Horst Herrlich

Bio: Horst Herrlich is an academic researcher from University of Bremen. The author has contributed to research in topics: Axiom of choice & Topological space. The author has an hindex of 21, co-authored 91 publications receiving 3794 citations.

Abstract and concrete categories : the joy of cats

Abstract: Motivation. Foundations. CATEGORIES, FUNCTORS, AND NATURAL TRANSFORMATIONS. Categories and Functors. Subcategories. Concrete Categories and Concrete Functors. Natural Transformations. OBJECTS AND MORPHISMS. Objects and Morphisms in Abstract Categories. Objects and Morphisms in Concrete Categories. Injective Objects and Essential Embeddings. SOURCES AND SINKS. Sources and Sinks. Limits and Colimits. Completeness and Cocompleteness. Functors and Limits. FACTORIZATION STRUCTURES. Factorization Structures for Morphisms. Factorization Structures for Sources. E-Reflective Subcategories. Factorization Structures for Functors. ADJOINTS AND MONADS. Adjoint Functors. Adjoint Situations. Monads. TOPOLOGICAL AND ALGEBRAIC CATEGORIES. Topological Categories. Topological Structure Theorems. Algebraic Categories. Algebraic Structure Theorems. Topologically Algebraic Categories. Topologically Algebraic Structure Theorems. CARTESIAN CLOSEDNESS AND PARTIAL MORPHISMS. Cartesian Closed Categories. Partial Morphisms, Quasitopoi, and Topological Universes. Bibliography. Tables. Table of Categories. Table of Symbols. Index.

A concept of nearness

Abstract: This paper offers solutions for two problems which have attracted many topologists over the years: 1. (1) It provides a natural and reasonably simple concept of “nearness” which unifies various concepts of “topological structures” in the sense that the category Near of all nearness spaces and nearness preserving maps contains the categories (a) of all topological Ro-spaces and continuous maps, (b) of all uniform spaces and uniformly continuous maps (Weil [34], Turkey [33]), (c) of all proximity spaces and δ-maps (Efremovic [9], Smirnov [28,29]), (d) of all contiguity spaces and contiguity maps (Ivanova and Ivanov [17]) as nicely embedded (either bireflective or bicoreflective) full (!) subcategories. 2. (2) It provides a general method by means of which as many T1-extension of a T1-space can be obtained as might be reasonably expected; namely, all strict extensions (in the sense of Banaschewski [3]).

Axiom of Choice

Abstract: Origins.- Choice Principles.- Elementary Observations.- Disasters without Choice.- Disasters with Choice.- Disasters either way.- Beauty without Choice.

Multi-index Mittag-Leffler Functions

Abstract: Consider the function defined for \(\alpha _{1},\ \alpha _{2} \in \mathbb{R}\) (α 1 2 +α 2 2 ≠ 0) and \(\beta _{1},\beta _{2} \in \mathbb{C}\) by the series $$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$ (6.1.1) Such a function with positive α 1 > 0, α 2 > 0 and real \(\beta _{1},\beta _{2} \in \mathbb{R}\) was introduced by Dzherbashian [Dzh60].

Large Networks and Graph Limits

Abstract: Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. To develop a mathematical theory of very large networks is an important challenge. This book describes one recent approach to this theory, the limit theory of graphs, which has emerged over the last decade. The theory has rich connections with other approaches to the study of large networks, such as "property testing" in computer science and regularity partition in graph theory. It has several applications in extremal graph theory, including the exact formulations and partial answers to very general questions, such as which problems in extremal graph theory are decidable. It also has less obvious connections with other parts of mathematics (classical and non-classical, like probability theory, measure theory, tensor algebras, and semidefinite optimization). This book explains many of these connections, first at an informal level to emphasize the need to apply more advanced mathematical methods, and then gives an exact development of the theory of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits. This is an amazing book: readable, deep, and lively. It sets out this emerging area, makes connections between old classical graph theory and graph limits, and charts the course of the future. --Persi Diaconis, Stanford University This book is a comprehensive study of the active topic of graph limits and an updated account of its present status. It is a beautiful volume written by an outstanding mathematician who is also a great expositor. --Noga Alon, Tel Aviv University, Israel Modern combinatorics is by no means an isolated subject in mathematics, but has many rich and interesting connections to almost every area of mathematics and computer science. The research presented in Lovasz's book exemplifies this phenomenon. This book presents a wonderful opportunity for a student in combinatorics to explore other fields of mathematics, or conversely for experts in other areas of mathematics to become acquainted with some aspects of graph theory. --Terence Tao, University of California, Los Angeles, CA Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks. It is an authoritative, masterful text that reflects Lovasz's position as the main architect of this rapidly developing theory. The book is a must for combinatorialists, network theorists, and theoretical computer scientists alike. --Bela Bollobas, Cambridge University, UK

Abstract and concrete categories : the joy of cats

Abstract: Motivation. Foundations. CATEGORIES, FUNCTORS, AND NATURAL TRANSFORMATIONS. Categories and Functors. Subcategories. Concrete Categories and Concrete Functors. Natural Transformations. OBJECTS AND MORPHISMS. Objects and Morphisms in Abstract Categories. Objects and Morphisms in Concrete Categories. Injective Objects and Essential Embeddings. SOURCES AND SINKS. Sources and Sinks. Limits and Colimits. Completeness and Cocompleteness. Functors and Limits. FACTORIZATION STRUCTURES. Factorization Structures for Morphisms. Factorization Structures for Sources. E-Reflective Subcategories. Factorization Structures for Functors. ADJOINTS AND MONADS. Adjoint Functors. Adjoint Situations. Monads. TOPOLOGICAL AND ALGEBRAIC CATEGORIES. Topological Categories. Topological Structure Theorems. Algebraic Categories. Algebraic Structure Theorems. Topologically Algebraic Categories. Topologically Algebraic Structure Theorems. CARTESIAN CLOSEDNESS AND PARTIAL MORPHISMS. Cartesian Closed Categories. Partial Morphisms, Quasitopoi, and Topological Universes. Bibliography. Tables. Table of Categories. Table of Symbols. Index.

Algebraic approaches to graph transformation. Part I: basic concepts and double pushout approach

Abstract: The algebraic approaches to graph transformation are based on the concept of gluing of graphs, modelled by pushouts in suitable categories of graphs and graph morphisms This allows one not only to give an explicit algebraic or set theoretical description of the constructions, but also to use concepts and results from category theory in order to build up a rich theory and to give elegant proofs even in complex situations In this chapter we start with an overwiev of the basic notions common to the two algebraic approaches, the "double-pushout (DPO) approach" and the "single-pushout (SPO) approach"; next we present the classical theory and some recent development of the double-pushout approach The next chapter is devoted instead to the single-pushout approach, and it is closed by a comparison between the two approaches -- This document will appear as a chapter of the "The Handbook of Graph Grammars Volume I: Foundations", G Rozenberg (Ed), World Scientific