In this paper, we describe a denotational model of Intuitionist Linear Logic which is also a differential category. Formulas are interpreted as Mackey-complete topological vector space and linear proofs are interpreted as bounded linear functions. So as to interpret non-linear proofs of Linear Logic, we use a notion of power series between Mackey-complete spaces, generalizing entire functions in $\mathbb{C}$ . Finally, we get a quantitative model of Intuitionist Differential Linear Logic, with usual syntactic differentiation and where interpretations of proofs decompose as a Taylor expansion.

The study of algebraic modelling of labelled non-deterministic concurrent processes leads us to consider a category LB, obtained from a complete meet-semilattice B and from B-valued equivalence relations. We prove that, if B has enough properties, then LB presents a two-fold internal logical structure, induced by two doctrines definable on it: one related to its families of subobjects and one to its families of regular subobjects. The first doctrine is Heyting and makes LB a Heyting category, the second one is Boolean. We will see that the difference between these two logical structures, namely the different behaviour of the negation operator, can be interpreted in terms of a distinction between non-deterministic and deterministic behaviours of agents able to perform computations in the context of the same process. Moreover, the sorted first-order logic naturally associated with LB can be extended to a modal/temporal logic, again using the doctrinal setting. Relations are also drawn to other computational models.

We introduce a new concept of continuity of posets, called θ-continuity. Topological characterizations of θ-continuous posets are put forward. We also present two types of dcpo-completion of posets which are Dθ-completion and Ds2-completion. Connections between these notions of continuity and dcpo-completions of posets are investigated. The main results are (1) a poset P is θ-continuous iff its θ-topology lattice is completely distributive iff it is a quasi θ-continuous and meet θ-continuous poset iff its Dθ-completion is a domain; (2) the Dθ-completion of a poset B is isomorphic to a domain L iff B is a θ-embedded basis of L; (3) if a poset P is θ-continuous, then the Dθ-completion Dθ(P) is isomorphic to the round ideal completion RI(P, ≪θ).

We characterize Yoneda completeness for non-symmetric distances by combinations of metric and directed completeness. One of these generalizes the Kostanek–Waszkiewicz theorem on formal balls.

Coinductive predicates express persisting ‘safety’ specifications of transition systems. Previous observations by Hermida and Jacobs identify coinductive predicates as suitable final coalgebras in a fibration – a categorical abstraction of predicate logic. In this paper, we follow the spirit of a seminal work by Worrell and study final sequences in a fibration. Our main contribution is to identify some categorical ‘size restriction’ axioms that guarantee stabilization of final sequences after ω steps. In its course, we develop a relevant categorical infrastructure that relates fibrations and locally presentable categories, a combination that does not seem to be studied a lot. The genericity of our fibrational framework can be exploited for binary relations (i.e. the logic of ‘binary predicates’) for which a coinductive predicate is bisimilarity, constructive logics (where interests are growing in coinductive predicates) and logics for name-passing processes.