Category (category theory)
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.
Intuitively we can think of the objects of as being sets (perhaps with some additional structure) and morphisms as being functions between these sets (perhaps satisfying some properties) and composition as being regular function composition, however there are examples of categories which do not satisfy this. Typically when studying category theory we deal with morphisms and composition completely abstractly (similarly to how we study multiplication abstractly in group theory), and never talk about 'plugging things in to' morphisms.
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