Although mathematical morphology and formal concept analysis are two lattice-based data analysis theories, they are still developed in two disconnected research communities. The aim of this paper is to contribute to fill this gap, beyond the classical relationship between the Galois connections defined by the derivation operators and the adjunctions underlying the algebraic mathematical morphol...

The theory of monads on categories equipped with a dagger (a contravariant identity-on-objects involutive endofunctor) works best when all structure respects the dagger: the monad and adjunctions should preserve the dagger, and the monad and its algebras should satisfy the so-called Frobenius law. Then any monad resolves as an adjunction, with extremal solutions given by the categories of Kleis...

We define a transformation of models of additivefree propositional linear logic, closely related to de Paiva’s dialectica categories, but with a modified exponential based on Oliva’s dialectica interpretation of higher-order linear logic. As a result of this modification we obtain an elegant factorisation of dialectica models as linear-nonlinear adjunctions. From the point of view of game seman...

This work provides a TAG account of gapping in English, based on a novel deletion-like operation that is referred to as de-anchoring. De-anchoring applies onto elementary trees, but it is licensed by the derivation tree in two ways. Firstly, de-anchored trees must be linked to the root of the derivation tree by a chain of adjunctions , and the sub-graph of de-anchored nodes in a derivation tree...

In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove tha...

In this sequel to [HM02], we explore some of the global aspects of the category of paracategories. We establish its (co)completeness and cartesian closure. From the closed structure we derive the relevant notion of transformation for paracategories. We set-up the relevant notion of adjunction between paracategories and apply it to define (co)completeness and cartesian closure, exemplified by th...

It is a well-known fact that a nondeterministic automaton can be transformed into an equivalent deterministic automaton via the powerset construction. From a categorical perspective this construction is the right adjoint to the inclusion functor from the category of deterministic automata to the category of nondeterministic automata. This is in fact an adjunction between two categories of coalg...

We introduce a generalisation of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed λ-calculus syntax and indexed containers. We show that the Kleisli and Eilenberg-Moore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions...

The general theory of randomly timed automata is developed: starting with the practical motivation and presentation of the envisaged notion, the categorical theory of minimization, aggregation, encapsulation, interconnection and realization of such automata is worked out. All these constructions are presented universally: minimization and realization as adjunctions, aggregation as product, inte...

Abstract Over the recent years, theory of rewriting has been used and extended in order to provide systematic techniques show coherence results for strict higher categories. Here, we investigate a further generalization Gray categories, which are known be equivalent tricategories. This requires us develop setting precategories, adapted mechanized computations include categories as particular ca...