Mathematical Morphology arose in 1964 by the work of George Matheron and Jean Serra, who developed its main concepts and tools. It uses concepts from algebra and geometry. (Set theory, complete lattice theory, convexity etc,). The notion of adjunction is very fundamental in Mathematical Morphology. Morphological systems is a broad class of nonlinear signal operators that have found many applica...

We present a categorical framework to study probabilistic automata starting by obtaining aggregation and interconnection as universal constructions. We also introduce the notion of probabilistic behavior in order to get adjunctions between probabilistic behavior and probabilistic automata. Thus we are able to extend to the probabilistic setting free and minimal realizations as universal constru...

We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functor.

Several categorical relationships (adjunctions) between models for concurrency have been established, allowing the translation of concepts and properties from one model to another. A central example is a coreflection between Petri nets and asynchronous transition systems. The purpose of the present paper is to illustrate the use of such relationships by transferring to Petri nets a general conc...

We show how the domain of locality in a TAG elementary tree, (Frank 1992), can be extended through adjunction to include optional arguments for a dass of motion verbs and how the adjunctions can be restricted appropriately through the use of semantic features. Same examples of motion verbs we consider are shown in Table 1, which categorizes the verbs according to Levin classes (Levin 1993). Not...

We provide a general framework for pattern structures by investigating adjunctions between posets and their morphisms. Our special interest is the impact of pattern morphisms on the induced concept lattices. In particular we are interested in conditions which are sufficient for the induced residuated maps to be injective, surjective or bijective. One application is that every representation con...

A large part of wide coverage Tree Adjoining Grammars (TAG) is formed by trees that satisfy the restrictions imposed by Tree Insertion Grammars (TIG). This characteristic can be used to reduce the practical complexity of TAG parsing, applying the standard adjunction operation only in those cases in which the simpler cubic-time TIG adjunction cannot be applied. In this paper, we describe a parsi...

In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory. Motivated by some key results of the 1970s, the paper develops all needed ingredients from the theory of quantaloids in order to place essential results of categori...