Functors. Part 5.

We have called L ∶ P ⇄ Q ∶ R an adjoint pair of functions, but of course they more than just functions. If L ⊣ R is an adjunction, then property (2) of Galois connections says that ∀ p1, p2 ∈ P and q1, q2 ∈ Q we have

p1 ≤ P p2 ⇒ L(p1) ≤ Q L(p2) and q1 ≤ Q q2 ⇒ R(q1) ≤ P R(q2)

That is, the functions L ∶ P → Q and R ∶ Q → P are actually homomorphisms of posets.

Definition of Functor: Let C and D be categories. A functor F ∶ C → D consists a family of functions:

• A function on objects F ∶ Obj(C) → Obj(D),

• For each pair of objects c1, c2 ∈ C a function on hom sets:

F ∶ HomC(c1,c2) → HomD(F(c1),F(c2)). These functions must preserve the category structure:

(i) Identity: For all objects c ∈ C we have F (idc) = idF(c).

(ii) Composition: For all arrows α, β ∈ C such that β ○ α is defined, we have

F (β ○ α) = F (β) ○ F (α)

Functors compose in an associative way, and for each category C there is a distinguished identity functor idC ∶ C → C. In other words, the collection of all categories with functors between them forms a (very big) category, which we denote by Cat.

This definition is not surprising. It basically says that a functor F ∶ C → D sends commutative diagrams in C to commutative diagrams in D. That is, for each diagram D ∶ I → C in C we have a diagram FI(D) ∶ I → D in D (defined by FI(D) ∶= F ○ D), which is commutative if and only if D is.

Now let’s try to guess the definition of an “adjunction of categories”. Let C and D be categories and let L ∶ C ⇄ D ∶ R be any two functors. When C is a poset, recall that ∀ x, y ∈ C we have ∣HomC (x, y)∣ ∈ {0, 1} and we use the notations

x ≤ y ⇐⇒ ∣HomC(x,y)∣ = 1

x ≤/ y ⇐⇒ ∣HomC(x,y)∣ = 0

Thus if C and D are posets, we can rephrase the definition of a poset adjunction L ∶ C ⇄ D ∶ R by stating that ∀ objects c ∈ C and d ∈ D there exists a bijection of hom sets:

HomC (c, R(d)) ←→ HomD (L(d), c)

In this form the definition now applies to any pair of functors between categories.

However, if we want to preserve the important theorems (uniqueness of adjoints and RAPL) then we need to impose some “naturality” condition on the family of bijections between hom sets. This condition is automatic for posets, so we will have to look elsewhere for motivation.

Let C and D be categories. We say that a pair of functors L ∶ C ⇄ D ∶ R is an adjunction if for all objects c ∈ C and d ∈ D there exists a bijection of hom sets

HomC (c, R(d)) ←→ HomD (L(d), c)

Furthermore, we require that these bijections fit together in the following “natural” way. For each arrow γ ∶ c2 → c1 in C and each arrow δ ∶ d1 → d2 in D we require that the following cube of functions commutes:

Natural Transformation:

Let C and D be categories and consider two parallel functors F1,F2 ∶ C → D. A natural transformation Φ ∶ F ⇒ G consists of a family of arrows Φc ∶F(c) → R(c), one for each object c ∈ C, such that for each arrow γ ∶ c1 → c2 in C the following square commutes:

The figure below the square is called a “2-cell diagram”. It hints at the close relationship between category theory and topology.

Let DC denote the collection of all functors from C to D and natural transformations between them. One can check that this forms a category (called a functor category). Given F1, F2 ∈ DC we say that F1 and F2 are naturally isomorphic if they are isomorphic in DC, i.e., if there exists a pair of natural transformations Φ ∶ F1 ⇒ F2 and Ψ ∶ F2 ⇒ F1 such that Ψ ○ Φ = idF1 and Ψ ○ Φ = idF2 are the identity natural transformations. In this case we will write F1 ≅ F2 and we will say that Φ and Φ−1 ∶= Ψ are natural isomorphisms.

To develop some intuition for this definition, let I be a small category and let C be any category. We have previously referred to functors D ∶ I → C as “diagrams of shape I in C“. Now we can think of CI as a category of diagrams. Given two such diagrams D1, D2 ∈ CI , we visualize a natural transformation Φ ∶ D1 ⇒ D2 as a “cylinder”:

The diagrams D1 and D2 need not be commutative, but if they are then the whole cylinder is commutative.

Limit/Colimit:

Consider a diagram D ∶ I → C. The limit of D, if it exists, consists of an object limID ∈ C and a canonical natural transformation Λ ∶ (limID)I ⇒ D such that for each object c ∈ C and natural transformation Φ ∶ cI ⇒ D there exists a unique natural transformation υI ∶ cI ⇒ (limID)I making the following diagram in CI commute:

Hom Functors:

Let C be a category. For each object c ∈ C the mapping d ↦ HomC(c, d) defines a functor from C to the category of sets Set. We denote it by

Hc ∶= HomC(c,−) ∶ C → Set

To define the action of Hc on arrows, consider any δ ∶ d1 → d2 in C. Then we must have a function Hc(δ) ∶ Hc(d1) → Hc(d2), i.e., a function Hc(δ) ∶ HomC(c,d1) → HomC(c,d2). There is only one way to define this:

H(δ) (φ) ∶= δ ○ φ

Similarly, for each arrow δ ∶ c1 → c2 we can define a function Hc(δ) ∶ HomC(d2,c) → HomC(d1,c) by H(δ) (φ) ∶= φ ○ δ. This again defines a functor into sets, but this time it is from the opposite category Cop (which is defined by reversing all arrows in C):

Hc ∶= HomC(−,c) ∶ Cop → Set

Finally, we can put these two functors together to obtain the hom bifunctor

HomC(−,−)∶ Cop ×C →Set

which sends each pair of arrows (γ ∶ c2 → c1, δ ∶ d1 → d2) to the function

HomC(γ,δ) ∶ HomC(c1,d1) → HomC(c2,d2)

defined by φ ↦ δ ○ φ ○ γ. The product category Cop × C is defined in the most obvious way.