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Transcript
```BASIC DEFINITIONS IN CATEGORY THEORY
MATH 250B
1. Categories
A category C consists of the following data:
(1) A class of objects obC, usually denoted by just C.
(2) For each A, B ∈ C, a set of morphisms HomC (A, B). An element f ∈ HomC (A, B) is
called a morphism between A and B, and will sometimes be denoted by f : A → B
f
or A →
− B.
(3) An associative composition rule for morphisms f : A → B and g : B → C. I.e. this
is a map
HomC (B, C) × HomC (A, B) → HomC (A, C)
denoted by (f, g) 7→ f ◦ g, such that ◦ is associated (when defined).
(4) For each object A ∈ C a distinguished identity morphism 1A , which acts as a twosided identity for composition of morphisms. I.e. for all f ∈ HomC (A, B), one has
f ◦ 1A = f , and for all g ∈ HomC (B, A) one has 1A ◦ g = g.
We went over several examples of categories in class, and one can find many other examples
all over the place. A trivial, but important construction, is the opposite of a category C,
defined as follows.
For a category C, we let C op denote the opposite category of C, which is defined as follows.
(1) One has obC = obC op .
(2) For all A, B objects in C op (equiv. objects in C), one has
op
HomC (A, B) = HomC (B, A).
(3) Composition is defined in the natural way: if f : A → B and g : B → C are
morphisms in C op , we define g ◦ f (the composition in C op ) as f ◦C g (the composition
in C).
Some other important definitions which were covered in class are summarised in the following list:
(1) A category C is small if obC is actually a set.
(2) A category C is a subcategory of the category D if one has obC ⊂ obD and
HomC (A, B) ⊂ HomD (A, B)
for all A, B ∈ C ⊂ D.
(3) In the context of (2) above, we say that C is a full subcategory of D if one further
has
HomC (A, B) ⊂ HomD (A, B)
for all A, B ∈ C.
1
2. Functors
Let C and D be two categories. A (covariant) Functor between C and D, denoted by
F
F : C → D or C −
→ D, consists of the following data:
(1)
(2)
(3)
(4)
For each A ∈ C, an object F (A) ∈ D.
For each morphism f : A → B in C, a morphism F (f ) : F (A) → F (B) in D.
F is compatible with compositions: F (f ◦C g) = F (f ) ◦D F (g).
F is compatible with identities: F (1A ) = 1F (A) for all A ∈ C.
A contravariant functor between C and D is just a covariant functor F : C op → D.
If F, G : C → D are two functors, then a natural transformation between F and G,
η
denoted by η : F → G or F →
− G, is defined by the following data:
(1) For each A ∈ C, a moprhism η(A) : F (A) → G(A).
(2) If f : A → B is a moprhism in C, then one has a commutative diagram (in D):
F (A)
F (f )
η(A)
F (B)
/
/
η(B)
G(A)
G(f )
G(B)
With this definition, we can consider Cat the category of (small) categories, whose objects
are small categories, and whose morphisms are covariant functors. Note that if C, D ∈
Cat, then the hom-set HomCat (C, D) can also be given the structure of a category, where
morphisms are natural transformations.
Some further definitions related to functors are summarized in the following list:
(1) A functor F : C → D is called injective if F (A) = F (B) implies A = B.
(2) A functor F : C → D is called full resp. faithful if the map
F : HomC (A, B) → HomD (F (A), F (B))
is surjective resp. injective.
(3) A fully faithful functor is a functor which is both full and faithful.
3. Representable Functors
Let Set denote the category of sets (i.e. objects are sets and morphisms are functions
between sets). For a category C and an object A ∈ C, we have a natural functor HomC (A, •) :
C → Set defined as follows:
(1) For B ∈ C, the object of Set associated to B is the set HomC (A, B).
(2) For a morphism f : B → B 0 in C, the map
HomC (A, B) → HomC (A, B 0 )
is simply composition with f , i.e. g 7→ f ◦ g.
Similarly, we obtain a contravariant functor HomC (•, A) : C → Set defined as follows:
(1) For B ∈ C, the object of Set associated to B is the set HomC (B, A).
2
(2) For a morphism f : B → B 0 in C, the map
HomC (B 0 , A) → HomC (B, A)
is simply pre-composition with f , i.e. g 7→ g ◦ f .
A covariant functor F : C → Set said to be representable by A ∈ C if one has an
isomorphism of functors:
F ∼
= HomC (A, •).
Similarly, a contravariant functor G : C → Set is said to be representable by A ∈ C if one
has an isomorphism of functors:
G∼
= HomC (•, A).
It is more-or-less standard to call a contravariant functor C → Set a presheaf. This
terminology comes from algebraic geometry and modern algebraic topology. We will use this
terminology in class.
4. The Image of a Functor
Let F : C → D be a functor. The image of F , denoted by imF is a subcategory of D
defined as follows:
(1) The objects of imF is the sub-class F (obC) of obD.
(2) For A, B ∈ imF (i.e. A = F (A0 ) and B = F (B0 ) for some A0 , B0 ∈ C), we defined
HomimF (A, B) = F (HomC (A0 , B0 )).
It is easy to see that imF is a subcategory of D. Moreover, imF is a full subcategory if
and only if F is full.
5. Isomorphisms and Equivalences of Categories
Let C be a category. An isomoprhism f : A → B in C is just an invertible morphism.
I.e f is an isomoprhism if there exists g : B → A such that f ◦ g = 1B and g ◦ f = 1A .
Following the idea that functors are morphisms between categories, we say that a functor
F : C → D is an isomoprhism of categories if F is invertible as a functor. I.e. if there
exists a functor G : D → C such that G ◦ F = 1C and F ◦ G = 1D .
A more subtle notion than that of isomorphism is the notion of an equivalence of categories.
We say that a functor F : C → D is an equivalence of categories if there exists a functor
G : D → C such that F ◦ G ∼
= 1D and G ◦ F ∼
= 1C , where ∼
= denotes isomorphism of functors
(in the category Fun(C, C) resp. Fun(D, D)).
The following is more-or-less obvious:
(1) If F : C → D is an injective functor which is faithful, then F induces an isomorphism
of categories F : C → imF .
(2) If F is an injective, fully faithful functor, then imF is a full subcategory of D, which
is isomorphic to C.
(3) We say that F is essentially surjective if every object of D is isomorphic to some
object in imF . It is easy to see that an equivalence of categories is fully faithful and
essentially surjective.
(4) Conversely, if F is fully faithful and essentially surjective, then F is an equivalence
of categories (exercise).
3
We conclude this note witha brief remark which should aid in intuition: The notion of
equivalence of categories is analogous to the notion of homotopy equivalence in algebraic
topology. The analogy goes like this: if we think of a category as a topological space,
then a functor should be considered as a continuous map between two spaces. Thus an
isomorphism of categories should be thought of as a homeomorphism of topological spaces.
An isomorphism between two functors should then be thought of as a homotopy equivalence
between two continuous maps. Two equivalent categories should therefore be thought of as
two homotopy-equivalent spaces. This analogy can be made very precise, but the details are
beyond the scope of this course.
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