Bicategorical Yoneda lemma

Igor Bakovi´c Rudjer Boˇskovi´cInstitute

Abstract We review the Yoneda lemma for bicategories and its connection to 2-descent and some universal constructions.

1 The 2-category of 2-presheaves

Definition 1.1. Let B be a bicategory. A 2-presheaf P on a bicategory B is a homomor- phism of bicategories P : Bop → Cat. Let P : Bop → Cat be the 2-presheaf. It takes an object B ∈ Ob(B) to the category P (B), a 1-morphism f : A → B to the functor P (f): P (B) → P (A), and any 2-morphism ψ : f → g to the natural transformation P (ψ): P (f) → P (g). We also have natural trans- formations µg,f : P (f) ◦ P (g) → P (g ◦ f) and ηA : iP (A) → P (iA), which satisfy following coherence laws

µ ◦P (h) µ (P (f) ◦ P (g)) ◦ P (h) g,f / P (g ◦ f) ◦ P (h) h,g◦f / P (h ◦ (g ◦ f)) O

P (ah,g,f )

/ / P (f) ◦ (P (g) ◦ P (h)) P (f) ◦ P (h ◦ g) µ P ((h ◦ g) ◦ f) P (f)◦µh,g h◦g,f

µi ,f µf,i P (f) ◦ P (i ) B / P (i ◦ f) P (i ) ◦ P (f) A / P (f ◦ i ) O B B A O A

P (f)◦ηB P (λf ) ηA◦P (f) P (ρf )

  P (f) P (f) P (f) P (f)

1 Definition 1.2. Let P,R: Bop → Cat be two 2-presheaves on B. A (left) pseudo natural transformation σ : P ⇒ R consists of the functors σA : P (A) → R(A) for each object A ∈ B and is given by the square

σ P (B) B / R(B)

P (f)  R(f) {Ó σf

  P (A) / R(A) σA which represents a natural transformation σf : R(f) ◦ σB ⇒ σA ◦ P (f) for any 1-morphism f : A → B in B. These natural transformations satisfy the coherence law given by the diagram P (B) ? ??  ?? P (g)  ?? P (f)  ??  ??  ??  µP ?   g,f P (C) / P (A) P (g◦f) σB σg ;C ???σf  #  σC R(B) σA ? ??  ?? R(g)  ?? R(f)  ??  µR ??   g,f ??   ?  R(C) / R(A) R(g◦f) which becomes a commutative diagram

R(f)◦σg / R(f) ◦ R(g) ◦ σC R(f) ◦ σB ◦ P (g)

R µg,f ◦σC  R(g ◦ f) ◦ σC σf ◦P (g)

σg◦f   σ ◦ P (g ◦ f) o σ ◦ P (f) ◦ P (g) A P A σz◦µg,f

2 f g for any composable pair A / B / C of 1-morphisms in B. The second coherence is given by the commutative diagram

σA  ??  ??  ??  ?? ηR◦σ  ? σ ◦ηP A A  ?? A A  ??  ??  ??  ??   ? / R(iA) ◦ σA σA ◦ P (iA) σiA of natural transformations.

Definition 1.3. A modification Γ: σ → τ consists of the natural transformation ΓA : σA → τA for each object A in B such that the following diagram

σB $ P (B) ΓB R(B)  :

τB

P (f) R(f)  τf f _ X l σA R  s K$  P (A) ΓA R(A)  :

τA commutes. This means that the above diagram becomes a commutative diagram

R(f)◦ΓB / R(f) ◦ σB R(f) ◦ τB

σf τf

  / σA ◦ P (f) τA ◦ P (f) ΓA◦P (f) of 2-morphisms in B.

3 Proposition 1.1. Let B be a bicategory. Then 2-preshevaes are the objects of the 2- op category P := HomBicat(B , Cat) in which 1-morphisms are pseudo natural transforma- tion and 2-morphisms are modifications.

σ ξ Proof. For any two pseudo natural transformations P +3 R +3 S the composition is defined by (ξ ◦ σ)A := ξA ◦ σA and (ξ ◦ σ)f := (ξA ◦ σf )(ξf ◦ σB) which is just the pasting

σ ξ P (B) B / R(B) B / S(B)

P (f)  R(f)  S(f) {Ó σf {Ó ξf

   / / P (A) σ R(A) S(A) A ξA composite of the above diagram. That this is well defined composition we deduce from the pasting composite P (B) ? ?  ?? P (f)  ? P (g)  µP ??   g,f ??  ? P (C) / P (A) P (k◦h) σB σf ;C ???σg  #  σC R(B) σA ? ?  ?? R(f)  ? R(g)  µR ??   g,f ??   ?  R(C) / R(A) R(k◦h) ξB ξf ;C ???ξg  #  ξC S(B) ξA ? ?  ?? S(f)  ? S(g)  µS ??   g,f ??   ?  S(C) / S(A) S(g◦f) which gives a coherence for a pseudo natural transformation. This composition is asso- σ ξ ω ciative since for any three pseudo natural transformations P +3 R +3 S +3 T we have the associativity

[(ω ◦ξ)◦σ]A = (ω ◦ξ)A ◦σA = (ωA ◦ξA)◦σA = ωA ◦(ξA ◦σA) = ωA ◦(ξ ◦σ)A = [ω ◦(ξ ◦σ)]A

4 for components indexed by the object B in B, and

[(ω ◦ ξ) ◦ σ]f = [(ω ◦ ξ)A ◦ σf ][(ω ◦ ξ)f ◦ σB] = [(ωA ◦ ξA) ◦ σf ][((ωA ◦ ξf )(ωf ◦ ξB)) ◦ σB] = = [(ωA ◦ ξA) ◦ σf ][(ωA ◦ ξf ) ◦ σB][(ωf ◦ ξB) ◦ σB] = [ωA ◦ (ξA ◦ σf )][ωA ◦ (ξf ◦ σB)][ωf ◦ (ξB ◦ σB)] = = [(ωA ◦ (ξA ◦ σf )(ξf ◦ σB)][(ωf ◦ (ξB ◦ σB))] = [ωA ◦ (ξ ◦ σ)f ][ωf ◦ (ξ ◦ σ)B] = [ω ◦ (ξ ◦ σ)]f

for components indexed by the morphism f : A → B in B. The vertical composition for any two modifications σ Γ _*4 τ Π _*4 π is defined simply by (ΠΓ)A := ΠAΓA. The horizontal composition for any two modifications

σ ξ

  P  Γ BR  Ω B S

τ ζ

is defined by (Ω ◦ Γ)A := ΩB ◦ ΓB. We need to show that pasting of the diagram

σB ξB ! ! P (B) ΓB R(B) ΩB S(B)  =  =

τ B ζB P (f) R(f) S(f) τf ζ h _  V h _  Vf p σA N p ξ N  y E!  y A E!  P (A) ΓA R(A) ΩA S(A)  =  =

τ A ζA

5 gives the coherence diagram for the composite modification Ω ◦ Γ: ξ ◦ σ → ζ ◦ τ.

(ξ◦σ)B ! P (B) (Ω◦Γ)B S(B)  =

(ζ◦τ)B P (f) S(f) (ζ◦τ) h _  V f p (ξ◦σ) N  y A E!  P (A) (Ω◦Γ)A S(A)  =

(ζ◦τ)A

The commutativity of the above diagram is deduced from the commutative diagram

S(f)◦(ξB ◦ΓB ) / S(f)◦(ΩB ◦τB ) / S(f) ◦ (ξB ◦ σB) S(f) ◦ (ξB ◦ τB) S(f) ◦ (ζB ◦ τB)

α α α S(f),ξB ,σB S(f),ξB ,τB S(f),ζB ,τB    (S(f)◦ξB )◦ΓB / (S(f)◦ΩB )◦τB / (S(f) ◦ ξB) ◦ σB (S(f) ◦ ξB) ◦ τB (S(f) ◦ ζB) ◦ τB

ξf ◦σB ξf ◦τB ζf ◦τB    (ξA◦R(f))◦ΓB / (ΩA◦R(f))◦τB / (ξA ◦ R(f)) ◦ σB (ξA ◦ R(f)) ◦ τB (ζA ◦ R(f)) ◦ τB

α α α ξA,R(f),σB ξA,R(f),τB ζA,R(f),τB    ξA◦(R(f)◦ΓB ) / ΩA◦(R(f)◦τB ) / ξA ◦ (R(f) ◦ σB) ξA ◦ (R(f) ◦ τB) ζA ◦ (R(f) ◦ τB)

ξA◦σf ξA◦τf ζA◦τf    ξA◦(ΓA◦P (f)) / ΩA◦(τA◦P (f)) / ξA ◦ (σA ◦ P (f)) ξA ◦ (τA ◦ P (f)) ζA ◦ (τA ◦ P (f)

α−1 α−1 α−1 ξA,σA,P (f) ξA,τA,P (f) ζA,τA,P (f)    / / (ξA ◦ σA) ◦ P (f) (ξA ◦ τA) ◦ P (f) (ζA ◦ τA) ◦ P (f) (ξA◦ΓA)◦P (f) (ΩA◦τA)◦P (f) where the first square in the second row and the second square in the the fourth row commute because they define horizontal compositions ξf ◦ΓB and ΩA ◦τf , respectively. The commutativity of the second square in the second row and the first square in the fourth row follows from the coherence for modifications Ω: ξ → ζ and Γ: σ → τ, respectively

6 and squares in the first, third and fifth row are associativity coherence. But the pasting composition of the top and front faces of the coherence for the modification Ω ◦ Γ is equal to the composition of the top and right edges of the above diagram, and the pasting composition of the back and bottom faces of the coherence is equal to the composition of the left and bottom edges of the diagram. The associativity and identities for horizontal composition follows immediately from the associativity of horizontal composition of natural transformations in Cat. Also, for any diagram of modifications

σ ξ  Γ   Ω  / / P τ BR B S ζ  Π  Λ

π χ we have Godement interchange law

(Λ ◦ Π)(Ω ◦ Γ) = (ΛΩ) ◦ (ΠΓ) since it holds for natural transformations and the composition were defined componentwise.

7 2 The Yoneda embedding

We will now take a more closer look to the representable 2-presheaves. Each object C in op B gives rise to the 2-presheaf YC : B → Cat on B, defined by YC (B) := B(B,C), for each object B in B, and for any two objects A, B ∈ Bop by the functor

A,B YC : B(A, B) → Cat(YC (B), YC (A)), for which we will usually omit indexing objects in the superscript. The value of this functor at any object f ∈ B(A, B) is a functor YC (f): YC (B) → YC (A) defined by

YC (f)(g) := g ◦ f, YC (f)(φ) := ψ ◦ f for any 1-morphism g ∈ B(B,C) and any 2-morphism ψ ∈ B(B,C). The value of the A,B C C 0 functor YC : B(A, B) → Cat(Y (B), Y (A)) for any 2-morphism φ: f → f (viewed as a morphism of the category B(A, B)), is the natural transformation

0 YC (φ): YC (f) → YC (f ) as in the diagram Y (φ) C g / 0 YC (f)(g) YC (f )(g)

0 YC (f)(ν) YC (f )(ν)

  0 / 0 0 YC (f)(g ) YC (f )(g ) YC (φ)g0 for any 2-morphism ν : g → g0 in B(B,C), whose component for any object g ∈ B(B,C) is defined by YC (φ)g := g ◦ φ. The commutativity of the diagram is assured by the definition of the horizontal composition of 2-morphisms and the Godement interchange law

(g0 ◦ φ)(ν ◦ f) = ν ◦ ψ = (ν ◦ f 0)(g ◦ φ).

Proposition 2.1. Let C be an object of the bicategory B. Then the above construction op defines a 2-presheaf YC : B → Cat.

Proof. The components of the natural transformation µg,f : YC (f) ◦ YC (g) → YC (g ◦ f) are given by µg,f (h) := αh,g,f :(YC (f) ◦ YC (g))(h) → YC (g ◦ f)(h) and the diagram for an

8 associativity coherence ((k ◦ h) ◦ g) ◦ f ((k ◦ h) ◦ g) ◦ f

ak◦h,g,f ak,h,g◦f   (k ◦ h) ◦ (g ◦ f) (k ◦ (h ◦ g)) ◦ f

ak,h,g◦f ak,h◦g,f   k ◦ (h ◦ (g ◦ f)) o k ◦ ((h ◦ g) ◦ f)) k◦ah,g,f becomes the commutative diagram

((YC (f) ◦ YC (g)) ◦ YC (h))(k) (YC (f) ◦ (YC (g) ◦ YC (h)))(k)

µg,f (YC (h)(k)) YC (f)(µh,g(k))   YC (g ◦ f) ◦ YC (h)(k) (YC (f) ◦ YC (h ◦ g))(k)

µh,g◦f (k) µh◦g,f (k)   o YC (h ◦ (g ◦ f))(k) YC ((h ◦ g) ◦ f)(k) YC (ah,g,f )(k) and since the last diagram is commutative for any morphism k : D → E (viewed as an object of the category B(D,E)), we obtain the coherence diagram

(YC (f) ◦ YC (g)) ◦ YC (h) YC (f) ◦ (YC (g) ◦ YC (h))

µg,f ◦YC (h) YC (f)◦µh,g   YC (g ◦ f) ◦ YC (h) YC (f) ◦ YC (h ◦ g)

µh,g◦f µh◦g,f   o YC (h ◦ (g ◦ f)) YC ((h ◦ g) ◦ f) YC (ah,g,f ) for the horizontal composition. The coherence for units follows also from the left and right identity coherence. The components of the identity natural transformation ηB : IYC (B) → YC (iB) are defined for each morphism g : B → C (again seen as an object of B(B,C)) by −1 ηg := ρg : g → g ◦iB. For any pair of 1-morphisms f : A → B and g : B → C the deformed triangle diagram for identity coherence α g,iB ,f / (g ◦ iB) ◦ f g ◦ (iB ◦ f)

ρg◦f g◦λf

  g ◦ f g ◦ f

9 becomes a commutative diagram

µi ,f (g) (Y (f) ◦ Y (i ))(g) B / Y (i ◦ f)(g) C O C B C B

−1 YC (f)(ρg ) YC (λf )(g)



(YC (f) ◦ IYC (B))(g) YC (f)(g) and since the last diagram is commutative for any morphism g : B → C (viewed as an object of the category B(B,C)), we obtain the coherence diagram

µi ,f Y (f) ◦ Y (i ) B / Y (i ◦ f) C O C B C B

YC (f)◦ηB YC (λf )



YC (f) ◦ IYC (B) YC (f)

For any 1-morphism h: C → D in B, there is a pseudo natural transformation of corresponding 2-presheaves Y(h): YC → YD whose component for any object B ∈ Ob(B) is a functor Y(h)B : B(B,C) → B(B,D) defined by Y(h)B(g) := h ◦ g, Y(h)B(ν) := h ◦ ν for any 1-morphism g : B → C, and any 2-morphism ν : g → g0 (viewed as an object and a morphism in B(B,C), respectively). The component of the pseudo natural transformation is given by the diagram Y(h)B / YC (B) YD(B)

 YC (f) {Ó YD(f) Y(h)f

  / YC (A) YD(A) Y(h)A

10 for any 1-morphism f : A → B. It is a natural transformation whose components are defined by the associativity isomorphism as Y(h)f (g) := αh,g,f :(h ◦ g) ◦ f → h ◦ (g ◦ f). Again, the associativity coherence gives a coherence for a pseudo natural transformation. Also for any 2-morphism β : h ⇒ h0 we have a modification Y(β): Y(h) → Y(h0) whose component indexed by the object B in B is given by the natural transformation 0 Y(β)B : Y(h)B → Y(h )B defined by Y(β)B := β ◦g for any 1-morphism g : B → C (viewed as the object in B(B,C)). Proposition 2.2. The above construction gives rise to the homomorphism

op Y : B → CatB

op from the bicategory B to the 2-category P = CatB of 2-presheaves which we call Yoneda embedding.

Proof. The above homomorphism sent an object C in B to the representable 2-presheaf YC , and any 1-morphism h: C → D to the pseudo natural transformation Y(h): YC → YD. h k For any composable pair of 1-morphisms C / D / E in B we define a modification µk,h : Y(k) ◦ Y(h) → Y(k ◦ h) whose component (µk,h)g indexed by the object g ∈ YC (B) −1 is given by the associativity isomorphism αk,h,g : k ◦ (h ◦ g) → (k ◦ h) ◦ g. The coherence for the associativity in B

H×Id B2 / B4 B3 ?? ?? ?? ?? ?? ?? ??  ?? ? Id ×H×Id α×{ÓId ? H×Id ?? B1 B1 B1 ?? B1 ?? ?? ?? ?? ?? ?? ?? ?? ? ? Id ×H / B2 B3 B2 H×IdB1

Id ×H  B1 {Ó ' {Ó IdB ×α  1 {Ó α  H×Id  B1 / B3 ? B2 ? H ?? ?? ?? ?? {Ó ?? ?? α ? IdB1 ×H ? ?? ?? H ?? ?? Id ×H ? ? B1 ?? ?? ?? ?? ?? {Ó ?? ?  α ?  B / B 2 H 1

11 is the equality αk,h,g◦f αk◦h,g,f = (k ◦ αh,g,f )αk,h◦g,f αk,h◦g,f of natural transformations indexed by composable quadruple of 1-morphisms in B. For fixed choice of 1-morphisms f, h, k the above cube becomes a coherence for the modification µk,h

YC (B) T ?? TTT ? TTTT ?? TTTT ?? TTYT(k◦h)B ? TTTT Y(h) ?? TTT B ? (µk,h);CB TTT ??  TTT TTT* / YC (f) YD(B) YE(B) Y(k)B

{Ó {Ó Y(h)f Y(k◦h)f   YC (A) YD(f) {Ó YE (f) ? T T Y(k)f ?? T T ?? T T ?? T YT(k◦h)A ? T T Y(h) ?? T A ? (µk,h);CA T T ??  T  T T*  / YD(A) YE(A) Y(k)A in which the back and right face of the cube became the back face of the prism. The component of the modification ηC : IYC → Y(iC ) indexed by the object B in B is the natural transformation (ηC )B :(IYC )B → (Y(iC ))B, whose component indexed by the 1- morphism g : B → C (seen as an object of the category YC (B)) is defined by (ηC )B(g) := −1 λg : g → iC ◦ g. The commutative diagram

o λg◦f g ◦ f (iC ◦ g) ◦ f

α iC ,g,f

 o g ◦ f iC ◦ (g ◦ f) λg◦f

12 obtained from the coherence for left and right identities, becomes a diagram

YC (f)◦(ηC )B (g/) (YC (f) ◦ (IYC )B)(g) (YC (f) ◦ Y(iC )B)(g)

(IYC )f (Y(iC )f )g

  / ((IYC )A ◦ YC (f))(g) (Y(iC )A ◦ YC (f))(g). (ηC )A◦(IYC )f (g)

Since the above diagram is commutative for any 1-morphism g : B → C (seen as an object of the category YC (B)) it gives a coherence

YC (f)◦(ηC )B / YC (f) ◦ (IYC )B YC (f) ◦ Y(iC )B

(IYC )f Y(iC )f

  / (IYC )A ◦ YC (f) Y(iC )A ◦ YC (f) (ηC )A◦(IYC )f

for the modification ηC : IYC → Y(iC ).

13 3 Bicategorical Yoneda lemma

An object of the category HomP (YC ,P ) is a pseudo natural transformation γ : YC → P which consists of a family of functors γB : YC (B) → P (B), indexed by objects B ∈ B, and of a natural transformations

γB / YC (B) P (B)

Y (f)  P (f) C {Ó γf

  Y (A) / P (A) C γA for each morphism f : A → B in B, such that the coherence diagram

P (f)◦γg/ P (f) ◦ P (g) ◦ γC P (f) ◦ γB ◦ YC (g)

P µg,f ◦γC  P (g ◦ f) ◦ γC γf ◦YC (g)

γg◦f

  o γA ◦ YC (g ◦ f) γA ◦ YC (f) ◦ YC (g) YC γA◦µg,f

f g for any composable pair A / B / C of 1-morphisms in B, commutes. Any modi- fication Γ: γ → δ consists of the family of natural transformations ΓB : γB → δB indexed by the objects B ∈ B. For any 2-presheaf P : Bop → Cat, we will define a new 2-presheaf SP : Bop → Cat by SP (C) := P(YC ,P ) for any object C ∈ B, and for any two objects C,D ∈ B we have a functor SPC,D : B(C,D) → [SP (D), SP (C)], which takes a morphism h: C → D to the functor SP (h) := P(Y(h),P ): SP (D) → SP (C), (where we omitted subscripts for convenience) defined by SP (h)(φ) := φ ◦ Y(h) and SP (h)(Ω) := Ω ◦ Y(h) for any pseudo natural transformation φ: YD → P and any modification Ω: φ → ψ, seen as an object and morphism of the category P(YD,P ), respectively. Moreover, the functor 0 SPC,D : B(C,D) → [SP (D), SP (C)] takes a 2-morphism β : h → h to a natural transfor- mation SP (β): SP (h) → SP (h0), whose component at the object φ ∈ SP (D) is given by SP (β)(φ) := φ ◦ Y(β): SP (h)(φ) → SP (h0)(φ).

14 Proposition 3.1. For any 2-presheaf P : Bop → Cat, the above construction defines a new 2-presheaf SP : Bop → Cat.

SP Proof. The component of the natural transformation µh,g : SP (g) ◦ SP (h) → SP (h ◦ g) SP indexed by the pseudo natural transformation φ: YD → P is the modification µh,g (φ) from the pseudo natural transformation

[SP (g) ◦ SP (h)](φ) = SP (g)(φ ◦ Y(h)) = (φ ◦ Y(h)) ◦ Y(g) = φ ◦ (Y(h) ◦ Y(g)): YB → P to the pseudo natural transformation

SP (h ◦ g)(φ) = φ ◦ Y(h ◦ g): YB → P in SP (B). Its component indexed by the object A in B is a natural transformation SP µh,g (φ)A :[φ◦(Y(h)◦Y(g))]A → [φ◦(Y(h◦g))]A whose component indexed by 1-morphism SP −1 f : A → B (seen as an object of the category YB(A)) is given by µh,g (φ)A(f) := φA(αh,g,f ). The corresponding coherence condition

µ ◦SP (k) µ (SP (g) ◦ SP (h)) ◦ SP (k) h,g / SP (h ◦ g) ◦ SP (k) k,h◦g / SP (k ◦ (h ◦ g)) O

SP (αk,h,g)

/ / SP (g) ◦ (SP (h) ◦ SP (k)) SP (g) ◦ SP (k ◦ h) µ SP ((k ◦ h) ◦ g) SP (g)◦µk,h k◦h,g evaluated by the pseudo natural transformation ²: YE → P is the diagram of modifications

µ (²◦Y(k)) µ (²) (² ◦ Y(k)) ◦ (Y(h) ◦ Y(g)) h,g / (² ◦ Y(k)) ◦ Y(h ◦ g) k,h◦g / ² ◦ Y(k ◦ (h ◦ g)) O

²◦Y(αk,h,g)

[² ◦ (Y(k) ◦ Y(h))] ◦ Y(g) / (² ◦ Y(k ◦ h)) ◦ Y(g) / ² ◦ Y((k ◦ h) ◦ g) µk,h(²)◦Y(g) µk◦h,g(²) which becomes an image by the functor ²A : YE(A) → P (A)

−1 −1 ²A(k◦α ) ²A(α ) ² (k ◦ [h ◦ (g ◦ f)]) h,g,f / ² (k ◦ [(h ◦ g) ◦ f]) k,h◦g,f / ² ([k ◦ (h ◦ g)] ◦ f) A A A O

²A(αk,h,g◦f)

² (k ◦ [h ◦ (g ◦ f)]) / ² ([k ◦ h] ◦ [g ◦ f]) / ² ([(k ◦ h) ◦ g] ◦ f) A −1 A −1 A ²A(αk,h,g◦f ) ²A(αk◦h,g,f )

15 of the pentagonal coherence for associativity when evaluated for any morphism f : A → B. The natural transformation ηC : ISP (C) → SP (iC ) indexed by the pseudo natural trans- formation γ : YC → P is given by the modification ηC (γ): γ → γ ◦Y(iC ), whose component indexed by the object B in B is a natural transformation ηC (γ)B : γB → γB ◦ Y(iC )B. Its component indexed by the 1-morphism g : B → C (seen as an object of the category YC (B)) is given by the morphism ηC (γ)B(g) := γB(λg): γB(iC ◦ g) → γB(g) in P (B). Two coherence conditions for natural transformation ηC : ISP (C) → SP (iC )

µi ,g µg,i SP (g) ◦ SP (i ) C / SP (i ◦ g) SP (i ) ◦ SP (g) B / SP (g ◦ i ) O C C B O B

SP (g)◦ηC SP (λg) ηB ◦SP (g) SP (ρg)

  SP (g) SP (g) SP (g) SP (g) are satisfied since when evaluated for pseudo natural transformation γ : YC → P , we have

µi ,g(γ) µg,i (γ) (γ ◦ Y (i )) ◦ Y(g) C / γ ◦ Y(i ◦ g) (γ ◦ Y (g)) ◦ Y(i ) B / γ ◦ Y (g ◦ i ) CO C O B B

ηC (γ)◦Y (g) γ◦Y(λg) ηB (γ◦Y(g)) γ◦Y(ρg)

  γ ◦ Y(g) γ ◦ Y(g) SP (g) SP (g) and this two diagrams for any 1-morphism f : A → B give the image by the functor γA : YC (A) → P (A)

−1 −1 γA(αi ,g,f ) γA(αi ,g,f ) γ (i ◦ (g ◦ f)) C / γ ((i ◦ g) ◦ f) γ (g ◦ (i ◦ f)) C / γ ((g ◦ i ) ◦ f) A C O A C A O B A B

−1 −1 γA(λg◦f ) γA(λg◦f) γA(g◦λf ) γA(ρg◦f)

  γA(g ◦ f) γA(g ◦ f) γA(g ◦ f) γA(g ◦ f) of two coherence diagrams for left and right identities.

16 This allows us to state the categorification of the Yoneda lemma. Theorem 3.1. (Bicategorical Yoneda lemma) There is an equivalence

θ : SP → P in P, (which is just a pseudo natural equivalence θ : P(Y,P ) → P between 2-presheaves).

Proof. The component of the pseudo natural transformation θ indexed by an object C ∈ B is a functor θC : P(YC ,P ) → P (C), defined by θC (γ) := γC (idC ) for any pseudo natural transformation γ : YC → P, and θC (Γ) := ΓC (idC ): γC (idC ) ⇒ δC (idC ) for any modifi- cation Γ: γ → δ in P(YC ,P ). This 2-morphism in P (C) is just a component indexed by the object idC ∈ YC (C) of the natural transformation ΓC : γC → δC , which is in turn component of the modification Γ: γ → δ, indexed by the object C ∈ B. For any morphism h: C → D, the natural transformation θh : P (h) ◦ θD → θC ◦ SP (h), given by the square

θ SP (D) D / P (D)

SP (h) {Ó P (h) θh

  SP (C) / P (C) θC has the morphism θh(φ):(P (h) ◦ θD)(φ) → (θC ◦ SP (h))(φ) in the category P (C) as the component indexed by the object φ: YD → P of the category SP (D). This morphism is defined by the composition

φ (ρ−1) φh(idD)/ φC (λh)/ C h / P (h)(φD(idD)) φC (idD ◦ h) φC (h) φC (h ◦ idC ) where the morphism φh(idD): P (h)(φD(idD)) → φC (idD ◦ h) is defined by the component of φD / YD(D) P (D)

 YD(h) {Ó P (h) φh

  / YD(C) P (C) φC given by φh(idD):(P (h) ◦ φD)(idD) → (φC ◦ YD(h))(idD), which is just the morphism φh(idD): P (h)(φD(idD)) → φC (idD ◦ h) in P (C).

17 To show that this really defines a pseudo natural transformation θ : SP → P , we need to check that the coherence diagram

P (f)◦θg / P (f) ◦ P (g) ◦ θC P (f) ◦ θB ◦ SP (g)

P µg,f ◦θC  P (g ◦ f) ◦ θC θf ◦SP (g)

θg◦f   θ ◦ SP (g ◦ f) o θ ◦ SP (f) ◦ SP (g) A SP A θA◦µg,f

f g for any composable pair A / B / C of 1-morphisms in B, commutes. But for any pseudo natural transformation γ : YC → P, we obtain the diagram

(P (f)◦θg)(γ) / (P (f) ◦ P (g) ◦ θC )(γ) (P (f) ◦ θB ◦ SP (g))(γ)

P (µg,f ◦θC )(γ)  (P (g ◦ f) ◦ θC )(γ) (θf ◦SP (g))(γ)

(θg◦f )(γ)   (θ ◦ SP (g ◦ f))(γ)(o θ ◦ SP (f) ◦ SP (g))(γ) A SP A (θA◦µg,f )(γ) and this diagram transforms into the diagram

−1 (P (f)◦γg)(/idC ) (P (f)◦γB )(/λg) (P (f)◦γB )(ρ/ g ) (P (f) ◦ P (g) ◦ γC )(idC ) (P (f) ◦ γB)(idC ◦ g) (P (f) ◦ γB)(g) (P (f) ◦ γB)(g ◦ idB)

P µg,f (γC (idC )) γf (g◦idB )   (P (g ◦ f) ◦ γC )(idC ) γA((g ◦ idB) ◦ f)

γ (α ) γg◦f (idC ) A g,idB ,f   γA(idC ◦ (g ◦ f)) γA(g ◦ (idB ◦ f))

γA(λg◦f ) γA(g◦λf )   γ (g ◦ f) / γ ((g ◦ f) ◦ id ) o γ (g ◦ (f ◦ id )) o γ (g ◦ f) A −1 A A −1 A A −1 A γA(ρ ) γA(α ) γA(g◦ρ ) g◦f g,f,idA f

whose bottom edge is the identity. This enable us to put it in the form of the diagram

18 (P (f)◦γg)(idC )/ (P (f)◦γB )(λg) / (P (f) ◦ P (g) ◦ γC )(idC ) (P (f) ◦ γB)(idC ◦ g) (P (f) ◦ γB)(g)

P −1 µg,f (γC (idC )) γf (idC ◦g) (P (f)◦γB )(ρg )    (P (g ◦ f) ◦ γC )(idC ) γA((idC ◦ g) ◦ f) (P (f) ◦ γB)(g ◦ idB) ggg γA(αid ,g,f ) ggg Cggggg γg◦f (idC ) gggg γA(λg◦f) γf (g◦idB )  gs gggg   / o γA(idC ◦ (g ◦ f)) γA(g ◦ f) γA((g ◦ idB) ◦ f) γA(λg◦f ) γA(ρg◦f) and this diagram commutes since the pentagon is the coherence

(P (f)◦γg)(idC )/ (P (f) ◦ P (g) ◦ γC )(idC ) (P (f) ◦ γB)(idC ◦ g)

P (µg,f ◦γC )(idC )  (P (g ◦ f) ◦ γC )(idC ) γf (idC ◦g)

(γg◦f )(idC )   o γA(idC ◦ (g ◦ f)) γA(idC ◦ g) ◦ f) γ (α ) A idC ,g,f for the pseudo natural transformation γ : YC → P, evaluated by the object idC in YC (C). The commutativity of the triangle is the consequence of the triangle coherence for left and right identities and the hexagon is the deformed square expressing the naturality of −1 γf : P (f) ◦ γB ⇒ γA ◦ YC (f) for the 2-morphism ρg ◦ λg : idC ◦ g ⇒ g ◦ idB.

The assertion of the theorem means that each component functor θC : SP → P (C) of the pseudo natural transformation is an equivalence and that each component natural transformation θh : P (h) ◦ θD ⇒ θC ◦ SP (h) is a natural isomorphism. In order to prove this, we will use the following result. Lemma 3.1. There exist a pseudo natural transformation ω : P → SP in P, which is a weak inverse to θ : SP → P .

Proof. The component of the pseudo natural transformation ω : P → SP indexed by the object D ∈ B, is the functor ωD : P (D) → SP (D), which sends any object Z ∈ P (D) to the Z Z pseudo natural transformation ωD : YD → P , whose component (ωD)C : YD(C) → P (C) indexed by the object C ∈ B, is a functor which sends any 1-morphism h: C → D (viewed Z as an object of the category B(C,D)), to the object (ωD)C (h) := P (h)(Z) in the cate- gory P (C), and which sends any 2-morphism ²: h → h0 (viewed as a morphism of the Z 0 category B(C,D)), to a morphism (ωD)C (²) := P (²)Z : P (h)(Z) → P (h )(Z) in P (C), which is just the component indexed by the object Z ∈ P (D) of the natural transfor- mation P (²): P (h) → P (h0) between the two functors P (h),P (h0): P (D) → P (C). The

19 Z component of the pseudo natural transformation ωD : YD → P indexed by the morphism g : B → C in B, is the natural transformation given by the diagram

(ωZ ) D C / YD(C) P (C)

 YD(g) {Ó Z P (g) (ωD)g

  Y (B) / P (B) D Z (ωD)B whose component indexed by the 1-morphism h: C → D (viewed as an object of the Z Z Z category B(C,D)), is the morphism (ωD)g(h):(P (g) ◦ (ωD)C )(h) → ((ωD)B ◦ YD(g))(h) in Z P the category P (B), defined by (ωD)g(h) := (µh,g)Z :(P (g) ◦ P (h))(Z) → P (h ◦ g)(Z). For f g any composable pair A / B / C of 1-morphisms in B, the coherence diagram for Z the pseudo natural transformation ωD : YD → P

P (f)◦(ωZ ) Z D g / Z P (f) ◦ P (g) ◦ (ωD)C P (f) ◦ (ωD)B ◦ YD(g)

P Z µg,f ◦(ωD)C  Z (ωZ ) ◦Y (g) P (g ◦ f) ◦ (ωD)C D f D

Z (ωD)g◦f   Z o Z (ω )A ◦ YD(g ◦ f)(ω )A ◦ YD(f) ◦ YD(g) D Z YD D (ωD)A◦µg,f evaluated by the morphism h: C → D in B, gives a diagram

(P (f)◦(ωZ ) )(h) Z D g / Z (P (f) ◦ P (g) ◦ (ωD)C )(h) (P (f) ◦ (ωD)B ◦ YD(g))(h)

P Z (µg,f ◦(ωD)C )(h)  Z ((ωZ ) ◦Y (g))(h) (P (g ◦ f) ◦ (ωD)C )(h) D f D

Z ((ωD)g◦f )(h)   Z o Z ((ω )A ◦ YD(g ◦ f))(h) ((ω )A ◦ YD(f) ◦ YD(g))(h) D Z YD D ((ωD)A◦µg,f )(h)

20 which is equal to the coherence diagram

P (P (f)◦µ )Z (P (f) ◦ P (g) ◦ P (h))(Z) h,g / (P (f) ◦ P (h ◦ g))(Z)

P (µg,f ◦P (h))Z  P (P (g ◦ f) ◦ P (h))(Z) (µh,g◦f )Z

P (µh,g◦f )Z   P (h ◦ (g ◦ f))(Z) o P ((h ◦ g) ◦ f)(Z) P (αh,g,f )Z for the 2-presheaf P : Bop → Cat, evaluated by the object Z in P (D). 0 The functor ωD : P (D) → SP (D) sends any morphism z : Z → Z in the category z Z Z0 P (D) to the modification ωD : ωD → ωD whose component indexed by the object C ∈ B z Z Z0 is a natural transformation (ωD)C :(ωD)C ⇒ (ωD )C . The component of the natural transformation indexed by the 1-morphism h: C → D (viewed as an object of the category Z Z Z0 B(C,D)), is the morphism (ωD)C (h):(ωD)C (h) → (ωD )C (h) in the category P (C), given by P (h)(z): P (h)(Z) → P (h)(Z0). The coherence for this modification

P (g)◦(ωz ) Z D C / Z0 P (g) ◦ (ωD)C P (g) ◦ (ωD )C

(ωZ ) Z0 D g (ωD )g

  Z / Z0 (ωD)B ◦ YD(g) z (ωD )B ◦ YD(g) (ωD)B ◦YD(g) evaluated by the morphism h: C → D gives a diagram

(P (g)◦(ωz ) )(h) Z D C / Z0 (P (g) ◦ (ωD)C )(h) (P (g) ◦ (ωD )C )(h)

((ωZ ) )(h) Z0 D g (ωD )g(h)

  Z / Z0 ((ωD)B ◦ YD(g))(h) z ((ωD )B ◦ YD(g))(h) ((ωD)B ◦YD(g))(h)

21 which is equal to the naturality diagram

(P (g)◦P (h))(z) (P (g) ◦ P (h))(Z) / (P (g) ◦ P (h))(Z0)

P P (µh,g)Z (µh,g)Z0

  P (h ◦ g)(Z) / P (h ◦ g)(Z0) P (h◦g)(z)

P 0 for the natural transformation µh,g : P (g) ◦ P (h) → P (h ◦ g), and the morphism z : Z → Z in the category P (D). The value of the pseudo natural transformation ω indexed by the morphism h: C → D is given by the natural transformation

ω P (D) D / SP (D)

P (h)  SP (h) {Ó ωh

  P (C) / SP (C) ωC

Z whose component ωh :(SP (h)◦ωD)(Z) → (ωC ◦P (h))(Z) indexed by the object Z ∈ P (D), Z Z P (h)(Z) is a modification ωh : ωD ◦ Y(h) → ωC , and its component indexed by the object Z Z P (h)(Z) B ∈ B is the natural transformation (ωh )B :(ωD ◦ Y(h))B → (ωC )B. Its component Z Z P (h)(Z) (ωh )B(g):(ωD◦Y(h))B(g) → (ωC )B(g) indexed by the 1-morphism g : B → C (viewed as an object of the category B(B,C)), is a morphism given by the inverse of component Z P −1 of the coherence (ωh )B(g) := (µh,g)Z : P (h◦g)(Z) → (P (g)◦P (h))(Z) for the composition.

The composition ω ◦θ : SP → SP is a pseudo natural transformation whose component indexed by any object D in B, is a functor (ω ◦ θ)D : SP (D) → SP (D) which takes any pseudo natural transformation φ: YD → P to the pseudo natural transformation (ωD ◦ φD(idD) φD(idD) θD)(φ) = ωD(φD(idD)) = ωD : YD → P . Its component (ωD )C : YD(C) → P (C) indexed by the object C ∈ B, is a functor which sends any 1-morphism h: C → D (viewed φD(idD) as an object of the category B(C,D)), to the object (ωD )C (h) = P (h)(φD(idD)) = 0 (P (h)◦φD)(idD) in the category P (C), and which sends any 2-morphism ²: h → h to a mor- φD(idD) 0 phism (ωD )C (²) = P (²)φD(idD) : P (h)(φD(idD)) → P (h )(φD(idD)) in P (C), which is just the component indexed by the object idD ∈ YD(D) of the natural transformation 0 P (²)◦φD : P (h)◦φD ⇒ P (h )◦φD. The value of the functor (ω◦θ)D : SP (D) → SP (D) for

22 any modification Ω: φ → ψ in SP (D) is the modification (ωD ◦ θD)(Ω) = ωD(ΩC (idD)) = ΩD(idD) φD(idD) ψD(idD) ωD : ωD → ωD , whose component indexed by the object C ∈ B is a natural ΩD(idD) φD(idD) ψD(idD) transformation (ωD )C :(ωD )C → (ωD )C . The component of this natural transformation indexed by the 1-morphism h: C → D (viewed as an object of the cate- ΩD(idD) φD(idD) ψD(idD) gory B(C,D)), is the morphism (ωD )C (h):(ωD )C (h) → (ωD )C (h) in the category P (C), given by P (h)(ΩD(idD)): P (h)(φD(idD)) → P (h)(ψD(idD)). The component of the pseudo natural transformation ω ◦ θ : SP → SP indexed by the 1-morphism h: C → D in B, is the natural transformation (ω ◦ θ)h given by the diagram

(ω◦θ) SP (D) D / SP (D)

SP (h) {Ó SP (h) (ω◦θ)h

  SP (C) / SP (C) (ω◦θ)C

and defined by the composition (ωC ◦θh)(ωh ◦θD): SP (h)◦(ω◦θ)D → (ω◦θ)C ◦SP (h). The component indexed by the pseudo natural transformation φ: YD → P is the modification

−1 φC (ρh ◦λh)φh(idD) φD(idD) (ω ◦θ)h(φ) = [(ωC ◦θh)(ωh ◦θD)](φ) = [ωC ◦θh](φ)[ωh ◦θD](φ) = ωC ωh

φD(idD) (φ◦Y(h))C (idC ) between pseudo natural transformation ωD ◦ Y(h) and ωC in SP (C). The component of the modification (ω ◦ θ)h(φ)B : YC (B) → P (B) indexed by the object B in φD(idD) (φ◦Y(h))C (idC ) B, is natural transformation (ω ◦ θ)h(φ)B :(ωD ◦ Y(h))B → (ωC )B. The equality

−1 −1 φC (ρh λh) φh(idD) φD(idD) φC (ρh λh) φh(idD) φD(idD) (ω ◦ θ)h(φ)B(g) = (ωC ωC ωh )B(g) = [(ωC )B(ωC )B(ωh )B](g) = −1 φC (ρh λh)) φh(idD) φD(idD) −1 P −1 = [(ω )B](g)[(ω )B](g)[(ω )B](g) = P (g)(φC (ρ λh)φh(idD))(µ ) C C h h h,g φD(idD)

gives the component of the natural transformation (ω ◦ θ)h(φ)B indexed by g : B → C

φD(idD) (φ◦Y(h))C (idC ) (ω ◦ θ)h(φ)B(g):(ωD ◦ Y(h))B(g) → (ωC )B(g) which is given by the composition

P −1 [P (g)(φh(idD))][(µ ) ]: P (h ◦ g)(φD(idD)) → P (g)(φC (h ◦ idC )) h,g φD(idD) of morphisms in the category P (B).

23 Lemma 3.2. There exists an invertible modification Υ: ω ◦ θ → ι to the identity pseudo natural transformation ι: SP → SP .

Proof. Its component indexed by the object D in B is a natural transformation ΥD :(ω ◦ θ)D → ιD whose component indexed by the pseudo natural transformation φ: YD → P is φD(idD) a modification ΥD(φ): ωD → φ. Its component indexed by the object C in B is the φD(idD) natural transformation ΥD(φ)C :(ωD )C → φC , such that morphism

[ΥD(φ)C ](h): P (h)(φD(idD)) → φC (h) defines which is its component indexed by the morphism h: C → D, is defined by

[ΥD(φ)C ](h) := φC (λh)φh(idD) . The coherence for modification Υ: ω ◦ θ → ι

SP (h)◦ΥD / SP (h) ◦ (ω ◦ θ)D SP (h) ◦ ιD

(ω◦θ)h ιh

  / (ω ◦ θ)C ◦ SP (h) ιC ◦ SP (h) ΥC ◦SP (h) is satisfied since by evaluating the above diagram for any φ: YD → P , the diagram

(SP (h)◦ΥD)(φ)/ (SP (h) ◦ (ω ◦ θ)D)(φ) (SP (h) ◦ ιD)(φ)

(ω◦θ)h(φ) ιh(φ)

  / (ω ◦ θ)C ◦ SP (h))(φ) (ιC ◦ SP (h))(φ) (ΥC ◦SP (h))(φ) becomes the diagram of modifications

ΥD(φ)◦Y(h) φD(idD) / φ ◦ Y(h) ωD ◦ Y(h)

(ω◦θ)h(φ)

 ω(φ◦Y(h))C (idC ) / φ ◦ Y(h) C ΥC (φ◦Y(h))

24 Evaluating this diagram by the morphism g : B → C

(ΥD(φ)◦Y(h))B (g) φD(idD) / (φ ◦ Y(h)) (g) (ωD ◦ Y(h))B(g) B

((ω◦θ)h(φ))B (g)

 (φ◦Y(h))C (idC ) / (ω )B(g) (φ ◦ Y(h))B(g) C (ΥC (φ◦Y(h)))B (g)

we obtain the diagram

φB (λh◦g)φh◦g(idD/ ) (P (h ◦ g) ◦ φD)(idD) φB(h ◦ g)

−1 P −1 P (g)(φC (ρ λh)φh(idD))(µ ) h h,g φD(idD)

 / (P (g) ◦ φC )(h ◦ idC ) φB(h ◦ g) (ΥC (φ◦Y(h)))B (g) Since we have the identities

[ΥD(φ)C ](h) := φC (λh)φh(idD)

(ΥC (φ ◦ Y(h)))B(g) = (φ ◦ Y(h))B(λg)(φ ◦ Y(h))g(idC ) = φB(h ◦ λg)(φB ◦ Y(h)g)(idC )(φg ◦ Y(h)C )(idC ) =

= φB(h ◦ λg)φB(αh,idC ,g)φg(h ◦ idC ) the last diagram is equal to the edge of the diagram

φh◦g(idD)/ φB (λh◦g) / (P (h ◦ g) ◦ φD)(idD) φB(idD ◦ (h ◦ g)) φB(h ◦ g) φB(h ◦ g) O jj5 Id I O jjj II ((µP )−1◦φ )(id ) φ (α ) jjj I φ (h◦λ ) h,g D D B idD,h,g jjj II B g jjj φB (λh◦g) II  j II II (P (g) ◦ P (h) ◦ φD)(idD) φB((idD ◦ h) ◦ g) II φB(h ◦ (idC ◦ g)) ii4 φB (ρh◦g) I O iiii II (P (g)◦φ )(id )) iii II φ (α ) h D iiii II B h,idC ,g  iii φg(idD◦h) I (P (g) ◦ φ )(id ◦ h) / (P (g) ◦ φ )(h) / (P (g) ◦ φ )(h ◦ id ) / φ ((h ◦ id ) ◦ g) C D C −1 C C B C (P (g)◦φ )(λ ) φg(h◦id ) C h (P (g)◦φC )(ρh ) C

in which the first pentagonal diagram commute since it is the coherence diagram for the pseudo natural transformation φ: YD → P

25 (P (g)◦φh)(idD)/ (P (g) ◦ P (h) ◦ φD)(idD) (P (g) ◦ φC )(idD ◦ h)

P (µh,g◦φD)(idD)  (P (h ◦ g) ◦ φD)(idD) φg(idD◦h)

φh◦g(idD)   o φB(idD ◦ (h ◦ g)) φB((idD ◦ h) ◦ g) φ (α ) B idD,h,g evaluated by the idD. The commutativity of two triangle diagrams is the consequence of the triangle coherence for left and right identities and the hexagonal diagram is the deformed square expressing the naturality of φg : P (g) ◦ φC ⇒ φB ◦ YD(g) for the 2- −1 morphism ρh λh : idD ◦ h ⇒ h ◦ idC .

26