This paper explain how the geometric notions of local contractibility and properness are related to the $\Sigma$-types and $\Pi$-types constructors of dependent type theory. We shall see how every Grothendieck fibration comes canonically with such a pair of notions—called smooth and proper maps—and how this recovers the previous examples and many more. This paper uses category theory to reveal a common structure between geometry and logic, with the hope that the parallel will be beneficial to both fields. The style is mostly expository, and the main results are proved in external references.

Up-to techniques are a well-known method for enhancing coinductive proofs of behavioural equivalences. We introduce up-to techniques for behavioural metrics between systems modelled as coalgebras, and we provide abstract results to prove their soundness in a compositional way.

In order to obtain a general framework, we need a systematic way to lift functors: we show that the Wasserstein lifting of a functor, introduced in a previous work, corresponds to a change of base in a fibrational sense. This observation enables us to reuse existing results about soundness of up-to techniques in a fibrational setting. We focus on the fibrations of predicates and relations valued in a quantale. To illustrate our approach, we provide an example on distances between regular languages.

Jacobs has proposed definitions for (weak, strong, split) generic objects for a fibered category; building on his definition of (split) generic objects, Jacobs develops a menagerie of important fibrational structures with applications to categorical logic and computer science, including higher order fibrations, polymorphic fibrations, $\lambda2$ -fibrations, triposes, and others. We observe that a split generic object need not in particular be a generic object under the given definitions, and that the definitions of polymorphic fibrations, triposes, etc. are strict enough to rule out some fundamental examples: for instance, the fibered preorder induced by a partial combinatory algebra in realizability is not a tripos in this sense. We propose a new alignment of terminology that emphasizes the forms of generic object appearing most commonly in nature, i.e. in the study of internal categories, triposes, and the denotational semantics of polymorphism. In addition, we propose a new class of acyclic generic objects inspired by recent developments in higher category theory and the semantics of homotopy type theory, generalizing the realignment property of universes to the setting of an arbitrary fibration.

Hoare Logic has a long tradition in formal verification and has been continuously developed and used to verify a broad class of programs, including sequential, object-oriented, and concurrent programs. Here we focus on partial and total correctness assertions within the framework of Hoare logic and show that a comprehensive categorical analysis of its axiomatic semantics needs the languages of indexed and fibered category theory. We consider Hoare formulas with local, finite contexts, of program and logical variables. The structural features of Hoare assertions are presented in an indexed setting, while the logical features of deduction are modeled in the fibered one.

Drawing on the analogy between any unary first-order quantifier and a “face operator,” this paper establishes several connections between model theory and homotopy theory. The concept of simplicial set is brought into play to describe the formulae of any first-order language L, the definable subsets of any L-structure, as well as the type spaces of any theory expressed in L. An adjunction result is then proved between the category of o-minimal structures and a subcategory of the category of linearly ordered simplicial sets with distinguished vertices.

Given a family of varieties $X\rightarrow \mathbb{P}^{n}$ over a number field, we determine conditions under which there is a Brauer–Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.

In this paper, an open problem in the multidimensional complex analysis is presented thatarises in the harmonic analysis related to the investigation of the regularity propertiesof Fourier integral operators and in the regularity theory for hyperbolic partialdifferential equations. The problem is discussed in a self-contained elementary way andsome results towards its resolution are presented. A conjecture concerning the structureof appearing affine fibrations is formulated.

Adapting the theory of the derived category to ordered groupoids, we prove that every ordered functor (and thus every inverse and regular semigroup homomorphism) factors as an enlargement followed by an ordered fibration. As an application, we obtain Lawson’s version of Ehresmann’s Maximum Enlargement Theorem, from which can be deduced the classical theory of idempotent-pure inverse semigroup homomorphisms and $E$-unitary inverse semigroups.

AMS 2000 Mathematics subject classification: Primary 20M18; 20L05; 20M17

Bertini's theorem on variable singular points may fail in characteristic $p$; there are algebraic fibrations that are pathological in the sense that all fibres are non-smooth though the total space admits only a finite number of singular points. We prove that if an algebraic fibration by plane projective quartic curves in characteristic 7 is pathological then, after an eventual cyclic base extension of degree 3, it is, up to birational equivalence, obtained by a base extension from the pencil of quartic curves cut out by the equations $ zx^3 + xy^3 + t yz^3 = 0$.

More generally, we classify pathological fibrations by canonical curves of arithmetic genus $g = \frac{p - 1}{2}$ in the projective space of dimension $\frac{p - 3}{2}$. We prove that the gap sequences of the smooth Weierstrass points of the fibres have the property that a positive integer $\ell$ is a gap if and only if $2g + 1 - \ell$ is a non-gap. In analogy to the Kodaira–Néron classification of special fibres of minimal fibrations by elliptic curves, we construct minimal proper regular models of pathological fibrations, determine the structure of the bad fibres, and study the global geometry of the total spaces.