Difference between revisions of "Category (category theory)"
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The class of all morphisms of <math>\mathcal{C}</math> is denoted <math>\text{Hom}(\mathcal{C})</math>. | 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 [[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. | + | 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. | 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. | ||
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| + | == 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]]. | ||
{{stub}} | {{stub}} | ||
[[Category:Category theory]] | [[Category:Category theory]] | ||
Revision as of 02:00, 2 September 2008
A category, 
, is a mathematical object consisting of:
- A class,
of objects. - For every pair of objects
, a class 
of morphisms from 
to 
. (We sometimes write 
to mean 
.) - For every three objects,
, a binary operation 
called composition, which satisfies:
- (associativity) Given
, 
and 
we have ![\[h\circ(g\circ f) = (h \circ g)\circ f.\]](//latex.artofproblemsolving.com/5/d/7/5d70e95a5f477d732eb7446c57b120cc8e5b2b7f.png)
- (identity) For and object
, there is an identity morphism 
such that for any : ![\[1_B\circ f = f = f\circ 1_A.\]](//latex.artofproblemsolving.com/a/e/c/aece5511fa01764dfc5258e44f54823bb93c2e2c.png)
- (associativity) Given
to 
. (We sometimes write 
to mean 
.)
, a binary operation 
called composition, which satisfies:
, 
and 
we have ![\[h\circ(g\circ f) = (h \circ g)\circ f.\]](http://latex.artofproblemsolving.com/5/d/7/5d70e95a5f477d732eb7446c57b120cc8e5b2b7f.png)
, there is an identity morphism 
such that for any ![\[1_B\circ f = f = f\circ 1_A.\]](http://latex.artofproblemsolving.com/a/e/c/aece5511fa01764dfc5258e44f54823bb93c2e2c.png)
The class of all morphisms of is denoted 
.
A category is called small if both 
and are sets. If is not small, then it is called large. is called locally small if 
is a set for all 
. Most important categories in math are not small, but are locally small.
Intuitively we can think of the objects of 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 groups, where morphisms are group homomorphisms.
- The category Ab of all abelian groups, where morphisms are group homomorphisms.
- The category Ring of all rings, where morphisms are ring homomorphisms.
- The category Field of all fields, where morphisms are field homomorphisms (notice that this means all morphisms are injective, and so they can be viewed as field extensions).
- The category Vect of all vector spaces, where morphisms are linear maps.
- The category Top of all topological spaces, where morphisms are continuous functions.
- The category Cat of all small categories, where morphisms are functors.
- For any categories
and 
, the functor category 
of functors 
where morphisms are natural transformations.
, the functor category 
of functors 
where morphisms are This article is a stub. Help us out by expanding it.
