CHAPTER 8 Toolbox

4

2. Functors

Definition 8.2.1. Let C and D be categories. Let F consist of

1. a map Ob C ∋ A ↦ F(A) ∈ Ob D,

2. a family of maps

{F_{A,B} : Mor_{C}(A, B) ∋ f ↦ F_{A,B}(f) ∈ Mor_{D}(F(A), F(B))|A, B ∈ C}

[or {F_{A,B} : Mor_{C}(A, B) ∋ f ↦ F_{A,B}(f) ∈ Mor_{D}(F(B), F(A))|A, B ∈ C}] F is called a covariant [contravariant] functor if

1. F_{A,A}(1_{A}) = 1_{F(A)} for all A ∈ Ob C,

2. F_{A,C}(gf) = F_{B,C}(g)F_{A,B}(f) for all A, B, C ∈ Ob C.

[F_{A,C}(gf) = F_{A,B}(f)F_{B,C}(g) for all A, B, C ∈ Ob C]. Notation: We write

A ∈ C instead of  A ∈ Ob C
f ∈ C instead of f ∈ Mor_{C}(A, B)
F(f) instead of  F_{A,B}(f).

Examples 8.2.2. 1. Id : Set ⟶ Set

2. Forget : R-Mod ⟶ Set

3. Forget : Ri ⟶ Ab

4. Forget : Ab ⟶ Gr

5. P : Set ⟶ Set, P(M) := power set of M. P(f)(X) := f^{−1}(X) for f : M ⟶ N, X ⊆ N is a contravariant functor.

6. Q : Set ⟶ Set, Q(M) := power set of M. Q(f)(X) := f(X) for f : M ⟶ N, X ⊆ M is a covariant functor.

Lemma 8.2.3. 1. Let X ∈ C. Then

Ob C ∋ A ↦ Mor_{C}(X, A) ∈ Ob Set

Mor_{C}(A, B) ∋ f ↦ Mor_{C}(X, f) ∈ Mor_{Set}(Mor_{C}(X, A), Mor_{C}(X, B)), with Mor_{C}(X, f) : Mor_{C}(X, A) ∋ g ↦ fg ∈ Mor_{C}(X, B) or Mor_{C}(X, f)(g) = fg is a covariant functor Mor_{C}(X, -).

2. Let X ∈ C. Then

Ob C ∋ A ↦ Mor_{C}(A, X) ∈ Ob Set

Mor_{C}(A, B) ∋ f ↦ Mor_{C}(f, X) ∈ Mor_{Set}(Mor_{C}(B, X), Mor_{C}(A, X)) with Mor_{C}(f, X) : Mor_{C}(B, X) ∋ g ↦ gf ∈ Mor_{C}(A, X) or Mor_{C}(f, X)(g) = gf is a contravariant functor Mor_{C}(-, X).

Proof. 1. Mor_{C}(X, 1_{A})(g) = 1_{A}g = g = id(g), Mor_{C}(X, f)Mor_{C}(X, g)(h) = fgh = Mor_{C}(X, fg)(h).

2. analogously.

Remark 8.2.4. The preceding lemma shows that Mor_{C}(-, -) is a functor in both arguments. A functor in two arguments is called a bifunctor. We can regards the bifunctor Mor_{C}(-, -) as a covariant functor

Mor_{C}(-, -) : C^{op} × C ⟶ Set.

The use of the dual category removes the fact that the bifunctor Mor_{C}(-, -) is contravariant in the first variable.

Obviously the composition of two functors is again a functor and this composition is associative. Furthermore for each category C there is an identity functor Id_{C}.

Functors of the form Mor_{C}(X, -) resp. Mor_{C}(-, X) are called representable functors (covariant resp. contravariant) and X is called the representing object (see also section 8.8).