# Difference between revisions of "Category (category theory)"

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** (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> | ** (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> | + | 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. | ||

{{stub}} | {{stub}} | ||

[[Category:Category theory]] | [[Category:Category theory]] |

## Revision as of 00:20, 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
- (identity) For and object , there is an identity morphism such that for any :

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.
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