Related papers: The Geometric Completion of a Doctrine

This paper aims to apply the tool of generalized existential completions of conjunctive doctrines, concerning a class $\Lambda$ of morphisms of their base category, to deepen the study of regular and exact completions of existential…

We suggest an ordering for the predicates in continuous logic so that the semantics of continuous logic can be formulated as a hyperdoctrine. We show that this hyperdoctrine can be embedded into the hyperdoctrine of subobjects of a suitable…

The standard approach to logic in the literature in philosophy and mathematics, which has also been adopted in computer science, is to define a language (the syntax), an appropriate class of models together with an interpretation of…

We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere…

We present a first-order logic equipped with an "asymmetric" directed notion of equality, which can be thought of as rewrites between terms, allowing for types to be interpreted as preorders. The logic is equipped with a precise syntactic…

Conjecturing and theorem proving are activities at the center of mathematical practice and are difficult to separate. In this paper, we propose a framework for completing incomplete conjectures and incomplete proofs. The framework can turn…

We provide a thorough algebraic analysis of three known completions having a central role in the exact completions of Lawvere's doctrines: the one adding comprehensive diagonals (i.e. forcing equality on terms to coincide with the equality…

Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces satisfying all the axioms of GLP are…

We argue that Godel's completeness theorem is equivalent to completability of consistent theories, and Godel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some…

We define the notion of sheaf in the context of doctrines. We prove the associate sheaf functor theorem. We show that grothendieck toposes and toposes obtained by the tripos to topos construction are instances of categories of sheaves for a…

We introduce the notion of a "category with path objects", as a slight strengthening of Kenneth Brown's classic notion of a "category of fibrant objects". We develop the basic properties of such a category and its associated homotopy…

We define the notion of exact completion with respect to an existential elementary doctrine. We observe that the forgetful functor from the 2-category exact categories to existential elementary doctrines has a left biadjoint that can be…

We establish a criterion for deciding whether a class of structures is the class of models of a geometric theory inside Grothendieck toposes; then we specialize this result to obtain a characterization of the infinitary first-order theories…

We prove a topological completeness theorem for the modal logic GLP containing operators $\langle\lambda\rangle$ for $\lambda \in$ Ord intended to capture progressively stronger notions of consistency in mathematical theories. We show that,…

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces---so-called ``topological semantics''. The first is classical higher-order logic, with…

We begin the study of categorical logic for continuous model theory. In particular, we 1. introduce the notions of metric logical categories and functors as categorical equivalents of a metric theory and interpretations, 2. prove a…

Although contemporary model theory has been called "algebraic geometry minus fields", the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes,…

We contribute to the knowledge of the quantifier completions and their applications by using the language of doctrines. This algebraic presentation allows us to properly analyse the behaviour of the existential and universal quantifiers. We…

We extend logical categories with fiberwise interior and closure operators so as to obtain an embedding theorem into powers of the category of topological spaces. The required axioms, besides the Kuratowski closure axioms, are a `product…

We investigate a recently-devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of…