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Bicategorical Yoneda Lemma

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Bicategorical Yoneda Lemma Bicategorical Yoneda lemma Igor Bakovi¶c Rudjer Bo·skovi¶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 De¯nition 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 2 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 à : f ! g to the natural transformation P (Ã): P (f) ! P (g): We also have natural trans- formations ¹g;f : P (f) ± P (g) ! P (g ± f) and ´A : iP (A) ! P (iA), which satisfy following coherence laws ¹ ±P (h) ¹ (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) ¹ P ((h ± g) ± f) P (f)±¹h;g h±g;f ¹i ;f ¹f;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)±´B P (¸f ) ´A±P (f) P (½f ) ² ² P (f) P (f) P (f) P (f) 1 De¯nition 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 2 B and is given by the square σ P (B) B / R(B) P (f) ÄÄÄ R(f) {¤ Äσf ² ² 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) Ä? ?? ÄÄ ?? P (g) ÄÄ ?? P (f) ÄÄ ?? ÄÄ ?? ÄÄ Â ?? Ä Â ¹P ? Ä ®¶ g;f  P (C) / P (A) P (g±f) σB σg ;C ???σf ÄÄÄ # ² σC R(B) σA Ä? ?? ÄÄ ?? R(g) ÄÄ ?? R(f) ÄÄ Â ?? ÄÄ ¹R ?? ÄÄ ®¶ g;f ?? ² ÄÄ ? ² 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 ¹g;f ±σC ² R(g ± f) ± σC σf ±P (g) σg±f ² ² σ ± P (g ± f) o σ ± P (f) ± P (g) A P A σz±¹g;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 Ä ?? ÄÄ ?? ÄÄ ?? ÄÄ ?? ´R±σ Ä ? σ ±´P A A ÄÄ ?? A A ÄÄ ?? ÄÄ ?? ÄÄ ?? ÄÄ ?? Ä Ä ? / R(iA) ± σA σA ± P (iA) σiA of natural transformations. De¯nition 1.3. A modi¯cation ¡: σ ! ¿ consists of the natural transformation ¡A : σA ! ¿A for each object A in B such that the following diagram σB $  P (B) ¡B R(B) ®¶ : ¿B P (f) R(f)  ®¶ ¿f f _ X l σA R ² s K$ ²  P (A) ¡A R(A) ®¶ : ¿A commutes. This means that the above diagram becomes a commutative diagram R(f)±¡B / R(f) ± σB R(f) ± ¿B σf ¿f ² ² / σA ± P (f) ¿A ± P (f) ¡A±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 modi¯cations. σ » Proof. For any two pseudo natural transformations P +3 R +3 S the composition is de¯ned by (» ± σ)A := »A ± σA and (» ± σ)f := (»A ± σf )(»f ± σB) which is just the pasting σ » P (B) B / R(B) B / S(B) P (f) ÄÄÄ R(f) ÄÄÄ S(f) {¤ σf {¤ »f ² ² ² / / P (A) σ R(A) S(A) A »A composite of the above diagram. That this is well de¯ned composition we deduce from the pasting composite P (B) ? ? ÄÄ ?? P (f) Ä Â ? P (g) ÄÄ ¹P ?? ÄÄ ®¶ g;f ?? ÄÄ ? P (C) / P (A) P (k±h) σB σf ;C ???σg ÄÄÄ # ² σC R(B) σA ? ? ÄÄ ?? R(f) Ä Â ? R(g) ÄÄ ¹R ?? ÄÄ ®¶ g;f ?? ² ÄÄ ? ² R(C) / R(A) R(k±h) »B »f ;C ???»g ÄÄÄ # ² »C S(B) »A ? ? ÄÄ ?? S(f) Ä Â ? S(g) ÄÄ ¹S ?? ÄÄ ®¶ g;f ?? ² ÄÄ ? ² S(C) / S(A) S(g±f) which gives a coherence for a pseudo natural transformation. This composition is asso- σ » ! ciative since for any three pseudo natural transformations P +3 R +3 S +3 T we have the associativity [(! ±»)±σ]A = (! ±»)A ±σA = (!A ±»A)±σA = !A ±(»A ±σA) = !A ±(» ±σ)A = [! ±(» ±σ)]A 4 for components indexed by the object B in B, and [(! ± ») ± σ]f = [(! ± »)A ± σf ][(! ± »)f ± σB] = [(!A ± »A) ± σf ][((!A ± »f )(!f ± »B)) ± σB] = = [(!A ± »A) ± σf ][(!A ± »f ) ± σB][(!f ± »B) ± σB] = [!A ± (»A ± σf )][!A ± (»f ± σB)][!f ± (»B ± σB)] = = [(!A ± (»A ± σf )(»f ± σB)][(!f ± (»B ± σB))] = [!A ± (» ± σ)f ][!f ± (» ± σ)B] = [! ± (» ± σ)]f for components indexed by the morphism f : A ! B in B. The vertical composition for any two modi¯cations σ ¡ _*4 ¿ ¦ _*4 ¼ is de¯ned simply by (¦¡)A := ¦A¡A. The horizontal composition for any two modi¯cations σ » Â ¿  ¿ P ®¶ ¡ BR ®¶ ­ B S ¿ ³ is de¯ned by (­ ± ¡)A := ­B ± ¡B. We need to show that pasting of the diagram σB »B ! !   P (B) ¡B R(B) ­B S(B) ®¶ = ®¶ = ¿ B ³B P (f) R(f) S(f)   ¿f ³ h _®¶ V h _®¶ Vf p σA N p » N ² y E! ² y A E! ²   P (A) ¡A R(A) ­A S(A) ®¶ = ®¶ = ¿ A ³A 5 gives the coherence diagram for the composite modi¯cation ­ ± ¡: » ± σ ! ³ ± ¿. (»±σ)B !  P (B) (­±¡)B S(B) ®¶ = (³±¿)B P (f) S(f)   (³±¿) h _®¶ V f p (»±σ) N ² y A E! ²  P (A) (­±¡)A S(A) ®¶ = (³±¿)A The commutativity of the above diagram is deduced from the commutative diagram S(f)±(»B ±¡B ) / S(f)±(­B ±¿B ) / S(f) ± (»B ± σB) S(f) ± (»B ± ¿B) S(f) ± (³B ± ¿B) ® ® ® S(f);»B ,σB S(f);»B ;¿B S(f);³B ;¿B ² ² ² (S(f)±»B )±¡B / (S(f)±­B )±¿B / (S(f) ± »B) ± σB (S(f) ± »B) ± ¿B (S(f) ± ³B) ± ¿B »f ±σB »f ±¿B ³f ±¿B ² ² ² (»A±R(f))±¡B / (­A±R(f))±¿B / (»A ± R(f)) ± σB (»A ± R(f)) ± ¿B (³A ± R(f)) ± ¿B ® ® ® »A;R(f),σB »A;R(f);¿B ³A;R(f);¿B ² ² ² »A±(R(f)±¡B ) / ­A±(R(f)±¿B ) / »A ± (R(f) ± σB) »A ± (R(f) ± ¿B) ³A ± (R(f) ± ¿B) »A±σf »A±¿f ³A±¿f ² ² ² »A±(¡A±P (f)) / ­A±(¿A±P (f)) / »A ± (σA ± P (f)) »A ± (¿A ± P (f)) ³A ± (¿A ± P (f) ®¡1 ®¡1 ®¡1 »A,σA;P (f) »A;¿A;P (f) ³A;¿A;P (f) ² ² ² / / (»A ± σA) ± P (f) (»A ± ¿A) ± P (f) (³A ± ¿A) ± P (f) (»A±¡A)±P (f) (­A±¿A)±P (f) where the ¯rst square in the second row and the second square in the the fourth row commute because they de¯ne horizontal compositions »f ±¡B and ­A ±¿f , respectively. The commutativity of the second square in the second row and the ¯rst square in the fourth row follows from the coherence for modi¯cations ­: » ! ³ and ¡: σ ! ¿, respectively 6 and squares in the ¯rst, third and ¯fth row are associativity coherence. But the pasting composition of the top and front faces of the coherence for the modi¯cation ­ ± ¡ 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 modi¯cations σ » Â Â ®¶ ¡ ¿ ®¶ ­ ¿ / / P ¿ BR B S  ³Â ®¶ ¦ ®¶ ¤ ¼  we have Godement interchange law (¤ ± ¦)(­ ± ¡) = (¤­) ± (¦¡) since it holds for natural transformations and the composition were de¯ned 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, de¯ned by YC (B) := B(B; C); for each object B in B, and for any two objects A; B 2 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 2 B(A; B) is a functor YC (f): YC (B) !YC (A) de¯ned by YC (f)(g) := g ± f; YC (f)(Á) := à ± f for any 1-morphism g 2 B(B; C) and any 2-morphism à 2 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 Á: f ! f (viewed as a morphism of the category B(A; B)), is the natural transformation 0 YC (Á): YC (f) !YC (f ) as in the diagram Y (Á) C g / 0 YC (f)(g) YC (f )(g) 0 YC (f)(º) YC (f )(º) ² ² 0 / 0 0 YC (f)(g ) YC (f )(g ) YC (Á)g0 for any 2-morphism º : g ! g0 in B(B; C), whose component for any object g 2 B(B; C) is de¯ned by YC (Á)g := g ± Á.
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