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 01:20, 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
:
- (associativity) Given
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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