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> | ||
{{stub}} | {{stub}} | ||
[[Category:Category theory]] | [[Category:Category theory]] |
Revision as of 01:12, 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
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