This document discusses topics in category theory, including set-functors, adjunctions, and limits. It begins by defining set-functors and natural transformations between them. It notes that a natural transformation is uniquely determined by its value on an initial element of a functor. It then introduces adjunctions and decomposes them into left and right adjoints. It shows that a left adjoint exists if and only if certain set-functors are representable. Finally, it defines limits of diagrams (I-systems) over an index category I. It shows that a limit exists if and only if the cone functor is representable.

1. TOPICS IN CATEGORY THEORY
HEINRICH HARTMANN
CONTENTS
1. Set-Functors 1
2. Adjunctions 3
3. Limits 4
3.1. Directed Index Categories 4
3.2. I-systems 5
3.3. Cones and Co-Cones 5
3.4. Limits in abelian categories 6
4. Appendix: Universal Cones 7
5. Appendix: Comma categories 7
1. SET-FUNCTORS
Let F;G : C !D be functors. A natural transformation h : F !G is a collection
of morphisms hX : FX !GX for all X 2 C such that
X
f
FX
hX
/
F f
GX
Gf
Y FY
hX
/ GY
commutes.
How can we effectively describe natural transformations? How strong is the
assumption of naturality? We study the important case D = (set).
Definition 1.1. A weak initial element for F : C !(set) is a pair (X;x) with X 2 C
and x 2 FX such that for all z 2 FZ there is a morphism f : X !Z with F f : x7! z.
We call x 2 FX (strongly) initial if the arrow f : X !Z is unique.
More generally an element of a functor F : C ! (set) is a pair (X;x) with x 2
FX. Elements form a category ( 2 F) with morphisms f : (X;x) ! (Y;y) are
morphisms f : X !Y in C with F f : x!y.
Lemma 1.2. If x 2 FX is weak initial then a natural transformation h : F !G is
uniquely determined by h(x) 2 GX.
If x 2 FX is strongly initial then for every u 2 GX there is a unique natural
transformation with h(x) = u.
Date: 29.05.2010.
1

2. 2 HEINRICH HARTMANN
Proof. Let x 2 FX be initial and x 2 FZ. Choose a f : X !Z with F f (x) = z then
it is necessarily h(z) = Gf (x).
Conversely an element u 2 GX comes from a natural transformation if and only
if for all z 2 FZ; f1; f2 : X !Y such that F fi(x) = z it is Gf1(u) = Gf2(u) thus
we can define h(z) = Gf1(u) without ambiguity. If x 2 FX is strongly initial this
condition is always satisfied.
Here is a reformulation of the above Lemma. Every element x 2 FX determines
a map (natural in G)
evx : Nat(F;G) !GX; h7! h(x)
The element x is (weak) initial if and only if evx is (injective) bijective.
Definition 1.3. Let C be a category and X 2 C be an object. Define the Yoneda-functors
gX :C !(Set);Z7! HomC (X;Z); f7! f _
X = hX :Cop !(Set);Z7! HomC (Z;X); f7! _ f
They come with initial elements idX 2 gX (X) and idX 2 hX (X).
Proof. Let z 2 gX (Z) = Hom(X;Z) then z : X ! Z is unique with property that
gX (z) : id7! z.
Let z 2 hX (Z) = Hom(Z;X) then zop : X !Z 2 Cop is unique with property that
hX (z) : id7! z.
Remark 1.4. A morphism f : X !Y induces a natural transformation
gY ( f ) : gY !gX ; (g : Y !Z)7! g f
and similarly hF
Y ( f ) : hFX
!hF
Y ;g7! f g. It is easy to check that this construction
promotes gX ;hx to functors g : Cop!Fun(C; (set)) and h : C !Fun(Cop; (set)).
Corollary 1.5. For a functor G : C !(set), there is a canonical bijection
evid : Nat(gX ;G) !GX:
Dually for G : Cop ! (set), there is a canonical bijection Nat(hX ;G) ! GX: In
particular we recover Yoneda’s lemma:
Nat(gX ;gY )=
gX (Y) = Hom(X;Y); Nat(hX ;hY )=
hY (X) = Hom(X;Y):
An explicit inverse is given by
q : GX !Nat(gX ;G); x 2 GX7! (qx : gX !G)
with qx( f ) = Gf (x) 2 GZ for f : X !Z.
The following Lemma is the reason why initial elements are not commonly used.
Proposition 1.6. A functor F : C !(set) has an initial element if and only if F is
representable, i.e. F =
gX for some X 2 C.
Proof. If gX =F then the image of idX in FX is initial.
Conversely, if x 2 FX is initial then
evx : Nat(F;gX )=
gX (X) and evid : Nat(gX ;F)=
FX:
Let h : F !gX correspond to idX and qx : gX !F correspond to x. Then h(qx(id))=
h(x) = id and q(h(x)) = q(x) = x. Hence both compositions map the initial ele-ments
to itself. If follows they are equal to the identity.

3. TOPICS IN CATEGORY THEORY 3
2. ADJUNCTIONS
Let F : C !D be a functor.
Definition 2.1. A left-adjoint of F is a functor C D : G with a bi-natural iso-morphism
h : Hom(GX;B)!Hom(X;FB) : DopC !(set):
Dually a right-adjoint is a functor C D : H with a bi-natural isomorphism
q : Hom(FA;Y)!Hom(A;HY) : CopD !(set):
We write G ` F ` H in this cases.
An adjoint is a very condensed notion. We decompose this definition into
smaller parts which we analyze in terms of set-functors studied in the previous
section.
Definition 2.2. To an object X 2 D we associate set-functors
gFX
: C !(set); Z7! HomD(X;FZ)
hFX
: Cop !(set); Z7! HomD(FZ;X)
Remark 2.3. A morphism r : X !Y induces a natural transformations gF
Y ( f ) : gF
Y !
gFX
and hF
Y ( f ) : hFX
!hF
Y . In this way we obtain functors
gF
: Dop !Fun(C; (set)); hF
: D !Fun(Cop; (set)):
FX
FX
We have seen this construction in the last section for the special case F = id.
Assume the functor gis represented by some g=
gG. This means that
Hom(G;B) = gG(B)=
gFX
(B) = Hom(X;FB)
naturally in B. In particular if F has a left-adjoint G then every gFX
is represented
FX
=
FX
by GX. The converse of this statement is also true, i.e. we get the naturality in X
for free!
Lemma 2.4. If for X 2 D the functor gis represented by (GX;uX : gGX g)
then we can extend the map of objects G7! GX uniquely to a functor G : D !C
which is a left-adjoint of F.
Dually if Y 2 D the functor hF
Y is represented by (HY;vY : hHY =
hF
Y ) then we
can extend the map of objects Y7! HY uniquely to a functor H : D !C which is
a right-adjoint of F.
Proof. Given a morphism f : X !Y we have to find a Gf : GX ! GY such that
the diagram (of functors in B)
Hom(GX;B) gGX(B)o
=
/ gFX
(B) Hom(X;FB)
_Gf
O
Hom(GY;B)
gGY (B)o
=
_ f
O
/ gF
Y (B) Hom(Y;FB)
is commutative. By Yoneda’s Lemma there is a unique choice for Gf . If g : Y !Z
is another map, then the commutativity of
Hom(Z;FB)
_g
/
_g f
2
Hom(Y;FB)
_ f
/ Hom(X;FB)

4. 4 HEINRICH HARTMANN
together with the uniqueness of G(g f ) shows that G(g f ) = GgGf .
Remark 2.5. Assume gFX
=
gG with universal element u 2 gFX
(G) = Hom(X;FG).
This means for every r : X !FC there is a unique f : G!C with r = F f u : X !
FG!FC.
G
f
X u
/
FG
B BB BB r
BB ! B
C FC
One may regard G as a “best approximation” for X inside C.
Dually if hFX
=
hH we get a universal v 2 hFX
(B) = Hom(FH;X). So that for all
s : FC!X there is a unique g : C!B with v F f : FC!FH !X.
H FH v
/ X
C
f
O
O
FC
s
={
{{ {{ {{ {
Lemma 2.6. An adjoint functor is unique up to unique isomorphism.
3. LIMITS
3.1. Directed Index Categories.
Definition 3.1. A partially ordered set (I;) is a set I together with a transitive
relation such that i j and j i implies i = j.
An monotone map p : (I;)!(J;) between partially ordered set is a map of
sets p : I !J such that i j; i; j 2 I implies p(i) p( j).
Partially ordered sets with monotone maps form a category (poset).
Lemma 3.2. A partially ordered set I is equivalent to the datum of a small category
J with the property that #Hom(i; j) 1 for all i; j 2 Ob(J).
Ob(J) = I; #Hom(i; j) = 1,i j:
A monotone map is the same as a functor between the categories.
For many constructions it is not more difficult to allow arbitrary small cate-gories
instead of po-sets as index sets. We will later consider special po-sets/index-categories
which we introduce now.
Definition 3.3. A partially ordered set (I;) is called directed if for all i; j 2 I
there is a k 2 I with i; j k.
Definition 3.4. The analogue of direced po-sets are directed categories. These are
small categories I with the following properties.
(1) For all i; j 2 I there is an object k 2 I with morphisms i!k; j!k.
(2) For two morphisms f ; f 0 : i! j there is a co-equalizer in I , i.e. a morphism
g : j!k with g f = g f 0.

5. TOPICS IN CATEGORY THEORY 5
3.2. I-systems. Let I be a small category which will serve as index set, and C an
arbitrary category.
Definition 3.5. An I-system in C is a functor A : I !C. The obtain the category
of directed systems as functor category Fun(I ;C) = CI .
We use the following notation
i7! Ai;
(i! j)7! nji : Ai !Aj
for the action of A on objects and morphisms.
Remark 3.6. If I is the category induced by a po-set then an I-system A = (Ai;μji)
consists of objects Ai 2 C indexed by i 2 I and if i j a morphism μji : Ai !Aj
satisfying the coherence conditions
(1) μii = idAi and
(2) if i j k then μki = μk j μji.
A morphism of I-systems A = (Ai;μji) ! B = (Bi;nji) is a collection of mor-phisms
fi : Ai !Bp(i) such that np( j)p(i) fi = f j μji.
Remark 3.7. If p : I ! J is a functor between small categories, then we get a
pullback functor
p : CJ !CI ; A!A p:
3.3. Cones and Co-Cones. There is a canonical embedding D : C !CI , mapping
A to the constant I-system (DA)i = A with nji = idA.
Definition 3.8. A cone (C;a) of a system A 2 CI is a morphism of functors
a : DC !A:
Dually, a co-cone (b;D) of A 2 CI is a morphism of functors b : A !DD.
Hence a cone is given by a coherent system of morphisms
C
ai
MM
MMM MMM aj
MMMMM / Ai
nji
A
j
There are two important natural structures we can define on cones. On the one
hand cones come as a functor
cone(A) : Cop !(set); Z7! HomCI (DZ;A)
on the other hand we have the category of cones cone(A) with objects cones
(C;a) and morphisms (C;a)!(D;b) are those f : C !D which satisfy ai =
bi f for all i 2 Ob(I).
C
ai
/
f
Ai
D
qq8qqq qqq bqq
qqq i
Dually we define co-cone(A) : C !(set) and the category co-cone(A).

6. 6 HEINRICH HARTMANN
Lemma 3.9. The functor cone(A) is representable cone(A)=
hC for some C 2 C
if and only if cone(A) has a terminal object.
The functor co-cone(A) is co-representable cone(A) =
gD for some D 2 C if
and only if co-cone(A) has an initial object.
Proof. It is
cone(A) = ( 2 cone(A))op
since morphisms in ( 2 cone(A)) from (C;a)!(D;b) are those f : D!C in
C with a7! a f = b, i.e. ai f = bi.
Hence an terminal object corresponds to an initial object of ( 2 cone(A))
which is equivalent to giving a natural isomorphism hC ! cone(A) by Proposi-tion
1.6.
The dual case is easier as directly co-cone(A)=
( 2 co-cone(A)).
Definition 3.10. A (projective/inverse) limit (lim (A);a) of A 2 CI is a terminal
cone.
A co-limit (b;lim !(A)) is an initial co-cone.
By the lemma this is equivalent to an giving an isomorphism
a : Hom(_;lim (A)) !cone(A) : Cop !(set):
b : Hom(lim !(A);_) !co-cone(A) : C !(set):
An explicit description is for all cones (C;b) there is a unique arrow C !
lim (A) such that
lim (A)
ai
/ Ai
O bi
C
oo7ooo ooo ooo ooo oo
commutes.
Remark 3.11. Limits as neutral extensions. If I has a final object then every I-system
has a limit. Conversely a limit can be seen as an extension of A to an
enlarged index category I+ by an terminal object.
Characterize limits a initial-neutral objects in the category of neutral exten-sions.
If I has a terminal object, then the limit is the image of the terminal object.
3.4. Limits in abelian categories. Direct limits are right exact, Projective limits
are left exact. This follows form the adjunction properties of the limits to the
diagonal embedding
lim
!
D lim
:
Direct limits along directed categories are exact.
Projective limits along directed categories along functors with Mittag-Leffler
condition are exact.
Example 3.12. Direct limits along non-directed categories are not always exact.
Consider the two projections R R2 !R. The direct limit of this diagram is 0.
The diagram 0 0!R is a sub-diagram with limit R.

7. TOPICS IN CATEGORY THEORY 7
4. APPENDIX: UNIVERSAL CONES
A natural transformation h : Hom(B;_) ! HomCI (A;D_) determines a cone
h(idB). Conversely a cone b : A ! D(B) determines a natural transformation
Hom(B;_)!Cone(A) mapping f : B!Z to f bi : Ai !B!Z.
Proposition 4.1. The natural transformation h uniquely determined by the univer-sal/
initial cone μ = h(idA) 2 Cone(A).
μ = (μi : Ai !A)i
Proof. We give a conceptional proof of a more general statement. For a set valued
functor F : C !(Set), we consider the comma category (fg!F) with
Objects : (A;a); A 2 C;a 2 F(A)
(A F
/ F(A) 3 a)
Hom((A;a); (B;b)) : f : A!B 2 C; with F( f ) : a7! b:
Every object (A;a) in (fg!F), defines a natural transformation
h : Hom(A;_)!F; ( f : A!B)7! F( f )(a) 2 F(B):
A
f
F
/ F(A)
3 a
_
B F(B) 3 f (a)
This natural transformation h : Hom(A;_)!F is an isomorphism if and only
if (A;a) is an initial object of (fg!F). Indeed, if (A;a) is initial, then there is a
unique f : A!B with b = F( f )(a), i.e. f : (A;a)!(B;b). Conversely if h is an
isomorphism, then we get an induced isomorphism of categories
(fg!Hom(A;_)) !(fg!F)
Clearly (A; idA) is an initial object of (fg ! Hom(A;_)). Given (B; f : A ! B)
there is a unique morphism (A; idA)!(B; f ) namely f : A!B.
As a corollary we see that the universal/initial cone (A;μ) initial the category
Cone(A) = (A !D) of cones under A which has
Objects : (B;n : A !D)
Hom((B;n); (B0;n0
)) : f : B!B0 such that n0
= D( f ) n:
Indeed there is a canonical isomorphisms of comma categories
(A !D)=
(fg!Cone(A))
by viewing n : D(B)!A as an element of Cone(A)(B), i.e. a morphism fg!
Cone(A)(B) of sets.
5. APPENDIX: COMMA CATEGORIES
Given a diagram of categories
A F
!C G
B
we define the comma category (F !G) with
Objects : (A; f ;B);A 2 A;B 2 B; f : F(A)!G(B) 2 C
Hom((A; f ;B); (A0; f 0;B0)) : (a : A!A0;b : B!B0) with f 0 F(a) = F(b) f

8. 8 HEINRICH HARTMANN
A
a
F(A)
f
/ G(B)
B
b
A0 F(A0)
f 0
/ G(B0) B0
If A = ¥ the category with one object and one morphism, the a functor F : A !
C is given by just one object A = F(1) 2 C. In this case the comma category is
denoted by (A!F). Similarly if B = ¥.
If A = B = ¥ and F;G are represented by objects A;B in C then (F ! G) =
(A!B) = Hom(A;B) as a discrete category (only identity morphisms). This mo-tivates
the notation.
A diagram of the form
A
a
F
/ C
b
NNN NNN NNN NNN N B G
q
N*NNN NNN NNN NNN
o
g
A0
pp4ppp ppp ppp pp
h
ppp ppp ppp ppp p F0
G0
/ C0 o
B0 with natural transformations h : F0 a!bF, q : gG!G0 b induces a functor
of comma categories
(F !G) !(F0 !G0):
Remark 5.1. The isomorphism in the above section
(A !D)=
(fg!Cone(A))
can be constructed in this context using the diagram
¥ A
/ CI
N#+NNN NNN NNN NNN
NNN NNN NNN NNN N
H
C D
o
¥
pp4ppp ppp ppp pp
ppp ppp ppp ppp p fg
Cone(A)
/ (Set) o
C:
Here H Here the upper row defines the comma category (A;D) and the second one
(fg!Cone(A). It is an isomorphism because of the fundamental relation
Hom(;Hom(A;B)) = Hom(A;B)
in (Set).