Mathematical Object - Category Theory

Category Theory

A variant of this approach replaces relations with operations, the basis of universal algebra. In this variant the axioms often take the form of equations, or implications between equations.

A more abstract variant is category theory, which abstracts sets as objects and the operations thereon as morphisms between those objects. At this level of abstraction mathematical objects reduce to mere vertices of a graph whose edges as the morphisms abstract the ways in which those objects can transform and whose structure is encoded in the composition law for morphisms. Categories may arise as the models of some axiomatic theory and the homomorphisms between them (in which case they are usually concrete, meaning equipped with a faithful forgetful functor to the category Set or more generally to a suitable topos), or they may be constructed from other more primitive categories, or they may be studied as abstract objects in their own right without regard for their provenance.

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Other articles related to "category theory, category, theory":

List Of Types Of Functions - Relation To Category Theory
... Category Theory is a branch of mathematics that formalizes the notion of a special function via arrows or morphisms ... A category is an algebraic object that (abstractly) consists of a class of objects, and for every pair of objects, a set of morphisms ... In a so-called concrete category, the objects are associated with mathematical structures like sets, magmas, groups, rings, topological spaces, vector spaces, metric ...
Saunders Mac Lane - Contributions
... After a thesis in mathematical logic, his early work was in field theory and valuation theory ... via the Eilenberg–Steenrod axioms, the abstract approach to homology theory, he and Eilenberg originated category theory in 1945 ... A recurring feature of category theory, abstract algebra, and of some other mathematics as well, is the use of diagrams, consisting of arrows (morphisms) linking objects, such ...
Category Theory - Higher-dimensional Categories
... For example, a (strict) 2-category is a category together with "morphisms between morphisms", i.e ... In this context, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply natural transformations of morphisms in the usual sense ... Another basic example is to consider a 2-category with a single object these are essentially monoidal categories ...
Timeline Of Category Theory And Related Mathematics - 1971–1980
... book Categories for the working mathematician, which became the standard reference in category theory 1971 Horst Herrlich-Oswald Wyler Categorical topology The study of ... study and uses structured sets in a topological category as general topology study and uses topological spaces ... spaces to structured sets in a topological category ...

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