a signature $\mathcal{S}$ by formulas constructed via the grammar: $$\varphi ::= R(t_1, \dotsc, t_n) \mid \bot \mid \varphi \wedge \psi \mid \bigvee_{i\in I} \varphi_i \mid \exists y.\varphi$$ where $I$ may be infinite, conjunction ($\wedge$) is always finite, and disjunction ($\bigvee$) may be infinitary. A geometric sequent consists of an entailment $\varphi \vdash \psi$, closed under substitution and geometric inference rules. Finitary fragments of geometric logic restrict $I$ to finite sets, admitting only finite disjunctions.

Categorically, the syntactic category of a finitary fragment is required to be closed under filtered colimits of representables. This entails that the associated doctrine on left-exact categories—typically a KZ-monad $T$ on Lex—preserves filtered pseudocolimits.

2. Subgeometric and Subfinitary Logics

Subgeometric logics generalize the notion of fragment by associating logic not with particular syntactic resources, but with collections of geometric morphisms. Following Di Liberti & Ye (2025), a fragment $$\mathcal{H} \subseteq \operatorname{Mor}(\text{Topoi})$$ defines the class of $\mathcal{H}$-injective topoi: those topoi $\mathcal{E}$ for which every $f \in \mathcal{H}$ into $\mathcal{E}$ admits a right Kan extension, forming a reflective sub-2-category of Topoi.

The corresponding relative KZ-monad $T^{\mathcal{H}}$ on Lex has algebras precisely corresponding to the syntactic small left-exact categories for $\mathcal{H}$. Fragments lying strictly between regular logic (direct images preserve epimorphisms) and coherent logic (direct images preserve epimorphisms and finite coproducts) are termed subgeometric. When $T^{\mathcal{H}}$ is finitary (i.e., $\mathcal{H}$ contains the regular fragment), such logics are finitary subgeometric.

3. Craig Interpolation in Categorical Frameworks

Interpolation finds an intermediate formula or subobject (the "interpolant") expressed solely in the common vocabulary between antecedent and consequent. Algebraically, for posets, a lax square

$$\begin{matrix} C \xrightarrow{f} A \ \downarrow^g \quad \downarrow^u \ B \xrightarrow{v} D \ \end{matrix}$$

with pointwise inequality $vg \leq uf$, has interpolation if for every $b \in B$, $a \in A$ with $v(b) \leq u(a)$, there exists $c \in C$ with $b \leq g(c)$ and $f(c) \leq a$. In categorical terms, for a lax square in Lex (left-exact categories), interpolation is tested on subobject lattices.

For doctrines, $T$ on Lex, interpolation is realized if every cocomma square in the 2-category Alg($T$) of algebras admits such interpolation. Syntactically: for any provable sequent $\varphi(p,r) \vdash \psi(p,s)$, an interpolant $\chi(p)$ exists in the common vocabulary such that both $\varphi(p,r) \vdash \chi(p)$ and $\chi(p) \vdash \psi(p,s)$ are provable in the fragment.

4. Main Theorems and the Role of Étale-Map Classifiers

The central result asserts that any finitary subgeometric logic $\mathcal{H}$ between the regular and coherent fragments, admitting an étale-map classifier, possesses Craig interpolation:

• If $$H_{\text{matte}} \subseteq \mathcal{H} \subseteq H_{\text{flat}}$$ is a fragment between regular and coherent logic, and $\mathcal{H}$ admits an étale-map classifier (enabling $T^{\mathcal{H}}$ to preserve slicing), then:
  • $T^{\mathcal{H}}$ has interpolation.
  • Consequently, every provable sequent in the finitary fragment $\mathcal{H}$ admits an interpolant in the same fragment.

An étale-map classifier in a logic $\mathcal{H}$ is realized as $Z_* \to Z$ in the category WRInj($\mathcal{H}$) (the $\mathcal{H}$-injective topoi), classifying all étale maps via pullbacks. If $\mathcal{H}$ admits such a classifier, $T^{\mathcal{H}}$ preserves slicing: for any left-exact category $\mathcal{A}$ and $A \in \mathcal{A}$,

$$(T^{\mathcal{H}}\mathcal{A})/\eta(A) \overset{\sim}{\to} T^{\mathcal{H}}(\mathcal{A}/A)$$

establishing closure under slices and cocartesian behavior of slice maps.

Table: Fragments and Properties

Fragment	Syntactic Closure	Categorical Property
Regular	Finite conjunctions	Direct images preserve epis
Coherent	+ finite disjunctions	+ finite coproducts
Finitary subgeo	Only finite disjunctions	Doctrine preserves slicing

5. Interactions of Algebraic and Categorical Logic

A principal innovation is a dictionary relating concepts from algebraic logic (interpolation, amalgamation, exactness) and categorical logic (doctrine properties on Lex):

• Lattice interpolation in Weldedt-Heyting algebras aligns with interpolation of posets.
• Syntactic formula interpolation matches lax cocomma interpolation in Alg($T$).
• Amalgamation of algebras corresponds to closure under cocomma for t-conservative maps: morphisms $A \to B$ in Alg($T$) reflecting truth of subterminal objects.

Theorem (6.1): For finitary doctrine $T$ preserving slicing,

$$T \text{ has interpolation} \iff t\text{-conservative maps in Alg}(T) \text{ are closed under cocomma squares}$$

where a $t$-conservative map preserves maximality of subterminal objects. Reduction corollary: When $T$ preserves slicing, interpolation for $T$ reduces to interpolation for the induced square on subobjects of the terminal object (propositional case).

6. Proof Strategy and Technical Ingredients

The strategy combines several categorical techniques:

• Slicing and cocartesian maps: Slicing as cocomma, and its preservation via $T$.
• Orthogonal factorization: Constructing filter quotients (left) and $t$-conservatives (right) in Alg($T$).
• Interpolation-exactness equivalence: Characterizing interpolation squares as lax cocomma squares, with stability under cocomma for $t$-conservative maps.
• Étale classifier mechanism: $T^{\mathcal{H}}$ sends canonical pullbacks for slices to étale maps in presheaf topoi, reconstructing slicing structure.

Combining these yields the interpolation theorem for finitary fragments admitting étale-map classifiers.

7. Corollaries, Applications, and Further Examples

These logical and categorical results have several important corollaries and concrete applications:

• Beth definability: Craig interpolation in subfragments gives conditions for Beth definability, even though geometric logic generally lacks implication and deduction theorems.
• Model theory: In topos-theoretic semantics, one gains finer control over intermediate definability and preservation under base change.
• New proofs: The paper provides a new proof that the doctrine of pretopoi (coherent logic) has interpolation, using only classifying topos and known facts about proper and dominant geometric morphisms in pullbacks.
• Examples: Coherent logic (via a simpler classifying topos argument), regular logic with bottom, and fragments obtained by freely adding finite coproducts.

Formally, for signatures $\Sigma_{0}, \Sigma_{1}$ and coherent formulas $\varphi_{0}(x), \varphi_{1}(x)$ in $\Sigma_{0}, \Sigma_{1}$ with common free variables $x$, if

$$\Sigma_{0} \cup \Sigma_{1} \vdash \varphi_{0}(x) \Rightarrow \varphi_{1}(x)$$

is provable in coherent logic, then a coherent formula $\chi(x)$ exists in the common signature $\Sigma_{0} \cap \Sigma_{1}$ satisfying

$$\Sigma_{0} \vdash \varphi_{0}(x) \Rightarrow \chi(x), \quad \Sigma_{1} \vdash \chi(x) \Rightarrow \varphi_{1}(x)$$

and categorically, $\chi$ appears as a subobject in the pushout of syntactic categories arranged in a cocomma square.

This suggests that the categorical and logical analysis of finitary fragments of geometric logic enables both a sharp syntactic understanding (through interpolation and definability properties) and a powerful semantic interpretation (via topoi, doctrines, and structural correspondences with algebraic logic), facilitating further research in constructive logic, topos theory, and categorical model theory.