Opposite category
Given a category 
, we may form another category 
, called the opposite category or the dual category by 'reversing all the morphisms of .
Formally we define by:
- Every morphism
in 
is associated to a unique morphism 
in 
(and every morphism 
in is equal to 
for some 
in ). - For any and
the composition of 
and is definied by:![\[f^{op}\circ g^{op} = (g\circ f)^{op}.\]](//latex.artofproblemsolving.com/f/b/d/fbd328c7710a5508098069fe51f521282282f805.png)
in 
in 
in 
for some 
in 
the composition of 
and ![\[f^{op}\circ g^{op} = (g\circ f)^{op}.\]](http://latex.artofproblemsolving.com/f/b/d/fbd328c7710a5508098069fe51f521282282f805.png)
It is know easy to verify that is a category:
- Given , and
we have:![\[f^{op}\circ(g^{op}\circ h^{op}) = f^{op}\circ(h\circ g)^{op} = [(h\circ g)\circ f]^{op} = [h\circ (g\circ f)]^{op} = (g\circ f)^{op}\circ h^{op} = (f^{op}\circ g^{op})\circ h^{op}.\]](//latex.artofproblemsolving.com/f/f/5/ff5653ac8c81bea9f0498aa7227e9eee607a5545.png)
- For any
we claim that the identity morphism on 
is just 
. Indeed, for any , we have:![\[1_A^{op}\circ f^{op} = (f\circ 1_A)^{op} = f^{op} = (1_B\circ f)^{op} = f^{op}\circ 1_B^{op}.\]](//latex.artofproblemsolving.com/3/4/3/343525e13989e20f2a92dec8b535009e038005ab.png)
we have:![\[f^{op}\circ(g^{op}\circ h^{op}) = f^{op}\circ(h\circ g)^{op} = [(h\circ g)\circ f]^{op} = [h\circ (g\circ f)]^{op} = (g\circ f)^{op}\circ h^{op} = (f^{op}\circ g^{op})\circ h^{op}.\]](http://latex.artofproblemsolving.com/f/f/5/ff5653ac8c81bea9f0498aa7227e9eee607a5545.png)
we claim that the identity morphism on 
is just 
. Indeed, for any ![\[1_A^{op}\circ f^{op} = (f\circ 1_A)^{op} = f^{op} = (1_B\circ f)^{op} = f^{op}\circ 1_B^{op}.\]](http://latex.artofproblemsolving.com/3/4/3/343525e13989e20f2a92dec8b535009e038005ab.png)
Duality Principle
The existence of the opposite category allows us to prove the duality principle which roughly states that: for any 'theorem' provable from the axioms of category theory, it's dual statement (i.e. the statement formed by 'replacing every morphism with a morphism pointing in the other direction') is also true (and can be proved by going through the exact same proof in the opposite category).
Similarly, given any construction or definition in category theory, we can from a dual concept in the same way (by defining that concept in the opposite category). The dual of a concept sometimes named by attaching the prefix co- to the name of the original concept (for instance, the dual of a limit is called a colimit).
