View source for Category (category theory)

A category, <math>\mathcal{C}</math>, is a mathematical object consisting of:

* A [[class]], <math>\text{Ob}(\mathcal{C})</math> of objects.
* For every pair of objects <math>A,B\in \text{Ob}(\mathcal{C})</math>, a class <math>\text{Hom}(A,B)</math> of morphisms from <math>A</math> to <math>B</math>. (We sometimes write <math>f:A \to B</math> to mean <math>f\in \text{Hom}(A,B)</math>.)
* For every three objects, <math>A,B,C \in \mathcal{C}</math>, a binary operation <math>\circ: \text{Hom}(B,C) \times \text{Hom}(A,B) \to \text{Hom}(A,C)</math> called composition, which satisfies:
** (associativity) Given <math>f:A\to B</math>, <math>g:B\to C</math> and <math>h:C \to D</math> we have <cmath>h\circ(g\circ f) = (h \circ g)\circ f.</cmath>
** (identity) For and object <math>X</math>, there is an identity morphism <math>1_X:X\to X</math> such that for any <math>f:A\to B</math>: <cmath>1_B\circ f = f = f\circ 1_A.</cmath>

The class of all morphisms of <math>\mathcal{C}</math> is denoted <math>\text{Hom}(\mathcal{C})</math>.

A category <math>\mathcal{C}</math> is called '''small''' if both <math>\text{Ob}(\mathcal{C})</math> and <math>\text{Hom}(\mathcal{C})</math> are [[set|sets]]. If <math>\mathcal{C}</math> is not small, then it is called '''large'''. <math>\mathcal{C}</math> is called '''locally small''' if <math>\text{Hom}(A,B)</math> is a set for all <math>A,B\in \text{Ob}(\mathcal{C})</math>. Most important categories in math are not small, but are locally small.

Intuitively we can think of the objects of <math>\mathcal{C}</math> as being sets (perhaps with some additional structure) and morphisms as being functions between these sets (perhaps satisfying some properties) and composition as being regular function composition, however there are examples of categories which do not satisfy this. Typically when studying category theory we deal with morphisms and composition completely abstractly (similarly to how we study multiplication abstractly in [[group theory]]), and never talk about 'plugging things in to' morphisms.

== Examples ==

Some common examples of categories are:

* The category '''Set''' of all sets, where morphisms are [[functions]].
* The category '''Grp''' of all [[group|groups]], where morphisms are [[group homomorphism|group homomorphisms]].
* The category '''Ab''' of all [[abelian group|abelian groups]], where morphisms are [[group homomorphism|group homomorphisms]].
* The category '''Ring''' of all [[ring|rings]], where morphisms are [[ring homomorphism|ring homomorphisms]].
* The category '''Field''' of all [[field|fields]], where morphisms are [[field homomorphism|field homomorphisms]] (notice that this means all morphisms are injective, and so they can be viewed as [[field extension|field extensions]]).
* The category '''Vect''' of all [[vector space|vector spaces]], where morphisms are [[linear map|linear maps]].
* The category '''Top''' of all [[topological space|topological spaces]], where morphisms are [[continuous function|continuous functions]].
* The category '''Cat''' of all small categories, where morphisms are [[functor|functors]].
* For any categories <math>\mathcal{C}</math> and <math>\mathcal{D}</math>, the '''functor category''' <math>\mathcal{D}^\mathcal{C}</math> of functors <math>\mathcal{C}\to \mathcal{D}</math> where morphisms are [[natural transformation|natural transformations]].
* For any set <math>S</math>, we can form the ''[[discrete category]] on <math>S</math>'' whose objects are elements of <math>S</math> and such that the only morphisms are the identity morphisms on each object.
* For any category <math>\mathcal{C}</math> we can form the ''[[opposite category]]'' or the ''dual category'', <math>\mathcal{C}^{op}</math> whose objects are the objects of <math>\mathcal{C}</math>, but where all the morphisms are 'reversed' (i.e. a morphism from <math>X</math> to <math>Y</math> would be replaced by a morphism from <math>Y</math> to <math>X</math>).
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[[Category:Category theory]]