Let us introduce first a basic lemma in category theory that links the equivalence of two abelian categories to certain fully faithful functors.
Abelian Category Equivalence Lemma.
Let
and
be any two Abelian categories, and also let
be an exact, fully faithful, essentially surjective
functor. faithful, essentially surjective functor. Then
is an equivalence of Abelian categories
and
.
The next step is to define the hom-functors. Let
be the category
of sets. The functors
, for any category
, form a functor category
(also written as
. Then, any object
gives rise to the functor
. One has also that the assignment
extends to a natural contravariant functor
.
One of the most commonly used results in category theory for establishing an equivalence of categories is provided by the following proposition.
Yoneda Lemma.The functor
is a
fully faithful functor
because it induces isomorphisms on the Hom sets.
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