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 CI consisting of diagrams and natural transformations between them. For each object c ∈ C we also have the constant diagram cI ∶ I → C that sends each object i ∈ Obj(I) to cI(i) ∶= c and each arrow δ ∈ Arr(I) to cI(δ) ∶= idc.
It is a general phenomenon that many categorical properties of CI 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 ↦ cI defines a fully faithful functor (−)I ∶ C → CI which we call the diagonal embedding.
(ii) For any functor F ∶ C → D the mapping FI(D)∶= F ○ D defines a functor FI ∶ CI → DI with the property that F (−)I = FI((−)I).
(iii) Any adjunction L ∶ C ⇄ D ∶ R induces an adjunction LI ∶ CI ⇄ DI ∶ RI That is, we have a natural isomorphism of bifunctors
HomCI (−, RI(−)) ≅ HomDI (LI(−), −)
from (CI)op × DI to Set.
(iv) In particular, naturality in DI tells us that for all objects l ∈ C and all natural transformations Λ ∶ lI ⇒ D we have a commutative square:
(i): For any arrow α ∶ c1 → c2 in C we want to define a natural transformation of diagrams αI ∶ cI1 ⇒ cI2, and there is only one way to do this. Since (cI1)i = c1 and (cI2)i = c2 ∀ i ∈ I, the arrow (αI)i ∶= (cI1)i → (cI2)i must be defined by (αI)i ∶= α. Then for any arrow δ ∶ i → j in I we have cI1(δ) = idc1 and cI2(δ) = idc2, so that
(αI)i ○ (cI1) (δ) = (α ○ idc1) = (idc2 ○ α) = (cI2)i (δ) ○ (αI)i
and hence we obtain a natural transformation αI ∶ cI1 ⇒ cI2. The assignment α ↦ αI is functorial since for all arrows α, β such that α ○ β exists and ∀ i ∈ I we have (α ○ β)Ii = α ○ β = (αI)i ○ (βI)i = (αI ○ βI)i,
and hence (α ○ β)I = αI ○ βI. Finally, note that we have a bijection of hom sets HomC (c1, c2) ↔ HomCI (cI1, cI2) given by α ↔ αI, and hence the functor (−)I ∶ C → CI is fully faithful.
(ii): Let F ∶ C → D be any functor. Then for any diagram D ∶ I → C we obtain a diagram FI(D) ∶ I → D by composition: FI(D) ∶= F ○ D. This assignment is functorial in D ∈ CI. To see this, consider any natural transformation Φ ∶ D1 ⇒ D2 in the category CI. 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 FI(Φ)i ∶= F(Φi) ∀ i ∈ I then this second commutative square says that FI(Φ) ∶ FI(D1) ⇒ FI(D2) is a natural transformation in DI. If Φ and Ψ are two arrows (natural transformations) in CI such that Φ ○ Ψ is defined, then ∀ i ∈ I we have FI(Φ ○ Ψ)i = F((Φ ○ Ψ)i) = F(Φi ○ Ψi) = F(Φi) ○ F(Ψi) = FI(Φ)i ○ FI(Ψ)i = (FI(Φ) ○ FI(Ψ))i and hence FI(Φ ○ Ψ) = FI(Φ) ○ FI(Ψ). Thus we have defined a functor FI ∶ CI → DI. Finally, note that ∀ i ∈ I, c ∈ C, and α ∈ Arr(C) we have
FI(cI)i = F((cI)i) = F(c) = ((F(c))I)i FI(αI)i = F((αI)i) = F(α) = ((F(α))I)i
and hence we have an equality of functors FI((−)I) = F (−)I from C to DI
(iii): Let L ∶ C ⇄ D ∶ R be any adjunction. We will denote each bijection HomC(−,R(−)) ↔ HomC(L(−), −) by φ ↦ φ–, so that φ= = φ. Now we want to define a natural family of bijections HomCI (−, RI (−)) ≅ HomDI (LI(−), −)
To do this, consider diagrams C ∈ CI, D ∈ DI, and a natural transformation Φ ∶ C ⇒ RI(D). Then for each index i ∈ I we have an arrow Φi ∶ 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– ∶ LI(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)) ○ Φi)– naturality of L ⊣ R
= (Φj ○ C(δ))– naturality of Φ
= Φj– ○ L(C(δ)) naturality of L ⊣ R
as desired. In a similar way one can check that for each natural transformation Ψ ∶ LI(C) ⇒ D, the arrows Ψi ∶ C(i) → R(D(i)) assemble into a natural transformation Ψ ∶ C ⇒ RI(D). Thus we have established the desired bijection of hom sets HomCI (C, RI (D)) ↔ HomDI (LI (C), D).
To prove that this bijection is natural in (C, D) ∈ (CI)op × DI, consider any pair of natural transformations Γ∶ C2 ⇒ C1 in CI and ∆ ∶ D1 ⇒ D2 in DI. We need to show that a certain cube of functions commutes. For a fixed diagram C ∈ CI the following square commutes:
First, recall that the natural transformation RI(∆) ∶ RI(D1) ⇒ RI(D2) is defined pointwise by RI(∆)i ∶= R(∆i) ∶ R(D1(i)) → R(D2(i)). Now consider any Φ ∶ C ⇒ RI(D1). The naturality of the original adjunction tells us that (R(∆i) ○ Φi)– = ∆i ○ Φi–, and hence we have
((RI(∆) ○ Φ)i)– = (RI(∆)i ○ Φi)–
= (R(∆i) ○ Φi)–
= ∆i ○ Φi–
= (∆ ○ Φ–)i
∀ i ∈ I. By definition this means that (RI(∆) ○ Φ)– = ∆ ○ Φ–, and hence the desired square commutes. It remains only to check that the cube is natural in (CI)op. This follows from a similar pointwise computation.
(iv): Now fix an element l ∈ C, a diagram D ∈ DI, and a natural transformation Λ ∶ lI ⇒ D. By substituting C = R(l)I, D1 = lI, 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 idIR(l) around the square in two ways gives
(RI(Λ) ○ idIR(l))– = Λ ○ (idIR(l))–
Finally, one can check pointwise that (idIR(l))– = ((idR(l))I)– and hence we obtain the identity
(RI(Λ) ○ idIR(l))– = ((idR(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 → CI : colimI ⊣(−)I ⊣ limI
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.