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NLab
The ''n''Lab is a wiki for research-level notes, expositions and collaborative work, including original research, in mathematics, physics, and philosophy, with a focus on methods from type theory, category theory, and homotopy theory. The ''n''Lab espouses the "''n''-point of view" (a deliberate pun on Wikipedia's "neutral point of view") that type theory, homotopy theory, category theory, and higher category theory provide a useful unifying viewpoint for mathematics, physics and philosophy. The ''n'' in ''n''-point of view could refer to either ''n''-categories as found in higher category theory, ''n''-groupoids as found in both homotopy theory and higher category theory, or ''n''-types as found in homotopy type theory. Overview The ''n''Lab was originally conceived to provide a repository for ideas (and even new research) generated in the comments on posts at the ''n''-Category Café, a group blog run (at the time) by John C. Baez, David Corfield and Urs Schreiber. Eventua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Theory
In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline. Applications to other fields of mathematics Besides algebraic topology, the theory has also been used in other areas of mathematics such as: * Algebraic geometry (e.g., A1 homotopy theory, A1 homotopy theory) * Category theory (specifically the study of higher category theory, higher categories) Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid Pathological (mathematics), pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being Category of compactly generated weak Hausdorff spaces, compactly generated weak Hausdorff or a CW complex. In the same vein as above, a "Map (mathematics), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Instiki
Instiki is a wiki software written in Ruby on Rails, created by David Heinemeier Hansson and maintained by physicist Jacques Distler. Instiki is free software under the Ruby license. Features Instiki includes a LaTeX plugin. Notable users A custom fork of it is run on the nLab The ''n''Lab is a wiki for research-level notes, expositions and collaborative work, including original research, in mathematics, physics, and philosophy, with a focus on methods from type theory, category theory, and homotopy theory. The ''n''Lab ... mathematics wiki. References External links * Free software programmed in Ruby Wiki software {{cms-software-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Theory
In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that have been proposed as foundations are: * Typed λ-calculus of Alonzo Church * Intuitionistic type theory of Per Martin-Löf Most computerized proof-writing systems use a type theory for their foundation. A common one is Thierry Coquand's Calculus of Inductive Constructions. History Type theory was created to avoid paradoxes in naive set theory and formal logic, such as Russell's paradox which demonstrates that, without proper axioms, it is possible to define the set of all sets that are not members of themselves; this set both contains itself and does not contain itself. Between 1902 and 1908, Bertrand Russell proposed various solutions to this problem. By 1908, Russell arrive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher Category Theory
In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic Invariant (mathematics), invariants of topological space, spaces, such as the Fundamental groupoid, fundamental . In higher category theory, the concept of higher categorical structures, such as (), allows for a more robust treatment of homotopy theory, enabling one to capture finer homotopical distinctions, such as differentiating two topological spaces that have the same fundamental group but differ in their higher homotopy groups. This approach is particularly valuable when dealing with spaces with intricate topological features, such as the Eilenberg-MacLane space. Strict higher categories An ordinary category (m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathOverflow
MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a part of the Stack Exchange Network, but distinct fromath.stackexchange.com It is primarily for asking questions on mathematics research – i.e. related to unsolved problems and the extension of knowledge of mathematics into areas that are not yet known – and does not welcome requests from non-mathematicians for instruction, for example homework exercises. It does welcome various questions on other topics that might normally be discussed among mathematicians, for example about publishing, refereeing, advising, getting tenure, etc. It is generally inhospitable to questions perceived as tendentious or argumentative. Origin and history The website was started by Berkeley graduate students and postdocs Anton Geraschenko, David ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-Category Café
John Carlos Baez ( ; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, applications of higher categories to physics, and applied category theory. Additionally, Baez is known on the World Wide Web as the author of the crackpot index. Education John C. Baez attended Princeton University where he graduated with an A.B. in mathematics in 1982; his senior thesis was titled "Recursivity in quantum mechanics", under the supervision of John P. Burgess. He earned his doctorate in 1986 from the Massachusetts Institute of Technology under the direction of Irving Segal. Career Baez was a post-doctoral researcher at Yale University. Since 1989, he has been a faculty member at UC Riverside. From 2010 to 2012, he was a visiting professor at the Centre for Quantum Technologies in Singapore and continued working there in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Corfield
David Neil Corfield is a British philosopher specializing in philosophy of mathematics and philosophy of psychology. He was Senior Lecturer in Philosophy at the University of Kent until 2024. Education Corfield studied mathematics at the University of Cambridge, and later earned his MSc and PhD in the philosophy of science and mathematics at King's College London. His doctoral advisor was Donald A. Gillies. Work Corfield is the author of ''Towards a Philosophy of Real Mathematics'' (2003), in which he argues that the philosophical implications of mathematics did not stop with Kurt Gödel's incompleteness theorems. He has also co-authored a book with Darian Leader about psychology and psychosomatic medicine, ''Why Do People Get Ill?'' (2007). He joined the University of Kent in September 2007 where he was Senior Lecturer in Philosophy until 2024. He is a member of the informal steering committee of ''n''Lab, a wiki-lab for collaborative work on mathematics, physics, and phi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Urs Schreiber
Urs Schreiber (born 1974) is a mathematician specializing in the connection between mathematics and theoretical physics (especially string theory) and currently working as a researcher at New York University Abu Dhabi. He was previously a researcher at the Czech Academy of Sciences, Institute of Mathematics, Department for Algebra, Geometry and Mathematical Physics. Education Schreiber obtained his doctorate from the University of Duisburg-Essen in 2005 with a thesis supervised by Robert Graham and titled ''From Loop Space Mechanics to Nonabelian Strings''. Work Schreiber's research fields include the mathematical foundation of quantum field theory. Schreiber is a co-creator of the ''n''Lab, a wiki for research mathematicians and physicists working in higher category theory In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly stu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiki
A wiki ( ) is a form of hypertext publication on the internet which is collaboratively edited and managed by its audience directly through a web browser. A typical wiki contains multiple pages that can either be edited by the public or limited to use within an organization for maintaining its internal knowledge base. Its name derives from the first user-editable website called " WikiWikiWeb," with "wiki" being a Hawaiian word meaning "quick." Wikis are powered by wiki software, also known as wiki engines. Being a form of content management system, these differ from other web-based systems such as blog software or static site generators in that the content is created without any defined owner or leader. Wikis have little inherent structure, allowing one to emerge according to the needs of the users. Wiki engines usually allow content to be written using a lightweight markup language and sometimes edited with the help of a rich-text editor. There are dozens of differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Copyright License
A copyright is a type of intellectual property that gives its owner the exclusive legal right to copy, distribute, adapt, display, and perform a creative work, usually for a limited time. The creative work may be in a literary, artistic, educational, or musical form. Copyright is intended to protect the original expression of an idea in the form of a creative work, but not the idea itself. A copyright is subject to limitations based on public interest considerations, such as the fair use doctrine in the United States and fair dealings doctrine in the United Kingdom. Some jurisdictions require "fixing" copyrighted works in a tangible form. It is often shared among multiple authors, each of whom holds a set of rights to use or license the work, and who are commonly referred to as rights holders. These rights normally include reproduction, control over derivative works, distribution, public performance, and moral rights such as attribution. Copyrights can be granted by publ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Notices Of The American Mathematical Society
''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume was published in 1953. Each issue of the magazine since January 1995 is available in its entirety on the journal web site. Articles are peer-reviewed by an editorial board of mathematical experts. Beginning with the January 2025 issue, the editor-in-chief is Mark C. Wilson, succeeding past editor Erica Flapan. The cover regularly features mathematical visualizations. The ''Notices'' is self-described to be the world's most widely read mathematical journal. As the membership journal of the American Mathematical Society, the ''Notices'' is sent to the approximately 30,000 AMS members worldwide, one-third of whom reside outside the United States. By publishing high-level exposition, the ''Notices'' provides opportunities for mathematicians to find out what is going on in the field. Each is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
