1. Prove that a functor F: C → D is an equivalence i it is
essentially surjective, full and faithful.
Hint: let me remind you of all the necessary definitions.
definition 1. A category C consists of:
• a collection Ob(C) of objects.
• for any pair of objects x, y, a set hom(x, y) of morphisms from x
to y. (If f ∈ hom(x, y) we
write f: x → y.)
equipped with:
•
for any object x, an identity morphism 1x: x → x.
•
for any pair of morphisms f: x → y and g: y → z, a morphism fg: x → z called the
composite
of f and g.
such that:
•
for any morphism f: x → y, the left and right unit laws hold: 1xf = f = f1y.
•
for any triple of morphisms f:w → x, g: x → y, h: y → z, the associative law
holds :
(fg)h = f(gh).
We usually write x ∈C as an abbreviation for x ∈ Ob(C). An isomorphism is a morphism f: x → y
with an inverse, i.e. a morphism g: y → x such that fg = 1x and gf = 1y.
definition 2. Given categories C,D, a functor F:C → D consists of:
•
a function F: Ob(C) → Ob(D).
• for any pair of objects x, y ∈ Ob(C), a function F: hom(x, y) → hom(F(x), F(y)).
such that:
•
F preserves identities : for any object x ∈C,
.
•
F preserves composition: for any pair of morphisms f: x → y, g: y → z in C, F(fg)
=
F(f)F(g).
It's not hard to define identity functors and composition of functors, and to
check the left and right
unit law and associative law for these.
definition 3. Given functors F,G:C → D, a natural transformation
consists of:
•
a function mapping each object x ∈ C to a morphism
: F(x) → G(x)
such that:
•
for any morphism f: x → y in C, this diagram commutes:
With a little thought you can figure out how to compose
natural transformations α: F → G and
and get a natural transformation
. We can also define identity
natural
transformations. Again, it's not hard to check the left and right unit law and
associativity for these.
definition 4. Given functors F,G:C → D, a natural isomorphism
is a
natural
transformation that has an inverse, i.e. a natural transformation
such
that 
and
It's not hard to see that a natural transformation
is a natural
isomorphism iff for every
object x ∈ C, the morphism
is invertible.
definition 5. A functor F:C → D is an equivalence if it has a weak inverse, that
is, a functor
G: D → C such that there exist natural isomorphisms
.
definition 6. A functor F:C → D is essentially surjective if for every object x
∈ D there is an
object 
such that
.
definition 7. A functor F:C → D is full if for every pair of objects x, y ∈ C,
the function
F: hom(x, y) → hom(F(x), F(y)) is onto.
definition 8. A functor F:C → D is faithful if for every pair of objects x, y ∈
C, the function
F: hom( x, y) → hom(F(x), F(y)) is one -to-one.
I will be glad to give you further hints if you need them. The fun part is
constructing the weak
inverse of a functor F using the fact that it's essentially surjective, full and
faithful. This is a
categorified version of constructing the inverse of a function using the fact
that it's surjective and
injective.
