RAPL (Right Adjoint Preserve Limits) Theorem. Part 8a.

To prove the RAPL theorem we must first translate the definition of limit/colimit into a language that is compatible with the definition of adjoint functors.

Recall that a diagram is a functor D ∶ I → C from a small category I. If C is locally small then we have a locally small category C^I consisting of diagrams and natural transformations between them. For each object c ∈ C we also have the constant diagram c^I ∶ I → C that sends each object i ∈ Obj(I) to c^I(i) ∶= c and each arrow δ ∈ Arr(I) to c^I(δ) ∶= id_c.

It is a general phenomenon that many categorical properties of C^I are inherited from C. The next lemma collects a few of these properties that we will need later.

Diagram Lemma. Fix a small category I and locally small categories C, D. Then:

(i) For any category C, the mapping c ↦ c^I defines a fully faithful functor (−)^I ∶ C → C^I which we call the diagonal embedding.

(ii)  For any functor F ∶ C → D the mapping F^I(D)∶= F ○ D defines a functor F^I ∶ C^I → D^I with the property that F (−)^I = F^I((−)^I).

(iii)  Any adjunction L ∶ C ⇄ D ∶ R induces an adjunction L^I ∶ C^I ⇄ D^I ∶ R^I That is, we have a natural isomorphism of bifunctors

Hom_C^I (−, R^I(−)) ≅ Hom_D^I (L^I(−), −)

from (C^I)^op × D^I to Set.

(iv) In particular, naturality in D^I tells us that for all objects l ∈ C and all natural transformations Λ ∶ l^I ⇒ D we have a commutative square:

Proof:

(i): For any arrow α ∶ c₁ → c₂ in C we want to define a natural transformation of diagrams α^I ∶ c^I₁ ⇒ c^I₂, and there is only one way to do this. Since (c^I₁)_i = c₁ and (c^I₂)_i = c₂ ∀ i ∈ I, the arrow (α^I)_i ∶= (c^I₁)_i → (c^I₂)_i must be defined by (α^I)_i ∶= α. Then for any arrow δ ∶ i → j in I we have c^I₁(δ) = id_c1 and c^I₂(δ) = id_c2, so that

(α^I)_i ○ (c^I₁) (δ) = (α ○ id_c1) = (id_c2 ○ α) = (c^I₂)_i (δ) ○ (α^I)_i

and hence we obtain a natural transformation α^I ∶ c^I₁ ⇒ c^I₂. The assignment α ↦ α^I is functorial since for all arrows α, β such that α ○ β exists and ∀ i ∈ I we have (α ○ β)^I_i = α ○ β = (α^I)_i ○ (β^I)_i = (α^I ○ β^I)_i,

and hence (α ○ β)^I = α^I ○ β^I. Finally, note that we have a bijection of hom sets Hom_C (c₁, c₂) ↔ Hom_C^I (c^I₁, c^I₂) given by α ↔ α^I, and hence the functor (−)^I ∶ C → C^I is fully faithful.

(ii): Let F ∶ C → D be any functor. Then for any diagram D ∶ I → C we obtain a diagram F^I(D) ∶ I → D by composition: F^I(D) ∶= F ○ D. This assignment is functorial in D ∈ C^I. To see this, consider any natural transformation Φ ∶ D₁ ⇒ D₂ in the category C^I. Then for any arrow δ ∶ i → j in I we can apply F to the naturality square for Φ to obtain another commutative square:

If we define F^I(Φ)_i ∶= F(Φ_i) ∀ i ∈ I then this second commutative square says that F^I(Φ) ∶ F^I(D₁) ⇒ F^I(D₂) is a natural transformation in D^I. If Φ and Ψ are two arrows (natural transformations) in C^I such that Φ ○ Ψ is defined, then ∀ i ∈ I we have F^I(Φ ○ Ψ)_i = F((Φ ○ Ψ)_i) = F(Φ_i ○ Ψ_i) = F(Φ_i) ○ F(Ψ_i) = F^I(Φ)_i ○ F^I(Ψ)_i = (F^I(Φ) ○ F^I(Ψ))_i and hence F^I(Φ ○ Ψ) = F^I(Φ) ○ F^I(Ψ). Thus we have defined a functor F^I ∶ C^I → D^I. Finally, note that ∀ i ∈ I, c ∈ C, and α ∈ Arr(C) we have

F^I(c^I)_i = F((c^I)_i) = F(c) = ((F(c))^I)_i F^I(α^I)_i = F((α^I)_i) = F(α) = ((F(α))^I)_i

and hence we have an equality of functors F^I((−)^I) = F (−)^I from C to D^I

(iii): Let L ∶ C ⇄ D ∶ R be any adjunction. We will denote each bijection Hom_C(−,R(−)) ↔ Hom_C(L(−), −) by φ ↦ φ^–, so that φ⁼ = φ. Now we want to define a natural family of bijections Hom_C^I (−, RI (−)) ≅ Hom_D^I (L^I(−), −)

To do this, consider diagrams C ∈ C^I, D ∈ D^I, and a natural transformation Φ ∶ C ⇒ R^I(D). Then for each index i ∈ I we have an arrow Φ_i ∶ C(i) → R(D(i)), which determines an arrow Φ^–_i ∶ L(C(i)) → D(i) by adjunction. The arrows Φ^–_i assemble into a natural transformation Φ^– ∶ L^I(C) ⇒ D. To see this, consider any arrow δ ∶ i ∈ j in I. Then from the naturality of Φ and the adjunction L ⊣ R we have

D(δ) ○ Φ^–_i = (R(D(δ)) ○ Φ_i)^–                             naturality of L ⊣ R

= (Φ_j ○ C(δ))^–                                                        naturality of Φ

= Φ_j^– ○ L(C(δ))                                                     naturality of L ⊣ R

as desired. In a similar way one can check that for each natural transformation Ψ ∶ L^I(C) ⇒ D, the arrows Ψ_i ∶ C(i) → R(D(i)) assemble into a natural transformation Ψ ∶ C ⇒ R^I(D). Thus we have established the desired bijection of hom sets Hom_C^I (C, R^I (D)) ↔ Hom_D^I (L^I (C), D).

To prove that this bijection is natural in (C, D) ∈ (C^I)^op × D^I, consider any pair of natural transformations Γ∶ C₂ ⇒ C₁ in C^I and ∆ ∶ D₁ ⇒ D₂ in D^I. We need to show that a certain cube of functions commutes. For a fixed diagram C ∈ C^I the following square commutes:

First, recall that the natural transformation R^I(∆) ∶ R^I(D₁) ⇒ R^I(D₂) is defined pointwise by R^I(∆)_i ∶= R(∆_i) ∶ R(D₁(i)) → R(D₂(i)). Now consider any Φ ∶ C ⇒ R^I(D₁). The naturality of the original adjunction tells us that (R(∆_i) ○ Φ_i)^– = ∆i ○ Φ_i^–, and hence we have

((R^I(∆) ○ Φ)_i)^– = (R^I(∆)_i ○ Φ_i)^–

= (R(∆_i) ○ Φ_i)^–

= ∆_i ○ Φ_i^–

= (∆ ○ Φ^–)_i

∀ i ∈ I. By definition this means that (R^I(∆) ○ Φ)^– = ∆ ○ Φ^–, and hence the desired square commutes. It remains only to check that the cube is natural in (C^I)^op. This follows from a similar pointwise computation.

(iv): Now fix an element l ∈ C, a diagram D ∈ D^I, and a natural transformation Λ ∶ l^I ⇒ D. By substituting C = R(l)^I, D₁ = l^I, D₂ = D, and ∆ = Λ into the above commutative square and using part (ii), we obtain the commutative square from the statement of the lemma. In particular, following the identity arrow id^I_R(l) around the square in two ways gives

(R^I(Λ) ○ id^I_R(l))^– = Λ ○ (id^I_R(l))^–

Finally, one can check pointwise that (id^I_R(l))^– = ((id_R(l))^I)^– and hence we obtain the identity

(R^I(Λ) ○ id^I_R(l))^– = ((id_R(l))^I)^–

Now we will reformulate the definition of limit/colimit in terms of adjoint functors. If all limits/colimits of shape I exist in some category C then it turns out (surprisingly) that we can think of limits/colimits as right/left adjoints to the diagonal embedding (−)^I ∶ C → C^I : colim_I ⊣(−)^I ⊣ lim_I

In the next section/part’s lemma we will prove something slightly more general. We will characterize a specific limit/colimit of shape I, without assuming that all limits/colimits of shape I exist.

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