arXiv subjects

Francesca Guffanti

Publications and source records attributed to Francesca Guffanti.

7 recordsLinked to original sources

arXiv2026

The doctrinal G\"odel's completeness theorem and the type space functor

We give a self-contained proof of G\"odel's completeness theorem entirely within the formalism of first-order Boolean doctrines (an algebraic approach to classical many-sorted first-order logic). Moreover, we show that G\"odel's completeness theorem entails that the fiberwise Stone dual of a first-order Boolean doctrine is its type space functor; roughly speaking, this means that the Stone dual of the Boolean algebra of formulas in context $X$ is the Stone space of $X$-pointed models modulo elementary equivalence.

math.LO↗

arXiv2026

On the Beck--Chevalley condition

Boolean hyperdoctrines provide an algebraic semantics for classical first-order logic with equality. In the definition of a Boolean hyperdoctrine, the Beck--Chevalley condition captures the commutativity of substitutions with quantifiers and with equality. Often, a generalization of these conditions is considered, which requires the commutativity of an appropriate square for every pullback square in the base category. A Boolean hyperdoctrine satisfying this condition is called full. Our contribution is twofold. On the negative side, we exhibit a non-full Boolean hyperdoctrine. On the positive side, we show that every Boolean hyperdoctrine $\mathsf{FinSet} \to \mathsf{BA}$ over $\mathsf{FinSet}^{\mathrm{op}}$ is full.

math.CT↗

arXiv2024

Freely adding one layer of quantifiers to a Boolean doctrine

We describe the layer of quantifier alternation depth at most one of the quantifier completion of a Boolean doctrine over a small category. This amounts to a doctrinal version of Herbrand's theorem for formulas with quantifier alternation depth at most one modulo a universal theory. The resulting construction satisfies a universal property that makes it the free QA-one-step Boolean doctrine. To achieve this version of Herbrand's theorem, we characterize, within the doctrinal setting, the classes $A$ of quantifier-free formulas for which there is a model $M$ such that $A$ is precisely the class of formulas whose universal closure is valid in $M$.

math.LO↗

arXiv2024

Quantifier-free formulas and quantifier alternation depth in doctrines

This paper aims to incorporate the notion of quantifier-free formulas modulo a first-order theory and the stratification of formulas by quantifier alternation depth modulo a first-order theory into the algebraic treatment of classical first-order logic. The set of quantifier-free formulas modulo a theory is axiomatized by what we call a quantifier-free fragment of a Boolean doctrine with quantifiers. Rather than being an intrinsic notion, a quantifier-free fragment is an additional structure on a Boolean doctrine with quantifiers. Under a smallness assumption, the structures occurring as quantifier-free fragments of some Boolean doctrine with quantifiers are precisely the Boolean doctrines (without quantifiers). In particular, every Boolean doctrine over a small category is a quantifier-free fragment of its quantifier completion. Furthermore, the sequences obtained by stratifying an algebra of formulas by quantifier alternation depth modulo a theory are axiomatized by what we call QA-stratified Boolean doctrines. While quantifier-free fragments are defined in relation to an "ambient" Boolean doctrine with quantifiers, a QA-stratified Boolean doctrine requires no such ambient doctrine, and it consists of a sequence of Boolean doctrines (without quantifiers) with connecting axioms. QA-stratified Boolean doctrines are in one-to-one correspondence with pairs consisting of a Boolean doctrine with quantifiers and a quantifier-free fragment of it.

math.LO↗

arXiv2024

Left adjoint to precomposition in elementary doctrines

It is well-known in universal algebra that adding structure and equational axioms generates forgetful functors between varieties, and such functors all have left adjoints. The category of elementary doctrines provides a natural framework for studying algebraic theories, since each algebraic theory can be described by some syntactic doctrine and its models are morphism from the syntactic doctrine into the doctrine of subsets. In this context, adding structure and axioms to a theory can be described by a morphism between the two corresponding syntactic doctrines, and the forgetful functor arises as precomposition with this last morphism. In this work, given any morphism of elementary doctrines, we prove the existence of a left adjoint of the functor induced by precomposition in the doctrine of subobjects of a Grothendieck topos.

math.CT↗

arXiv2023

Adding a constant and an axiom to a doctrine

We study the meaning of "adding a constant to a language" for any doctrine, and "adding an axiom to a theory" for a primary doctrine, by showing how these are actually two instances of the same construction. We prove their universal properties, and how these constructions are compatible with additional structure on the doctrine. Existence of Kleisli object for comonads in the 2-category of indexed poset is proved in order to build these constructions.

math.CT↗

arXiv2023

Rich doctrines and Henkin's Theorem

We find a possible interpretation of Henkin's Theorem in the language of existential implicational doctrines. Under some smallness assumption, starting from an implicational existential doctrine, with non-trivial fibers, we construct a new doctrine which is rich -- meaning that for every formula $\varphi(x)$ there is a constant $c$ such that $\exists x\varphi(x)$ has the same truth-value of $\varphi(c)$ -- and consistent. To obtain this result, we add a suitable amount of constants and axioms to the starting doctrine. We then show that a rich consistent doctrine admits an appropriate morphism towards the doctrine of subsets -- a model. Henkin's Theorem for doctrines follows from these two results, modeling our proof on the main lines of the original theorem.

math.CT↗