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A category consists of a collection of objects, together with a collection of maps between those objects, called "morphisms", and a way to compose morphisms. Many familiar mathematical objects may be fit into a category:
A category consists of a collection of objects, together with a collection of maps between those objects, called "morphisms", and a way to compose morphisms. Many familiar mathematical objects may be fit into a category:
:'''Set''' is the category of [[set|sets]]. The objects of '''Set''' are just sets, in the usual sense. A morphism between two objects '''A''' and '''B''' in set is just a function from '''A''' to '''B'''. The composition of morphisms is just the composition of set functions.
:'''Set''' is the category of [[set]]s. The objects of '''Set''' are just sets, in the usual sense. A morphism between two objects '''A''' and '''B''' in set is just a function from '''A''' to '''B'''. The composition of morphisms is just the composition of set functions.
:'''Vect'''<math>{}_k</math> is the category of [[vector space|vector spaces]] over a [[field]] k. A morphism in this category is a linear transformation between two vector spaces.
:'''Vect'''<math>{}_k</math> is the category of [[vector space]]s over a [[field]] k. A morphism in this category is a linear transformation between two vector spaces.
:'''Top''' is the category of [[topological space|topological spaces]], with maps as [[continuous]] functions.
:'''Top''' is the category of [[topological space]]s, with maps as [[continuous]] functions.
Many concepts in mathematics, computer science and mathematical physics can be phrased in the language of categories and morphisms between categories. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942 as a tool in the study of [[algebraic topology]].
Many concepts in mathematics, computer science and mathematical physics can be phrased in the language of categories and morphisms between categories. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942 as a tool in the study of [[algebraic topology]].
Category theory is considered very powerful by some mathematicians. But is considered unnecessarily abstract by other mathematicians, so much so that they jokingly refer to it as "[[abstract nonsense]]".
Category theory is considered very powerful by some mathematicians. But is considered unnecessarily abstract by other mathematicians, so much so that they jokingly refer to it as "[[abstract nonsense]]".
[[category:mathematics]]
[[Category:Mathematics]]