Double adjunctions

We characterize double adjunctions in terms of presheaves and universal squares, and then apply these characterizations to free monads and Eilenberg-Moore objects in double categories. We improve upon an earlier result of Fiore-Gambino-Kock in [7] to conclude: if a double category with cofolding admits the construction of free monads in its horizontal 2-category, then it also admits the construction of free monads as a double category horizontally and vertically, and also in its vertical 2-category. We also prove that a double category admits Eilenberg-Moore objects if and only if a certain parameterized presheaf is representable. Along the way, we develop parameterized presheaves on double categories and prove a double Yoneda Lemma.

Fuzzy Adjunctions revisited

En este trabajo se intenta obtener la noción de adjunción más débil entre estructuras difusas. Este trabajo continúa la línea de investigación en el estudio y construcción d adjunciones que han realizado los autores en contribuciones anteriores. Nos centraremos ahora en la noción de relación difusa que es en cierto sentido interpretable como una función difusa. Existen varios trabajos en la literatura relacionados con este tema. Entre todos ellos, trabajaremos con un enfoque próximo al de Ciric et al cuando definen las denominadas funciones parciales difusas. El nuevo concepto estudiado es el de relaciones difusas funcionales y la construcción de adjunciones entre ellas.

Weighted limits in simplicial homotopy theory

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.

Monads in double categories

We extend the basic concepts of Street's formal theory of monads from the setting of 2-categories to that of double categories. In particular, we introduce the double category Mnd(C) of monads in a double category C and dene what it means for a double category to admit the construction of free monads. Our main theorem shows that, under some mild conditions, a double category that is a framed bicategory admits the construction of free monads if its horizontal 2-category does. We apply this result to obtain double adjunctions which extend the adjunction between graphs and categories and the adjunction between polynomial endofunctors and polynomial monads.

Mathematical Morphology on Hypergraphs Using Vertex-Hyperedge Correspondence

The focus of this paper is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of dual adjunctions between the vertex set and the hyperedge set of a hypergraph , by defining a vertex-hyperedge correspondence. This allows us to recover the classical notion of a dilation/erosion of a subset of vertices and to extend it to subhypergraphs of . This paper also studies the concept of morphological adjunction on hypergraphs for which both the input and the output are hypergraphs

Morphological filtering on hypergraphs

The focus of this article is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of dual adjunctions between the vertex set and the hyper edge set of a hypergraph H, by defining a vertex-hyperedge correspondence. This allows us to recover the classical notion of a dilation/erosion of a subset of vertices and to extend it to subhypergraphs of H. Afterward, we propose several new openings, closings, granulometries and alternate sequential filters acting (i) on the subsets of the vertex and hyperedge set of H and (ii) on the subhypergraphs of a hypergraph

A New Approach to Fuzzy Arithmetic

This work shows an application of a generalized approach for constructing dilation-erosion adjunctions on fuzzy sets. More precisely, operations on fuzzy quantities and fuzzy numbers are considered. By the generalized approach an analogy with the well known interval computations could be drawn and thus we can define outer and inner operations on fuzzy objects. These operations are found to be useful in the control of bioprocesses, ecology and other domains where data uncertainties exist.

2-categories and cyclic homology

The topic of this thesis is the application of distributive laws between comonads to the theory of cyclic homology. The work herein is based on the three papers 'Cyclic homology arising from adjunctions', 'Factorisations of distributive laws', and 'Hochschild homology, lax codescent,and duplicial structure', to which the current author has contributed. Explicitly, our main aims are: 1) To study how the cyclic homology of associative algebras and of Hopf algebras in the original sense of Connes and Moscovici arises from a distributive law, and to clarify the role of different notions of bimonad in this generalisation. 2) To extend the procedure of twisting the cyclic homology of a unital associative algebra to any duplicial object defined by a distributive law. 3) To study the universality of Bohm and Stefan’s approach to constructing duplicial objects, which we do in terms of a 2-categorical generalisation of Hochschild (co)homology. 4) To characterise those categories whose nerve admits a duplicial structure.

On the existence of right adjoints for surjective mappings between fuzzy structures0

En este trabajo los autores continúan su estudio de la caracterización de la existencia de adjunciones (conexiones de Galois isótonas) cuyo codominio no está dotado de estructura en principio. En este artículo se considera el caso difuso en el que se tiene un orden difuso R definido en un conjunto A y una aplicación sobreyectiva f:A-> B compatible respecto de dos relaciones de similaridad definidas en el dominio A y en el condominio B, respectivamente. Concretamente, el problema es encontrar un orden difuso S en B y una aplicación g:B-> A compatible también con las correspondientes similaridades definidas en A y en B, de tal forma que el par (f,g) constituya un adjunción.