Mathematic Category Theory

Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942-1945, in connection with algebraic topology.
The term "abstract nonsense" has been used by some critics to refer to its high level of abstraction, compared to more classical branches of mathematics. Homological algebra is category theory in its aspect of organising and suggesting calculations in abstract algebra. Diagram chasing is a visual method of arguing with abstract 'arrows'. Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology.

In mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships. A category is composed of a collection of abstract "objects" of any kind, linked together by a collection of abstract "arrows" of any kind that have a few basic properties (the ability to compose the arrows associatively and the existence of an identity arrow for each object). Many well-known categories are conventionally identified by a short capitalized word or abbreviation in bold or italics such as Set (category of sets) or Ring (category of rings).

In homological algebra, the snake lemma, a statement valid in every abelian category, is the crucial tool used to construct the long exact sequences.

Did you know?

• ... that in higher category theory, there are two major notions of higher categories, the strict one and the weak one ?
• ... that factorization systems generalize the fact that every function is the composite of a surjection followed by an injection ?
• ... that in a multicategory, morphisms are allowed to have a multiple arity, and that multicategories with one object are operads ?
• ... that it is possible to define the end and the coend of certain functors ?
• ... that in the category of rings, the coproduct of two rings is their tensor product ?
• ... that the Yoneda lemma proves that any small category can be embedded in a presheaf category ?
• ... that it is possible to compose profunctors so that they form a bicategory?

Mathematic Analysis

Analysis

Analysis is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle, though analysis as a formal concept is a relatively recent development. As a formal concept, the method has variously been ascribed to Ibn al-Haytham, Descartes (Discourse on the Method), Galileo, and Newton, as a practical method of physical discovery. 


In mathematics, the recurring decimal 0.999…, denotes a real number equal to 1. In other words, the notations "0.999…" and "1" represent the same real number. The equality has long been accepted by professional mathematicians and taught in textbooks. Various proofs of this identity have been formulated with varying rigour, preferred development of the real numbers, background assumptions, historical context, and target audience.
In the last few decades, researchers of mathematics education have studied the reception of this equality among students. A great many question or reject the equality, at least initially. Many are swayed by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example that each real number has a unique decimal expansion, that nonzero infinitesimal real numbers should exist, or that the expansion of 0.999… eventually terminates.

Did you know?

• ...that the Gudermannian function relates the circular trigonometric functions and the hyperbolic trigonometric functions without the use of complex numbers?

Mathematics algebra

   Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the muslim Persian mathematician, astronomer, astrologer and geographer, Muhammad bin Mūsā al-Khwārizmī titled Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided symbolic operations for the systematic solution of linear and quadratic equations.
Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.
Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.

The Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as

     

well-behaved and useful, at least in certain contexts. It is named after 19th century mathematician Bernhard Riemann. It is also called the complex projective line, denoted CP¹.
On a purely algebraic level, the complex numbers with an extra infinity element constitute a number system known as the extended complex numbers. Arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field. However, the Riemann sphere is geometrically and analytically well-behaved, even near infinity; it is a one-dimensional complex manifold, also called a Riemann surface.
In complex analysis, the Riemann sphere facilitates an elegant theory of meromorphic functions. The Riemann sphere is ubiquitous in projective geometry and algebraic geometry as a fundamental example of a complex manifold, projective space, and algebraic variety. It also finds utility in other disciplines that depend on analysis and geometry, such as quantum mechanics and other branches of physics.  




A tetrahedron can be placed in 12 distinct positions by rotation alone. These are illustrated above in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (reddish arrows) rotations that permute the tetrahedron through those positions.

Did you know?

• ...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?
• ...that it is possible for a three dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.
• ...that the Gudermannian function relates the regular trigonometric functions and the hyperbolic trigonometric functions without the use of complex numbers?
• ...that the classification of finite simple groups was not completed until the mid 1980s?
• ...that a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers?