Dedicated to the memory of Horst Herrlich, a great mathematician and a great friend.Abstract. Probability metrics constitute an important tool in probability theory and statistics [DKS91], [R91], [Z83] as they are specific metrics on spaces of random variables which, by satisfying an extra condition, concord well with the randomness structure. But probability metrics suffer from the same instability under constructions as metrics. In [L15], as well as in former and related work which can be found in the references of [L15], a comprehensive setting was developed to deal with this. It is the purpose of this note to point out that these ideas can also be applied to probability metrics thus embedding them in a natural categorical framework, showing that certain constructions performed in the setting of probability theory are in fact categorical in nature. This allows us to deduce various separate results in the literature from a unified approach.
1971 he moved to the University of Bremen where he was a full professor until he retired in 2002. Horst's mathematical work ranges over a large body of material, mostly in Topology, Algebra and Category Theory.In Category Theory he can be seen as the father of Categorical Topology, a beautiful field which inspired so many other researchers, and the genesis of which could already be seen in his monumental work "Topologische Reflexionen und Coreflexionen" which appeared in the Springer Lecture Notes in 1968. Starting in the seventies and throughout his career he developed the fundamentals of Categorical Topology, including topological functors, initial and final completions, factorization structures, cartesian closedness of topological constructs, and natural extensions. He also focused on the intuitive concept of nearness between arbitrary collections of sets, thus axiomatizing some new foundation for topology which he, together with one of his co-authors H. L. Bentley, proved to be extremely useful in the study of extensions of spaces.His visits to the U.S. near the end of the 60's and his collaboration with G. E. Strecker culminated in 1973 in the joint book "Category theory" which was very influential in the field. Later in the eighties together with J. Adámek the book was extended to "Abstract and Concrete Categories". This book has the subtitle "The Joy of Cats", reflecting Horst's humour and love for both categories and cats. It differs from other books in the field by emphasising concrete categories because of their importance for applying categorical methods to topological problems, and became "the" reference work for generations of researchers and students working in Categorical Topology.In the beginning of the eighties he published a paper entitled "Categorical Topology 1971-1981" with 700 references and containing many open problems. Together with
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At this moment in time, when I step down as Editor-in-Chief of Applied Categorical Structures, I want to look back on the years from its inception till today and at the same time stress the excellent perspectives for the future. It was at a conference in Bremen in the late eighties that I launched the idea to create a journal devoted to category theory and its applications in mathematics, and this idea was immediately received with great enthusiasm by the many colleagues present. From that moment onwards it took almost two years, many trips back and forth to the headquarters of Kluwer in Dordrecht, meetings with Kluwer representatives both in Dordrecht and in Antwerp, and several extensive surveys in the community before the decision to launch the journal was made and the contracts were signed. It still took quite a while to finalise the initial Editorial Board, decide on the design of the cover of the journal and on many other practical matters. The first issue finally appeared in 1993. From then until 2004 the journal was published under Kluwer, and in 2005, after the merger of Kluwer with Springer the journal was published under the Springer banner. I take this opportunity to thank all people at Kluwer and Springer with whom I dealt those many years for the always pleasant and fruitful cooperation. Over the years the journal underwent fundamental changes, going from strength to strength with an ever widening author-and readership. It started with a scope and Editorial Board geared towards mainstream algebra, order, topology and theoretical computer science. That scope was gradually extended to include homological and homotopical algebra, derived categories, categorical investigations in theoretical physics and higher category theory. Of course these changes could only be made provided they were reflected by the constitution of the Editorial Board. I was fortunate, at all times, to be able to attract excellent Editors.
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