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category (Definition)

A category $ \mathcal{C}$ consists of the following data:

  1. a class $ \operatorname{ob}(\mathcal{C})$ of objects (of $ \mathcal{C}$)
  2. for each ordered pair $ (A,B)$ of objects of $ \mathcal{C}$, a collection (we will assume it is a set) $ \hom(A,B)$ of morphisms from the domain $ A$ to the codomain $ B$
  3. a function $ \circ:\hom(A,B)\times\hom(B,C)\to\hom(A,C)$ called composition.

We normally denote $ \circ(f,g)$ by $ g \circ f$ for morphisms $ f,g$. The above data must satisfy the following axioms: for objects $ A,B,C,D$,

A1: $ \hom(A,B) \cap \hom(C,D)=\emptyset$ whenever $ (A,B)\neq (C,D)$, i.e. the intersection is non-trivial only when $ A=C$ and $ B=D$.

A2: (Associativity) if $ f \in \hom(A,B)$, $ g\in\hom(B,C)$ and $ h\in\hom(C,D)$, $ h\circ (g\circ f)=(h\circ g)\circ f$

A3: (Existence of an identity morphism) for each object $ A$ there exists an identity morphism $ {}id_{A}\in\hom(A,A)$ such that for every $ f\in\hom(A,B)$, $ f\circ id_{A}=f$ and $ {}id_{A}\circ g=g$ for every $ g \in \hom(B,A)$.

Some examples of categories:

  • 0 is the empty category with no objects or morphisms, 1 is the category with one object and one (identity) morphism.
  • If we assume we have a universe $ U$ which contains all sets encountered in “everyday” mathematics, Set is the category of all such small sets with morphisms being set functions
  • Top is the category of all small topological spaces with morphisms continuous functions
  • Grp is the category of all small groups whose morphisms are group homomorphisms



"category" is owned by mathcam. [ full author list (3) | owner history (1) ]
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See Also: category of sets, monad, group object, group scheme, inverse limit, $\mathcal{U}$-small, endomorphism, subcategory, precategory, monoidal category

Also defines:  morphism, identity, object

Attachments:
arrow category (Example) by CWoo
commutative diagram (Definition) by Dr_Absentius
category associated to a partial order (Example) by archibal
monoid as a category (Definition) by kompik
algebra formed from a category (Definition) by rspuzio
category of matrices (Example) by rspuzio
product of categories (Definition) by CWoo
alternative definition of category (Definition) by rspuzio
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Cross-references: group homomorphisms, groups, topological spaces, set functions, contains, universe, empty category, associativity, intersection, axioms, composition, function, codomain, domain, collection, ordered pair
There are 446 references to this entry.

This is version 20 of category, born on 2001-11-19, modified 2007-06-24.
Object id is 965, canonical name is Category.
Accessed 27088 times total.

Classification:
AMS MSC: 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)
Pending Errata and Addenda
None.
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link to 'class' is incorrect by guffin on 2007-05-21 11:42:39
The word "class" links to "Vizing's theorem", instead of "Class": http://planetmath.org/encyclopedia/Class.html
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What is an object? by dataweaver on 2006-10-22 10:10:51
Supposedly, the definition of "category" also defines "object". Reading through the definition, it is not clear to me how this is done, or what the definition of "object" is. All I know for certain is that a class of them is one of the things that a category consists of, and that morphisms are defined in terms of pairs of them. 
[ reply | up ]
"small" by archibal on 2004-02-11 21:28:15
What is meant by the qualifier "small" in the examples? That is, when is a group "small" in this sense? A topological space? Does this just mean that their underlying object must be a set? (I had assumed that was part of their definition, but I suppose it need not be... for a topological space, this would seem to pose problems...)

Andrew
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