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 ![\[h\circ(g\circ f) = (h \circ g)\circ f.\]](//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.
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