Contents
Categories and Functors
"general abstract nonsense", "functors leave complexes
complex"
 category
 A category C consists of a family Ob C
of objects, and for each A,B∈Ob
C a set Mor_{C}(A,B)
of morphisms between A and B
such that
r 
∀A∈Ob C
∃id_{A}∈Mor_{C}(A,A) 
reflexive, identity 
t 
∀A,B,C∈Ob
C ∃∘:Mor_{C}(B,C)×Mor_{C}(A,B)→Mor_{C}(A,C)
with 
transitive, composition 
n 
∀A,B∈Ob
C ∀g∈Mor_{C}(A,B)
g∘id_{A} = g 
neutral 
∀A,B∈Ob
C ∀f∈Mor_{C}(B,A)
id_{A}∘f = f 
a 
∀A,B,C,D∈Ob
C ∀f∈Mor_{C}(A,B)∀g∈Mor_{C}(B,C)∀h∈Mor_{C}(C,D)
(h∘g)∘f = h∘(g∘f) 
associative 
We usually assume that Mor_{C}(A,B)
∩ Mor_{C}(C,D)
= ∅ unless A=C ∧
B=D.
Write: f:A→B for
f∈Mor_{C}(A,B)
☡ Beware: Unlike Mor_{C}(A,B),
Ob C does not need to be a set of objects,
but only a class of objects. Nevertheless we formally write ∈,⊆ etc.
 covariant
functor
 A covariant functor F:C→D
between two categories C and D
consists of a map F:Ob C→Ob
D and for each A,B∈Ob
C a map F:Mor_{C}(A,B)→Mor_{D}(F(A),F(B))
that satisfies the following conditions
1 
∀A∈Ob C
F(id_{A}) = id_{F(A)} 
unital 
m 
∀f∈Mor_{C}(A,B)∀g∈Mor_{C}(B,C)
F(g∘f) = F(g)
∘ F(f) 
morph 
☡ Beware: F:C→D
is not a map, since C,D
are no sets (they are categories) and even Ob C,Ob
D need not be sets.
 contra
variant
functor
 A contravariant functor F:C→D
between two categories C and D
consists of a map F:Ob C→Ob
D and for each A,B∈Ob
C a map F:Mor_{C}(A,B)→Mor_{D}(F(B),F(A))
that satisfies the following conditions
1 
∀A∈Ob C
F(id_{A})
= id_{F(}_{A}_{)} 
unital 
m° 
∀f∈Mor_{C}(A,B)∀g∈Mor_{C}(B,C)
F(g∘f) = F(f)
∘ F(g) 
contramorph 
A contravariant functor F:C→D
is a covariant functor F:C→D°
(or F:C°→D).
 natural
trans
formation

A natural transformation (or morphism of functors) α:F→G
between the functors F,G:C→D
is a family (α_{C}∈Mor_{D}(F(C),G(C)))_{C}_{∈Ob
C} of morphisms such that
∀f∈Mor_{C}(C,C')
G(f)∘α_{C}
= α_{C'}∘F(f)
The functors C→D
form a category with natural transformations being the morphisms.
Remark: an isomorphism of functors is just a family of isomorphisms (α_{C}∈Iso_{D}(F(C),G(C)))_{C}_{∈Ob
C} with the above property (⇒
especially F(C)≅G(C)
∧ G(f) = α_{C'}∘F(f)∘α_{C}^{1}).
The elements of the monoid Mor_{C}(A,A)
are called endomorphisms of A. The elements of Mor_{C}(A,B)
that have an inverse in Mor_{C}(B,A)
are called isomorphisms. The elements of the group of isomorphisms in
Mor_{C}(A,A)
are called automorphisms.
Let f∈Mor_{C}(A,A')
be a morphism. 
monomorphism 
∀B∈Ob C
m_{B}(f) injective 
f is left cancelable 

⇔ f injective 
(if C=Ens
) 
epimorphism 
∀B∈Ob C
m^{B}(f) injective 
f is right cancelable 

⇔ f surjective 
(if C=Ens
) 
 kernel ≤ equalizer
 i:X→A is a kernel of
f:A→B :⇔
f∘i = 0, and ∀j:X'→A with f∘j=0
∃!j':X'→X such that the diagram commutes
.
Write: i = ker(f)
 cokernel ≤ coequalizer
 p:B→Y is a kernel of
f:A→B :⇔
p∘f = 0, and ∀j:B→Y' with j∘f=0
∃!j':Y→Y' such that the diagram commutes
.
Write: p = coker(f)
 additive
 A category C is additive, if

∀A,B∈Ob
C Hom_{C}(A,B)
:= Mor_{C}(A,B)
is an Abelian(!)
group 

d 
∘ is distributive over +, i.e. Hom_{C}(·,B)
and Hom_{C}(B,·)
are functors C→Ab
 ∀A,B,C∈Ob C ∀f∈Hom_{C}(B,C)
∀g,h∈Hom_{C}(A,B)
f∘(g+h) = (f∘g) + (f∘h)
 ∀A,B,C∈Ob C ∀f,g∈Hom_{C}(B,C)
∀h∈Hom_{C}(A,B)
(f+g)∘h = (f∘h) + (g∘h)

distributive 
 additive
 A functor F:C→D
between two additive categories C and D
is additive, if
∀A,B∈Ob C
F:Hom_{C}(A,B)→Hom_{D}(F(A),F(B))
homomorphism of groups
⇒ ∀G≅F G
is additive
 faithful
 A functor F is faithful, if it is injective on maps.
 full
 A functor F is full, if it is surjective on maps.
 Abelian
 A category C is Abelian, if

C is additive 


all coproducts, kernels and cokernels exist. (⇒ short exact sequences exist)



Isomorphism Theorem.



every monomorphism is the kernel of its cokernel (m = ker coker m),
every epimorphism the cokernel of its kernel (f = coker ker f).

normal 
⇔

C has a zero object 


all finitary limits and colimits exist



every monomorphism is the kernel of its cokernel (m = ker coker m),
every epimorphism the cokernel of its kernel (f = coker ker f).

normal 
Important Examples

For every category C, there is the opposite
(dual) category C° obtained by "turning
the arrows around", per Ob C° := Ob C,
and Mor_{C°}(A,B) := Mor_{C}(B,A),
whereas the identity morphisms are the same, and the composition is f∘_{C°}g
:= g∘_{C}f.
Duality Principle: Any theorem holding for a category C
also holds (in C°) with all
arrows reversed.

Categories together with functors form a category.

Additive categories form a category together with additive functors.

Functors of fixed categories together with their natural transformations
form a category.

Let C be a category. For each X∈Ob C
there are two functors
 covariant homfunctor
m_{X}:=Mor_{C}(X,·) 
C 
→ 
Ens 
Y 
↦ 
Y_{∗}:=Mor_{C}(X,Y) 
(f∈Mor_{C}(Y,Z)) 
↦ 
(Mor_{C}(X,Y)→Mor_{C}(X,Z);
g↦f∘g) 
 contravariant homfunctor
m^{X}:=Mor_{C}(·,X) 
C 
→ 
Ens 
Y 
↦ 
Y^{∗}:=Mor_{C}(Y,X) 
(f∈Mor_{C}(Y,Z)) 
↦ 
(Mor_{C}(Z,X)→Mor_{C}(Y,X);
g↦g∘f) 
They result from currying ∘. The properties of being a functor
coincides with the properties of morphisms (n) and (a), here. m^{X}
is the opposite of m_{X}. If C
is additive we write Hom_{C}(X,·):=h_{X}:=m_{X},
and Hom_{C}(·,X):=h^{X}:=m^{X}
for these (now additive) functors in C→Ab.
The properties of being additive coincides with (d).

Let C be a category, and S∈Ob C.
Then C/S is the category of Sobjects in C
with
Ob C/S := {(X,f) ¦ X∈Ob C, f ∈Mor_{C}(X,S)}
Mor_{C/S}((X,f),(Y,g)) := {φ∈Mor_{C}(X,Y) ¦ f = g ∘ φ}
Further Terminology
 YonedaLemma
 Let F be the category of functors C→Ens
⇒ ∀A,B∈Ob C the following map is
bijective
α 
Mor_{C}(A,B) 
→̃ 
Mor_{F}(m_{B},m_{A}) 
τ_{B}(id_{B}) 
↤ 
τ 
φ 
↦ 
α_{φ} 
m_{B}
 → 
m_{A} 
(α_{φ})_{C} 
Mor_{C}(B,C)
 → 
Mor_{C}(A,C) 
f 
↦ 
f∘φ 

If C is additive and F
the category of (additive?) functors C→Ab,
then α is an isomorphism of groups.
 adjoint

F:C→C'
leftadjoint to G:C'→C
(and G rightadjoint to F)
:⇔
m_{F}
:= Mor_{C'}(F(·),·)
≅ Mor_{C}(·,G(·))
=: m^{G}
in C°×C'→Ens
"what F does to the source (adding primes) is what G does to the domain
(removing primes)"
 equivalent
categories
 F:C→C'
is an equivalence of categories :⇔
∃G:C'→C
with F∘G≅id_{C'}
∧ G∘F≅id_{C}
⇒ F leftadjoint to G
and conversely
Ismomorphic categories are equivalent.
For the following definition, A and A'
must be categories that have exact sequences (more precisely: Abelian
categories) .
 exact
 F:A→A'
is exact :⇔ F:A→A'
is a (covariant) additive functor, and
If 
0 
→ 
A' 
→^{α} 
A 
→^{β} 
A'' 
→ 
0 
is an exact sequence in A 
Then 
0 
→ 
F(A') 
→^{F(α)} 
F(A) 
→^{F(β)} 
F(A'') 
→ 
0 
is an exact sequence in A' 
That the image under F of the sequence (or any
complex) is a complex, is always true since F
is a functor.
 left exact
 F:A→A'
is left exact :⇔ F:A→A'
is a (covariant) additive functor, and
If 
0 
→ 
A' 
→^{α} 
A 
→^{β} 
A'' 
(→ 
0) 
is an exact sequence in A 
Then 
0 
→ 
F(A') 
→^{F(α)} 
F(A) 
→^{F(β)} 
F(A'') 


is an exact sequence in A'. 
 right exact
 F:A→A'
is right exact :⇔ F:A→A'
is a (covariant) additive functor, and
If 
(0 
→) 
A' 
→^{α} 
A 
→^{β} 
A'' 
→ 
0 
is an exact sequence in A 
Then 


F(A') 
→^{F(α)} 
F(A) 
→^{F(β)} 
F(A'') 
→ 
0 
is an exact sequence in A'. 
F:A→A'
is right exact ⇔ F:A°→A'°
is left exact.
 injective
 I∈Ob A is injective :⇔ Hom_{A}(·,I) exact
⇔ ∀A'⊆A ∈ A ∀α':A'→I ∃α:A→I which is a continuation of α'.
A has sufficiently many injective objects :⇔ each object is a subobject of an injective object.
 projective
 P∈Ob A is projective :⇔ Hom_{A}(P,·) exact
⇔ ∀β:B→B'' surjective ∀γ:P→B''
∃γ̃:P→B which is a lifting, i.e. β∘γ̃ = γ.
A has sufficiently many projective objects :⇔ each object is a quotient of a projective object.
Universal Elements
Let C be a category.
 terminal
object

A∈Ob C is a terminal object :⇔ ∀B∈Ob C
∃!φ∈Mor_{C}(B,A) ⇔ A is an
initial object in C°.
terminal objects are uniquely isomorph, i.e. each two terminal objects A,B
have a unique isomorphism A→B.
Terminal objects are the limits of the empty category.
 initial
object

A∈Ob C is an initial object :⇔ ∀B∈Ob C
∃!φ∈Mor_{C}(A,B)
initial objects are uniquely isomorph, i.e. each two initial objects A,B
have a unique isomorphism A→B.
Initial objects are the colimits of the empty category.
 zero
object
 A∈Ob C is a zero object :⇔ A is an initial and terminal object.
 presentable
 a covariant (resp. contravariant) functor F:C→Ens
is presentable :⇔ ∃A∈Ob C F≅m_{A}
(resp. F≅m^{A})
A is called presenting object for F.
 universal element
 If F:C→Ens
is a presentable covariant functor with a corresponding isomorphism of
functors α:m_{A}→F,
then u_{F} := α_{A}(id_{A})
∈ F(A) is called universal element of F .
⇒ ∀B∈Ob
C ∀x∈
F(B)
∃!f∈Mor
_{C}(A,B)=
m_{A}(B)
x =
F(f)(u
_{F})
 (The converse is also true)
 if C and Ens
(then Ab) are additive, then (m_{A},F
are and) the α_{B}
are isomorphisms of groups.
 More generally, if F (on morphisms) is
a homomorphism of a law + that ∘
is distributive over, then the α_{B}
are isomorphisms of +.
Let F:C→Ens
be a presentable covariant functor.
 presenting objects for F are uniquely
isomorph. Precisely: for all isomorphisms of functors α:m_{A}→F,
α':m_{A}_{'}→F
∃!φ:A_{}→A' isomorphism with α∘α_{φ}
= α'. (Because of YonedaLemma).
 an isomorphism of functors α:m_{A}→F
is uniquely determined by its universal element, i.e. ∀u∈F(A)
there is at most one α:m_{A}→F
with α_{A}(id_{A})=u.
Product and Coproduct
Let C be a category, and I be a set.
 product ≤ limit

P∈Ob C, with the projectors π_{i}∈Mor(P,A_{i})
for i∈I, is a product of the objects (A_{i})_{i∈I}⊆Ob
C :⇔
(∀C∈Ob C ∀(g_{i}_{}∈Mor_{C}(C,A_{i}))_{i∈I}
∃!g∈Mor_{C}(C,P) ∀i∈I π_{i}∘g
= g_{i} )
Write: P = ∏_{i∈I}A_{i}
"product is universally attracting", and
product is the presenting object of the contravariant functor C→Ens; C ↦ Mor(C,A_{1})×Mor(C,A_{2})
with universal projector maps π_{i}. Furthermore, it is just a terminal
object in a suitable category. Products are the limits of a functor from a
discrete category (i.e. which only has identity morphisms and thus
reduces to a family of objects).
 coproduct ≤ colimit

S∈Ob C, with the inclusions ι_{i}∈Mor(A_{i},S)
for i∈I, is a coproduct of the objects (A_{i})_{i∈I}⊆Ob
C :⇔
(∀C∈Ob C ∀(g_{i}_{}∈Mor_{C}(A_{i},C))_{i∈I}
∃!g∈Mor_{C}(S,C) ∀i∈I g∘ι_{i}
= g_{i} )
Write: S = ∐_{i∈I}A_{i} (= ∑_{i∈I}A_{i},
sometimes ⊕_{i∈I}A_{i})
"coproduct is universally repelling", and it is just an initial
object in a suitable category. A coproduct in C
is just a product in C°.
 fibredproduct
 X ×_{S} Y is a fibred product of X ∈ Ob C and Y∈Ob C
over S∈Ob C if it is a product of (X,f) and (Y,g) in the category C/S
of Sobjects. Also called pullback:
"parallel translation".
Pullbacks are the limits of a threeobject category with
f:X→S, g:Y→S as nonidentity morphisms. They visualize as commutative
squares with diagonal.
 equalizer ≤ limit

the map e∈Mor(E,X), with E∈Ob C, is an
equalizer of the maps f,g∈Mor(X,Y) :⇔
f∘e
= g∘e_{
}_{}∧ (∀E'∈Ob C ∀e'_{}∈Mor_{C}(E',X)
f∘e'
= g∘e' ⇒
∃!η∈Mor_{C}(E',E) e'
= e∘η )
e is a unique monomorphism (up to isomorphism). f.ex. The kernel of f is the
equalizer of f and 0. Equalizers are the limits of the identity functor
from a twoobject category with two parallel morphisms in between.
 coequalizer ≤ colimit

the map c∈Mor(Y,C), with C∈Ob C, is a coequalizer of the
maps f,g∈Mor(X,Y) :⇔
c∘f
= c∘g_{
}∧ (∀C'∈Ob C ∀c'∈Mor_{C}(Y,C')
c'∘f
= c'∘g ⇒
∃!γ∈Mor_{C}(C,C') c'
= γ∘c )
c is a unique epimorphism (up to isomorphism). f.ex. The cokernel of f is the
coequalizer of f and 0.
As terminal or initial objects, products and coproducts are uniquely
determined up to a unique isomorphism. Even more so as they are special cases of
limits and colimits.
Limits and Colimits
Let C, J be
categories.
 limit

L∈Ob C, together with φ_{X}∈Mor_{C}(L,F(X))
for each X∈Ob J, is a cone of the covariant
functor F:J→C
:⇔
∀f∈Mor_{J}(X,Y)
F(f)_{}∘φ_{X}=φ_{Y}
A limit of the covariant functor F:J→C
is a universal cone
(L,(φ_{X})_{X∈Ob
J}), i.e.
the morphisms of any cone factor through L with the unique factorization u, i.e.
∀(N,(ψ_{X})_{X∈Ob
J}) cone of F
∃!u∈Mor_{C}(N,L) ∀X∈Ob J
φ_{X}∘u = ψ_{X}
Write: L = lim F
= lim_{←} F "glue together related objects by morphisms".
Limits relativate products to the situation where the morphisms commute
over the F(f).
Limits are sometimes also known as inverse limit or projective
limit. Examples of limits include products, terminal objects, equalizers,
kernels, pullbacks.
 colimit

L∈Ob C, together with a family φ_{X}∈Mor_{C}(F(X),L)
for X∈Ob J, is a cocone of the covariant
functor F:J→C
:⇔
∀f∈Mor_{J}(X,Y)
φ_{X}∘F(f)=φ_{Y}
A colimit of the covariant functor F:J→C is a universal
cocone (L,φ_{X}), i.e.
∀(N,(ψ_{X})_{X∈Ob
J}) cocone of F
∃!u∈Mor_{C}(L,N) ∀X∈Ob J
u∘φ_{X} = ψ_{X}
Write: L = colim F = lim_{→} F
F:J→C has a colimit ⇔
∀N∈Ob C the covariant functor X↦Mor_{C}(F(X),N)
in J^{op} has a limit.
⇒ Mor_{C}(colim F,N)
= lim Mor_{C}(F(·),N)
"Colimits are for glueing together mathematical objects."
Colimits are sometimes also known as direct limit or inductive limit.
Examples of colimits include coproducts, inital objects, coequalizers,
cokernels, pushouts.
Limits and colimits are uniquely
determined up to a unique isomorphism, because they are inital (resp. terminal)
objects in the category of cones of F.