CAT E G O RY T H E O RY I I
by gabriel drummond-cole
transcribed by damien lejay
Hi, so as you may remember, I have volunteer to do this because no
one wanted to do it. But basically this talk is for no one because either
you already know all of this or either you no nothing.
functors
I am not doing serious computations or proofs. We are only going to
assert examples and not prove that they are examples. So remember
la time Damien has defined some categories:
Set
,
Vect
R
, etc. And
here you see I already make a big abuse, I have only talked about the
objes in the categories. But that is because in all those examples, the
maps are the only possible ones you can think of. La week Damien
has made the diinion between set theory and category theory by
saying that points and
∈
are replaced by objes and morphisms. But
what are the morphisms inside category theory itself? Funors are
‘morphisms between categories’. Let
C
and
D
be two categories. A
funor F : C → D is
• an assignement on objes F : Ob(C) → Ob(D );
•
an assignement on morphisms
F : Hom
C
(c, c
0
) → Hom
D
(Fc, Fc
0
)
.
satisfying F(g ◦ f ) = F(g) ◦ F(f ) and F(id) = id.
A warning on ‘covariant/contravariant’: in old literature you could
encounter the term ‘contravariant funor’ from
C
to
D
. It means a
funor from
C
op
to
D
. You shouldn’t use this terminology and only
speak of opposite categories.
Definition. — A funor F : C → D is
• faithful, if F is injeive on hom-sets;
• full, if it is sujeive on hom-sets;
• essentially surjeive, if for every obje d ∈ D, there exis an obje
c ∈ C such that F(c) ' d.
Examples of funors
• id
C
;
The Derived Seminar, March , Pohang, Korea.
©
Damien Lejay. All rights
reserved.
• const
d
;
• a very important class: forgetful funors.
U : Vect
R
→ Set
this one is faithful, not full and not essentially surjeive. There is
also a forgetful funor for groups, topological spaces and abelian
groups. These are faithful, not full and essentially surjeive.
•
if
y
is an obje of
C
then
Hom
C
(y, −) : C → Set
is a funor. This
example is extremely important and will come up all the time. It
also comes in contravariant way: Hom
C
(−, y) : C
op
→ Set;
•
How do we endow a set with a topology in a natural way? Well,
there are the discrete funor and the indiscrete funor. The real
problem in describing such a funor is not to define it on the
objes but to make sure that this definition is consient with all
the funions and compositions. The discrete funor is faithful,
full and not essentially surjeive as not every topological space
is homeomorphic to a discrete topological space. The indiscrete
funor has the same properties.
•
What about more algebraic contexts? How shall I build a funor
from the category of sets to the category of groups? There is the
Free funor:
S → F(S)
. The same can be done for abelian groups
or veor spaces (free veor space generated by a set)
•
La weak Damien has defined the category
BG
. What is a funor
from this category to a category
C
? There is only one obje to
map so such a funor is described by one element
c ∈ C
and
automorphisms
F(g) : c → c
for every
g ∈ G
. If
C = Set
, such a
funor is a group morphism
G → Aut(X)
. The same would be
true for veor spaces G → Aut(V).
natural transformations
Sometimes you want to compare funors, like the discrete and indis-
crete funors for example. For this we have natural transformations.
Natural transformations are ‘the morphisms between funors’.
Definition. — A natural transformation between two funors
F, G : C →
D
is an assignement for each
c ∈ C
of a morphism
φ
c
: F(c) → G(c)
satisfying the commutativity relation
F(c) G(c)
F(c
0
) G(c
0
)
φ
c
F(f ) G(f )
φ
c
0
As an example, a natural transformation between two funors
from BG is a G-equivariant map.
Another example: take the funor
∗∗ : V 7→ V
∗∗
. There is a natural
transformation from
id → ∗∗
. This comes from the andard map
V → V
∗∗
: there is a natural choice to build such a map, it is canonical.
eq uivalences
Definition. — An isomorphism of categories
F : C → D
is a funor such
that there exis another funor
G
such that the composition of the two is
equal to the identity. But this concept is USELESS. A funor
F : C → D
is
an equivalence of categories is full, faithful and essentially surjeive.
An example is the equivalence between the category of metric
spaces and the category of metrisable spaces. Clearly those two are
definitly not isomorphic (we can pick two diin metrics on the same
metrisable space). So there is no way we are going to build an inverse
funor. Here I want to give you another defintion for pedagogical
reasons: a funor
F
is an equivalence if there exis another funor
G :
D → C
such that we have natural transformations
F ◦ G → id
and
id →
G ◦ F
. Compared to the definition of isomorphisms, we have gotten rid
of the equality sign to replace it by an isomorphism. This recalls what
Damien said la weak: we replaced a bad categorical concept (equality
of funors) by a good categorical concept (isomorphism of funors).
adjunction
So I have given this alternate definition where you are naturally iso-
morphic. But it is possible to even weaken the notion of equivalence
where you only have natural transformations (not isomorphism). This
would define the notion of ajoint funors. But an alternate definition
for adjoint funors is quite convenient.
Definition. — A pair of funors
C D
L
R
is an adjoint pair if
Hom(Lc , d) ' Hom(c , Rd)
The name comes from the definition of ajoint operators. Adjoint
funors are ubiquituous in category theory. We use the symbol
L a R
to say that L is the left ajoint to R.
Example: does the forgetful funor
U : Top → Set
has a left or
right adjoint? Yes, by computation it has a left adjoint. If
S
is a set,
then
L(S)
should have the bigge number of open sets possible:
L
is
the discrete funor. Following the same idea, the indiscrete funor is
the right ajoint to the forgetful funor.
Example: what about the case of
U : Gp → Set
? A left adjoint
exis, it is given by the free group funor. Does it have a right ajoint?
That would mean that we have
Hom
Set
(UG, S) ' Hom
Gp
(G, RS)
. Take
S = ∅
then the left hand side is empty but the right hand side is never
empty. So there cannot be a right adjoint.
La thing, remember the funor
Hom
Set
(y, −) : Set → Set
, does
this have an adjoint? The answer is that it has a left ajoint because we
have
Hom
Set
(X × Y, Z) ' Hom
Set
(X, Hom(Y, Z))
This is called Curification in computer science. In other contexts, for
example abelian groups, the produ would be replaced by the tensor
produ.
