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 almost all 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 spaces, direct products, completion, and duality.
The Jupiter Ace by Jupiter Cantab was a British home computer released in 1982. The Ace differed from other microcomputers of the time in that its programming environment used Forth instead of the more popular BASIC. This difference, along with limited available software and poor character based graphic display, limited sales and the machine was not a success.
In mathematics, pointless topology, also called point-free topology and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this approach it becomes possible to construct topologically interesting spaces from purely algebraic data.
The Sinclair ZX80 is a home computer launched on 29 January 1980 by Science of Cambridge Ltd.. It is notable for being one of the first computers available in the United Kingdom for less than a hundred pounds. It was available in kit form for £79.95, where purchasers had to assemble and solder it together, and as a ready-built version at £99.95.
The ZX Spectrum is an 8-bit home computer developed and marketed by Sinclair Research. Considered one of the most influential computers ever made, it is also one of the best-selling British computers ever, with over five million units sold. It was released in the United Kingdom on 23 April 1982, and around the world in the following years, most notably in Europe, the United States, and Eastern Bloc countries.
The ZX81 is a home computer that was produced by Sinclair Research and manufactured in Dundee, Scotland, by Timex Corporation. It was launched in the United Kingdom in March 1981 as the successor to Sinclair's ZX80 and designed to be a low-cost introduction to home computing for the general public. It was hugely successful; more than 1.5 million units were sold. In the United States it was initially sold as the ZX-81 under licence by Timex. Timex later produced its own versions of the ZX81: the Timex Sinclair 1000 and Timex Sinclair 1500. Unauthorized ZX81 clones were produced in several countries.
Jupiter Cantab Limited was a Cambridge based home computer company. Its main product was the 1983 Forth-based Jupiter Ace.
This article gives some very general background to the mathematical idea of topos. This is an aspect of category theory, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory.
Sinclair BASIC is a dialect of the programming language BASIC used in the 8-bit home computers from Sinclair Research, Timex Sinclair and Amstrad. The Sinclair BASIC interpreter was written by Nine Tiles Networks Ltd.
In mathematics, given two partially ordered sets P and Q, a function f: P → Q between them is Scott-continuous if it preserves all directed suprema. That is, for every directed subset D of P with supremum in P, its image has a supremum in Q, and that supremum is the image of the supremum of D, i.e. , where is the directed join. When is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets.
Francis William Lawvere was an American mathematician known for his work in category theory, topos theory and the philosophy of mathematics.
Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor. The categorical framework provides a rich conceptual background for logical and type-theoretic constructions. The subject has been recognisable in these terms since around 1970.
This is a timeline of category theory and related mathematics. Its scope is taken as:
In mathematics, a topos is a category that behaves like the category of sheaves of sets on a topological space. Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic.
Colin McLarty is an American logician whose publications have ranged widely in philosophy and the foundations of mathematics, as well as in the history of science and of mathematics.
Richard Francis Altwasser is a British engineer and inventor, responsible for the hardware design of the ZX Spectrum.
In mathematics, a ringed topos is a generalization of a ringed space; that is, the notion is obtained by replacing a "topological space" by a "topos". The notion of a ringed topos has applications to deformation theory in algebraic geometry and the mathematical foundation of quantum mechanics. In the latter subject, a Bohr topos is a ringed topos that plays the role of a quantum phase space.
In mathematics, a generalized space is a generalization of a topological space. Impetuses for such a generalization comes at least in two forms:
- A desire to apply concepts like cohomology for objects that are not traditionally viewed as spaces. For example, a topos was originally introduced for this reason.
- A practical need to remedy the deficiencies that some naturally-occurring categories of spaces tend not to be abelian, a standard requirement to do homological algebra.
