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In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object.
Category theory is a field of mathematics which deals in an abstract way with mathematical structures and relationships between them. Arising as an abstraction of homological algebra , which itself was affectionately called " abstract nonsense ", category theory is sometimes called " generalized abstract nonsense ".
For example, geometry has its origins in the calculation of distances and areas in the real world, and algebra started with methods of solving problems in arithmetic. Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract.
At the very least, category theoretic language clarifies what exactly these related areas have in common (in some abstract sense). Category theory has been applied in other fields as well, see applied category theory. For example, John Baez has shown a link between Feynman diagrams in physics and monoidal categories. [7]
Categories are distinct collections of concrete or abstract instances (category members) that are considered equivalent by the cognitive system. Using category knowledge requires one to access mental representations that define the core features of category members (cognitive psychologists refer to these category-specific mental representations as concepts).
An example of this abstraction process is the generational development of programming language from the machine language to the assembly language and the high-level language. Each stage can be used as a stepping stone for the next stage. The language abstraction continues for example in scripting languages and domain-specific programming languages.
Abstract objects are most commonly used in philosophy, particularly metaphysics, and semantics. They are sometimes called abstracta in contrast to concreta. The term abstract object is said to have been coined by Willard Van Orman Quine. [5] Abstract object theory is a discipline that studies the nature and role of abstract objects. It holds ...
The headings used were the three objective categories of Abstract Relation, Space (including Motion) and Matter and the three subjective categories of Intellect, Feeling and Volition, and he found that under these six headings all the words of the English language, and hence any possible predicate, could be assembled. [41]