CHAPTER 8 Toolbox

1. Categories

Definition 8.1.1. Let C consist of

1. a class Ob C whose elements A, B, C, … ∈ Ob C are called objects,

2. a family {Mor_{C}(A, B)|A, B ∈ Ob C} of mutually disjoint sets whose elements f, g, … ∈ Mor_{C}(A, B) are called morphisms, and

3. a family {Mor_{C}(A, B) × Mor_{C}(B, C) ∋ (f, g) ↦ gf ∈ Mor_{C}(A, C)|A, B, C ∈

Ob C} of maps called compositions. C is called a category if the following axioms hold for C

1. Associative Law:

∀A, B, C, D ∈ Ob C, f ∈ Mor_{C}(A, B), g ∈ Mor_{C}(B, C), h ∈ Mor_{C}(C, D) :

h(gf) = (hg)f;

2. Identity Law:

∀A ∈ Ob C ∃1_{A} ∈ Mor_{C}(A, A) ∀B, C ∈ Ob C, ∀f ∈ Mor_{C}(A, B), ∀g ∈ Mor_{C}(C, A) :

1_{A}g = g  and  f1_{A} = f.

Examples 8.1.2. 1. The category of sets Set.

2. The categories of R-modules R-Mod, k-vector spaces k-Vec or k-Mod, groups Gr, abelian groups Ab, monoids Mon, commutative monoids cMon, rings Ri, fields Fld, topological spaces Top.

Since modules are highly important for all what follows, we recall the definition and some basic properties.

Definition and Remark 8.1.3. Let R be a ring (always associative with unit). A left R-module _{R}M is an (additively written) abelian group M together with an operation R × M ∋ (r, m) ↦ rm ∈ M such that

1. (rs)m = r(sm),

2. (r + s)m = rm + sm,

3. r(m + m′) = rm + rm′,

4. 1m = m for all r, s ∈ R, m, m′ ∈ M.

Each abelian group is a ℤ-module in a unique way.

A homomorphism of left R-modules f : _{R}M ⟶ _{R}N is a group homomorphism such that f(rm) = rf(m).

Analogously we define right R-modules M_{R} and their homomorphisms.

We denote by Hom_{R}(.M, .N) the set of homomorphisms of left R-modules_{R}M and RN. Similarly Hom_{R}(M., N.) denotes the set of homomorphisms of right R-modules M_{R} and N_{R}. Both sets are abelian groups by (f + g)(m) := f(m) + g(m).

For arbitrary categories we adopt many of the customary notations.

Notation 8.1.4. f ∈ Mor_{C}(A, B) will be written as f : A ⟶ B or A ⟶^{f} B. A is called the domain, B the range of f.

The composition of two morphisms f : A ⟶ B and g : B ⟶ C is written as gf : A ⟶ C or as g ∘ f : A ⟶ C.

Definition and Remark 8.1.5. A morphism f : A ⟶ B is called an isomorphism if there exists a morphism g : B ⟶ A in C such that fg = 1_{B} and gf = 1_{A}. The morphism g is uniquely determined by f since g′ = g′fg = g. We write f^{−1} := g.

An object A is said to be isomorphic to an object B if there exists an isomorphism f : A ⟶ B. If f is an isomorphism the so is f^{−1}. If f : A ⟶ B and g : B ⟶ C are isomorphisms in C then so is gf : A ⟶ C. We have: (f^{−1})^{−1} = f and (gf)^{−1} = f^{−1}g^{−1}. The relation of being isomorphic between objects is an equivalence relation.

Example 8.1.6. In the categories Set, R-Mod, k-Vec, Gr, Ab, Mon, cMon, Ri, Fld the isomorphisms are exactly those morphisms which are bijective as set maps.

In Top the set M = {a, b} with 𝔗_{1} = {∅, {a}, {b}, {a, b}} and with 𝔗_{2} = {∅, M} defines two different topological spaces. The map f = id : (M, 𝔗_{1}) ⟶ (M, 𝔗_{2}) is bijective and continuous. The inverse map, however, is not continuous, hence f is no isomorphism (homeomorphism).

Many well known concepts can be defined for arbitrary categories. We are going to apply some of them. Here are two examples.

Definition 8.1.7. 1. A morphism f : A ⟶ B is called a monomorphism if ∀C ∈ Ob C, ∀g, h ∈ Mor_{C}(C, A) :

fg = fh ⟹ g = h (f is left cancellable).

2. A morphism f : A ⟶ B is called an epimorphism if ∀C ∈ Ob C, ∀g, h ∈ Mor_{C}(B, C) :

gf = hf ⟹ g = h (f is right cancellable).

Definition 8.1.8. Given A, B ∈ C. An object A × B in C together with morphisms p_{A} : A × B ⟶ A and p_{B} : A × B ⟶ B is called a (categorical) product of A and B if for every object T ∈ C and every pair of morphisms f : T ⟶ A and g : T ⟶ B there exists a unique morphism (f, g) : T ⟶ A × B such that the diagram

[figure: T; f; g; A; A × B; p_{A}; B; p_{B}; (f,g)]

commutes.

An object E ∈ C is called a final object if for every object T ∈ C there exists a unique morphism e : T ⟶ E (i.e. Mor_{C}(T, E) consists of exactly one element).

A category C which has a product for any two objects A and B and which has a final object is called a category with finite products.

Remark 8.1.9. If the product (A×B, p_{A}, p_{B}) of two objects A and B in C exists then it is unique up to isomorphism.

If the final object E in C exists then it is unique up to isomorphism.

Problem 8.1.1. Let C be a category with finite products. Give a definition of a product of a family A_{1}, …, A_{n} (n ≥ 0). Show that products of such families exist in C.

Definition and Remark 8.1.10. Let C be a category. Then C^{op} with the following data Ob C^{op} := Ob C, Mor_{C^{op}}(A, B) := Mor_{C}(B, A), and f ∘_{op} g := g ∘ f defines a new category, the dual category to C.

Remark 8.1.11. Any notion expressed in categorical terms (with objects, morphisms, and their composition) has a dual notion, i.e. the given notion in the dual category.

Monomorphisms f in the dual category C^{op} are epimorphisms in the original category C and conversely. A final objects I in the dual category C^{op} is an initial object in the original category C.

Definition 8.1.12. The coproduct of two objects in the category C is defined to be a product of the objects in the dual category C^{op}.

Remark 8.1.13. Equivalent to the preceding definition is the following definition.

Given A, B ∈ C. An object A ⨿ B in C together with morphisms j_{A} : A ⟶ A ⨿ B and j_{B} : B ⟶ A ⨿ B ⟶ B is a (categorical) coproduct of A and B if for every object T ∈ C and every pair of morphisms f : A ⟶ T and g : B ⟶ T there exists a unique morphism [f, g] : A ⨿ B ⟶ T such that the diagram

[figure: T; f; g; A; A ⨿ B; j_{A}; B; j_{B}; [f,g]]

commutes.

The category C is said to have finite coproducts if C^{op} is a category with finite products. In particular coproducts are unique up to isomorphism.