We study the functorial characterisation of bisimulation-based equivalences over a categorical model of labelled trees. We show that in a setting where all labels are visible, strong bisimilarity can be characterised in terms of enriched functors by relying on the reflection of paths with their factorisations. For an enriched functor F, this notion requires that a path (an internal morphism in our framework) π going from F(A) to C corresponds to a path p going from A to K, with F(K) = C, such that every possible factorisation of π can be lifted in an appropriate factorisation of p. This last property corresponds to a Conduché property for enriched functors, and a very rigid formulation of it has been used by Lawvere to characterise the determinacy of physical systems. We also consider the setting where some labels are not visible, and provide characterisations for weak and branching bisimilarity. Both equivalences are still characterised in terms of enriched functors that reflect paths with their factorisations: for branching bisimilarity, the property is the same as the one used to characterise strong bisimilarity when all labels are visible; for weak bisimilarity, a weaker form of path factorisation lifting is needed. This fact can be seen as evidence that strong and branching bisimilarity are strictly related and that, unlike weak bisimilarity, they preserve process determinacy in the sense of Milner.

We introduce pairwise Stone spaces as a bitopological generalisation of Stone spaces – the duals of Boolean algebras – and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakia's duality.

We introduce a strict version of the concept of a load/store instruction set architecture in the setting of Maurer machines. We take the view that transformations on the states of a Maurer machine are achieved by applying threads as considered in thread algebra to the Maurer machine. We study how the transformations on the states of the main memory of a strict load/store instruction set architecture that can be achieved by applying threads depend on the operating unit size, the cardinality of the instruction set and the maximal number of states of the threads.

Iterative monads were introduced by Calvin Elgot in the 1970's and are those ideal monads in which every guarded system of recursive equations has a unique solution. We prove that every ideal monad has an iterative reflection, that is, an embedding into an iterative monad with the expected universal property. We also introduce the concept of iterativity for algebras for the monad , following in the footsteps of Evelyn Nelson and Jerzy Tiuryn, and prove that is iterative if and only if all free algebras for are iterative algebras.

We characterise those quasi-metric spaces (X, d) whose poset BX of formal balls satisfies the condition (*) From this characterisation, we then deduce that a quasi-metric space (X, d) is Smyth-complete if and only if BX is a dcpo satisfying condition (*). We also give characterisations in terms of formal balls for sequentially Yoneda complete quasi-metric spaces and for Yoneda complete T1 quasi-metric spaces. Finally, we discuss several properties of the Heckmann quasi-metric on the formal balls of any quasi-metric space.

Girard's Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cut elimination. We introduce a typed version of GoI, called Multiobject GoI for both multiplicative linear logic (MLL) and multiplicative exponential linear logic (MELL) with units. We present a categorical setting that includes our previous (untyped) GoI models, as well as more general models based on monoidal *-categories. Our development of multiobject GoI depends on a new theory of partial traces and trace classes, which we believe is of independent interest, as well as an abstract notion of orthogonality (which is related to work of Hyland and Schalk). We develop Girard's original theory of types, data and algorithms in our setting, and show his execution formula to be an invariant of cut elimination (under some restrictions). We prove soundness theorems for the MGoI interpretation (for Multiplicative and Multiplicative Exponential Linear Logic) in partially traced *-categories with an orthogonality. Finally, we briefly discuss the relationship between our GoI interpretation and other categorical interpretations of GoI.