Haskell/Category theory
This article attempts to give an overview of category theory, in so far as it applies to Haskell. To this end, Haskell code will be given alongside the mathematical definitions. Absolute rigour is not followed; in its place, we seek to give the reader an intuitive feel for what the concepts of category theory are and how they relate to Haskell. Introduction to categories[edit] A simple category, with three objects A, B and C, three identity morphisms and , and two other morphisms . A category is, in essence, a simple collection. A collection of objects.A collection of morphisms, each of which ties two objects (a source object and a target object) together. Lots of things form categories. is a morphism in Grp iff: It may seem that morphisms are always functions, but this needn't be the case. ) defines a category where the objects are the elements of P, and there is a morphism between any two objects A and B iff . are both functions with source object and target object Category laws[edit] . ? . .
Category Theory
First published Fri Dec 6, 1996; substantive revision Fri Apr 26, 2013 Category theory has come to occupy a central position in contemporary mathematics and theoretical computer science, and is also applied to mathematical physics. Roughly, it is a general mathematical theory of structures and of systems of structures. 1. 1.1 Definitions Categories are algebraic structures with many complementary natures, e.g., geometric, logical, computational, combinatorial, just as groups are many-faceted algebraic structures. An alternative approach, that of Lawvere (1963, 1966), begins by characterizing the category of categories, and then stipulates that a category is an object of that universe. (C1) Given three mappings α1, α2 and α3, the triple product α3(α2α1) is defined if and only if (α3α2)α1 is defined. α3(α2α1) = (α3α2)α1 holds. As Eilenberg & Mac Lane promptly remark, objects play a secondary role and could be entirely omitted from the definition. (R1) idX : X → X, 1.2 Examples
Kategorientheorie
Bedeutung[Bearbeiten] Aufgrund ihres hohen Grades an Abstraktion wird die Kategorientheorie gelegentlich – selbst von den Mathematikern, die sie entwickelten – als allgemeiner Unsinn bezeichnet.[1][2] Definitionen[Bearbeiten] Kategorie[Bearbeiten] Eine Kategorie besteht aus folgendem: die im offensichtlichen Sinne assoziativ sind: sofern und (Gelegentlich wird das weggelassen und als angeschrieben.) einem Identitätsmorphismus zu jedem Objekt , der neutrales Element für die Verknüpfung mit Morphismen mit Quelle oder Ziel ist, d. h. es gilt , falls ist, und , falls . Die Klasse aller Morphismen wird auch mit oder bezeichnet (von englisch arrow, französisch flèche, deutsch Pfeil). Unterkategorie[Bearbeiten] Eine Unterkategorie einer Kategorie ist eine Kategorie , so dass eine Teilklasse von ist und für je zwei Objekte in die Morphismenmenge Teilmenge von ist. gleich denen von , ist eine volle Unterkategorie. Duale Kategorie[Bearbeiten] Die duale Kategorie zu einer Kategorie ist die Kategorie mit . ist gleich mit nach
Mathematical and theoretical biology
Mathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications in biology, biotechnology, and medicine.[1] The field may be referred to as mathematical biology or biomathematics to stress the mathematical side, or as theoretical biology to stress the biological side.[2] It includes at least four major subfields: biological mathematical modeling, relational biology/complex systems biology (CSB), bioinformatics and computational biomodeling/biocomputing. Mathematical biology aims at the mathematical representation, treatment and modeling of biological processes, using a variety of applied mathematical techniques and tools. It has both theoretical and practical applications in biological, biomedical and biotechnology research. Importance[edit] Applying mathematics to biology has a long history, but only recently has there been an explosion of interest in the field. Areas of research[edit] Evolutionary biology[edit] Spatial modelling[edit]
reference request - Good books and lecture notes about category theory.
Groupoid
In mathematics , especially in category theory and homotopy theory , a groupoid (less often Brandt groupoid or virtual group ) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: Special cases include: Setoids , that is: sets that come with an equivalence relation ; G-sets , sets equipped with an action of a group G . Groupoids are often used to reason about geometrical objects such as manifolds . Definitions [ edit ] Algebraic [ edit ] A groupoid is a set G with a unary operation and a partial function Here * is not a binary operation because it is not necessarily defined for all possible pairs of G -elements. and −1 have the following axiomatic properties. Associativity : If a * b and b * c are defined, then ( a * b ) * c and a * ( b * c ) are defined and equal. In short: ( a * b ) * c = a * ( b * c ); ∀a ∃ a −1 * a ∃ a * a −1 ; a * b * b −1 = a and a −1 * a * b = b . From these axioms, two easy and convenient properties follow: Category theoretic [ edit ]
Robert Rosen (theoretical biologist)
Robert Rosen (June 27, 1934 – December 28, 1998) was an American theoretical biologist and Professor of Biophysics at Dalhousie University.[1] Rosen was born on June 27, 1934 in Brownsville (a section of Brooklyn), in New York City. He studied biology, mathematics, physics, philosophy, and history; particularly, the history of science. In 1959 he obtained a PhD in relational biology, a specialization within the broader field of Mathematical Biology, under the guidance of Professor Nicolas Rashevsky at the University of Chicago. His year-long sabbatical in 1970 as a Visiting Fellow at Robert Hutchins' Center for the Study of Democratic Institutions in Santa Barbara, California was seminal, leading to the conception and development of what he later called Anticipatory Systems Theory, itself a corollary of his larger theoretical work on relational complexity. He served as president of the Society for General Systems Research, (now known as ISSS), in 1980-81. stands for the metabolic and
Category Theory: Paperback: Steve Awodey
Important growing area of mathematicsClear definitions of all basic concepts Combines rigour with an appealing informalityContains precise statements of all essential theorems, with full proofs of all theorems, propositions and lemmasBased on courses given at Carnegie Mellon UniversityNumerous exercises provided New to this edition Nearly a hundred new exercisesMany more examples and diagramsWorked solutions to almost half the exercisesNew coverage of monoidal categoriesCategory theory is a branch of abstract algebra with incredibly diverse applications. Although assuming few mathematical pre-requisites, the standard of mathematical rigour is not compromised. This Second Edition contains numerous revisions to the original text, including expanding the exposition, revising and elaborating the proofs, providing additional diagrams, correcting typographical errors and, finally, adding an entirely new section on monoidal categories.
