What are and naturally equivalent to?

Okay I'm stuck in this one, I know what a natural equivalence is, but my problem is I don't really understand what does, I know it reverses all arrows in the Category but I don't get what changes by doing that.

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- Sep 27th 2009, 05:20 PMJose27Category Theory
What are and naturally equivalent to?

Okay I'm stuck in this one, I know what a natural equivalence is, but my problem is I don't really understand what does, I know it reverses all arrows in the Category but I don't get what changes by doing that. - Sep 27th 2009, 07:01 PMaliceinwonderland
In wiki (link), an opposite category or dual category of of a given category C is formed by reversing the morphisms. It means the objects of are just the objects of C and the arrows of are arrows in a one to one correspondence manner (if an arrow of C is , then in is . I assume you already know this one.

Now consider some examples. Let S be a functor from the category of to category of B such that . It assigns to each object an object Sc of B and to each arrow of an arrow of B.

Here is another example. If is a functor from the category of C to the category of B, then we can define a functor from . In this case, can be applied to both, where c is an object of both C and C^{op} (Remember that objects remain same in the dual category). However, should be changed to for the latter one. - Sep 27th 2009, 10:41 PMNonCommAlg
this question is not easy to answer! right now i have no idea what category is equivalent to but it is known that is equivalent to the category of compact abelian groups (continuous

homomorphisms are the morphisms of this category). the idea is to associate to any abelian group the abelian group of its characters, i.e. where is the circle group.

we also associate to any homomorphism the morphism defined by the topology on is the pointwise convergence topology on where is the set

of all functions there are so much details here to be proved and you probably can find find them in some textbooks.