Enrichments over symmetric Picard categories

Enric hmen ts o v er symmetr ic Picard categori es Vincen t Sc hmitt email: vrrsc hmitt @ yahoo.fr Octob er 30, 2018 Abstract Categorica l rings w ere i ntrod uced in [JiPi07], w h ich w e call 2-rings. In these notes w e present basic d eﬁnitions and results regarding 2-mo du les. This is w ork in progress. 1 In tro d uction Categoric a l rings were intro duced by M.Jibladze and T.Pir ashvili in [JiPi07]. W e ca ll tho s e 2-r ings. The pr esent s et of notes contains basic results a b o ut 2-mo dules and this work is in prog ress. Section 2 contains t echnical preliminaries, namely reminders o n symmetric Picard categories as w ell conv enient references to pr evious works. In particular the development s in [Sc h0 8] for symmetric monoidal categories tra ns po se well to symmetric Pica rd categ o ries and a suitable tensor pro duct can b e deﬁned for the latter. Section 3 and 4 treat enrichmen ts over symmetric Picar d ca tegories . Those w er e introduced in [Dup08 ] a nd deﬁned by means of mu ltilinea r maps. W e also deﬁne them using the tenso r pro duct. Section 5 c o ntains exp ected examples of 2-ring s a nd 2 -enrichmen ts. Section 6 c ontains basic results regarding ca tegories of A -mo dules for a 2- ring A . In particular we show that A -mo dules a re particular alge bras for the endo-2-functor A ⊗ − of the 2-catego ry of symmetric Pic ard ca tegories. A la rge app endix contains the more technical dev elo pments. 2 Preliminaries A categor ical group structure ( A , j ) consis ts of a monoida l catego ry A and an ass ignment for every ob ject a of A of an ob ject a • with an isomorphism j a : I → a • ⊗ a , ( a • is a n inverse of a ). W e are concerne d in this pap er with symmetric Pic ar d c ate gories which are the ca tegorical groups ( A , j ) for which A has a symmetric mono idal str ucture a nd its underlying category is a group oid. S P C de no tes the 2- categor y with ob jects symmetric Picar d catego ries, a rrows symmet- ric mono ida l functors and 2-cells monoida l natural tra nsformations. Ther e is a forge tful 2 -functor S P C → S M C forgetting the g roup structure whe r e S M C denotes the 2-c ategory with ob jects symmetric monoidal categ ories, ar r ows symmetric monoidal functors, a nd monoidal natural trans - formations as 2- cells. The 2-categ orical prop erties of S P C are s imilar to those of S M C , the la tter 2-catego ry has b een studied in diﬀerent works in par ticular in [HyPo02] and in [Sch08]. W e refer the reader to this last work for basic notations, a nd more elab ora te results. In this ﬁr s t section, we des c rib e br ieﬂy and compare the imp orta nt pro pe r ties of S M C and S P C . The 2-categ ory S M C admits a n in ternal hom a nd the same holds for S P C whic h hom is inherited fr om S M C . The following was mentioned in [Dup08] with a r ather co ncise explana tion. Lemma 2.1 Given any t wo obje cts in A and B in S M C with B b eing a symmetric Pic ar d c ate gory, the internal hom [ A , B ] in S M C admits a symmetric Pic ar d stru ctur e given p ointwise by that of B . 1 W e pr esent a pro o f that relies on cohere nc e results for ca tegorica l gro ups from [Lap83]. (This work also contains r e fer ences to earlier works on the topic.) Let us reca ll these. F or a categorica l g roup A with family of isomorphisms j a : I → a • ⊗ a for each ob ject a there is a unique w ay of extending the assignment a 7→ a • int o a functor A op → A tha t makes the j a natural in a . It is a n equiv alence. W e will write it ( − ) • and write f • : b • → a • for the image of any arrow f : a → b by this functor . There is a coher ence theo r em stating that any pa ir of a and b of ob jects of A ther e is at most one “canonica l” arrow a → b , those canonical a rrows b eing the ones generated in an exp ected wa y from the cano nical ar rows from the monoidal structures a nd the j a ’s. Even tually Laplaza ’s pap er also pr ovides a combinatorial description of free catego r ical groups. Let us r e call also the following kno wn facts for any symmetric Pic a rd categor ies A and B . F or any symmetric monoidal functor ( F, F 2 , F 0 ) : A → B the comp onent F 0 is determined by F 2 . Actually a monoidal str uc tur e on a functor F : A → B is g iven by a natura l F 2 a,b : F a ⊗ F b → F ( a ⊗ b ) in B satisfying the o nly axio m that F a ⊗ ( F b ⊗ F c ) 1 ⊗ F 2 b,c   ∼ = ( F a ⊗ F b ) ∼ = F c F 2 a,b ⊗ 1   F a ⊗ F ( b ⊗ c ) F 2 a,b ⊗ c   F ( a ⊗ b ) ⊗ F c F 2 a ⊗ b, 1   F ( a ⊗ ( b ⊗ c )) F ( ∼ = ) F (( a ⊗ b ) ⊗ c ) commutes for any ob jects a , b and c of A . Also any natural tra nsformation σ : F → G : A → B betw ee n mono idal functors is monoida l if it sa tis ﬁe s the only axio m that the diag ram in B F a ⊗ F b F 2 a,b / / σ a ⊗ σ b   F ( a ⊗ b ) σ a ⊗ b   Ga ⊗ Gb G 2 a,b / / G ( a ⊗ b ) commutes fo r any ob jects a , b o f A . Let us co nsider a symmetric Picar d ca tegory ( A , j ). Since A is a group oid, o ne has a functor A op → A which is the iden tity on ob jects and sends arrows to their in verses. The functor inv : A → A is obtained b y comp os ing the previo us functors and ( − ) • ab ov e. Let us denote b y ! the unique ca nonical arr ow b etw een tw o ob jects of A , when it exists. Lemma 2.2 F or any symmetric Pic ar d c ate gory ( A , j ) , the asso ciate d functor inv admits a sym- metric monoidal structur e wher e inv 2 has c omp onent inv 2 a,b : a • ⊗ b • → ( a ⊗ b ) • in any ( a, b ) the c omp osite a • ⊗ b • s / / b • ⊗ a • ! / / ( a ⊗ b ) • (which is also a • ⊗ b • ! / / ( b ⊗ a ) • s • / / ( a ⊗ b ) • ac c or ding to L emma 7.1 in App endix). PROOF:See Appendix 7.2. 2.3 F or any symmetric Pic ar d c ate gory ( A , j ) one has a monoidal natur al isomorphism I → inv ✷ id : A → A which c omp onent in any obje ct a is j a : I → a • ⊗ a . PROOF:See Appendix 7.4. 2 Now given ob jects A and B in S M C with ( B , j ) symmetric P icard, a group structur e is obtained on the hom [ A , B ] in S M C , which is a group oid, as follows. The strict symmetric monoidal functor [ A , − ] : [ B , B ] → [[ A , B ] , [ A , B ]] sends the monoidal transfor mation j : I → inv ✷ id : B → B of 2.3 to a monoidal transforma tion I [ A , I ] [ A ,j ] / / [ A , inv ✷ i d ] [ A , inv ] ✷ [ A , id ] [ A , inv ] ✷ id : [ A , B ] → [ A , B ] . which we deﬁne as the j on [ A , B ]. This is to say that F • for any symmetric monoidal F is the comp osite A F / / B inv / / B . and the natural isomorphisms j F : I ∼ = F • ✷ F are p o int wis e ( j F ) a : I → ( F a ) • ⊗ F a . Then for any monoidal σ : F → G : A → B one has that σ • : G • → F • is p oint wise ( σ a ) • : ( Ga ) • → ( F a ) • . W e will a lwa ys consider that this is the c ho sen group structure on [ A , B ] when cons ide r ed as an ob ject of S P C . This structure is deter mined by that o f B . One has a notion of strictness for arrows in S P C , which is diﬀer ent from the no tion o f strictness in S M C . Let us consider any symmetric Picar d categor ies A and B . F or a symmetric monoidal functor F : A → B one ha s the natural isomorphism 2.4 F ( a • ) ∼ = a ( F a ) • . deﬁned precisely in App endix-7 .5. A s ymmetric monoidal functor F : A → B is called a strict arrow in S P C when it preser ves strictly the monoidal structure and moreover preserves strictly the isomo rphisms j meaning that the natural isomo rphism 2.4 is an identit y o r equiv alently that for an y ob ject a in A , F se nds a • to ( F a ) • and j a : I → a • ⊗ a to j F a : I → ( F a ) • ⊗ F a . W e wr ite S trS P C for the sub-2 - categor y of S P C with same ob jects, strict arrows and 2-cells inherited fro m S P C . The r esults from [Sch08] r egarding S M C transp os e rather straig htf o r wardly to S P C as follows. Given an arbitr a ry symmetr ic P icard categor y C , it happ ens that the isomorphism D A , B , C : [ A , [ B , C ]] → [ B , [ A , C ]] deﬁned in chapter 6 is a lso a strict a rrow in S P C . F or an y ar r ow F : A → [ B , C ] in S P C , its image F ∗ by D , called it “dual”, is strict in S P C if and only for any ob ject b of B , the arrow F ∗ ( b ) : A → C is strict in S P C . F or a ny ob jects A , B and C in S P C , the ar row [ A , − ] B , C : [ B , C ] → [[ A , B ] , [ A , C ]] deﬁned in chapter 8 is str ict in S P C . The hom 2 -functor o f S M C deﬁned in c hapter 9 induces b y r estriction a hom 2-functor S P C op × S P C → S P C for S P C . The statement s regar ding the 2 -naturality o f D (chapter 10) and the ev aluation functor s (chapter 11) still hold when replacing for mally S M C b y S P C . In particular the ev aluation functor s a re strict arr ows in S P C . Similarly to the ca se of the 2-catego ry S M C , one has a tensor pro duct in S P C . F or any symmetric Pica rd catego ries A and B , their tensor A ⊗ B satisﬁes the universal prop erty of the existence of a 2-natural iso morphism 2.5 S P C ( A , [ B , C ]) ∼ = C S trS P C ( A ⊗ B , C ) betw ee n 2-functors S trS P C → Cat in the ar gument C . Note that the 2-naturality in question inv olves only strict mor phisms in S P C . W e brieﬂy sketc h a descr iption of the ab ov e tensor A ⊗ B by g enerator and relations. It is similar to that g iven with more details in [Sc h0 8] for the tensor 3 in S M C . W e consider a gr aph H with vertices the terms o f the fr ee { I , ( − ) • , ⊗} - a lgebra ov er the set Ob j ( A ) × O bj ( B ), i.e. they are w o rds of the for mal language containing a ll pa ir s ( a, b ) – which we write a ⊗ b – for ob jects a o f A and b of B , the one-symbol word I , the w or ds X • for any vertex X , a nd X ⊗ Y for any vertices X and Y . The set of edges of H cons ists of: - The “canonica l” edges for the sy mmetric monoida l structure which ar e the ass X,Y ,Z : X ⊗ ( Y ⊗ Z ) → ( X ⊗ Y ) ⊗ Z , r X : X ⊗ I → X , l X : I ⊗ X → X , s X,Y : X ⊗ Y → Y ⊗ X for all v er tices X , Y , Z ; - E dges j X : I → X • ⊗ X , one for each v er tex X ; - E dges γ a,a ′ ,b : ( a ⊗ b ) ⊗ ( a ′ ⊗ b ) → ( a ⊗ a ′ ) ⊗ b and δ a,b,b ′ : ( a ⊗ b ) ⊗ ( a ⊗ b ′ ) → a ⊗ ( b ⊗ b ′ ) indexed by ob jects a , a ′ of A and b , b ′ of B ; - E dges a ⊗ f : a ⊗ b → a ⊗ b ′ indexed by ob jects a of A and arrows f : b → b ′ of B ; - E dges f ⊗ b : a ⊗ b → a ′ ⊗ b indexed by ob jects b of B and arrows f : a → a ′ of A ; - Edges X ⊗ p : X ⊗ Y → X ⊗ Z a nd p ⊗ X : Y ⊗ X → Z ⊗ X for any vertex X and any edge p : Y → Z ; with the conv ention that edges ab ov e with diﬀerent names ar e diﬀerent. Let us consider F G ( H ) the free gro up o id on H , i.e its ar rows are mere concatenatio ns o f edges of H and their formal in verses. F or any vertex X , one ha s tw o gra ph endomorphisms of H , na mely X ⊗ − and − ⊗ X sending resp ectively an ar bitr ary edg e f : Y → Z to X ⊗ Y → X ⊗ Z , re s p. Y ⊗ X → Z ⊗ X . These tw o extend uniq ue ly to endofunctors of F G ( H ) and w e extend the nota tion X ⊗ f and f ⊗ X to denote the image s o f a r rows of F G ( H ) by these functor s. The tensor A ⊗ B is the quotient of F G ( H ) by the congruence genera ted by the following rela- tions ∼ on its ar rows from 2.6 to 2.2 0 b elow. F or all edg es X t / / Y and Z s / / W of H , 2.6 X ⊗ W t ⊗ W % % L L L L L L L L L L X ⊗ Z X ⊗ s 9 9 s s s s s s s s s s t ⊗ Z % % K K K K K K K K K K ∼ Y ⊗ W. Y ⊗ Z Y ⊗ s 9 9 r r r r r r r r r r 2.7 R elations giving the c oher enc e c onditions for ass , r , l and s in A ⊗ B . These are the following. - F or any vertices X , Y , Z and T , X ⊗ ( Y ⊗ ( Z ⊗ T )) ∼ ass / / 1 ⊗ ass   ( X ⊗ Y ) ⊗ ( Z ⊗ T ) ass / / (( X ⊗ Y ) ⊗ Z ) ⊗ T X ⊗ (( Y ⊗ Z ) ⊗ T ) ass / / ( X ⊗ ( Y ⊗ Z )) ⊗ T . ass ⊗ 1 O O - F or any vertices X a nd Y , X ⊗ ( I ⊗ Y ) ass / / 1 ⊗ l & & N N N N N N N N N N N ∼ ( X ⊗ I ) ⊗ Y r ⊗ 1 x x p p p p p p p p p p p X ⊗ Y . 4 - F or any vertex X , X ⊗ I s / / r # # G G G G G G G G ∼ I ⊗ X l { { w w w w w w w w X . - F or any vertices X , Y and Z , X ⊗ ( Y ⊗ Z ) ass / / 1 ⊗ s   ( X ⊗ Y ) ⊗ Z s / / ∼ Z ⊗ ( X ⊗ Y ) ass   X ⊗ ( Z ⊗ Y ) ass / / ( X ⊗ Z ) ⊗ Y ( Z ⊗ X ) ⊗ Y . s ⊗ 1 o o 2.8 R elations for t he natur alities of ass , r , l , and s in A ⊗ B . F or instance, one ha s for any edge f : X → X ′ of H , and any vertices Y a nd Z , X ⊗ ( Y ⊗ Z ) ass X,Y ,Z / / ∼ f ⊗ 1   ( X ⊗ Y ) ⊗ Z ( f ⊗ 1) ⊗ 1   X ′ ⊗ ( Y ⊗ Z ) ass X ′ ,Y ,Z / / ( X ′ ⊗ Y ) ⊗ Z . W e will no t wr ite here the other relatio ns. Ther e are tw o mor e for the na turalities of ass X,Y ,Z in Y and Z , one for that of l X in X , one for that of r X in X a nd tw o for tho s e of s X,Y in X a nd Y . F or any ob ject a in A and any arrows b f / / b ′ g / / b ′′ in B , 2.9 a ⊗ b a ⊗ ( g ◦ f ) / / a ⊗ f # # H H H H H H H H H ∼ a ⊗ b ′′ a ⊗ b ′ a ⊗ g : : u u u u u u u u u F or any ob ject b in B a nd any arrows a f / / a ′ g / / a ′′ in A , 2.10 a ⊗ b ( g ◦ f ) ⊗ b / / f ⊗ b # # H H H H H H H H H ∼ a ′′ ⊗ b a ′ ⊗ b g ⊗ b : : u u u u u u u u u F or any ob jects a in A and b in B , 2.11 a ⊗ id b ∼ id a ⊗ b and 2.12 id a ⊗ b ∼ id a ⊗ b . where id b , id a and id a ⊗ b ab ov e a re the iden tities resp ectively a t b in B , at a in A and at a ⊗ b in F G ( H ). F or any arrows f : a → a ′ in A and g : b → b ′ in B , 5 2.13 a ⊗ b a ⊗ g   f ⊗ b / / ∼ a ′ ⊗ b a ′ ⊗ g   a ⊗ b ′ f ⊗ b ′ / / a ′ ⊗ b ′ . 2.14 R elations for the “natur alities” of γ a,a ′ ,b in a , a ′ and b and δ a,b,b ′ in a , b and b ′ . F or ins ta nce by the relations for the “naturality” of γ a,a ′ ,b in b it is mea nt that fo r an y ob jects a, a ′ in A and a ny arr ow g : b → b ′ in B , ( a ⊗ b ) ⊗ ( a ′ ⊗ b ) (1 ⊗ g ) ⊗ ( 1 ⊗ g )   γ a,a ′ ,b / / ∼ ( a ⊗ a ′ ) ⊗ b 1 ⊗ g   ( a ⊗ b ′ ) ⊗ ( a ′ ⊗ b ′ ) γ a,a ′ ,b ′ / / ( a ⊗ a ′ ) ⊗ b ′ . W e will not wr ite explicitly now the ﬁve other r elations. F or any ob jects a in A and b , b ′ , b ′′ in B , 2.15 ( a ⊗ b ) ⊗ (( a ⊗ b ′ ) ⊗ ( a ⊗ b ′′ )) ass / / 1 ⊗ δ a,b ′ ,b ′′   ∼ (( a ⊗ b ) ⊗ ( a ⊗ b ′ )) ⊗ ( a ⊗ b ′′ ) δ a,b,b ′ ⊗ 1   ( a ⊗ b ) ⊗ ( a ⊗ ( b ′ ⊗ b ′′ )) δ a,b,b ′ ⊗ b ′′   ( a ⊗ ( b ⊗ b ′ )) ⊗ ( a ⊗ b ′′ ) δ a,b ⊗ b ′ ,b ′′   a ⊗ ( b ⊗ ( b ′ ⊗ b ′′ )) 1 ⊗ ass b,b ′ ,b ′′ / / a ⊗ (( b ⊗ b ′ ) ⊗ b ′′ ) . F or any ob jects a in A and b , b ′ in B , 2.16 ( a ⊗ b ) ⊗ ( a ⊗ b ′ ) δ a,b,b ′ / / s a ⊗ b,a ⊗ b ′   ∼ a ⊗ ( b ⊗ b ′ ) 1 ⊗ s b,b ′   ( a ⊗ b ′ ) ⊗ ( a ⊗ b ) δ a,b ′ ,b / / a ⊗ ( b ′ ⊗ b ) . F or any ob jects a , a ′ , a ′′ in A and b in B , 2.17 ( a ⊗ b ) ⊗ (( a ′ ⊗ b ) ⊗ ( a ′′ ⊗ b )) ass / / 1 ⊗ γ a ′ ,a ′′ ,b   ∼ (( a ⊗ b ) ⊗ ( a ′ ⊗ b )) ⊗ ( a ′′ ⊗ b ) γ a,a ′ ,b ⊗ 1   ( a ⊗ b ) ⊗ (( a ′ ⊗ a ′′ ) ⊗ b )) γ a,a ′ ⊗ a ′′ ,b   (( a ⊗ a ′ ) ⊗ b ) ⊗ ( a ′′ ⊗ b ) γ a ⊗ a ′ ,a ′′ ,b   ( a ⊗ ( a ′ ⊗ a ′′ )) ⊗ b ass ⊗ 1 / / (( a ⊗ a ′ ) ⊗ a ′′ ) ⊗ b. F or any ob jects a , a ′ in A and b in B , 6 2.18 ( a ⊗ b ) ⊗ ( a ′ ⊗ b ) γ a,a ′ ,b / / s a ⊗ b,a ′ ⊗ b   ∼ ( a ⊗ a ′ ) ⊗ b s a,a ′ ⊗ 1   ( a ′ ⊗ b ) ⊗ ( a ⊗ b ) γ a ′ ,a,b / / ( a ′ ⊗ a ) ⊗ b. F or any ob jects a , a ′ in A and b , b ′ in B , 2.19 (( a ⊗ b ) ⊗ ( a ⊗ b ′ )) ⊗ (( a ′ ⊗ b ) ⊗ ( a ′ ⊗ b ′ )) δ a,b,b ′ ⊗ δ a ′ ,b,b ′   ∼ (( a ⊗ b ) ⊗ ( a ′ ⊗ b )) ⊗ (( a ⊗ b ′ ) ⊗ ( a ′ ⊗ b ′ )) γ a,a ′ ,b ⊗ γ a,a ′ ,b ′   ( a ⊗ ( b ⊗ b ′ )) ⊗ ( a ′ ⊗ ( b ⊗ b ′ )) γ a,a ′ ,b ⊗ b ′   (( a ⊗ a ′ ) ⊗ b ) ⊗ (( a ⊗ a ′ ) ⊗ b ′ ) δ a ⊗ a ′ ,b,b ′ r r e e e e e e e e e e e e e e e e e e e e e e e e e e ( a ⊗ a ′ ) ⊗ ( b ⊗ b ′ ) . wher e the t op arr ow is t he c onc atenation ass X ⊗ Y ,Z,T / / ass − 1 X,Y ,Z ⊗ 1 / / (1 ⊗ s Y ,Z ) ⊗ T / / ass X,Z,Y ⊗ T / / ass X ⊗ Z,Y ,T − 1 / / with the ass − 1 b eing the formal inverses of e dges ass and X , Y , Z and T standing r esp e ct ively for a ⊗ b , a ⊗ b ′ , a ′ ⊗ b and a ′ ⊗ b ′ . 2.20 Exp ansions of al l r elations ab ove by iter ations of X ⊗ − and − ⊗ X for al l vertic es X . Whic h means precisely that the set of relations ∼ is the s ma llest set of r elations o n arrows of F G ( H ) co ntaining the pr evious relations (2.6 to 2.19) and satisfying the closure prop erties that for any relation f ∼ g : Y → Z tha t it contains and any vertex X , it contains also the relations X ⊗ f ∼ X ⊗ g : X ⊗ Y → X ⊗ Z and f ⊗ X ∼ g ⊗ X : Y ⊗ X → Z ⊗ X . The pro ofs that the above ca tegory A ⊗ B is a well deﬁned symmetric Pic a rd categor y and that it satisﬁes the universal prop er ty 2.5, are similar to those in [Sch08] for the well deﬁnition and universal pro p er ty of the tenso r pro duct in S M C . W e will therefore not r eplicate them. F or any ob jects A , B and C in S P C , one obtains an adjunction E n ⊣ R n : [ A ⊗ B , C ] → [ A , [ B , C ]] in the 2- categor y S P C (in this case it is an equiv alence) with R n ◦ E n = 1 and wher e the ar- rows Rn A , B , C are strict. The isomorphism 2.5 b eco mes 2 - natural in A a nd B for a unique tenso r 2-functor S P C × S P C → S P C . There exists a free s ymmetric Picard ca tegory on one generator , which we s ha ll write I , that diﬀers obviously fro m the “unit” for S M C wr itten I a nd deﬁned in [Sch08]-chapter 1 8, but tha t has a very similar presentation by generator s and relation. The only diﬀerences are the following. Its set ob jects is now the free { I , ⊗ , ( − ) • } -algebr a ov er one g enerator ⋆ . It is a quotient of fr e e gr oup oid generated b y the gr aph containing the usual canonica l edges for the symmetric mono idal structur es 7 (the ass X,Y ,Z , r X , l X , s X,Y ) and moreov er containing one edge j X : I → X • ⊗ X for each ob ject X . The relations on this free gr oup oid deﬁning I are just those for the naturalities of the collections ass , r , l and s , those expressing the coherence axioms for the s ymmetric monoida l s tructure, and relations expr essing the bifunctoriality of − ⊗ − : I × I → I . The universal prop erty deﬁning I is that for any symmetric P icard categor y A , there exits a unique strict arrow v : I → [ A , A ] such that v ( ⋆ ) is the identit y arr ow at A → A with its stric t structure in S P C . One obtains with similar pr o ofs, simila r results. Namely: 2.21 F or any symmetric Pic ar d c ate gory A , the dual v ∗ : A → [ I , A ] of v : I → [ A , A ] has right adjoint in S P C the evaluation at ⋆ functor ev ⋆ : [ I , A ] → A . F ro m this one can exhibit a kind of “symmetric monoidal closed 2-structure” o n S P C in the same wa y as done in [Sch08] chapters 19, 2 0 a nd 21 for S M C . Namely one can deﬁne the canonical ar rows A ′ A , B , C : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ), R ′ A : A → A ⊗ I , L ′ A : A → I ⊗ A and S A , B : A ⊗ B → B ⊗ A in this case with resp ective inv er se equiv a lences A A , B , C , R A , L A and S B , A and s atisfying the (strict) cohe r ence ax io ms given in c hapter 20. Ev entually this 2- categoric al structure induces a symmetric mono idal c lo sed structure on S P C / ∼ where ∼ denotes the co ng ruence genera ted by the 2-c e lls of S P C . 3 S P C -categories, S P C -functors and S P C -natural transfor- mations S P C -ca tegories , S P C -functor s and S P C -natural transforma tions hav e b een cons ide r ed by M.Dupo nt in his thesis [Dup08 ]. They are resp ectively bicatego ries with homs in S P C , a nd pseudo -functors and ps eudo-natural tra nsformations with linea r comp one nts, i.e with arrows and 2- c e lls in the 2-catego ry S P C rather than in Cat . As such they obviously for m a 2-catego ry with a forg e tful 2-functor to the 2-catego r y of bicateg o ries, pseudo-functors and pseudo natura l transformations . W e start by recalling these no tio ns which a re actua lly slight (enriched) v ariations of the usual notions of bicatego ries, pseudo-functor s and pseudo- natural transfo r mations. By a n-linear natura l transfor mation σ b etw een n-linear ma ps F and G , written σ : F → G : A 1 × ... × A n → B we will mean a 2- cell of [ A 1 , [ A 2 , ..., [ A n , B ] ... ]. Multi-linear maps and m ultilinear natural tra nsformation comp os e in an evident wa y , which can b e justiﬁed by the 2 - natural isomor - phism D A , B , C : [ A , [ B , C ]] ∼ = [ B , [ A , C ]] and the existence o f a forg etful 2-functor S P C → Cat (see [Sch08] c ha pter 6). An enrichment ( A , c , j , α, ρ, λ ), over S P C also named an S P C -c ate gory and which we migh t sometimes denote simply by A , consis ts of the following data: - A small set with elements x , y , z ... ca lled the obje cts of A . - A map A sending any pair x , y o f o b jects to an ob ject A ( x, y ) of S P C sometimes a lso written A x,y for conv enience and ca lled the hom of x and y . - A collection of bilinea r maps c x,y ,z : A y ,z × A x,y → A x,z , the c omp osition maps index ed by ob jects x , y , z of A and we wr ite g ◦ f for c x,y ,z ( g , f ) for any o b jects g of A y ,z and f of A x,y and τ ∗ σ for c x,y ,z ( τ , σ ) for any arrows τ of A y ,z and σ of A x,y . - A collectio n of ob jects 1 x of A x,x indexed by ob jects x of A ; - Co llections of natural tra ns formations α x,y ,z ,t , whic h ar e triline ar , indexe d by ob jects x , y , z , t , and ρ x,y and λ x,y bo th line ar , as follows 3.1 ( α x,y ,z ,t ) h,g ,f : h ◦ ( g ◦ f ) → h ◦ ( g ◦ f ) which lies in A x,t for ob jects h in A z ,t , g in A y ,z and f in A x,y 8 3.2 ( ρ x,y ) f : f → f ◦ 1 x which lies in A x,y for ob jects f in A x,y ; 3.3 ( λ x,y ) f : f → 1 y ◦ f which lies in A x,y for o b jects f in A x,y ; and those are sub jects to the cohere nc e axioms 3.4 and 3.5 b elow. 3.4 F or any obje cts x , y , t and u of A , any obje cts f of A x,y , g of A y ,z , h of A z ,t and k of A t,u , the diagr am in A x,u k ◦ ( h ◦ ( g ◦ f ))) ( α x,z ,t,u ) k,h,g ◦ f / / k ∗ ( α x,y,z ,t ) h,g,f   ( k ◦ h ) ◦ ( g ◦ f ) ( α x,y,z ,u ) k ◦ h,g,f   k ◦ (( h ◦ g ) ◦ f ) ( α x,y,t,u ) k,h ◦ g,f / / ( k ◦ ( h ◦ g )) ◦ f ( α y,z ,t,u ) k,h,g ∗ f / / (( k ◦ h ) ◦ g ) ◦ f c ommu tes. 3.5 F or any obje cts x , y , z in A and any obje cts f of A ( x, y ) and g of A ( y , z ) the diagr am in A ( x, y ) g ◦ f g ∗ ( λ x,y ) f y y s s s s s s s s s s ( ρ y,z ) g ∗ f % % K K K K K K K K K K g ◦ (1 y ◦ f ) ( α x,y,y,z ) g, 1 y ,f / / ( g ◦ 1 y ) ◦ f c ommu tes. F or any S P C -ca tegory A , its underlying bicategory is denoted A 0 . Given tw o arbitra ry S P C -catego ries A and B , a S P C -fun ctor F : A → B consists of the fol- lowing data. - A map F sending ob jects of A to ob jects of B ; - Ar rows F x,y : A x,y → B F x,F y in S P C for each pair of ob jects x , y in A . - A co llection of biline ar natural transforma tions F 2 x,y ,z indexed b y ob jects x , y , z of A with com- po nents ( F 2 x,y ,z ) g,f : F y ,z ( g ) ◦ F x,y ( f ) → F x,z ( g ◦ f ) in B F x,F z for ob jects g in A y ,z and f in A x,y . - A collectio n of ar rows F 0 x : 1 F x → F x,x (1 x ) in B F x,F x indexed by ob jects x of A . Those ar e sub jects to the coherence axio ms 3.6, 3.7 and 3.8 b elow. 3.6 F or any obje cts x , y , z , t of A , and any obje cts f of A x,y , g of A y ,z and h of A z ,t the diagr am in B ( F x, F t ) F z ,t h ◦ ( F y ,z g ◦ F x,y f ) ( α F x ,F y,F z ,F t ) F h,F g,F f / / 1 ∗ ( F 2 x,y,z ) g,f   ( F z ,t h ◦ F y ,z g ) ◦ F x,y f ( F 2 y,z ,t ) h,g ∗ 1   F z ,t h ◦ F x,z ( g ◦ f ) ( F 2 x,z ,t ) h,g ◦ f   F y ,t ( h ◦ g ) ◦ F x,y f ( F 2 x,y,t ) h ◦ g,f   F x,t ( h ◦ ( g ◦ f )) F x,t (( α x,y,z ,t ) h,f,g ) / / F x,t (( h ◦ g ) ◦ f ) 9 c ommu tes. 3.7 F or any obje cts x , y of A and any obje ct f of A x,y , the diagr am in B F x,F y F f F ( ρ x,y f )   ( ρ F x,F y ) F f / / F f ◦ 1 F x 1 ∗ F 0 x   F ( f ◦ 1 x ) F f ◦ F (1 x ) ( F 2 x,x,y ) f, 1 x o o c ommu tes. 3.8 F or any obje cts x , y of A and any obje ct f of A x,y , the diagr am in B F x,F y F f F ( λ x,y f )   ( λ F x ,F y ) F f / / 1 F y ◦ F f F 0 y ∗ 1   F (1 y ◦ f ) F (1 y ) ◦ F f ( F 2 x,y,y ) 1 y ,f o o c ommu tes. Given tw o S P C -functors F , G : A → B a S P C -natura l tr ansformation ( σ , κ ) co nsists in a family of arr ows σ x of B F x,Gx indexed b y o b jects x of A together with a collection of linear natural transformatio ns κ x,y indexed by ob jects x and y as follows ( κ x,y ) f : Gf ◦ σ x → σ y ◦ F f lies in B F x,Gy for ob jects f of A x,y , a nd these satisfy the coherence axioms 3.9 and 3 .10 b elow. 3.9 F or any obje cts f in A x,y and g in A y ,z , the diagr am in B F x,Gz ( Gg ◦ Gf ) ◦ σ x ( G 2 x,y,z ) g,f ∗ σ x / / G ( g ◦ f ) ◦ σ x ( κ x,z ) g ◦ f / / σ z ◦ F ( g ◦ f ) Gg ◦ ( Gf ◦ σ x ) ( α F x,Gx,Gy,Gz ) Gg,Gf ,σ x O O 1 ∗ ( κ x,y ) f   σ z ◦ ( F g ◦ F f ) ( α F x ,F x,F z ,Gz ) σ z ,F g ,F f   1 ∗ ( F 2 x,y,z ) g,f O O Gg ◦ ( σ y ◦ F f ) ( α F x,F y ,Gy,Gz ) Gg,σ y ,F f / / ( Gg ◦ σ y ) ◦ F f ( κ y,z ) g ∗ 1 / / ( σ z ◦ F g ) ◦ F f c ommu tes. 3.10 F or any obje ct x of A , the diagr am in B F x,Gx σ x ( ρ F x,Gx ) σ x y y s s s s s s s s s s ( λ F x ,Gx ) σ x % % K K K K K K K K K K σ x ◦ 1 F x σ x ∗ F 0 x   1 Gx ◦ σ x G 0 x ∗ σ x   σ x ◦ F (1 x ) G (1 x ) ◦ σ x ( κ x,x ) 1 x o o c ommu tes. 10 W e want to give a lternative deﬁnitions o f S P C -categor ies, S P C -functors and S P C -natural transformatio ns by means of commuting diagr ams in S P C in a ﬁrst instance without using the tensor. They ar e o btained by r eplacing multilinear maps and multilinear natural tra ns formations from the previous deﬁnition by cor resp onding ar rows and 2-cells in S P C . This yields the following. One can deﬁne a S P C -catego ry as a collectio n of ob jects A , with homs A x,y in S P C as befor e, with colle ctions of arr ows - A ( x, − ) y ,z : A y ,z → [ A x,y , A x,z ] in S P C , indexed b y ob jects x , y , z o f A with dual A − ,y : A x,y → [ A y ,z , A x,z ] wr itten A ( − , y ); - u x : I → A x,x indexed by x , w hich are strict; and c o llections of 2-cells: - α ′ x,y ,z ,t in S P C , indexed by ob jects x , y , z a nd t o f S P C as follows 3.11 A z ,t A ( y , − ) ' ' N N N N N N N N N N N N A ( x, − ) v v m m m m m m m m m m m m m m [ A x,z , A x,t ] [ A x,y , − ]   [ A y ,z , A y ,t ] [1 , A ( x, − )]   [[ A x,y , A x,z ] , [ A x,y , A x,t ]] [ A ( x, − ) , 1] / / α ′ / 7 g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g [ A y ,z , [ A x,y , A x,t ]] - ρ ′ x,y and λ ′ x,y indexed by ob jects x and y resp ectively as fo llows 3.12 ρ ′ x,y : A x,y A ( x, − ) / / id [ A x,x , A x,y ] [ u x , 1]   + 3 A x,y [ I , A x,y ] ev ⋆ o o and 3.13 λ ′ x,y : A x,y A ( − ,y ) / / id [ A y ,y , A x,y ] [ u y , 1]   + 3 A x,y [ I , A x,y ] ev ⋆ o o those satis fying cohere nce axioms 3.1 4 and 3.15 b elow 11 3.14 The 2-c el ls in S P C A t,u A ( x, − ) / / id α ′ x,z ,t,u   [ A x,t , A x,u ] [ A x,z , − ] / / [[ A x,z , A x,t ] , [ A x,z , A x,u ]] [ A ( x, − ) , 1] / / [ A z ,t , [ A x,z , A x,u ]] id [1 , [ A x,y , − ]] / / = [ A z ,t , [[ A x,y , A x,z ] , [ A x,y , A x,u ]]] [1 , [ A ( x, − ) , 1]] / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] id A t,u A ( z, − ) / / id = [ A z ,t , A z ,u ] [1 , A ( x, − )] / / id [1 ,α ′ x,y,z ,u ]   [ A z ,t , [ A x,z , A x,u ]] [1 , [ A x,y , − ]] / / [ A z ,t , [[ A x,y , A x,z ] , [ A x,y , A x,u ]]] [1 , [ A ( x, − ) , 1]] / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] id A t,u A ( z, − ) / / [ A z ,t , A z ,u ] [1 , A ( y, − )] / / [ A z ,t , [ A y,z , A y,u ]] [1 , [1 , A ( x, − )]] / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] and A t,u = A ( x, − ) / / id [ A x,t , A x,u ] [ A x,y , − ] / / [[ A x,y , A x,t ] , [ A x,y , A x,u ]] [ A y,z , − ] / / [[ A y,z , [ A x,y , A x,t ]] , [ A y,z , [ A x,y , A x,u ]]] [ α ′ x,y,z ,t , 1]   [[ A ( x, − ) , 1] , 1] / / id [[ A x,y , A x,z ] , [ A x,y , A x,t ]] , [ A y,z , [ A x,y , A x,u ]]] [[ A x,y , − ] , 1] / / [[ A x,z , A x,t ] , [ A y,z , [ A x,y , A x,u ]]] [ A ( x, − ) , 1]   A t,u id = A ( x, − ) / / [ A x,t , A x,u ] [ A x,y , − ] / / [[ A x,y , A x,t ] , [ A x,y , A x,u ]] [ A y,z , − ] / / id = [[ A y,z , [ A x,y , A x,t ]] , [ A y,z , [ A x,y , A x,u ]]] [[1 , A ( x, − )] , 1] / / [[ A y,z , A y,t ] , [ A y,z , [ A x,y , A x,u ]]] [ A ( y, − ) , 1] / / id = [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] id A t,u A ( x, − ) / / id α ′ x,y,t,u   [ A x,t , A x,u ] [ A x,y , − ] / / [[ A x,y , A x,t ] , [ A x,y , A x,u ]] [ A ( x, − ) , 1] / / [ A y,t , [ A x,y , A x,u ]] [ A y,z , − ] / / id = [[ A y,z , A y,t ] , [ A y,z , [ A x,y , A x,u ]]] [ A ( y, − ) , 1] / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] id A t,u A ( y, − ) / / id = [ A y,t , A y,u ] id [1 , A ( x, − )] / / = [ A y,t , [ A x,y , A x,u ]] [ A y,z , − ] / / [[ A y,z , A y,t ] , [ A y,z , [ A x,y , A x,u ]]] id [ A ( y, − ) , 1] / / = [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] id A t,u A ( y, − ) / / id = [ A y,t , A y,u ] [ A y,z , − ] / / [[ A y,z , A y,t ] , [ A y,z , A y,u ]] id [1 , [1 , A ( x, − )]] / / = [[ A y,z , A y,t ] , [ A y,z , [ A x,y , A x,u ]]] [ A ( y, − ) , 1] / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] id A t,u id A ( y, − ) / / α ′ y,z ,t,u   [ A y,t , A y,u ] [ A y,z , − ] / / [[ A y,z , A y,t ] , [ A y,z , A y,u ]] [ A ( y, − ) , 1] / / [ A z ,t , [ A y,z , A y,u ]] id [1 , [1 , A ( x, − )]] / / = [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] id A t,u A ( z, − ) / / [ A z ,t , A z ,u ] [1 , A ( y, − )] / / [ A z ,t , [ A y,z , A y,u ]] [1 , [1 , A ( x, − )]] / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] ar e e qu al. 3.15 The 2-c el l A y ,z A ( y , − )   id ρ ′   A y ,z A ( x, − ) / / [ A x,y , A x,z ] [ A y ,y , A y ,z ] [ u y , 1] / / [ I , A y ,z ] ev ⋆ O O 12 is e qual t o A y ,z A ( x, − )   id = A y ,z id A ( x, − )   = A y ,z A ( x, − )   id A y ,z A ( x, − )   = id A y ,z A ( x, − )   [ A x,y , A x,z ] id id [ A x,y , A x,z ] [ ev ⋆ , 1]   id [ A x,y , A x,z ] [ A x,y , − ]   α ′ x,y,y,z + 3 [ A y ,y , A y ,z ] [1 , A ( x, − )]   id [ A y ,y , A y ,z ] [ u y , 1]   [[ I , A x,y ] , A x,z ] [[ u y , 1] , 1]   [[ A x,y , A x,y ] , [ A x,y , A x,z ]] [ A ( x, − ) , 1]   [ I , A y ,z ] ev ⋆   [ λ ′ x,y , 1] + 3 [[ A y ,y , A x,y ] , A x,z ] [ A ( − ,y ) , 1]   = ( I ) [ A y ,y , [ A x,y , A x,z ]] [ u y , 1]   = id [ A y ,y , [ A x,y , A x,z ]] [ u y , 1]   = ( I I ) A y ,z A ( x, − )   [ I , [ A x,y , A x,z ]] ev ⋆   [ I , [ A x,y , A x,z ]] ev ⋆   [ A x,y , A x,z ] id [ A x,y , A x,z ] id [ A x,y , A x,z ] id [ A x,y , A x,z ] id [ A x,y , A x,z ] . Note: Equality ( I ) in the ﬁrst of the pa stings ab ov e is established in 7 in Appe ndix . The equality ( I I ) results straightforwardly from Lemma [Sch08]-11.2. Let us justify the equiv a lence with the previo us deﬁnition of S P C -ca tegory . F o r ob jects x , y and z the arr ows of S P C A ( x, − ) : A ( y , z ) → [ A ( x, y ) , A ( x, z )] corres p o nd to the bilinear c x,y ,z : A ( y , z ) × A ( x, y ) → A ( x, z ) and for ob jects x , the o b jects 1 x in A x,x corres p o nd to strict the strict ar rows u x : I → A x,x . The trilinear 2-cells α x,y ,z ,t corres p o nd to the 2-cells α ′ x,y ,z ,t and the linear natural tr ansformations ρ x,y and λ x,y corres p o nd resp ectively to 2-cells ρ ′ x,y and λ ′ x,y in S P C . F or data as a b ove, since the for getful functor s S P C ( X , Y ) → Cat ( X , Y ) are faithful, the eq ualities of 2-cells of Axio m 3.14 given below are equiv ale nt to the equality o f natural tr ansformations of Axioms 3.4 , and similarly Axiom 3.15 given below and 3.5 are equiv alent. Given tw o S P C -ca tegories , a S P C -functor A → B consists o f a map F sending ob jects of A to ob jects o f B , with a collection of arr ows in S P C F x,y : A x,y → B F x,F y indexed b y o b jects x a nd y of A and co llections o f 2-cells in S P C : - F ′ 2 x,y ,z indexed by ob jects x , y , z and as follows 3.16 A y ,z F y,z / / A ( x, − )   B F y, F z B ( F x, − ) / / [ B F x,F y , B F x,F z ] [ F x,y , 1]   F ′ 2 x,y,z o w h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h [ A x,y , A x,z ] [1 ,F x,z ] / / [ A x,y , B F x,F z ] . - F 0 x indexed by ob jects x as follows 13 3.17 I u x / / u F x ! ! C C C C C C C C C C C C C C C C C A x,x F x,x   F 0 x 7 ? x x x x x x x x x x x x x x x x B F x,F x Those sa tisfy the coherence conditions 3.1 8, 3.19 and 3.2 0 b e low. 3.18 The 2-c el ls in S P C [ B F x,F z , B F x,F t ] [ B F x,F y , − ] / / α ′ F x,F y ,F z ,F t   [[ B F x,F y , B F x ,F z ] , [ B F x,F y , B F x,F t ]] [ B ( F x, − ) , 1]   A z ,t A ( y, − )   F z ,t / / B F z ,F t B ( F x, − ) 9 9 s s s s s s s s s s s s s B ( F y, − ) / / [ B F y,F z , B F y,F t ] F ′ 2 y,z ,t p x j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j [1 , B ( F x, − )] / / [ F y,z , 1]   = [ B F y,F z , [ B F x,F y , B F x,F t ]] [ F y,z , 1]   [ A y,z , A y,t ] [1 ,F y,t ] / / [1 , B ( F x, − )] * * U U U U U U U U U U U U U U U U U U U [ A y,z , B F y,F t ] [1 , B ( F x, − )] / / [1 ,F ′ 2 x,y,t ]   [ A y,z , [ B F x,F y , B F x,F t ]] [1 , [ F x,y , 1]] / / [ A y,z , [ A x,y , B F x,F t ]] [ A y,z , [ A x,y , A x,t ]] [1 , [1 ,F x,t ]] 3 3 g g g g g g g g g g g g g g g g g g g g g g g g g g g and B F z ,F t B ( F x, − ) / / [ B F x,F z , B F x,F t ] [ F x,z , 1]   [ A x,y , − ] / / F ′ 2 x,z ,t ~                                        = [[ A x,y , B F x,F z ] , [ A x,y , B F x,F t ]] [[ F x,y , 1] , 1] / / [[1 ,F x,z ] , 1]   [[ B F x,F y , B F x ,F z ] , [ A x,y , B F x ,F t ]] [ B ( F x, − ) , 1] / / [ F ′ 2 x,y,z , 1]   [ B F y,F z , [ A x,y , B F x,F t ]] [ F y,z , 1]   [ A y,z , B F x,F t ] [ A x,y , − ] / / = [[ A x,y , A x,z ] , [ A x,y , B F x,F t ]] [ A ( x, − ) , 1] / / [ A y,z , [ A x,y , B F x,F t ]] A z ,t A ( x, − ) / / F z ,t O O A ( y, − ) + + W W W W W W W W W W W W W W W W W W W W W W W W W W [ A x,z , A x,t ] [ A x,y , − ] / / [1 ,F x,t ] O O [[ A x,y , A x,z ] , [ A x,y , A x,t ]] [ A ( x, − ) , 1] / / [1 , [1 ,F x,t ]] O O = α ′ x,y,z ,t   [ A y,z , [ A x,y , A x,t ]] [1 , [1 ,F x,t ]] O O [ A y,z , A y,t ] [1 , A ( x, − )] 3 3 f f f f f f f f f f f f f f f f f f f f f f f f f f f f ar e e qu al. 14 3.19 The 2-c el ls A x,y A ( x, − )   F x,y / / B F x ,F y B ( F x, − )   F ′ 2 x,x,y o w f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f 1 x x [ A x,x , A x,y ] [ u x , 1]   [1 ,F x,y ] ' ' O O O O O O O O O O O O [ B F x,F x , B F x,F y ] [ u F x , 1]   [ F x,x , 1] v v m m m m m m m m m m m m m [ A x,x , B F x,F y ] [ u x , 1] ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q [ F 0 x , 1] k s ρ ′ F x,F y k s [ I , A x,y ] = = ev ⋆   [1 ,F x,y ] / / [ I , B F x,F y ] ev ⋆   A x,y F x,y / / B F x ,F y and A x,y id A ( x, − )   ρ ′ x,y   A x,y F x,y / / B F x ,F y [ A x,x , A x,y ] [ u x , 1] / / [ I , A x,y ] ev ⋆ O O ar e e qu al. 3.20 The 2-c el ls in S P C A x,y A ( − ,y )   F x,y / / B F x,F y B ( − ,F y )   ( F ′ 2 x,y,y ) ∗ o w f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f 1 x x [ A y,y , A x,y ] [ u y , 1]   [1 ,F x,y ] ' ' O O O O O O O O O O O O [ B F y,F y , B F x,F y ] [ u F y , 1]   [ F x,y , 1] v v m m m m m m m m m m m m m [ A y,y , B F x ,F y ] [ u y , 1] ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q [ F 0 y , 1] k s λ ′ F x ,F y k s [ I , A x,y ] = = ev ⋆   [1 ,F x,y ] / / [ I , B F x ,F y ] ev ⋆   A x,y F x,y / / B F x,F y and A x,y id A ( − ,y )   λ ′ x,y   A x,y F x,y / / B F x,F y [ A y,y , A x,y ] [ u y , 1] / / [ I , A x,y ] ev ⋆ O O ar e e qu al. One ca n also deﬁne the S P C -natural transfo rmations, in a similar wa y . F or this purp ose, we need so me notation. Let A b e a n arbitra ry S P C -categor y . T o give a strict arr ow I → A ( x, y ) is equiv alent to g ive an ar row x → y of the under lying bicatego ry A 0 and we might co nfuse the tw o. W e therefore deﬁne for any ob ject z of A and any arr ow f : x → y o f A 0 the ar rows 15 3.21 A ( f , 1 ) as the c omp osite arr ow in S P C A x,y A ( x, − ) / / [ A x,y , A x,z ] [ F , 1] / / [ I , A x,z ] ev ⋆ / / A x,z and 3.22 A (1 , f ) as A z ,x A ( − ,y ) / / [ A x,y , A z ,y ] [ F , 1] / / [ I , A z ,y ] ev ⋆ / / A z ,y . Given a ny a rrows x f / / y g / / z h / / t in A 0 , we deﬁne the tw o cells 3.23 c 1 f ,y ,z A z ,t A ( y , − ) / / A ( x, − )   [ A y ,z , A y ,t ] [1 , A ( f , 1)]   [ A x,z , A x,t ] [ A ( f , 1) , 1] / / 2 : m m m m m m m m m m m m m m m m m m m m m m m m [ A y ,z , A x,t ] 3.24 c 2 x,g ,t A z ,t A ( g, 1)   A ( x, − ) / / [ A x,z , A x,t ] [ A (1 ,g ) , 1]   u } s s s s s s s s s s s s s s s s s s A y ,t A ( x, − ) / / [ A x,y , A x,t ] and e ventually 3.25 c 3 x,y ,h A y ,z A (1 ,h ) / / A ( x, − )   A y ,t A ( x, − )   [ A x,y , A x,z ] [1 , A (1 ,h )] / / 2 : m m m m m m m m m m m m m m m m m m m m m m m m m m [ A x,y , A x,t ] which are obtained from the trilinear natural tra nsformation 3.1 ( α x,y ,z ,t ) h,g ,f : h ◦ ( g ◦ f ) → ( h ◦ g ) ◦ f by ﬁxing one of its argument. F or c 1 , c 2 and c 3 ﬁx r esp ectively f , g and h . F or mally c 1 f ,z ,t is the comp os ite A z ,t α ′ x,y,z ,t + 3 [ A y ,z , [ A x,y , A x,t ]] [1 , [ f , 1]] / / [ A y ,z , [ I , A x,t ]] [1 ,ev ⋆ ] / / [ A y ,z , A x,t ] (see 7 .23 in Appendix), c 2 x,g ,t it is the comp osite A z ,t α ′ x,y,z ,t / / [ A y ,z , [ A x,y , A x,t ]] [ g, 1] / / [ I , [ A x,y , A x,t ]] ev ⋆ / / [ A x,y , A x,t ] and c 3 x,y ,h is the image by ev ⋆ of the comp osite I h / / A z ,t α ′ x,y,z ,t + 3 [ A y ,z , [ A x,y , A x,t ]] 16 (see 7 .26 in Appendix). By deﬁnition a ll 2 -cells c 1 , c 2 and c 3 are identities when A is strict. One has also for any ob jects x a nd y of A the 2 -cell in S P C 3.26 r ′ x,y : I u x / / v " " E E E E E E E E E E E E E E E E E E E A x,x A ( − ,y )   5 = t t t t t t t t t t t t t t t t t t [ A x,y , A x,y ] , which a c cording to Remarks 7.10 is determined b y its v a lue in ⋆ which is the linear natura l tra ns- formation ρ x,y f : f ◦ 1 x → f of 3 .2, and corresp o nds b y the bijection 7.1 6 / 7.18 in App endix to the 2 -cell ρ ′ x,y A x,y A ( x, − ) / / id [ A x,x , A x,y ] [ u x , 1]   + 3 A x,y [ I , A x,y ] . ev ⋆ o o Similarly for any ob jects x and y of A o ne ha s the 2-cell in S P C 3.27 l ′ x,y : I u y / / v " " E E E E E E E E E E E E E E E E E E E A y ,y A ( x, − )   5 = t t t t t t t t t t t t t t t t t t [ A x,y , A x,y ] that co rresp onds to the linea r natura l transfo r mation ( λ x,y ) f : 1 y ◦ f → f of 3 .3 and which co rresp onds by the bijection 7.1 6 / 7.18 in Appe ndix to the 2-cell λ ′ x,y A x,y A ( − ,y ) / / id [ A y ,y , A x,y ] [ u y , 1]   + 3 A x,y [ I , A x,y ] . ev ⋆ o o Given a ny strict arrow ˜ f : I → A x,y , with corr e sp onding arrow f : x → y in A 0 , one has the 2-cell in S P C 17 3.28 u 1 f : I u y / / ˜ f " " E E E E E E E E E E E E E E E E E E E A y ,y A ( f, 1)   5 = t t t t t t t t t t t t t t t t t t [ A x,y , A x,y ] which corresp onds to the 2-c el l ρ f : f → 1 y ◦ f : x → y of A 0 . It is the pasting = I u y / / v C C ˜ f B B A y ,y A ( x, − ) / / A ( f, 1) # # [ A x,y , A x,y ] [ ˜ f , 1] / / [ I , A x,y ] ev ⋆ / / A x,y l ′ K S = where the b ottom identit y 2-cell ab ov e is established in Lemma 7.11 in App endix. Similarly o ne has the 2-cell in S P C 3.29 u 2 f : I u x / / ˜ f E E E E E E E E E E E E E E E E E E E A x,x A (1 ,f )   5 = t t t t t t t t t t t t t t t t t t [ A x,y , A x,y ] that co rresp onds the 2-cell ( λ x,y ) f : f → f ◦ 1 x and is the pasting = I u x / / v C C ˜ f B B A x,x A ( − ,y ) / / A (1 ,f ) # # [ A x,y , A x,y ] [ ˜ f , 1] / / [ I , A x,y ] ev ⋆ / / A x,y r ′ K S = Given tw o S P C -functors F , G : A → B , a S P C -natura l tr ansformation ( σ, κ ) : F → G : A → B consists o f a c o llection o f strict ar rows σ x : I → B ( F x, Gx ) (or 1-c ells σ x : F x → Gx in B 0 ), 18 indexed by o b jects x o f A together with a collection of 2-ce lls κ x,y in S P C for ob jects x , y of A as follows 3.30 A x,y F x,y / / G x,y   B F x,F y B (1 ,σ y )   B Gx,Gy 5 = t t t t t t t t t t t t t t t t t t B ( σ x , 1) / / B F x,Gy and tha t sa tisﬁes the tw o coherence conditions 3 .31, 3.32 a nd b elow. 3.31 F or any obje ct x , y and z in A , the 2-c el ls Ξ 1 , Ξ 2 , Ξ 3 , Ξ 4 , Ξ 5 , Ξ 6 , Ξ 7 and Ξ 8 b elow satisfy the e quality Ξ 2 ◦ Ξ 1 = Ξ 8 ◦ (Ξ 7 ) − 1 ◦ Ξ 6 ◦ Ξ 5 ◦ Ξ 4 ◦ (Ξ 3 ) − 1 . Ξ 1 is A y ,z A ( x, − )   G y,z / / B Gy ,Gz G ′ 2 x,y,z   B ( Gx, − ) / / [ B Gx,Gy , B Gx,Gz ] [ G x,y , 1]   [ A x,y , A x,z ] [1 ,G x,z ] / / [ A x,y , B Gx,Gz ] [1 , B ( σ x , 1)] / / [ A x,y , B F x,Gz ] Ξ 2 is [ A x,y , B Gx,Gz ] [1 , B ( σ x , 1)] ( ( Q Q Q Q Q Q Q Q Q Q Q Q [1 ,κ x,z ]   A y ,z A ( x, − ) / / [ A x,y , A x,z ] [1 ,G x,z ] 7 7 n n n n n n n n n n n n [1 ,F x,z ] ' ' P P P P P P P P P P P P [ A x,y , B F x,Gz ] [ A x,y , B F x,F z ] [1 , B (1 ,σ z )] 6 6 m m m m m m m m m m m m Ξ 3 is [ B Gx,Gy , B Gx,Gz ] [1 , B ( σ x , 1)] ) ) R R R R R R R R R R R R R R A y ,z G y,z / / B Gy ,Gz B ( Gx, − ) 7 7 o o o o o o o o o o o B ( F x, − ) ' ' O O O O O O O O O O O [ B Gx,Gy , B F x,Gz ] [ G x,y , 1] / / [ A x,y , B F x,Gz ] [ B F x,Gx , B F x,Gz ] [ B ( σ x , 1) , 1] 5 5 l l l l l l l l l l l l l l c 1 B ( σ x , 1) ,y,z K S Ξ 4 is [ B Gx,Gy , B F x,Gz ] [ κ x,y , 1]   [ G x,y , 1] ( ( R R R R R R R R R R R R R A y ,z G y,z / / B Gy ,Gz B ( F x, − ) / / [ B F x,Gy , B F x,Gz ] [ B ( σ x , 1) , 1] 5 5 l l l l l l l l l l l l l [ B (1 ,σ y ) , 1] ) ) R R R R R R R R R R R R R [ A x,y , B F x,Gz ] [ B F x,F y , B F x,Gz ] [ F x,y , 1] 6 6 m m m m m m m m m m m m m 19 Ξ 5 is [ B F x,Gy , B F x,Gz ] [ B (1 ,σ y ) , 1] ) ) R R R R R R R R R R R R R c 2 F x,σ y ,Gz   A y ,z G y,z / / B Gy ,Gz B ( σ y , 1) ' ' O O O O O O O O O O O B ( F x, − ) 7 7 o o o o o o o o o o o [ B F x,F y , B F x,Gz ] [ F x,y , 1] / / [ A x,y , B F x,Gz ] B F y ,Gz B ( F x, − ) 5 5 l l l l l l l l l l l l l l Ξ 6 is B Gy ,Gz B ( σ y , 1) $ $ J J J J J J J J J κ y,z   A y ,z G y,z ; ; v v v v v v v v v F y,z # # H H H H H H H H H B F y ,Gz B ( F x, − ) / / [ B F x,F y , B F x,Gz ] [ F x,y , 1] / / [ A x,y , B F x,Gz ] B F y ,F z B (1 ,σ z ) : : t t t t t t t t t Ξ 7 is B F y ,Gz B ( F x, − ) ) ) R R R R R R R R R R R R R R A y ,z F y,z / / B F y ,F z B (1 ,σ z ) 7 7 o o o o o o o o o o o B ( F x, − ) ' ' O O O O O O O O O O O [ B F x,F y , B F x,Gz ] [ F x,y , 1] / / [ A x,y , B F x,Gz ] [ B F x,F y , B F x,F z ] [1 , B (1 ,σ z )] 5 5 l l l l l l l l l l l l l c 3 F x,F y ,σ z K S Ξ 8 is A y ,z F y,z / / A ( x, − )   B F y ,F z B ( F x, − ) / / F ′ 2 x,y,z   [ B F x,F y , B F x,F z ] [ F x,y , 1]   [ A x,y , A x,z ] [1 ,F ] / / [ A x,y , B F x,F z ] [1 , B (1 ,σ z )] / / [ A x,y , B F x,Gz ] 3.32 F or any obje ct x of A , the 2-c el ls u 2 σ x   F 0   I u   u / / σ x   A x,x F x,x / / B F x,F x B (1 ,σ x ) / / B F x,Gx 20 and u 1 σ x   I id u Gx / / σ x " " G 0   B Gx,Gx B ( σ x , 1) / / κ $ , Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q B F x,Gx I u x / / A x,x G x,x O O F x,x / / B F x,F x B (1 ,σ x ) O O ar e e qu al. 4 S P C -categories via the tensor In this section we g ive deﬁnitions of S P C -catego ries and S P C -functors that r ely on the tensor pro duct in S P C . A S P C -c ate gory ( A , u , c, α, ρ, λ ) co nsists of the following data: - As befor e: a small set of ob jects with a map sending any pair x , y of ob jects to an ob ject A x,y of S P C ; - Co llections o f strict mor phisms u x : I → A x,x and c x,y ,z : A y ,z ⊗ A x,y → A x,z indexed by ob jects of A with co lle ctions of 2-ce lls α x,y ,z ,t , ρ x,y and λ x,y in S P C indexed by ob jects of A and as follows 4.1 ( A z ,t ⊗ A y ,z ) ⊗ A x,y A ′ / / c y,z ,t ⊗ 1   A z ,t ⊗ ( A y ,z ⊗ A x,y ) 1 ⊗ c x,y,z   α x,y,z ,t o w g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g A y ,t ⊗ A x,y c x,y,t ' ' P P P P P P P P P P P P P A z ,t ⊗ A x,z c x,z ,t w w n n n n n n n n n n n n n A x,t 4.2 A x,y R ′ / / id A x,y ⊗ I 1 ⊗ u x   ρ x,y + 3 A x,y A x,y ⊗ A x,x c x,x,y o o and 4.3 A x,y L ′ / / id I ⊗ A x,y u y ⊗ 1   λ x,y + 3 A x,y A y ,y ⊗ A x,y c x,y,y o o 21 Those sa tisfy the coherence Axioms 4.4 and 4.5 b elow. 4.4 F or any obje cts x , y , z , t and u of A , the 2-c el ls (( A t,u A z ,t ) A y ,z ) A x,y A ′ / / ( c z ,t,u ⊗ 1) ⊗ 1   = ( A t,u A z ,t )( A y ,z A x,y ) id c z ,t,u ⊗ 1   = ( A t,u A z ,t )( A y ,z A x,y ) A ′ / / 1 ⊗ c x,y,z   = A t,u ( A z ,t ( A y ,z A x,y )) 1 ⊗ (1 ⊗ c x,y,z )   ( A z ,u A y ,z ) A x,y A ′ / / c y,z ,t ⊗ 1   A z ,u ( A y ,z A x,y ) 1 ⊗ c x,y,z   α x,y,z ,u v v v v v v v v v v v v v v v v v v v v w  v v v v v v v v v v v v v v v v v v v v ( A t,u A z ,t ) A x,z A ′ / / c z ,t,u ⊗ 1   A t,u ( A z ,t A x,y ) 1 ⊗ c x,y,t   α x,z ,t,u v v v v v v v v v v v v v v v v v v v v w  v v v v v v v v v v v v v v v v v v v v A y ,u A x,y c x,y,u   A z ,u A x,z c x,z ,u   id = A z ,u A x,z c x,z ,u   A t,u A x,t c x,t,u   A x,u id A x,u id A x,u id A x,u and (( A t,u A z ,t ) A y ,z ) A x,y A ′ ⊗ 1 / / ( c z ,t,u ⊗ 1) ⊗ 1   (( A t,u ( A z ,t A y ,z )) A x,y A ′ / / (1 ⊗ c y,z ,t ) ⊗ 1   = α y,z ,t,u ⊗ 1 v v v v v v v v v v v v v v v v v v v v w  v v v v v v v v v v v v v v v v v v v v A t,u (( A z ,t A y ,z ) A x,y ) 1 ⊗ A ′ / / 1 ⊗ ( c y,z ,t ⊗ 1)   A t,u ( A z ,t ( A y ,z A x,y )) 1 ⊗ (1 ⊗ c x,y,z )   1 ⊗ α x,y,z ,t w w w w w w w w w w w w w w w w w w w w w  w w w w w w w w w w w w w w w w w w w w ( A z ,u A y ,z ) A x,y c y,z ,u ⊗ 1   ( A t,u A y ,t ) A x,y c y,t,u ⊗ 1   A ′ / / A t,u ( A y ,t A x,y ) 1 ⊗ c x,y,t   α x,y,t,u v v v v v v v v v v v v v v v v v v v v v v v ~ v v v v v v v v v v v v v v v v v v v v A t,u ( A z ,t A x,z ) 1 ⊗ c x,z ,t   A y ,u A x,y c x,y,u   id = A y ,u A x,y c x,y,u   A t,u A x,t id c x,t,u   = A t,u A x,t c x,t,u   A x,u id A x,u id A x,u id A x,u ar e e qual. Note that the domains of the ab ove 2-c el ls ar e e qual sinc e (1 ⊗ A ′ ) ◦ A ′ ◦ ( A ′ ⊗ 1) = A ′ ◦ A ′ by L emma [Sch08 ]-19.10. 4.5 F or any obje cts x , y , z of A , the 2-c el ls Ξ 1 = A y ,z ⊗ A x,y R ′ ⊗ 1   id ρ y,z ⊗ 1   A y ,z ⊗ A x,y c x,y,z / / A x,z ( A y ,z ⊗ I ) ⊗ A x,y (1 ⊗ u y ) ⊗ 1 / / ( A y ,z ⊗ A y ,y ) ⊗ A x,y c y,y,z ⊗ 1 O O Ξ 2 = A y ,z ⊗ A x,y 1 ⊗ L ′   id 1 ⊗ λ x,y   A y ,z ⊗ A x,y c x,y,z / / A x,z A y ,z ⊗ ( I ⊗ A x,y ) 1 ⊗ ( u y ⊗ 1) / / A y ,z ⊗ ( A y ,y ⊗ A x,y ) 1 ⊗ c x,y,y O O and 22 Ξ 3 = ( A y ,z ⊗ I ) ⊗A x,y (1 ⊗ u y ) ⊗ 1 / / A ′   ( A y ,z ⊗ A y ,y ) ⊗A x,y c y,y,z ⊗ 1 / / A ′   A y ,z ⊗ A x,y c x,y,z & & L L L L L L L L L L L L A y ,z ⊗ A x,y R ′ ⊗ 1 7 7 o o o o o o o o o o o 1 ⊗ L ′ ' ' O O O O O O O O O O O O = = A x,z A y ,z ⊗ ( I ⊗ A x,y ) 1 ⊗ ( u y ⊗ 1) / / A y ,z ⊗ ( A y ,y ⊗ A x,y ) 1 ⊗ c x,y,y / / α x,y,y,z } } } } } } } } } } } } } } } } } } : B } } } } } } } } } } } } } } } } } } } } A y ,z ⊗ A x,y c x,y,z 8 8 r r r r r r r r r r r r satisfy the e quality Ξ 1 = Ξ 3 ∗ Ξ 2 Let us justify the eq uiv ale nc e of the deﬁnitions of S P C -categ ories. W e deﬁne the following bijec- tive corr esp ondence b etw een data in volved the deﬁnitions. Arrows A ( x, − ) y ,z : A y ,z → [ A x,y , A x,z ] and c x,y ,z : A y ,z ⊗ A x,y → A x,z corres p o nd via the adjunction 2.5. By Lemma [Sc h08]-19 . 6 one has a bijectiv e cor resp ondence b etw een 2-cells of the kind α x,y ,z ,t and 2 - cells α ′ x,y ,z ,t , the later b eing images b y Rn ◦ Rn of the ﬁr s t one s . The co domains of the 2-cells ρ x,y and ρ ′ x,y are equal b y Lemma 7.9, and these 2-cells corresp ond when are eq ua l. The co do mains o f the 2-cells λ x,y and λ ′ x,y are equal b y Lemma 7.8 and these 2-cells corre s po nd when they a re equal. F or such corr e sp onding data, the pro ofs of the equiv ale nce o f Axioms 4.4 and 3.14 rely on the adjunction 2.5. The 2-cells of Axiom 4.4 hav e ima ges by Rn ◦ Rn ◦ R n the t wo 2- cells of Axioms 3.1 4 and their commo n domain is a strict ar row with s trict images by Rn and Rn ◦ R n . Computation details a re in App endix in 7.27. The pro o f of the equiv alence of Axioms 4.5 and Axioms 3.1 5 is simila r . The 2-cells Ξ 1 of Axiom 4.5 has a strict domain and its imag e by R n is ρ ′ whereas the 2-ce ll Ξ 3 ◦ Ξ 2 has image by Rn the second 2-cell o f Axio m 3.15. Computatio n details a r e in Appe ndix in 7.28. W e hav e an alternative deﬁnition for the S P C -functor s with the tenso r in S P C . Given tw o a rbitrary S P C -ca tegories A a nd B , a S P C -functor F : A → B consists o f the following data: - A map F sending ob jects of A to ob jects of B ; - F or any o b jects x , y o f A , and ar row F x,y : A ( x, y ) → B ( F x, F y ) in S P C ; - Collectio ns of 2- cells of S P C : the F 2 x,y , indexed by pa ir of ob jects x , y of A and the F 0 x , indexed by ob jects x of A , as follows 4.6 A y ,z ⊗ A x,y F y,z ⊗ F x,y / / c   B F y, F z ⊗ B F x,F y c   F 2 x,y,z q y k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k A x,z F x,z / / B F x,F z and I u x / / u F x ! ! C C C C C C C C C C C C C C C C C A x,x F x,x   F 0 x 7 ? x x x x x x x x x x x x x x x x B F x,F x and that satisfy the coher ence conditions 4.7, 4 .8 and 4.9 b elow. 23 4.7 F or any obje cts x , y , z , t of A , the 2-c el ls ( A z ,t ⊗ A y,z ) ⊗A x,y c ⊗ 1 / / ( F z ,t ⊗ F y,z ) ⊗ F x,y   A y,t ⊗ A x,y c / / F y,t ⊗ F x,y   A x,t F x,t   ( B F z ,F t ⊗ B F y,F z ) ⊗B F x ,F y F 2 y,z ,t ⊗ 1 1 9 k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k c ⊗ 1 / / A ′   B F y,F t ⊗ B F x ,F y F 2 x,y,t 2 : m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m c / / B F x,F t id B F z ,F t ⊗ ( B F y ,F z ⊗ B F x,F y ) 1 ⊗ c / / α F x ,F y,F z ,F t . 6 f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f B F z ,F t ⊗ B F x ,F z c / / B F x,F t and ( A z ,t ⊗ A y,z ) ⊗A x,y c ⊗ 1 / / A ′   A y,t ⊗ A x,y c / / A id A z ,t ⊗ ( A y,z ⊗ A x,y ) F z ,t ⊗ ( F y,z ⊗ F x,y )   1 ⊗ c / / α x,y,z ,t . 6 e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e A z ,t ⊗ A x,z c / / F z ,t ⊗ F x,z   A x,t F x,t   B F z ,F t ⊗ ( B F y,F z ⊗ B F x,F y ) 1 ⊗ c / / 1 ⊗ F 2 x,y,z 1 9 k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k B F z ,F t ⊗ B F x,F z c / / F 2 x,z ,t 2 : m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m B F x,F t ar e e qu al. 4.8 F or any obje cts x , y of A , the 2-c el ls A x,y F x,y / / R ′   = B F x ,F y R ′   id A x,y ⊗ I 1 ⊗ u   F x,y ⊗ 1 / / B F x ,F y ⊗ I 1 ⊗ u   1 ⊗ F 0 x q y j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j A x,y ⊗ A x,x c   F x,y ⊗ F x,x / / B F x,F y ⊗ B F x,F x F 2 x,x,y q y j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j c   ⇐ ρ F x,F y A x,y F x,y / / B F x ,F y and A x,y id R ′   ρ x,y   A x,y F / / B F x,F y A x,y ⊗ I 1 ⊗ u x / / A x,y ⊗ A x,x c O O ar e e qu al. 24 4.9 F or any obje cts x , y of A , the 2-c el ls A x,y F / / L ′   = B F x ,F y id L ′   I ⊗ A x,y u y ⊗ 1   1 ⊗ F x,y / / I ⊗ B F x ,F y u F y ⊗ 1   F 0 x,y ⊗ 1 q y j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j A y,y ⊗ A x,y c   F y,y ⊗ F x,y / / B F y ,F y ⊗ B F x,F y F 2 x,y,y q y j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j c   ⇐ λ F x,F y A F x,y / / B F x ,F y and A x,y id L ′   λ   A x,y F x,y / / B F x,F y I ⊗ A x,y u y ⊗ 1 / / A y,y ⊗ A x,y c O O ar e e qu al. The adjunction 2 .5 gives a bijective corresp o ndence b etw een 2-cells F 2 as in 4.6 and 2-cells F ′ 2 as in 3.16. F o r such cor esp onding data, it turns out that Axio ms 3.18 and 4.7 are equiv alent, this is prov ed in 7.29 in Appendix, Axioms 3.1 9 and 4 .8 are equiv alent, this is proved in 7.3 0 in Appendix, and Axioms 3.2 0 and 4.9 a re equiv alent, this is prov ed in 7.31 in App endix. W e say that a S P C -categ ory A as ab ov e, is strict if and o nly if the arr ows A ( x, − ) are strict in S P C and the 2-cells α , ρ , λ , o r equiv alently the 2-cells α ′ , ρ ′ and λ ′ , are all identities. W e hav e also the notio n of strict S P C -functors ( F, F 0 , F 2 ) : A → B : they are the ones for whic h the co mpo nents F x,y : A x,y → B F x,F y are strict a rrows in S P C a nd for whic h the 2 - cells of the collections F 0 and F ′ 2 are identities. Note tha t for an F as ab ove with A strict the F ′ 2 are identities if a nd only if the F 2 are. 5 First examples One-p oint enrichmen ts are of par ticular interest and named 2-rings. As men tioned by M.Dupont in his thesis [Dup08], they are also the c ate goric al rings deﬁned b y Jibladze and Pir ashvili [JiPi0 7], those are also known to b e the Ann-c ate gories of [Qu87]. Given a 2-ring A , we shall write sim- ply A for the hom A ( ∗ , ∗ ) o f its unique ob ject ∗ . Therefore the forma l deﬁnition of a 2-ring ( A , c, u, α, ρ, λ ), a s a “weak” monoid in S P C , namely a Pica r d ca tegory A with a multiplic ation c : A ⊗ A → A and a unit u : I → A (which ar e s trict a rrows!) and appropriate 2-cells α , ρ and λ , is obtained b y removing the subscr ipts x, y , z , ... from the deﬁnitions o f S P C -categor ies. W e shall use the alternative deﬁnition of a 2-ring ( A , c ′ , u, α ′ , ρ ′ , λ ′ ) not using the tensor obtained simila rly by forgetting subscripts and where multiplication c ′ : A → [ A , A ] deno tes the unique a rrow A ∗ , − . The following deﬁnition of 2 -rings can be obta ined by written explicitly all linea r ity conditions from the deﬁnition of enriched categor ies a nd functors. It is equiv ale nt and very close to that of Jibladze a nd Pirashvili [JiP i07] (for their categorica l r ings). Detailed explana tions that the 2-rings with their morphisms in the sense b elow are just one- po int S P C -ca tegories with their functors is given in App endix-7.33 and 7 .34. 25 Deﬁnition 5.1 A 2- ring c onsists of a s ymm et ric Pic ar d c ate gory ( A , j ) , wher e A is denote d ad- ditively ( A , + , 0 , ass, r, l , s ) , to gether with a functor A × A → A , denote d by a multiplic ation “.”, an obje ct 1 of A and natur al isomorphisms ˜ α a,b,c : ( a.b ) .c → a . ( b.c ) , ˜ ρ a : a. 1 → a, ˜ λ a : 1 .a → a, a b,b ′ : a.b + a.b ′ → a. ( b + b ′ ) , b a,a ′ : a.b + a ′ .b → ( a + a ′ ) .b, such that t he data ( A , ., 1 , ˜ α, ˜ ρ, ˜ λ ) deﬁnes a m onoidal structur e on A and the diagr ams b elow fr om 5.2 to 5.11 c ommute for al l p ossible obje cts of A . 5.2 a.b + ( a .b ′ + a.b ′′ ) id + a b ′ ,b ′′   ass / / ( a.b + a .b ′ ) + a.b ′′ a b,b ′ + id   a.b + a. ( b ′ + b ′′ ) a b,b ′ + b ′′   a. ( b + b ′ ) + a.d a b + b ′ ,b ′′   a. ( b. ( b ′ + b ′′ )) a.ass / / a. (( b + b ′ ) + b ′′ ) 5.3 a.b + ( a ′ .b + a ′′ .b ) id + b a ′ ,a ′′   ass / / ( a.b + a ′ .b ) + a ′′ .b b a,a ′ + id   a.b + ( a ′ + a ′′ ) .b b a,a ′ + a ′′   ( a + a ′ ) .b + a ′′ .b b a + a ′ ,a ′′   ( a + ( a ′ + a ′′ )) .b ass.b / / (( a + a ′ ) + a ′′ ) .b 5.4 a.b + a.b ′ a b,b ′ / / s   a. ( b + b ′ ) a.s   a.b ′ + a.b a b ′ ,b / / a. ( b ′ + b ) . 5.5 a.b + a ′ .b b a,a ′ / / s   ( a + a ′ ) .b s.b   a ′ .b + a.b b a ′ ,a / / ( a ′ + a ) .b 26 5.6 ( a.b + a.b ′ ) + ( a ′ .b + a ′ .b ′ ) ∼ = a b,b ′ + a ′ b,b ′   ( a.b + a ′ .b ) + ( a ′ .b + a ′ .b ′ ) b a,a ′ + b ′ a,a ′   a. ( b + b ′ ) + a ′ . ( b + b ′ ) b + b ′ a,a ′ ) ) T T T T T T T T T T T T T T T ( a + a ′ ) .b + ( a + a ′ ) .b ′ a + a ′ b,b ′ u u j j j j j j j j j j j j j j j ( a + a ′ ) . ( b + b ′ ) 5.7 ( a.b ) .c + ( a ′ .b ) .c c a.b,a ′ .b / / ˜ α a,b,c + ˜ α a ′ ,b,c   (( a.b ) + ( a ′ .b )) .c b a,a ′ .c / / (( a + a ′ ) .b ) .c ˜ α a + a ′ ,b,c   a. ( b.c ) + a ′ . ( b.c ) b.c a,a ′ / / ( a + a ′ ) . ( b.c ) 5.8 ( a.b ) .c + ( a.b ′ ) .c c a.b,a.b ′ / / ˜ α a,b,c + ˜ α a,b ′ ,c   (( a.b ) + ( a.b ′ )) .c a b,b ′ .c / / ( a. ( b + b ′ )) .c ˜ α a,b + b ′ ,c   a. ( b.c ) + a. ( b.c ′ ) a b.c,b.c ′ / / a. ( b.c + b.c ′ ) a. c b,b ′ / / a. (( b + b ′ ) .c ) 5.9 ( a.b ) .c + ( a.b ) .c ′ a.b c,c ′ / / ˜ α a,b,c + ˜ α a,b,c ′   ( a.b ) . ( c + c ′ ) ˜ α a,b,c + c ′   a. ( b.c ) + a. ( b.c ′ ) a b.c,b.c ′ / / a. ( b.c + b.c ′ ) a.b c,c ′ / / a. ( b. ( c + c ′ )) 5.10 ( a + b ) . 1 1 a,b   ˜ ρ a + b / / a + b a + b t t t t t t t t t t t t t t t t t t t t 5.11 1 . ( a + b ) 1 a,b   ˜ λ a + b / / a + b a + b t t t t t t t t t t t t t t t t t t t t Note that in the deﬁnition given in [JiPi0 7] inv erses of maps a b,c and a b,c rather than the maps themselves are cons ide r ed and diagrams 5.2 and 5 .4 ar e replaced by 27 5.12 a. ( b + b ′ ) + a. ( c + c ′ ) a b,b ′ + a c,c ′ * * U U U U U U U U U U U U U U U U a. (( b + b ′ ) + ( c + c ′ )) a b + b ′ ,c + c ′ 5 5 j j j j j j j j j j j j j j j a. ∼ = ( a.b + a .b ′ ) + ( a.c + a .c ′ ) ∼ = a. (( b + c ) + ( b ′ + c ′ )) a b + c,b ′ + c ′ ) ) T T T T T T T T T T T T T T T ( a.b + a .c ) + ( a .b ′ + a.c ′ ) a. ( b + c ) + a . ( b ′ + c ′ ) a b,c + a b ′ ,c ′ 4 4 i i i i i i i i i i i i i i i i and s imilarly diagr ams 5.3 a nd 5.5 are r eplaced by 5.13 ( a + a ′ ) .c + ( b + b ′ ) .c c a,a ′ + c b,b ′ * * U U U U U U U U U U U U U U U U (( a + a ′ ) + ( b + b ′ )) .c c a + a ′ ,b + b ′ 4 4 j j j j j j j j j j j j j j j ∼ = .c ( a.c + a ′ .c ) + ( b.c + b ′ .c ) ∼ = (( a + b ) + ( a ′ + b ′ )) .c c a + b,a ′ + b ′ * * T T T T T T T T T T T T T T T ( a.c + b.c ) + ( a ′ .c + b ′ .c ) ( a + b ) .c + ( a ′ + b ′ ) .c ′ c a,b + c a ′ ,b ′ 4 4 i i i i i i i i i i i i i i i i The deﬁnitio ns her e and in [J iPi07] are indeed equiv alent (T o see this use for insta nce Lemma [Sch08]-7.1.) Deﬁnition 5.14 A morphism of c ate goric al ring A → B c onsists of a functor H : A → B with a symmetric monoidal structur e b etwe en the symmet ric c ate goric al gr oups H + : ( A , + , 0 , ass, r, l , s ) → ( B , + , 0 , ass, r, l , s ) and a monoidal structu r e b etwe en the monoidal c ate gories H × : ( A , ., 1 , ˜ α, ˜ ρ, ˜ λ ) → ( B , ., 1 , ˜ α, ˜ ρ, ˜ λ ) such that the fol lowing diagr ams 5.15 H ( a ) .H ( b ) + H ( a ) .H ( b ′ ) H × 2 a,b + H × 2 a,b ′ / / H ( a.b ) + H ( a.b ′ ) H + 2 a.b,a.b ′ ( ( Q Q Q Q Q Q Q Q Q Q Q Q H ( a ) . ( H ( b ) + H ( b ′ )) H ( a ) H ( b ) ,H ( b ′ ) 4 4 i i i i i i i i i i i i i i i i H ( a ) .H 2 + b,b ′ * * U U U U U U U U U U U U U U U U H ( a.b + a.b ′ ) H ( a ) .H ( b + b ′ ) H × 2 a,b + b ′ / / H ( a. ( b + b ′ )) H ( a b,b ′ ) 6 6 m m m m m m m m m m m m 28 and 5.16 H ( a ) .H ( b ) + H ( a ′ ) .H ( b ) H × 2 a,b + H × 2 a ′ ,b / / H ( a.b ) + H ( a ′ .b ) H + 2 a.b,a.b ′ ( ( Q Q Q Q Q Q Q Q Q Q Q Q ( H ( a ) + H ( a ′ )) .H ( b ) H + 2 a,a ′ .H ( b ) * * U U U U U U U U U U U U U U U U H ( b ) H ( a ) ,H ( a ′ ) 4 4 i i i i i i i i i i i i i i i i H ( a.b + a ′ .b ) H ( a + a ′ ) .H ( b ) H × 2 a + a ′ ,b / / H (( a + a ′ ) .b ) H ( b a,a ′ ) 6 6 m m m m m m m m m m m m c ommu te for al l p ossible obje cts involve d. The following is a cruc ia l example of S P C -catego ry . Prop ositi o n 5. 17 The 2-c ate gory S P C gets str ictly enriche d over itself, i.e. it admits a strict enriche d stru ctur e as fol lows. The hom m ap sends any p air A , B of obje cts to [ A , B ] , the c omp osition maps ar e the [ A , − ] B , C : [ B , C ] → [[ A , B ] , [ A , C ]] and t he unit arr ows u A ar e the v : I → [ A , A ] . PROOF:See 7.37 in App e ndix. Let us make the following remark ab o ut the terminology . If S P C ′ denotes just for the purpo se of this explanation the enriched structure of S P C o ver itself then for any A in S P C and any 1-cell F : B → C in S P C , o ne has that: - S P C ′ ( A , − ) B , C is [ A , − ] B , C : [ B , C ] → [[ A , B ] , [ A , C ]]; - S P C ′ ( − , C ) A ,B is [ − , C ] A , B : [ A , B ] → [[ B , C ] , [ A , C ]]; - S P C ′ (1 , F ) is [1 , F ] : [ A , B ] → [ A , C ]; - S P C ′ ( F, 1) is [ F, 1] : [ B , C ] → [ A , C ]. The ﬁrs t tw o p oints re s ults fro m the deﬁnitio ns . The other tw o po ints are the following lemma prov ed in Appendix 7.39. Lemma 5.18 F or any A and any arr ows F : B → C and ˜ F : I → [ B , C ] strict with ev ⋆ ( ˜ F ) = F the diagr ams in S P C [ C , A ] [ F , A ] / / R ′   [ B , A ] [ C , A ] ⊗ I 1 ⊗ ˜ F / / [ C , A ] ⊗ [ B , C ] c O O and [ A , B ] [ A ,F ] / / L ′   [ A , C ] I ⊗ [ A , B ] ˜ F ⊗ 1 / / [ B , C ] ⊗ [ A , B ] c O O b oth c ommu te. Given any S P C -ca teg ory A , 2- rings are obtained by r estriction of A to a ny one of its p oints. The particular case of the s trict enriched str ucture on S P C yields a strict 2 -ring structure o n [ A , A ] fo r a ny Pica r d catego ry A . Another imp ortant example of 2- ring is provided by the unit I of S P C . 29 Prop ositi o n 5. 19 The unit I of S P C admits a strict 2-ring structu r e with multiplic ation given by L I : I ⊗ I → I (or v : I → [ I , I ] ) and u nit the identity at I . PROOF:See Appendix 7. 6 Mo d ules and their morphisms An y 2-ring A yields a category A -mo d of A -mo dules and their morphisms. F or mally A -mo d is the category of S P C -functors A → S P C a nd S P C -na tur al tr ansformatio ns b etw een them. In this section w e present a lternative descriptions of A -mo dules and their mo rphisms. In pa rticular we show that the categ ory A -mo d is iso morphic to a categor y o f T -alg ebras a nd their mor phisms for the do c tr ine T = A ⊗ − over S P C . Even tually we prove in Pro p o sition 6.36 that the categ ory I − mod of mo dules over the unit 2 -ring I is equiv alent to S P C . In this section A stand for a 2- ring with m ultiplication c ′ : A → [ A , A ]/ c : A ⊗ A → A with unit u : I → A and co herence 2 -cell α ′ / α (3.11 / 4.1), ρ / ρ ′ (4.2 / 3.1 2) and λ / λ ′ (4.3 / 3.13). Considering an arbitrar y S P C -functor F : A → S P C , let us write M for the ob ject F ( ⋆ ) o f S P C image by F of the uniq ue point ⋆ of A , ϕ ′ for the arr ow unique co mpo nent F ⋆,⋆ : A → [ M , M ] of F , β ′ for the 2 -cell F ′ 2 ⋆,⋆,⋆ and γ ′ for the 2 -cell F 0 ⋆ . Then one obtains the following ﬁrs t deﬁnition of A -mo dules by rewriting the data 3.1 6 and 3 .17 and Axio ms 3.18, 3.19 a nd 3.20 with these new notations. A A -mo dule M = ( M , ϕ ′ , β ′ , γ ′ ) consists of the following data in S P C : an ob ject M , with an arrow ϕ ′ : A → [ M , M ], ca lled its action , and tw o 2 -cells β ′ and γ ′ as follows 6.1 A c ′ % % J J J J J J J J J J ϕ ′ x x p p p p p p p p p p p p [ M , M ] [ M , − ]   [ A , A ] [1 ,ϕ ′ ]   [[ M , M ] , [ M , M ]] [ ϕ ′ , 1] / / β ′ 0 8 i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i [ A , [ M , M ]] 6.2 I u / / v ! ! C C C C C C C C C C C C C C C C C C A ϕ ′   γ ′ 7 ? v v v v v v v v v v v v v v v v v v [ M , M ] and tho se satisfy the co herence a xioms 6.3, 6.4 and 6.5 b elow. 30 6.3 The 2-c el ls [[ M , M ] , [ M , M ]] [[ M , M ] , − ] / / α ′ M , M , M , M   [[[ M , M ] , [ M , M ]] , [[ M , M ] , [ M , M ]]] [[ M , − ] , 1]   A c ′   ϕ ′ / / [ M , M ] [ M , − ] ; ; v v v v v v v v v v v v v [ M , − ] / / [[ M , M ] , [ M , M ]] β ′ q y j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j [1 , [ M , − ]] / / [ ϕ ′ , 1]   = [[ M , M ] , [[ M , M ] , [ M , M ]]] [ ϕ ′ , 1]   [ A , A ] [1 ,ϕ ′ ] / / [1 ,c ′ ] * * U U U U U U U U U U U U U U U U U U U U U U [ A , [ M , M ]] [1 , [ M , − ]] / / [1 ,β ′ ]   [ A , [[ M , M ] , [ M , M ]]] [1 , [ ϕ ′ , 1]] / / [ A , [ A , [ M , M ]]] [ A , [ A , A ]] [1 , [1 ,ϕ ′ ]] 4 4 h h h h h h h h h h h h h h h h h h h h h h h and [ M , M ] [ M , − ] / / [[ M , M ] , [ M , M ]] [ ϕ ′ , 1]   [ A , − ] / / β ′ }                                        = [[ A , [ M , M ]] , [ A , [ M , M ]]] [[ ϕ ′ , 1] , 1] / / [[1 ,ϕ ′ ] , 1]   [[[ M , M ] , [ M , M ]] , [ A , [ M , M ]]] [[ M , − ] , 1] / / [ β ′ , 1]   [[ M , M ] , [ A , [ M , M ]]] [ ϕ ′ , 1]   [ A , [ M , M ]] [ A , − ] / / = [[ A , A ] , [ A , [ M , M ]]] [ c ′ , 1] / / [ A , [ A , [ M , M ]]] A c ′ / / ϕ ′ O O c ′ + + W W W W W W W W W W W W W W W W W W W W W W W W W W W W [ A , A ] [ A , − ] / / [1 ,ϕ ′ ] O O [[ A , A ] , [ A , A ]] [ c ′ , 1] / / [1 , [1 ,ϕ ′ ]] O O = α ′   [ A , [ A , A ]] [1 , [1 ,ϕ ′ ]] O O [ A , A ] [1 ,c ′ ] 3 3 f f f f f f f f f f f f f f f f f f f f f f f f f f f ar e e qu al. 6.4 The 2-c el ls in S P C A c ′   ϕ ′ / / [ M , M ] [ M , − ]   β ′ o w g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g 1 x x [ A , A ] [ u, 1]   [1 ,ϕ ′ ] % % L L L L L L L L L L [[ M , M ] , [ M , M ]] [ v, 1]   [ ϕ ′ , 1] v v n n n n n n n n n n n n [ A , [ M , M ]] [ v, 1] ( ( P P P P P P P P P P P P [ γ ′ , 1] k s = [ I , A ] = = ev ⋆   [1 ,ϕ ′ ] / / [ I , [ M , M ]] ev ⋆   A ϕ ′ / / [ M , M ] and A id c ′   ρ ′   A ϕ ′ / / [ M , M ] [ A , A ] [ u, 1] / / [ I , A ] ev ⋆ O O ar e e qu al. 31 6.5 The 2-c el ls in S P C A c ′ ∗   ϕ ′ / / [ M , M ] [ − , M ]   ( β ′ ) ∗ o w g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g 1 x x [ A , A ] [ u, 1]   [1 ,ϕ ′ ] % % L L L L L L L L L L [[ M , M ] , [ M , M ]] [ v, 1]   [ ϕ ′ , 1] v v n n n n n n n n n n n n [ A , [ M , M ]] [ v, 1] ( ( P P P P P P P P P P P P [ γ ′ , 1] k s = [ I , A ] = = ev ⋆   [1 ,ϕ ′ ] / / [ I , [ M , M ]] ev ⋆   A ϕ ′ / / [ M , M ] and A id c ′   λ ′   A ϕ ′ / / [ M , M ] [ A , A ] [ u, 1] / / [ I , A ] ev ⋆ O O ar e e qu al. W e shall a lso deno te by γ ′′ for the 2-cell 6.6 M ϕ ′∗ / / id [ A , M ] [ u, 1]   γ ′′ + 3 M [ I , M ] ev ⋆ o o that co rresp onds to γ ′ via the bijection 7.16 / 7.17. Consider now a S P C -natura l transfor mation b e tween pr esheav es with domain a o ne p oint category A ( σ , κ ) : F → G : A → S P C. Let us write M = ( M , ϕ ′ , β ′ , γ ′′ ) and N = ( N , ψ ′ , β ′ , γ ′′ ) for the tw o mo dules corresp onding resp ectively to F and G . The S P C -natural tra nsformation σ has a unique comp onent at the unique ob ject ⋆ of A , which is a s trict a r row I → [ M , N ] in S P C or equiv alently an a rrow H : M → N . T he colle ction κ co nsists of a unique 2 - cell 6.7 A ϕ ′ / / ψ ′   [ M , M ] [1 ,H ]   [ N , N ] [ H, 1] / / 5 = t t t t t t t t t t t t t t t t t t [ M , N ] . 32 which we name δ ′ . W e obtain therefor e the following deﬁnition of morphism of A -mo dules by rewriting Axioms 3.31 and 3 .32 with these new notations. A mor phism o f A -mo dule ( H, δ ′ ) : ( M , ϕ ′ , β ′ , γ ′ ) → ( N , ψ ′ , β ′ , γ ′ ) consists of an arrow H : M → N in S P C with a 2 -cell δ ′ as in 6 .7, those satisfying Axioms 6.8 and 6.9 b elow. 6.8 The 2-c el ls Ξ 1 , Ξ 2 , Ξ 3 , Ξ 4 , Ξ 5 , Ξ 6 , Ξ 7 and Ξ 8 in S P C b elow satisfy the e quality Ξ 2 ◦ Ξ 1 = Ξ 8 ◦ Ξ 7 ◦ Ξ 6 ◦ Ξ 5 ◦ Ξ 4 ◦ Ξ 3 . Ξ 1 is A c ′   ψ ′ / / [ N , N ] β ′   [ N , − ] / / [[ N , N ] , [ N , N ]] [ ψ ′ , 1]   [ A , A ] [1 ,ψ ′ ] / / [ A , [ N , N ]] [1 , [ H, 1]] / / [ A , [ M , N ]] Ξ 2 is [ A , [ N , N ]] [1 , [ H, 1]] ' ' O O O O O O O O O O O [1 ,δ ′ ]   A c ′ / / [ A , A ] [1 ,ψ ′ ] 8 8 r r r r r r r r r r [1 ,ϕ ′ ] & & L L L L L L L L L L [ A , [ M , N ]] [ A , [ M , M ]] [1 , [1 ,H ]] 7 7 o o o o o o o o o o o Ξ 3 is the identity [[ N , N ] , [ N , N ]] [1 , [ H, 1]] ) ) R R R R R R R R R R R R R R A ψ ′ / / [ N , N ] [ N , − ] 7 7 o o o o o o o o o o o [ M , − ] ' ' O O O O O O O O O O O [[ N , N ] , [ M , N ]] [ ψ ′ , 1] / / [ A , [ M , N ]] [[ M , N ] , [ M , N ]] [[ H, 1] , 1] 5 5 l l l l l l l l l l l l l l = Ξ 4 is [[ N , N ] , [ N , N ]] [ δ ′ , 1]   [ ψ ′ , 1] ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q A ψ ′ / / [ N , N ] [ M , − ] / / [[ M , N ] , [ M , N ]] [[ H, 1] , 1] 5 5 k k k k k k k k k k k k k k [[1 ,H ] , 1] ) ) S S S S S S S S S S S S S S [ A , [ M , N ]] [[ M , M ] , [ M , N ]] [ ϕ ′ , 1] 6 6 m m m m m m m m m m m m m 33 Ξ 5 is the identity 2-c el l [[ M , N ] , [ M , N ]] [[1 ,H ] , 1] ) ) S S S S S S S S S S S S S S = A ψ ′ / / [ N , N ] [ H, 1] ' ' O O O O O O O O O O O [ M , − ] 7 7 o o o o o o o o o o o [[ M , M ] , [ M , N ]] [ ϕ ′ , 1] / / [ A , [ M , N ]] [ M , N ] [ M , − ] 5 5 k k k k k k k k k k k k k k Ξ 6 is [ N , N ] [ H, 1] % % K K K K K K K K K δ ′   A ψ ′ < < x x x x x x x x x ϕ ′ " " F F F F F F F F F [ M , N ] [ M , − ] / / [[ M , M ] , [ M , N ]] [ ϕ ′ , 1] / / [ A , [ M , N ]] [ M , M ] [1 ,H ] 9 9 s s s s s s s s s Ξ 7 is the identity 2-c el l [ M , N ] [ M , − ] ) ) S S S S S S S S S S S S S S S = A ϕ ′ / / [ M , M ] [1 ,H ] 6 6 n n n n n n n n n n n n [ M , − ] ( ( P P P P P P P P P P P P [[ M , M ] , [ M , N ]] [ ϕ ′ , 1] / / [ A , [ M , N ]] [[ M , M ] , [ M , M ]] [1 , [1 ,H ]] 5 5 k k k k k k k k k k k k k k and Ξ 8 is A ϕ ′ / / c ′   [ M , M ] [ M , − ] / / β ′   [[ M , M ] , [ M , M ]] [ ϕ ′ , 1]   [ A , A ] [1 ,ϕ ′ ] / / [ A , [ M , M ]] [1 , [1 ,H ]] / / [ A , [ M , N ]] . 6.9 The 2-c el ls in S P C u 2 H   γ ′   I v   u / / H   A ϕ ′ / / [ M , M ] [1 ,H ] / / [ M , N ] 34 and = I id v / / H " " γ ′   [ N , N ] [ H, 1] / / δ ′ $ , Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q [ M , N ] I u / / A ψ ′ O O ϕ ′ / / [ M , M ] [1 ,H ] O O ar e e qu al. F ro m these ﬁrst deﬁnitions of A -mo dules and A -mo dule morphis ms , one obtains immediately the following simple other deﬁnitions involving multilinear maps and multilinear natural transfor - mations. A A -mo dule ( M , ϕ, β , γ ) consists of a bilinea r map ϕ : A × M → M , which we write as a m ultiplica tio n ϕ ( a, m ) = a.m , with two natura l tra nsformations , β which is trilinear, and γ which is linea r, as follows. 6.10 β a 1 ,a 2 ,m : a 1 . ( a 2 .m ) → ( a 1 .a 2 ) .m lies in M , for a 1 , a 2 obje cts of A and m obje ct of M . 6.11 γ m : m → 1 A .m lies in M for m obje ct of M . Those sa tisfy the coherence conditions 6.1 2, 6.13 and 6.1 4 b e low. 6.12 F or any obje cts a 1 , a 2 , a 3 of A and m of M the diag r am in M a 1 . ( a 2 . ( a 3 .m ))) a 1 .β a 2 ,a 3 ,m   β a 1 ,a 2 ,a 3 .m / / ( a 1 .a 2 ) . ( a 3 .a ) β a 1 .a 2 ,a 3 ,m   a 1 . (( a 2 .a 3 ) .m ) β a 1 ,a 2 .a 3 ,m   ( a 1 . ( a 2 .a 3 )) .m α a 1 ,a 2 ,a 3 .m / / (( a 1 .a 2 ) .a 3 ) .m c ommu tes. 6.13 F or any obje cts a of A and m of M the diagr am in M a.m a.γ m / / ρ a .m   : : : : : : : : : : : : : : : a. (1 A .m ) β a, 1 A ,m                   ( a. 1 A ) .m c ommu tes. 35 6.14 F or any obje cts a of A and m of M the diagr am in M a.m λ a .m   : : : : : : : : : : : : : : : γ a.m / / 1 A . ( a.m ) β 1 A ,a,m                   (1 A .a ) .m c ommu tes. A A - mo dule morphism ( H , δ ) : M → N cons ists o f a n arrow H : M → N in S P C with a bilinear na tur al trans formation 6.15 δ a,m : a.H ( m ) → H ( a.m ) which lies in N for a obje ct of A and m obje ct of M which satisfy the Axioms 6 .1 6 and 6.17 b elow. 6.16 F or any obje ct m of M the fol lowi n g diagr am in N H m H ( γ m ) / / γ H ( m )   7 7 7 7 7 7 7 7 7 7 7 7 7 7 H ( ⋆.m ) ⋆.H m δ ⋆,m A A                c ommu tes. 6.17 F or any obje cts a 1 , a 2 in A and m in M , the fol lowing diagr am in N a 1 . ( a 2 .H m ) β a 1 ,a 2 ,m / / a 1 .δ a 2 ,m   ( a 1 .a 2 ) .H m δ a 1 .a 2 ,m   a 1 .H ( a 2 .m ) δ a 1 ,a 2 .m / / H ( a 1 . ( a 2 .m )) H ( β a 1 ,a 2 ,m ) / / H (( a 1 .a 2 ) .m ) c ommu tes. Note that Axiom 6.17 is obta ined from Axiom 6.9 by ev aluation at the gener a tor ⋆ s ince the com- po nent in ⋆ of u 2 H is an identit y . By Remar k 7.1 0 Axioms 6.9 and 6.17 ar e equiv a le n t. Even tually we g ive deﬁnitions of A -mo dules and their morphis ms using the tensor in S P C . This will show that in whic h sense A -modules o ccur as a lgebras for the do ctrine A ⊗ − ov er S P C . A A -mo dule ( M , ϕ, β , γ ) consists o f a strict arr ow ϕ : A ⊗ M → M in S P C in with 2-cells β and γ in S P C as follows 36 6.18 ( A ⊗ A ) ⊗ M A ′ / / c ⊗ 1   A ⊗ ( A ⊗ M ) 1 ⊗ ϕ   β p x h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h A ⊗ M ϕ & & M M M M M M M M M M M A ⊗ M ϕ x x q q q q q q q q q q q M and 6.19 M L ′ / / id I ⊗ M u ⊗ 1   γ + 3 M A ⊗ M ϕ o o satisfying the coher ence conditions 6.20, 6.2 1 and 6.22 b elow. 6.20 The 2-c el ls (( AA ) A ) M A ′ / / ( c ⊗ 1) ⊗ 1   = ( AA )( AM ) id c ⊗ 1   = ( AA )( AM ) A ′ / / 1 ⊗ ϕ   = A ( A ( AM )) 1 ⊗ (1 ⊗ ϕ )   ( AA ) M A ′ / / c ⊗ 1   A ( AM ) 1 ⊗ ϕ   β |                                  ( AA ) M A ′ / / c ⊗ 1   A ( AM ) 1 ⊗ ϕ   β |                                  AM ϕ   AM ϕ   id = AM ϕ   AM ϕ   M id M id M id M and (( AA ) A ) M A ′ ⊗ 1 / / ( c ⊗ 1) ⊗ 1   (( A ( AA ) ) M A ′ / / (1 ⊗ c ) ⊗ 1   = α ⊗ 1                 |                  A (( AA ) M ) 1 ⊗ A ′ / / 1 ⊗ ( c ⊗ 1)   A ( A ( AM )) 1 ⊗ (1 ⊗ ϕ )   1 ⊗ β |                                  ( AA ) M c ⊗ 1   ( AA ) M c ⊗ 1   A ′ / / A ( AM ) 1 ⊗ ϕ   β {                                  A ( AM ) 1 ⊗ ϕ   AM ϕ   id = AM ϕ   AM id ϕ   = AM ϕ   M id M id M id M ar e e qu al. 6.21 The 2-c el ls Ξ 1 = A ⊗ M R ′ ⊗ 1   id ρ ⊗ 1   A ⊗ M ϕ / / M ( A ⊗ I ) ⊗ M (1 ⊗ u ) ⊗ 1 / / ( A ⊗ A ) ⊗ M c ⊗ 1 O O 37 Ξ 2 = A ⊗ M 1 ⊗ L ′   id 1 ⊗ γ   A ⊗ M ϕ / / M A ⊗ ( I ⊗ M ) 1 ⊗ ( u ⊗ 1) / / A ⊗ ( A ⊗ M ) 1 ⊗ ϕ O O and Ξ 3 = ( A ⊗ I ) ⊗ M (1 ⊗ u ) ⊗ 1 / / A ′   ( A ⊗ A ) ⊗ M c ⊗ 1 / / A ′   A ⊗ M ϕ % % L L L L L L L L L L A ⊗ M R ′ ⊗ 1 7 7 n n n n n n n n n n n n 1 ⊗ L ′ ' ' P P P P P P P P P P P P = = M A ⊗ ( I ⊗ M ) 1 ⊗ ( u ⊗ 1) / / A ⊗ ( A ⊗ M ) 1 ⊗ ϕ / / β : B } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } A ⊗ M ϕ 9 9 r r r r r r r r r r satisfy the e quality Ξ 1 = Ξ 3 ◦ Ξ 2 . 6.22 The 2-c el ls Ξ 1 = A ⊗ M L ′ ⊗ 1   id λ ⊗ 1   A ⊗ M ϕ / / M ( I ⊗ A ) ⊗ M ( u ⊗ 1) ⊗ 1 / / ( A ⊗ A ) ⊗ M c ⊗ 1 O O Ξ 2 = A ⊗ M = ϕ / / L ′   M L ′   id γ   M I ⊗ ( A ⊗ M ) 1 ⊗ ϕ / / I ⊗ M u ⊗ 1 / / A ⊗ M ϕ O O Ξ 3 = A ⊗ M L ′ / / L ′ ⊗ 1 & & M M M M M M M M M M I ⊗ ( A ⊗ M ) u ⊗ 1 / / A ⊗ ( A ⊗ M ) 1 ⊗ ϕ / / A ⊗ M ϕ / / M ( I ⊗ A ) ⊗ M A ′ 7 7 o o o o o o o o o o o θ K S wher e the “c anonic al” 2-c el l θ is deﬁne d in App endix in 7.40 and has image by Rn an identity, and Ξ 4 = A ⊗ M L ′ ⊗ 1 / / ( I ⊗ A ) ⊗ M ( u ⊗ 1) ⊗ 1 / / A ′   = ( A ⊗ A ) ⊗ M ϕ ⊗ 1 / / A ′   A ⊗ M ϕ # # G G G G G G G G G M I ⊗ ( A ⊗ M ) u ⊗ 1 / / A ⊗ ( A ⊗ M ) 1 ⊗ ϕ / / β = E                                 A ⊗ M ϕ ; ; w w w w w w w w w satisfy the e quality Ξ 1 = Ξ 4 ◦ (Ξ 3 ) − 1 ◦ Ξ 2 . With the previous deﬁnition of A -mo dules , a mor phis m of A -mo dules ( H , δ ) : ( M , ϕ, β , γ ) → ( N , ψ , β , γ ) co nsists of an arr ow H : M → N with a 2-cell δ in S P C 38 6.23 A ⊗ M 1 ⊗ H / / ϕ   A ⊗ N u } s s s s s s s s s s s s s s s s s s s s ψ   M H / / N that sa tisfy Axioms 6.24 a nd 6.25 b elow. 6.24 The 2-c el ls A ( AM ) 1 ⊗ (1 ⊗ H ) / / A ( AN ) 1 ⊗ ψ   β                                                 ( AA ) M 1 ⊗ H / / A ′ 9 9 r r r r r r r r r r c ⊗ 1   ( AA ) N c ⊗ 1   A ′ 9 9 s s s s s s s s s s = = AM ϕ   1 ⊗ H / / AN δ t | q q q q q q q q q q q q q q q q q q q q ψ   AN ψ x x q q q q q q q q q q q M H / / N and ( AA ) M c ⊗ 1   A ′ / / A ( AM ) 1 ⊗ ϕ   1 ⊗ (1 ⊗ H ) / / β }                                  A ( AN ) 1 ⊗ ψ   1 ⊗ δ r r r r r r r r u } r r r r r r r r AM ϕ   AM ϕ   1 ⊗ H / / AN δ t | q q q q q q q q q q q q q q q q q q q q ψ   M id M H / / N ar e e qu al. 6.25 The 2-c el ls γ   M L ′ / / 1 ! ! I ⊗ M u ⊗ 1 / / A ⊗ M ϕ / / M H / / N and γ   N L ′ / / 1 ! ! I ⊗ N u ⊗ 1 / / A ⊗ N ψ / / δ  ' H H H H H H H H H H H H H H H H N M H O O L ′ / / = I ⊗ M u ⊗ 1 / / 1 ⊗ H O O = A ⊗ M ϕ / / 1 ⊗ H O O M H O O ar e e qu al. T o justify these new deﬁnitio ns let us consider again an arbitra r y S P C -functor F : A → S P C which deﬁnes a A -mo dule M with m ultiplica tio n ϕ ′ : A → [ M , M ] a nd 2-cells β ′ as in 6.1 and γ ′ as in 6.2. The multiplication corres p o nds by a djunction 2.5 to a strict ar row ϕ : A ⊗ M → M . 39 According to the adjunction 2.5, Lemmas 7.21 and [Sch08]-19.6, the map R n ◦ Rn deﬁnes a bijection betw ee n the sets o f 2- cells of the following kinds ( AA ) M A ′ / / c ⊗ 1   A ( AM ) 1 ⊗ ϕ   q y j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j AM ϕ $ $ I I I I I I I I I AM ϕ z z u u u u u u u u u M and A ϕ ′ / / c ′   [ M , M ] [ M , − ] / / [[ M , M ] , [ M , M ]] [ ϕ ′ , 1]   o w h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h [ A , A ] [ A , [ M , M ]] [1 ,ϕ ′ ] o o and o ne has a 2-cell β corresp o nding to β ′ / F ′ 2 ∗ , ∗ , ∗ via the ab ov e bijection. Note then that β has image b y Rn the 2-cell F 2 ⋆,⋆,⋆ which w e also write β ′′ . One obtains a 2-cell γ as in 6.19 that is equal to the 2-cell γ ′′ according Lemma 7.8. W e ar e go ing to chec k that the p oints 6.2 6, 6.27, 6.28, 6 .29, 6.30 and 6.31 b elow hold for a S P C -functor F and the r e lated data as ab ov e. Since the ar r ows doma ins of the 2-cells of the equalities of Axioms 6.20, 6.21 a nd 6.22 a re strict, it w ill r esult from the adjunction 2.5 that these axioms a re equiv alent resp ectively to Axioms 4.7, 4 .8 and 4.9 for the S P C -functor F . 6.26 The 2-c el l ( A ⊗ A ) ⊗ A ( ϕ ′ ⊗ ϕ ′ ) ⊗ ϕ ′   c ⊗ 1 / / A ⊗ A ϕ ′ ⊗ ϕ ′   c / / A ϕ ′   ([ M , M ] ⊗ [ M , M ]) ⊗ [ M , M ] A ′   c ⊗ 1 / / β ′′ ⊗ 1 0 8 i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i [ M , M ] ⊗ [ M , M ] c / / β ′′ 3 ; n n n n n n n n n n n n n n n n n n n n n n n n n n [ M , M ] id   [ M , M ] ⊗ ([ M , M ] ⊗ [ M , M ]) 1 ⊗ c / / = [ M , M ] ⊗ [ M , M ] c / / [ M , M ] is the image by R n of the ﬁrst of the 2-c el ls of Ax iom 6.20. PROOF:See 7.41 in App e ndix. 6.27 The 2-c el l ( AA ) A c ⊗ 1 / / A ′   AA c / / A id A ( AA ) α . 6 e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e 1 ⊗ c / / ϕ ′ ⊗ ( ϕ ′ ⊗ ϕ ′ )   AA ϕ ′ ⊗ ϕ ′   c / / A ϕ ′   [ M , M ] ⊗ ([ M , M ] ⊗ [ M , M ]) 1 ⊗ β ′′ 0 8 i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 1 ⊗ c / / [ M , M ] ⊗ [ M , M ] β ′′ 3 ; n n n n n n n n n n n n n n n n n n n n n n n n n n c / / [ M , M ] is the image by R n of the se c ond 2-c el l of Axiom 6.21 40 PROOF:See 7.42 in App e ndix. 6.28 The 2-c el l A id / / R ′   ρ   A ϕ ′ / / [ M , M ] A ⊗ I 1 ⊗ u / / A ⊗ A c O O is the image by R n of the 2-c el l Ξ 1 of Axiom 6.21 . PROOF:Immediate. 6.29 The 2-c el l A = ϕ ′ / / R ′   [ M , M ] R ′   id w w A ⊗ I 1 ⊗ u   ϕ ′ ⊗ 1 / / [ M , M ] ⊗ I 1 ⊗ γ ′ s { o o o o o o o o o o o o o o o o o o o o o o 1 ⊗ v   A ⊗ A ϕ ′ ⊗ ϕ ′ / / c   [ M , M ] ⊗ [ M , M ] β ′′ s { o o o o o o o o o o o o o o o o o o o o o o o o o o c   = A ϕ ′ / / [ M , M ] is the image by R n of the 2-c el l Ξ 3 ◦ Ξ 2 of Axiom 6.21 . PROOF:See 7.44 in App e ndix. 6.30 The 2-c el l A id / / L ′   λ   A ϕ ′ / / [ M , M ] I ⊗ A u ⊗ 1 / / A ⊗ A c O O is the image by R n of the 2-c el l Ξ 1 of Axiom 6.22 . PROOF:immediate. 6.31 The 2-c el l A = ϕ ′ / / L ′   [ M , M ] L ′   id w w I ⊗ A u ⊗ 1   1 ⊗ ϕ ′ / / I ⊗ [ M , M ] γ ′ ⊗ 1 s { o o o o o o o o o o o o o o o o o o o o o o v ⊗ 1   A ⊗ A ϕ ′ ⊗ ϕ ′ / / c   [ M , M ] ⊗ [ M , M ] β ′′ s { o o o o o o o o o o o o o o o o o o o o o o o o o o = c   A ϕ ′ / / [ M , M ] is the image by R n of the 2-c el l Ξ 4 ◦ (Ξ 3 ) − 1 ◦ Ξ 2 of Ax iom 6.22. 41 PROOF:See 7.45 in App e ndix. Let us co nsider now tw o mo dules M and N with r esp ective m ultiplicatio ns denoted by ϕ : A ⊗ M → M / ϕ ′ : A → [ M , M ] and ψ : A ⊗ N → N / ψ ′ : A → [ N , N ], with sets of coher ence t wo-cells for both written β / β ′′ / β ′ and γ / γ ′′ / γ ′ , and an arrow H : M → N in S P C . A 2-cell δ as in 6.2 3 cor resp onds by the adjunction 2.5 to a 2 -cell δ ′ . F or the ab ove r elated data the equiv alence of Axioms 6.24 and 6.8 r esults for m the tw o fo llowing po int s 6.32 and 6.33 b elow wherea s the equiv a lence 6.25 and 6 .9 follows from Remark 7.10 and p oints 6.3 4 and 6.3 5. 6.32 The ﬁrst of the 2-c el l of 6.24 has image by Rn ◦ Rn the p asting Ξ 2 ◦ Ξ 1 of 6.8, it has a strict domain which has a strict image by Rn . PROOF:See 7.46 in App e ndix. 6.33 The image by Rn ◦ Rn of the se c ond 2-c el l of Axiom 6.24 is the p asting Ξ 7 ◦ Ξ 6 ◦ Ξ 5 ◦ Ξ 4 ◦ Ξ 3 ◦ Ξ 2 ◦ Ξ 1 of 6.8. PROOF:See 7.47 in App e ndix. 6.34 The ﬁrst 2-c el l of Axiom 6.9 has image by ev ⋆ the ﬁrst 2-c el l of Axiom 6.25 and has a s t rict domain. PROOF:See 7.48 in App e ndix. 6.35 The se c ond 2-c el l of Axiom 6.9 has image by ev ⋆ the se c ond c el l of Axiom 6.25. PROOF:See 7.49 in App e ndix. F or a ny 2 -ring A , a A -mo dule M is said strict when the corr esp onding S P C -pr esheaf is strict which is to say that its action ϕ ′ : A → [ M , M ] is strict and the 2-cells β ′ and γ ′ are identities. A few remarks ar e in order . Consider any A -module M . If its 2-cell γ ′ is a n iden tity then cer- tainly its 2-cell γ ′′ is a ls o a n iden tity . Con versely if the a ction ϕ ′ is strict then for any m in M , ϕ ′∗ ( m ) : A → M is stric t, and the c omp onent ǫ ϕ ′∗ ( m ) ◦ u is an identit y and from this fact one ha s that if γ ′′ is strict then also is γ ′ . If the 2-cells β are iden tities then certainly are the 2-cells β ′ . Conv ersely if A is a s trict 2 -ring and the 2- cells β ′ are identities then the 2-cells β als o are. Prop ositi o n 6. 36 The for getful functor I − M od → S P C is p art of an e quivalenc e of c ate gories. Its e quivalenc e inverse factors as S P C ∼ = / / I − M od s inc / / I − M od wher e the left funct or is an isomorphism b etwe en S P C and the ful l sub-c ate gory I − M od s of I − M od gener ate d by the strict mo dules and in c is the inclusion functor. PROOF:One has a forg e tful 2 -functor I − M od → S P C and the result follows then from Lemmas 6.37, 6.38 and 6 .40 b elow. Lemma 6.37 Any obje ct A of S P C admits a un ique strict I -m o dule st ructur e, its mu ltiplic ation is given by the arr ow ϕ ′ = v : I → [ A , A ] (or e qu ivalently ϕ = L A : I ⊗ A → A ). 42 PROOF:If A ha s a strict I -mo dule structure with multiplication ϕ ′ = v : I → [ A , A ] the 2- cell γ ′′ (6.6) in this case b eing an iden tity one has that the comp osite A ϕ ′ ∗ / / [ I , A ] ev ⋆ / / A in S P C is necessar ily the identit y at A . Actually there is a unique a rrow f : A → [ I , A ] in S P C with stric t imag es in [ I , A ] – or equiv alently such that f ∗ is s trict – and such that the co mpo site A f / / [ I , A ] ev ⋆ / / A is the identit y a nd this arr ow is v ∗ . Now to establish that ϕ ′ = v : I → [ A , A ] gives a strict I -module structure o n A , it remains to chec k that o ne ha s in this case a n identit y 2-ce ll β ′ (6.1) which is the comm utativity of the external diagram in the pasting b elow I v & & M M M M M M M M M M M M v / / v   [ A , A ] [ A , − ]   [[ A , A ] , [ A , A ]] [ v, 1]   [ I , I ] [1 ,v ] / / [ I , [ A , A ]] where the top right diag r am commutes according to Lemma [Sch08]-18.5and the b ottom left a lso do es acco rding to Lemma [Sch08]-18.8. Lemma 6.38 F or any I -mo dule A and any arr ow H : A → B in S P C the re is a unique I - mo dule morphism fr om A to the str ict I -mo dule st ru ctur e on the symmetric Pic ar d c ate gory B which underlying map in S P C is H . If the multiplic ation of A is given by ϕ ′ : I → [ A , A ] this morphism has 2-c el l δ ′ as in 6.7 I v / / ϕ ′   [ B , B ] [ H, 1]   v ~ v v v v v v v v v v v v v v v v [ A , A ] [1 ,H ] / / [ A , B ] , which is determine d by its value in ⋆ by R emark 7.10 and such t hat ( δ ′ ⋆ ) a = ⋆.H a id H a H ( γ a ) / / H ( ϕ ′ ( ⋆ )( a )) . PROOF:Let us write t × a for ϕ ′ ( t )( a ) fo r any ob jects t of I and a of A . The c o herence Axiom 6 .16 fo r the pair H and δ ′ , with corr esp onding bilinear δ as in 6.15, amounts to the commutativit y o f the dia gram in B H a H ( γ a ) / / id 6 6 6 6 6 6 6 6 6 6 6 6 6 6 H ( ⋆ × a ) ⋆.H a. δ ⋆,a A A                43 That the ar row H : A → B in S P C tog ether with the 2- cell δ deﬁned by the co ndition ab ove satisﬁes Axio m 6 .1 7 amounts to the commutativit y of the diagram in B 6.39 t 1 . ( t 2 .H a ) id t 1 .δ t 2 ,a   ( t 1 .t 2 ) .H a δ t 1 .t 2 ,a   t 1 .H ( t 2 × a ) δ t 1 ,t 2 × a / / H ( t 1 × ( t 2 × a )) H ( β t 1 ,t 2 ,a ) / / H (( t 1 .t 2 ) × a ) for all ob jects t 1 , t 2 in I and a in A . W e prov e this las t p oint by induction o n the structure of the ob jects t 1 in I for ar bitr ary ob jects t 2 and a . F or t 1 = I dia gram 6.3 9 is the exter na l diagr am in the pasting I id id I δ I ,a   H 0 v v n n n n n n n n n n n n n n n H ( I ) H ( ϕ ′ 0 t 2 × a )   H ( ϕ ′ 0 a ) ' ' P P P P P P P P P P P P I H 0 9 9 r r r r r r r r r r r r δ I ,t 2 × a / / H ( I × ( t 2 × a )) H ( β I ,t 2 ,a ) / / H ( I × a ) in which all diagra ms commute a nd in particular the b o tto m- right tria ngle since the na tur al trans- formation β − ,t 2 ,a : − × ( t 2 × a ) → ( − .t 2 ) × a : I → A is monoida l. F or t 1 = ⋆ diag r am 6.39 is the external diag r am in the pasting ⋆. ( t 2 .H a ) id ⋆.δ t 2 ,a   ( ⋆.t 2 ) .H a δ ⋆.t 2 ,a   ⋆.H ( t 2 × a ) id id H (( ⋆.t 2 ) × a ) H ( t 2 × a ) H ( γ t 2 × a ) / / H ( ⋆ × ( t 2 × a )) H ( β ⋆,t 2 ,a ) O O where the b ottom diagram co mmutes by the co herence Axiom 6.14 for the I -mo dule A . 44 F or t 1 = t ′ 1 ⊗ t ′′ 1 diagram 6.39 is the exter nal diagr am in the pasting ( t ′ 1 ⊗ t ′′ 1 ) . ( t 2 .H a ) id id (( t ′ 1 ⊗ t ′′ 1 ) .t 2 ) .H a id ( t ′ 1 . ( t 2 .H a )) ⊗ ( t ′′ 1 . ( t 2 .H a )) id t ′ 1 .δ t 2 ,a ⊗ t ′′ 1 .δ t 2 ,a   (( t ′ 1 .t 2 ) .H a ) ⊗ (( t ′′ 1 .t 2 ) .H a ) δ t ′ 1 .t 2 ,a ⊗ δ t ′′ 1 .t 2 ,a   ( t ′ 1 .H ( t 2 × a )) ⊗ ( t ′′ 1 .H ( t 2 × a )) δ t ′ 1 ,t 2 × a ⊗ δ t ′′ 1 ,t 2 × a   H (( t ′ 1 .t 2 ) × a ) ⊗ H (( t ′′ 1 .t 2 ) × a )) H 2 ( t ′ 1 .t 2 ) × a, ( t ′′ 1 .t 2 ) × a   H ( t ′ 1 × ( t 2 × a )) ⊗ H ( t ′′ 1 × ( t 2 × a )) H 2 t ′ 1 × ( t 2 × a ) ,t ′′ 1 × ( t 2 × a )   H ((( t ′ 1 .t 2 ) × a ) ⊗ (( t ′′ 1 .t 2 ) × a )) H (( ϕ ′ 2 t ′ 1 .t 2 ,t ′′ 1 .t 2 ) a )   H (( t ′ 1 × ( t 2 × a )) ⊗ ( t ′′ 1 × ( t 2 × a ))) H (( ϕ ′ 2 t ′ 1 ,t ′′ 1 ) t 2 × a )   H ( β t ′ 1 ,t 2 ,a ) ⊗ H ( β t ′′ 1 ,t 2 ,a ) 2 2 e e e e e e e e e e e e e e e e e e e e e e e e e e H ((( t ′ 1 .t 2 ) ⊗ ( t ′′ 1 .t 2 )) × a ) id H (( t ′ 1 ⊗ t ′′ 1 ) × ( t 2 × a )) H ( β t ′ 1 ⊗ t ′′ 1 ,t 2 ,a ) / / H ((( t ′ 1 ⊗ t ′′ 1 ) .t 2 ) × a ) where the middle diag ram is comm utative if the diagram 6 .39 commutes for the v alues t 1 = t ′ 1 and t 1 = t ′′ 1 and the b ottom diagram commutes s ince the natural trans formation β − ,t 2 ,a : − × ( t 2 × a ) → ( − .t 2 ) × a : I → A is monoida l. F or t 1 = t • , the diagr am 6.39 is t • . ( t 2 .H a ) id t • .δ t 2 ,a   ( t • .t 2 ) .H a δ t • .t 2 ,a   t • .H ( t 2 × a ) δ t • ,t 2 × a / / H ( t • × ( t 2 × a )) H ( β t • ,t 2 ,a ) / / H (( t • .t 2 ) × a ) Note that acco rding to Lemma s 7.7 and 7.6 the a rrow δ t • ,a is t • .H a ( t.H a ) • H ( t × a ) • ∼ = ( δ t,a ) • o o H (( t × a ) • ) H ( ∼ = ) H ( t • × a ) . Therefore the left-b ottom leg rewrites 1. t • . ( t 2 .H a ) t • .δ t 2 ,a / / t • .H ( t 2 × a ) δ t • ,t 2 × a / / H ( t • × ( t 2 × a )) H ( β t • ,t 2 ,a ) / / H (( t • .t 2 ) × a ) 2. t • . ( t 2 .H a ) id ( t. ( t 2 .H a )) • ( t.H ( t 2 × a )) • ( t.δ t 2 ,a ) • o o H ( t × ( t 2 × a )) • ( δ t,t 2 × a ) • o o ∼ = H (( t × ( t 2 × a )) • ) ... ... H ( ∼ = ) H ( t • × ( t 2 × a )) H ( β t • ,t 2 ,a ) / / H (( t • .t 2 ) × a ) 3. t • . ( t 2 .H a ) id ( t. ( t 2 .H a )) • ( t.H ( t 2 × a )) • ( t.δ t 2 ,a ) • o o H ( t × ( t 2 × a )) • ( δ t,t 2 × a ) • o o ∼ = H (( t × ( t 2 × a )) • ) ... ... H ((( t.t 2 ) × a ) • ) H ( β t,t 2 ,a • ) H ( ∼ = ) H (( t • .t 2 ) × a ) 4. t • . ( t 2 .H a ) id ( t. ( t 2 .H a )) • ( t.H ( t 2 × a )) • ( t.δ t 2 ,a ) • o o H ( t × ( t 2 × a )) • ( δ t,t 2 × a ) • o o H ((( t.t 2 ) × a )) • ... ( H ( β t,t 2 ,a )) • o o ... ∼ = H ((( t.t 2 ) × a ) • ) H ( ∼ = ) H (( t • .t 2 ) × a ) 45 In the deriv atio n ab ove arrows 2 . and 3. are equal due to Lemma 7.7 and arrows 3 . and 4. are equal due to the na turality of the isomorphism 2.4. The top-rig ht leg of the diag ram ab ov e re w r ites 1. t • . ( t 2 .H a ) id ( t • .t 2 ) .H a δ t • .t 2 ,a / / H (( t • .t 2 ) × a ) 2. t • . ( t 2 .H a ) id ( t • .t 2 ) .H a id ( t.t 2 ) • .H a δ ( t.t 2 ) • ,a / / H (( t.t 2 ) • × a ) id H (( t • .t 2 ) × a ) 3. t • . ( t 2 .H a ) id ( t.t 2 ) • .H a id (( t.t 2 ) .H a ) • H (( t.t 2 ) × a ) • ( δ t.t 2 ,a ) • o o ∼ = H ((( t.t 2 ) × a ) • ) ... ... H ( ∼ = ) H (( t.t 2 ) • × a ) id H (( t • .t 2 ) × a ) Therefore the tw o leg s ab ov e are equal when dia gram 6.39 commut es for t 1 = t . Lemma 6.40 F or any I -mo dule A the un ique I -mo dule morphism fr om A to the strict I -mo dule structur e on the symmetric Pic ar d c ate gory A and which underlying m ap is the identity map 1 A at A in S P C – given by L emma 6.38 – is invertible in I -m o d. PROOF:Let ϕ ′ : I → [ A , A ] denote the multiplication of A , a nd t × a s ta nd for ϕ ( t )( a ) for any ob jects t o f I and a of A . The 2 -cell δ ′ given by 6.38 for the mor phism from A to the s trict I -mo dule on A in S P C , is in this ca se of the form I v   ⇓ ϕ ′ B B [ A , A ] . Its inv erse is therefor e of the exp ected form to b e par t of a mo dule morphism fr o m the s trict I -mo dule on A to the mo dule A with multiplication ϕ ′ . Recall that the coherence Axiom 6.16 for the pair (1 A , δ ′ ) as a morphism from A with m ulti- plication ϕ ′ to the strict I -mo dule o n A is the commutativit y of the diagram in A 6.41 a γ a / / id 2 2 2 2 2 2 2 2 2 2 2 2 2 ⋆ × a ⋆.a δ ⋆,a D D               for an y o b ject a where the bilinear δ corres p o nds to δ ′ , wherea s Axiom 6.17 a mounts to the commutation of the diagra m in A 6.42 t 1 . ( t 2 .a ) id t 1 .δ t 2 ,a   ( t 1 .t 2 ) .a δ t 1 .t 2 ,a   t 1 . ( t 2 × a ) δ t 1 ,t 2 × a / / t 1 × ( t 2 × a ) β t 1 ,t 2 ,a / / ( t 1 .t 2 ) × a for all ob jects t 1 , t 2 in I and a in A . 46 Axiom 6 .16 for the pair (1 A , δ ′− 1 ) amounts the c ommut a tion fo r any ob ject a in A of the diagram a γ a   4 4 4 4 4 4 4 4 4 4 4 4 4 id ⋆.a ⋆ × a δ − 1 ⋆,a C C               which do e s commute since dia gram 6.4 1 do es. Axiom 6 .17 fo r the pair (1 A , δ − 1 ) is the comm utation for any ob jects t 1 , t 2 in I and a in A o f the diag ram t 1 × ( t 2 × a ) β ′ t 1 ,t 2 ,a / / t 1 × δ − 1 t 2 ,a   ( t 1 .t 2 ) × a δ − 1 t 1 .t 2 ,a   t 1 × ( t 2 .a ) δ − 1 t 1 ,t 2 .a / / t 1 . ( t 2 .a ) id ( t 1 .t 2 ) .a which is equiv alent to the commutation of the external diagram in the pasting t 1 . ( t 2 .a ) id δ t 1 ,t 2 .a w w o o o o o o o o o o o t 1 .δ t 2 ,a   ( t 1 .t 2 ) .a δ t 1 .t 2 ,a   t 1 × ( t 2 .a ) t 1 × δ t 2 ,a ' ' O O O O O O O O O O O t 1 . ( t 2 × a ) δ t 1 ,t 2 × a   t 1 × ( t 2 × a ) β t 1 ,t 2 ,a / / ( t 1 .t 2 ) × a in which the left diagra m commutes b y natur ality of δ ′ t 1 : v ( t 1 ) → ϕ ′ ( t 1 ) : I → A and the right diagram is diagr am 6.42. 7 App endix This s ection contains v arious technical developmen ts. Section 2. Lemma 7.1 In any symmetric Pic ar d c ate gory t he diagr am a • ⊗ b • ! / / s   ( b ⊗ a ) • s •   b • ⊗ a • ! / / ( a ⊗ b ) • c ommu tes for any obje cts a and b . 47 PROOF:By deﬁnition the canonica l a rrow a • ⊗ b • → ( b ⊗ a ) • is the only arr ow f making the diagram ( a • ⊗ b • ) ⊗ ( b ⊗ a ) f ⊗ 1 / / ∼ =   ( b ⊗ a ) • ⊗ ( b ⊗ a ) j   a • ⊗ (( b • ⊗ b ) ⊗ a ) 1 ⊗ ( j ⊗ 1)   a • ⊗ ( I ⊗ a ) ∼ =   a • ⊗ a j / / I commute (see [Lap83]-p.3 1 0). In the following pasting all diagrams co mmu te, ( a • ⊗ b • ) ⊗ ( b ⊗ a ) 1 ⊗ s   s ⊗ 1 / / ( b • ⊗ a • ) ⊗ ( b ⊗ a ) ! ⊗ 1 / / 1 ⊗ s   ( a ⊗ b ) • ⊗ ( b ⊗ a ) s • ⊗ 1 / / 1 ⊗ s   ( b ⊗ a ) • ⊗ ( b ⊗ a ) j   ( a • ⊗ b • ) ⊗ ( a ⊗ b ) s ⊗ 1 / / ( b • ⊗ a • ) ⊗ ( a ⊗ b ) ! ⊗ 1 / / ∼ =   ( a ⊗ b ) • ⊗ ( a ⊗ b ) j / / I b • ⊗ (( a • ⊗ a ) ⊗ b ) 1 ⊗ ( j ⊗ 1) / / b • ⊗ ( I ⊗ b ) 1 ⊗ r / / b • ⊗ b j O O and by the coher ence theorem for symmetric monoidal categorie s the t wo left legs o f the tw o dia- grams ab ove a re equal. 7.2 Pr o of of L emma 2.2. PROOF:That the functor inv : A → A is monoidal result from po int s 7 .3 that it is symmetric amounts to Lemma 7.1. 7.3 In any symmet r ic Pic ar d c ate gory the diagr am a • ⊗ ( b • ⊗ c • ) ass / / 1 ⊗ s   ( a • ⊗ b • ) ⊗ c • s ⊗ 1   a • ⊗ ( c • ⊗ b • ) 1 ⊗ !   ( b • ⊗ a • ) ⊗ c • ! ⊗ 1   a • ⊗ ( b ⊗ c ) • s   ( a ⊗ b ) • ⊗ c • s   ( b ⊗ c ) • ⊗ a • !   c • ⊗ ( a ⊗ b ) • !   ( a ⊗ ( b ⊗ c )) • ass • / / (( a ⊗ b ) ⊗ c ) • c ommu tes for any obje cts a , b and c . 48 PROOF:By the naturality of s , the lemma is e quiv a le nt to the commutation of the external diagram in the pasting a • ⊗ ( b • ⊗ c • ) ass / / 1 ⊗ s   ( a • ⊗ b • ) ⊗ c • s ⊗ 1   a • ⊗ ( c • ⊗ b • ) s   ( b • ⊗ a • ) ⊗ c • s   ( c • ⊗ b • ) ⊗ a • ! ⊗ 1   ass / / c • ⊗ ( b • ⊗ a • ) 1 ⊗ !   ( b ⊗ c ) • ⊗ a • !   c • ⊗ ( a ⊗ b ) • !   ( a ⊗ ( b ⊗ c )) • ass • / / (( a ⊗ b ) ⊗ c ) • where the top diagram commutes according to the coherence for the s ymmetric monoidal structure and the b ottom theor em commutes acco rding to the coher ence theo rem for the gr oup structure. 7.4 Pr o of of L emma 2.3. PROOF:The collection j is natura l by deﬁnition of the functor inv . That it is monoida l amounts to the commutation of the diagra m I ∼ = j a ⊗ b / / ( a ⊗ b ) • ⊗ ( a ⊗ b ) ! ⊗ 1   ( b • ⊗ a • ) ⊗ ( a ⊗ b ) s ⊗ 1   ( a • ⊗ b • ) ⊗ ( a ⊗ b ) ∼ = I ⊗ I j a ⊗ j b / / ( a • ⊗ a ) ⊗ ( b • ⊗ b ) for any ob jects a and b of A , which holds b y deﬁnition of the a rrow ! : I : ( a ⊗ b ) • → b • ⊗ a • see [Lap83] p.310. 7.5 Deﬁnition of the isomorphism 2.4. 49 The iso morphism 2.4 is deﬁned p oint wise in a as the only arrow in B making the diagram I F 0   j F a / / ( F a ) • ⊗ F a ∼ = ⊗ 1 F I F ( j a ) % % J J J J J J J J J J F ( a • ) ⊗ F a F 2 a • ,a   F ( a • ⊗ a ) commute. Lemma 7.6 Given arr ows A F / / B G / / C in S P C the diagr am in C ( F G ( a )) • ∼ = ∼ = F ( G ( a ) • ) F ( ∼ = ) F G ( a • ) wher e al l the ∼ = ar e of typ e 2.4 , is c ommutative. PROOF:Consider the pa sting of c ommut a tive dia grams b elow where a ll the ∼ = are of type 2.4 I ( F G ) 0 ! ! j F Ga / / F 0   ( F Ga ) • ⊗ F Ga ∼ = ⊗ 1 v v m m m m m m m m m m m m m ∼ = ⊗ 1 ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q F ( Ga ) • ⊗ F Ga F ( ∼ = ) ⊗ 1 / / F 2 ( Ga ) • ,Ga   F G ( a • ) ⊗ F Ga F 2 G ( a • ) ,Ga   ( F G ) 2 a • ,a y y F I F j Ga / / F ( G 0 )   F (( Ga ) • ⊗ Ga ) F ( ∼ = ⊗ 1) / / F ( G ( a • ) ⊗ Ga ) F ( G 2 a • ,a )   F GI F Gj a / / F G ( a • ⊗ a ) Lemma 7.7 F or any m onoidal tr ansformation σ : F → G : A → B wher e A and B ar e obje cts of S P C , for any obje ct a of A , the diagr am in B ( F a ) • ∼ =   ( Ga ) • ( σ a ) • o o ∼ =   F ( a • ) σ a • / / G ( a • ) wher e the ∼ = denote isomorphisms of typ e 2.4, is c ommutative. 50 PROOF:Consider the pa sting of dia grams in B I j F a / / F 0   id W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W ( F a ) • ⊗ F a ∼ = ⊗ 1   [( σ a ) • ] − 1 ⊗ σ a + + F ( a • ) ⊗ F a F 2 a • ,a   σ a • ⊗ σ a + + I j Ga / / G 0   ( Ga ) • ⊗ Ga ∼ = ⊗ 1   F I F j a / / σ I + + W W W W W W W W W W W W W W W W W W W W W W W W W W W W F ( a • ⊗ a ) σ a • ⊗ a + + W W W W W W W W W W W W W W W W W W W W W W G ( a • ) ⊗ Ga G 2 a • ,a   GI Gj a / / G ( a • ⊗ a ) where the ∼ = denote isomorphisms of type 2.4. All diagra ms ab ov e commute a part fr om the one consisting of the four dotted a rrows. Since all arrows are inv ertible this last diag ram comm utes. The r esult follows since tenso ring with F a is a n equiv alence B → B . A few computational lemm as for S P C . W e present here a couple of results not stated in [Sch08]. Lemma 7.8 The 2-c el ls in S P C A L ′ / / I ⊗ A U ⊗ 1 / / B ⊗ A τ + 3 C and A Rn ( τ ) ∗ + 3 [ B , C ] [ U, 1] / / [ I , C ] ev ⋆ / / C ar e e qu al for any 2-c el l τ : B ⊗ A → C and U : I → B . PROOF:The 2-cell τ ◦ ( U ⊗ 1) ◦ L ′ ab ov e rewrites s uccessively 1. A η ∗ / / [ I , I ⊗ A ] ev ⋆ / / I ⊗ A U ⊗ 1 / / B ⊗ A τ + 3 C 2. A η ∗ / / [ I , I ⊗ A ] [1 ,U ⊗ 1] / / [ I , B ⊗ A ] [1 ,τ ] + 3 [ I , B ⊗ A ] ev ⋆ + 3 C 3. A η ∗ / / [ B , B ⊗ A ] [ U, 1] / / [ I , B ⊗ A ] [1 ,τ ] + 3 [ I , C ] ev ⋆ / / C 4. A η ∗ / / [ B , B ⊗ A ] [1 ,τ ] + 3 [ B , C ] [ U, 1] / / [ I , C ] ev ⋆ / / C 5. A ( Rn ( τ )) ∗ + 3 [ B , C ] [ U, 1] / / [ I , C ] ev ⋆ / / C . Lemma 7.9 The 2-c el ls in S P C A R ′ / / A ⊗ I 1 ⊗ U / / A ⊗ B τ + 3 C and A Rn ( τ ) + 3 [ B , C ] [ U, 1] / / [ I , C ] ev ⋆ / / C ar e e qu al for any 2-c el l τ : A ⊗ B → C and any arr ow U : I → B . 51 PROOF:The 2-cell τ ◦ (1 ⊗ U ) ◦ R ′ ab ov e rewrites successively 1. A η / / [ I , A ⊗ I ] ev ⋆ / / A ⊗ I 1 ⊗ U / / A ⊗ B τ + 3 C 2. A η / / [ I , A ⊗ I ] [1 , 1 ⊗ U ] / / [ I , A ⊗ B ] [1 ,τ ] + 3 [ I , C ] ev ⋆ / / C 3. A η / / [ B , A ⊗ B ] [ U, 1] / / [ I , A ⊗ B ] [1 ,τ ] + 3 [ I , C ] ev ⋆ / / C 4. A η / / [ B , A ⊗ B ] [1 ,τ ] + 3 [ A , C ] [ U, 1] / / [ I , C ] ev ⋆ / / C 5. A Rn ( τ ) + 3 [ B , C ] [ U, 1] / / [ I , C ] ev ⋆ / / C where in the ab ov e deriv ation a rrows 1 . and 2. are equal acc o rding to Co rollar y [Sch08]-11.2. Remark 7.10 Any 2-c el l σ : F → G : I → A with F strict in S P C is ful ly determine d by its c omp onent in ⋆ sinc e F b eing st rict the c omp onent at F of c ounit of the adjunction ǫ F : v ∗ ◦ ev ⋆ ( F ) → F is an identity and by n atur ality of ǫ one has t he c ommutation of v ∗ ( F ( ⋆ )) v ∗ ( σ ⋆ ) / / id v ∗ ( G ( ⋆ )) ǫ G   F σ / / G in S P C . Lemma 7.11 F or any s trict arr ow F : I → A the diag r am in S P C I v / / F   [ A , A ] [ F , 1]   A [ I , A ] ev ⋆ o o c ommu tes. PROOF:The arrow I v / / [ A , A ] [ F , 1] / / [ I , A ] has dual I F / / A v ∗ / / [ I , A ] . Since the arrow F is s trict it is equal to v ∗ ◦ ev ⋆ ( F ) a nd the result follows then fro m Lemma [Sch08]-11.9. Remark 7.12 Sinc e the units of the adjunctions v ∗ ⊣ ev ⋆ ar e identities in S P C , one has a bije ction for any A and B b et we en sets of 2-c el ls in S P C of the fol lowing kind 7.13 B v ∗ ! ! D D D D D D D D   A F ? ?         G / / [ I , B ] 52 and 7.14 A F / / G " " D D D D D D D D   B [ I , A ] ev ⋆ = = z z z z z z z z sending any 2-c el l Ξ of the typ e 7.13 to B v ∗ ! ! D D D D D D D D Ξ   A F ? ?         G / / [ I , B ] ev ⋆ / / B with inverse sending any 2-c el l Ξ ′ of typ e 7.14 to A F / / G " " D D D D D D D D Ξ ′   B v ∗   > > > > > > > > ǫ   [ I , A ] ev ⋆ z z z = = z z z z id A W e shall describ e for any arr ow ϕ : A ⊗ M → M and U : I → A of S P C some bijections betw ee n sets of 2-cells of the fo llowing kind: 7.15 IM U ⊗ 1 / / L ! ! C C C C C C C C C C C C C C C C C AM ϕ   9 A { { { { { { { { { { { { { { { { M 7.16 I U / / v ! ! C C C C C C C C C C C C C C C C C A Rn ( ϕ )   7 ? w w w w w w w w w w w w w w w w [ M , M ] 7.17 M ( Rn ( ϕ )) ∗ / / v ∗ ! ! C C C C C C C C C C C C C C C C C [ A , M ] [ U, 1]   8 @ y y y y y y y y y y y y y y y y [ I , M ] 53 7.18 M id ( Rn ( ϕ )) ∗ / / [ A , M ] [ U, 1]   + 3 M [ I , M ] ev ⋆ o o and 7.19 M L ′ / / id IM U ⊗ 1   + 3 M AM ϕ o o as follows. - Since L is the image b y E n of v : I → [ M , M ] and the image by Rn o f I M U ⊗ 1 / / AM ϕ / / M is I U / / A Rn ( ϕ ) / / [ M , M ] the maps Rn / E n deﬁne the bijection (a nd its inverse) betw een sets of 2-cells 7.15 a nd 7.16. - 2 -cells 7.16 a nd 7 .17 corr esp ond by dualit y . - The bijection betw een sets of 2 -cells 7.17 a nd 7 .18 sends any 2 -cell Ξ : v ∗ → [ j, 1] ◦ ( Rn ( ϕ )) ∗ to M ( Rn ( ϕ )) ∗ / / v ∗ ! ! C C C C C C C C C C C C C C C C C [ A , M ] [ U, 1]   Ξ 8 @ x x x x x x x x x x x x x x x x [ I , M ] ev ⋆ / / M which is as exp ected a 2-ce ll id → ev ⋆ ◦ [ U, 1] ◦ ( R n ( ϕ )) ∗ according to the adjunction 2.21. Its inv erse se nds a ny 2-cell Ξ : 1 → ev ⋆ ◦ [ U, 1 ] ◦ ( Rn ( ϕ )) ∗ to the pasting M id / / ( Rn ( ϕ )) ∗   Ξ   M v ∗ / / ǫ  & E E E E E E E E E E E E E E E E [ I , M ] [ A , M ] [ U, 1] / / [ I , M ] id < < y y y y y y y y y y y y y y y y y y ev ⋆ O O - E ven tually 2-c e lls 7.18 and 7 .1 9 ar e the sa me since their co domains a rrows resp ectively ev ⋆ ◦ [ U, 1] ◦ R n ( ϕ ) ∗ and ϕ ◦ ( U ⊗ 1) ◦ L ′ are eq ual acco rding to Lemma 7.8. 54 Lemma 7.20 The ab ove bije ction b etwe en sets of 2-c el ls 7.15/7.19 sends any Ξ : L → ϕ ◦ ( U ⊗ 1) : I M → M to the p asting IM U ⊗ 1 / / L ! ! C C C C C C C C C C C C C C C C C AM ϕ   Ξ 9 A { { { { { { { { { { { { { { { { M L ′ O O id / / = M PROOF:According to Lemma 7.8, the ab ov e 2 - cell A L ′ / / I A Ξ + 3 A is A Rn (Ξ) ∗ + 3 [ I , A ] ev ⋆ / / A . Consider then the image o f the ab ove 2-cell by the bijection 7.1 8 → 7.1 7, it is [ A , A ] ( Rn (Ξ) ∗   [ U, 1] # # H H H H H H H H H A v ∗ ! ! C C C C C C C C ǫ   A v ∗ / / Rn ( ϕ ) = = z z z z z z z z z [ I , A ] id ev ⋆ = = { { { { { { { { [ I , A ] which is just ( Rn ( ¯ λ )) ∗ : A → [ I , A ] since A v ∗ / / [ I , A ] ǫ + 3 A is an ident ity 2-cell, which dual has imag e by E n the 2 - cell ¯ λ . Easy computatio n also gives the following. Remark 7.21 F or any arr ows F : A → [ B , X ] and G : X → [ C , D ] of S P C the arr ow ( A ⊗ B ) ⊗ C E n ( F ) ⊗ 1 / / X ⊗ C E n ( G ) / / D has image by Rn A ⊗ B E n ( F ) / / X G / / [ C , D ] and has image by R n ◦ Rn the arr ow A F / / [ B , X ] [1 ,G ] / / [ B , [ C , D ]] . Sections 3 and 4. 7.22 Pr o of of Equality (I) in se c ond p asting of Ax iom 3.15. PROOF:T o chec k the equa lit y ( I ) of a r rows, consider the deriv ation o f equal comp osite arrows in S P C for any arrows d : A → [ A , A ] and j : I → A . 1. [ A , B ] [ A , − ] / / [[ A , A ] , [ A , B ]] [ d, 1] / / [ A , [ A , B ]] [ j, 1] / / [ I , [ A , B ]] ev ⋆ / / [ A , B ] 2. [ A , B ] [ A , − ] / / [[ A , A ] , [ A , B ]] [ d, 1] / / [ A , [ A , B ]] ev j ( ⋆ ) / / [ A , B ] 3. [ A , B ] [ A , − ] / / [[ A , A ] , [ A , B ]] [ d ∗ , 1] / / [ A , [ A , B ]] D / / [ A , [ A , B ]] ev j ( ⋆ ) / / [ A , B ] 55 4. [ A , B ] [ A , − ] / / [[ A , A ] , [ A , B ]] [ d ∗ , 1] / / [ A , [ A , B ]] [1 ,ev j ( ⋆ ) ] / / [ A , B ] 5. [ A , B ] [ A , − ] / / [[ A , A ] , [ A , B ]] [1 ,ev j ( ⋆ ) ] / / [ A , [ A , B ]] [ d ∗ , 1] / / [ A , B ] 6. [ A , B ] [ ev j ( ⋆ ) , 1] / / [[ A , A ] , B ] [ d ∗ , 1] / / [ A , B ] 7. [ A , B ] [ ev ⋆ , 1] / / [[ I , A ] , B ] [[ j, 1] , 1] / / [[ A , A ] , B ] [ d ∗ , 1] / / [ A , B ] where the equalities betw ee n ar rows stand for the following reaso ns: - 1 . and 2. : by the naturality in A of the colle ction of arrows q A : A → [[ A , C ] , C ] - 2 . and 3. : Lemma [Sch08]-10.9; - 3 . and 4. : Lemma [Sch08]-11.9; - 5 . and 6. : Lemma [Sch08]-11.3; - 6 . and 7. : by the naturality of the collections of arrows q ; 7.23 Deﬁnition of the 2-c el l c 1 f : x ′ → x,y ,z . The do main of this 2-cell rewrites 1. A y,z A ( x ′ , − ) / / [ A x ′ ,y , A x ′ ,z ] [ A x ′ ,x , − ] / / [[ A x ′ ,x , A x ′ ,y ] , [ A x ′ ,x , A x ′ ,z ]] [ A ( x ′ , − ) , 1] / / [ A x,y , [ A x ′ ,x , A x ′ ,z ]] [1 , [ f, 1]] / / [ A x,y , [ I , A x ′ ,z ]] ... ... [1 ,ev ⋆ ] / / [ A x,y , A x ′ ,z ] 2. A y,z A ( x ′ , − ) / / [ A x ′ ,y , A x ′ ,z ] [ A x ′ ,x , − ] / / [[ A x ′ ,x , A x ′ ,y ] , [ A x ′ ,x , A x ′ ,z ]] [1 , [ f, 1]] / / [[ A x ′ ,x , A x ′ ,y ] , [ I , A x ′ ,z ]] [1 ,ev ⋆ ] / / ... ... [[ A x ′ ,x , A x ′ ,y ] , A x ′ ,z ] [ A ( x ′ , − ) , 1] / / [ A x,y , A x ′ ,z ] 3. A y,z A ( x ′ , − ) / / [ A x ′ ,y , A x ′ ,z ] [ I , − ] / / [[ I , A x ′ ,y ] , [ I , A x ′ ,z ]] [[ f , 1] , 1] / / [[ A x ′ ,x , A x ′ ,y ] , [ I , A x ′ ,z ]] [1 ,ev ⋆ ] / / ... ... [[ A x ′ ,x , A x ′ ,y ] , A x ′ ,z ] [ A ( x ′ , − ) , 1] / / [ A x,y , A x ′ ,z ] 4. A y,z A ( x ′ , − ) / / [ A x ′ ,y , A x ′ ,z ] [ I , − ] / / [[ I , A x ′ ,y ] , [ I , A x ′ ,z ]] [[1 ,ev ⋆ ] / / [[ I , A x ′ ,y ] , A x ′ ,z ] [[ f , 1] , 1] / / [[ A x ′ ,x , A x ′ ,y ] , A x ′ ,z ] ... ... [ A ( x ′ , − ) , 1] / / [ A x,y , A x ′ ,z ] 5. A y,z A ( x ′ , − ) / / [ A x ′ ,y , A x ′ ,z ] [ ev ⋆ , 1] / / [[ I , A x ′ ,y ] , A x ′ ,z ] [[ f , 1] , 1] / / [[ A x ′ ,x , A x ′ ,y ] , A x ′ ,z ] [ A ( x ′ , − ) , 1] / / [ A x,y , A x ′ ,z ] 6. A y,z A ( x ′ , − ) / / [ A x ′ ,y , A x ′ ,z ] [ A ( f, 1) , 1] / / [ A x,y , A x ′ ,z ] . In the pre v ious deriv ation a rrows 2. and 3. are equa l acco r ding to Lemma [Sch08 ]-9.1 1and arrows 4 . and 5 . are eq ua l acco r ding to Lemma [Sch08]-11.3. The co domain of c 1 f ,y ,z rewrites A y ,z A ( x, − ) / / [ A x,y , A x,z ] [1 , A ( x ′ , − )] / / [ A x,y , [ A x ′ ,x , A x ′ ,z ]] [1 , [ f , 1]] / / [ A x,y , [ I , A x ′ ,z ]] [1 ,ev ⋆ ] / / [ A x,y , A x ′ ,z ] A x,y A ( x, − ) / / [ A x,y , A x,z ] [1 , A ( f , 1)] / / [ A x,y , A x ′ ,z ] Remark 7.24 F or any arr ows f : A → B and ˜ f : I → [ A , B ] such that ev ⋆ ( ˜ f ) = f and any obje ct D in S P C , al l the diagr ams in the p asting b elow c ommute [ A , B ] [ D , − ] / / [ f , B ]   [[ D , A ] , [ D , B ]] [ ˜ f , 1]   ev f q q q q x x q q q q [ D , B ] [ I , [ D , B ]] . ev ⋆ o o 56 The top left one c ommutes ac c or ding to Cor ol lary [Sch08]-11.6and the b ottom right one do es ac- c or ding to t he 2-natu r ality of the c ol le ction of arr ows q . 7.25 Deﬁnition of the 2-c el l c 2 x,g : y ′ → y ,z . Observe that the image by ev ⋆ : S P C ( I , [ A x,y ′ , A x,y ]) → S P C ( A x,y ′ , A x,y ) of I ˜ g / / A y ′ ,y A ( x, − ) / / [ A x,y ′ , A x,y ] is equa l to A (1 , g ) : A x,y ′ → A x,y . Therefor e according to Remark 7 .24 the domain o f this 2-cell which is A y ,z A ( x, − ) / / [ A x,y , A x,z ] [ A x,y ′ , − ] / / [[ A x,y ′ , A x,y ] , [ A x,y ′ , A x,z ]] [ A ( x, − ) , 1] / / [ A y ,y , [ A x,y ′ , A x,z ]] [ g , 1] / / [ I , [ A x,y ′ , A x,z ]] ... ... ev ⋆ / / [ A x,y ′ , A x,z ] is equal to A y ,z A ( x, − ) / / [ A x,y , A x,z ] [ A (1 ,g ) , 1] / / [ A x,y ′ , A x,z ] The co domain of the 2 -cell c 2 x,g : y ′ → y ,z rewrites success ively 1. A y ,z A ( y ′ − ) / / [ A y ′ ,y , A y ′ ,z ] [1 , A ( x, − )] / / [ A y ′ ,y , [ A x,y ′ , A x,z ]] [ g, 1] / / [ I , [ A x,y ′ , A x,z ]] ev ⋆ / / [ A x,y ′ , A x,z ] 2. A y ,z A ( y ′ − ) / / [ A y ′ ,y , A y ′ ,z ] [ g, 1] / / [ I , A y ′ ,z ] [1 , A ( x, − )] / / [ I , [ A x,y ′ , A x,z ]] ev ⋆ / / [ A x,y ′ , A x,z ] 3. A y ,z A ( y ′ − ) / / [ A y ′ ,y , A y ′ ,z ] [ g, 1] / / [ I , A y ′ ,z ] ev ⋆ / / A y ′ ,z A ( x, − ) / / [ A x,y ′ , A x,z ] 4. A y ,z A ( g, 1) / / A y ′ ,z A ( x, − ) / / [ A x,y ′ , A x,z ] where in the ab ov e deriv ation a rrows 2 . and 3. are equal due to Lemma [Sch08]-11.2. 7.26 Deﬁnition of the 2-c el l c 3 x,y ,h : z → z ′ . The 2 -cell I h / / A z ,z ′ A ( x, − ) / / [ A x,z , A x,z ′ ] [ A x,y , − ] / / [[ A x,y , A x,z ] , [ A x,y , A x,z ′ ]] [ A ( x, − ) , 1] / / [ A y ,z , [ A x,y , A x,z ′ ]] has imag e by ev ⋆ A y ,z A ( x, − ) / / [ A x,y , A x,z ] [1 , A (1 ,h )] / / [ A x,y , A x,z ′ ] which is the domain of c 3 x,y ,h : z → z ′ . The 2 -cell I h / / A z ,z ′ A ( y , − ) / / [ A y ,z , A y ,z ′ ] [1 , A ( x, − )] / / [ A y ,z , [ A x,y , A x,z ′ ]] has imag e by ev ⋆ A y ,z A (1 ,h ) / / A y ,z ′ A ( x, − ) / / [ A x,y , A x,z ′ ] which is the co domain o f c 3 x,y ,h : z → z ′ . 7.27 Pr o of of the e qu ivalenc e of Axioms 4.4/3. 14 . 57 The ﬁr st of the 2 -cell of Axiom 4 .4 decomp oses as the pro duct Ξ 2 ◦ Ξ 1 where Ξ 1 is (( A t,u A z ,t ) A y,z ) A x,y A ′ / / ( A t,u A z ,t )( A y,z A x,y ) 1 ⊗ c x,y,z / / ( A t,u A z ,t ) A x,z α x,z ,t,u + 3 A x,u and Ξ 2 is (( A t,u A z ,t ) A y,z ) A x,y ( c z ,t,u ⊗ 1) ⊗ 1 / / ( A z ,u A y,z ) A x,y α x,y,z ,u + 3 A x,u The 2-cell Ξ 1 has a strict domain which image by Rn is strict since the arr ows A ′ , Rn ( A ′ ) and c a re strict. Acco r ding to Lemma [Sc h0 8]-19.6, the 2-cell Ξ 1 has imag e b y R n ◦ Rn A t,u ⊗ A z ,t Rn ( α x,z ,t,u ) + 3 [ A x,z , A x,u ] [ A x,y , − ] / / [[ A x,y , A x,z ] , [ A x,y , A x,u ]] [ A ( x, − ) , 1] / / [ A y,z , [ A x,y , A x,u ]] . This 2-cell has a gain str ict domain a nd its image by R n is the 2- cell A t,u α ′ + 3 [ A z ,t , [ A x,z , A x,u ]] [1 , [ A x,y , − ]] / / [ A z ,t , [[ A x,y , A x,z ] , [ A x,y , A x,u ]]] [1 , [ A ( x, − ) , 1]] / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] . The ima g e by Rn o f Ξ 2 is ( A t,u ⊗ A z ,t ) ⊗ A y,z c z ,t,u ⊗ 1 / / A z ,u ⊗ A y,z Rn ( α x,y,z ,u ) + 3 [ A x,y , A x,u ] which has image b y R n the 2-cell A t,u ⊗ A z ,t c z ,t,u / / A z ,u α ′ x,y,z ,u + 3 [ A y,z , [ A x,y , A x,u ]] which has image b y R n the 2-cell A t,u A ( z, − ) / / [ A z ,t , A z ,u ] [1 ,α ′ x,y,z ,u ] + 3 [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] . The se cond 2-c e ll from Axio m 4.4 decomp oses as Ξ 5 ◦ Ξ 4 ◦ Ξ 3 where Ξ 3 is (( A t,u A z ,t ) A y,z ) A x,y A ′ ⊗ 1 / / ( A t,u ( A z ,t A y,z )) A x,y A ′ / / A t,u (( A z ,t A y,z )) A x,y ) 1 ⊗ α x,y,z ,t + 3 A t,u A x,t c / / A x,u Ξ 4 is (( A t,u A z ,t ) A y,z ) A x,y A ′ ⊗ 1 / / ( A t,u ( A z ,t A y,z )) A x,y (1 ⊗ c y,z ,t ) ⊗ 1 / / ( A t,u A y,t ) A x,y α x,y,t,u + 3 A x,u and Ξ 5 is (( A t,u A z ,t ) A y,z ) A x,y α y,z ,t,u ⊗ 1 + 3 A y,u A x,y c x,y,u / / A x,u The ima g e by Rn ◦ R n ◦ Rn of the 2-cell Ξ 3 is A t,u η / / [ A z ,t A y,z , A t,u ( A z ,t A y,z )] Rn / / [ A z ,t , [ A y,z , A t,u ( A z ,t A y,z )] [1 , [1 ,Rn ( A ′ )]] / / ... ... [ A z ,t , [ A y,z , [ A x,y , A t,u (( A z ,t A y,z ) A x,y )]]] [1 , [1 , [1 , 1 ⊗ α x,y,z ,t ]]] + 3 [ A z ,t , [ A y,z , [ A x,y , A t,u A x,t ]]] [1 , [1 , [1 ,c x,t,u ]]] / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] which r ewrites 1. A t,u η / / [ A z ,t A y,z , A t,u ( A z ,t A y,z )] [1 ,Rn ( A ′ )] / / [ A z ,t A y,z , [ A x,y , A t,u (( A z ,t A y,z ) A x,y )]] [1 , [1 , 1 ⊗ α x,y,z ,t ]] + 3 ... ... [ A z ,t A y,z , [ A x,y , A t,u A x,t ]] [1 , [1 ,c x,t,u ]] / / [ A z ,t A y,z , [ A x,y , A x,u ]] Rn / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] 2. A t,u η / / [( A z ,t A y,z ) A x,y , A t,u (( A z ,t A y,z ) A x,y )] Rn / / [ A z ,t A y,z , [ A x,y , A t,u (( A z ,t A y,z ) A x,y )]] [1 , [1 , 1 ⊗ α x,y,z ,t ]] + 3 ... ... [ A z ,t A y,z , [ A x,y , A t,u A x,t ]] [1 , [1 ,c x,t,u ]] / / [ A z ,t A y,z , [ A x,y , A x,u ]] Rn / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]] 58 3. A t,u η / / [( A z ,t A y,z ) A x,y , A t,u (( A z ,t A y,z ) A x,y )] [1 , 1 ⊗ α x,y,z ,t ] + 3 [( A z ,t A y,z ) A x,y , A t,u A x,t ] [1 ,c x,t,u ] / / ... ... [ ( A z ,t A y,z ) A x,y , A x,u ] Rn / / [ A z ,t A y,z , [ A x,y , A x,u ]] Rn / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] 4. A t,u η / / [ A x,t , A t,u A x,t ] [ α x,y,z ,t , 1] + 3 [( A z ,t A y,z ) A x,y , A t,u A x,t ] [1 ,c x,t,u ] / / [( A z ,t A y,z ) A x,y , A x,u ] Rn / / ... ... [ A z ,t A y,z , [ A x,y , A x,u ]] Rn / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] 5. A t,u A ( x, − ) / / [ A x,t , A x,u ] [ α x,y,z ,t , 1] / / [( A z ,t A y,z ) A x,y , A x,u ] [ A x,y , − ] + 3 [[ A x,y , ( A z ,t A y,z ) A x,y ] , [ A x,y , A x,u ]] [ η, 1] / / ... ... [ A z ,t A y,z , [ A x,y , A x,u ]] Rn / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] 6. A t,u A ( x, − ) / / [ A x,t , A x,u ] [ A x,y , − ] / / [[ A x,y , A x,t ] , [ A x,y , A x,u ]] [[1 ,α x,y,z ,t ] , 1] + 3 [[ A x,y , ( A z ,t A y,z ) A x,y ] , [ A x,y , A x,u ]] [ η, 1] / / ... ... [ A z ,t A y,z , [ A x,y , A x,u ]] Rn / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] 7. A t,u A ( x, − ) / / [ A x,t , A x,u ] [ A x,y , − ] / / [[ A x,y , A x,t ] , [ A x,y , A x,u ]] [ Rn ( α x,y,z ,t ) , 1] + 3 [ A z ,t A y,z , [ A x,y , A x,u ]] [ A y,z , − ] / / ... ... [ [ A y,z , A z ,t A y,z ] , [ A y,z , [ A x,y , A x,u ]]] [ η, 1] / / [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] 8. A t,u A ( x, − ) / / [ A x,t , A x,u ] [ A x,y , − ] / / [[ A x,y , A x,t ] , [ A x,y , A x,u ]] [ A y,z , − ] / / [[ A y,z , [ A x,y , A x,t ]] , [ A y,z , [ A x,y , A x,u ]]] ... ... [ α ′ x,y,z ,t , 1] + 3 [ A z ,t , [ A y,z , [ A x,y , A x,u ]]] The image by Rn o f the 2 -cell Ξ 4 is ( A t,u A z ,t ) A y ,z A ′ / / A t,u ( A z ,t A y ,z ) 1 ⊗ c y,z ,t / / A t,u A y ,t Rn ( α x,y,t,u ) + 3 [ A x,y , A x,u ] which according to Lemma [Sch08]-19.6as imag e b y R n ◦ Rn the 2 -cell A t,u α ′ x,y,t,u + 3 [ A y ,t , [ A x,y , A x,u ]] [ A y,z , − ] / / [[ A y ,z , A y ,t ] , [ A y ,z , [ A x,y , A x,u ]]] [ A ( y , − ) , 1] / / [ A z ,t , [ A y ,z , [ A x,y , A x,u ]]] . The image by Rn ◦ Rn ◦ R n of the 2-cell Ξ 5 is A t,u α ′ y,z ,t,u + 3 [ A z ,t , [ A y ,z , A y ,u ]] [1 , [1 , A ( x, − )]] / / [ A z ,t , [ A y ,z , [ A x,y , A x,u ]]] 7.28 Pr o of of the e qu ivalenc e of Axioms 4.5 and 3.15. PROOF:It is easy to chec k that the image by Rn of the 2 -cell Ξ 1 of Axiom 4.5 is A ( x, − ) y ,z ∗ ρ ′ y ,z the ﬁrs t 2- cell of the Axiom 3.15. The image by Rn o f the 2 -cell Ξ 2 is A y ,z A ( x, − ) / / [ A x,y , A x,z ] [ ev ⋆ , 1]   id [ λ ′ , 1]   [ A x,y , A x,z ] [[ I , A x,y ] , A x,z ] [[ u y , 1] , 1] / / [[ A y ,y , A x,y ] , A x,z ] [ A ( − ,y ) , 1] O O The 2 -cell Ξ 3 , namely A y ,z ⊗ A x,y R ′ ⊗ 1 / / ( A y ,z ⊗ I ) ⊗ A x,y (1 ⊗ u y ) ⊗ 1 / / A y ,z ⊗ A y ,y α x,y,y,z + 3 A x,z has imag e by Rn the 2 - cell A y ,z R ′ / / A y ,z ⊗ I 1 ⊗ u y / / A y ,z ⊗ A y ,y Rn ( α x,y,y,z ) / / [ A x,y , A x,z ] 59 which is accor ding to Lemma 7.9 A y ,z α ′ x,y,y,z + 3 [ A y ,y , [ A x,y , A x,z ]] [ u y , 1] / / [ I , [ A x,y , A x,z ]] [ ev ⋆ , 1] / / [ A x,y , A x,z ] 7.29 Equivale n c e of Axioms 4.7 and 3.18. First let us remark that the tw o 2 -cells of Axiom 3.18 hav e the s ame domain. This results from the following sequenc e of equa l arrows. 1. [ B F x ,F z , B F x ,F t ] [ B F x ,F y , − ] / / [[ B F x,F y , B F x,F z ] , [ B F x,F y , B F x,F t ]] [ B ( F x, − ) , 1] / / [ B F y,F z , [ B F x,F y , B F x,F t ]] [ F y,z , 1] / / ... ... [ A y,z , [ B F x,F y , B F x ,F t ]] [1 , [ F x,y , 1]] / / [ A y,z , [ A x,y , B F x,F t ]] 2. [ B F x ,F z , B F x ,F t ] [ B F x ,F y , − ] / / [[ B F x,F y , B F x,F z ] , [ B F x,F y , B F x,F t ]] [ B ( F x, − ) , 1] / / [ B F y,F z , [ B F x,F y , B F x,F t ]] [1 , [ F x,y , 1]] / / ... ... [ B F y,F z , [ A x,y , B F x ,F t ]] [ F y,z , 1] / / [ A y,z , [ A x,y , B F x,F t ]] 3. [ B F x ,F z , B F x ,F t ] [ B F x ,F y , − ] / / [[ B F x,F y , B F x,F z ] , [ B F x,F y , B F x,F t ]] [1 , [ F x,y , 1]] / / [[ B F x,F y , B F x,F z ] , [ A x,y , B F x ,F t ]] [ B ( F x, − ) , 1] / / ... ... [ B F y,F z , [ A x,y , B F x ,F t ]] [ F y,z , 1] / / [ A y,z , [ A x,y , B F x,F t ]] 4. [ B F x ,F z , B F x ,F t ] [ A x,y , − ] / / [[ A x,y , B F x,F z ] , [ A x,y , B F x ,F t ]] [[ F x,y , 1] , 1] / / [[ B F x,F y , B F x,F z ] , [ A x,y , B F x,F t ]] [ B ( F x, − ) , 1] / / ... ... [ B F y,F z , [ A x,y , B F x ,F t ]] [ F y,z , 1] / / [ A y,z , [ A x,y , B F x,F t ]] . In the ab ov e der iv atio n arr ows 3. a nd 4 . a re equa l accor ding to Lemma [Sch08]-9.11. The 2-cell ( A z ,t ⊗ A y ,z ) ⊗ A x,y ( F z ,t ⊗ F y,z ) ⊗ 1 / / ( B F z,F t ⊗ B F y, F z ) ⊗ A x,y 1 ⊗ F x,y / / ( B F z,F t ⊗ B F y, F z ) ⊗ B F x,F y α F x,F y ,F z ,F t + 3 B F x,F t has a strict domain and image by Rn A z ,t ⊗ A y ,z F z ,t ⊗ F y,z / / B F z,F t ⊗ B F y, F z Rn ( α ′ F x ,F y,F z ,F t ) / / [ B F x,F y , B F x,F t ] [ F x,y , 1] + 3 [ A x,y , B F x,F t ] which has a strict do ma in and image by Rn A z ,t F z ,t / / B F z,F t α ′ + 3 [ B F y, F z , [ B F x,F y , B F x,F t ]] [ F y,z , 1] / / [ A y ,z , [ B F x,F y , B F x,F t ]] [1 , [ F x,y , 1]] / / [ A y ,z , [ A x,y , B F x,F t ]] . The 2 -cell ( A z ,t ⊗ A y ,z ) ⊗ A x,y F 2 y,z ,t ⊗ 1 + 3 B F y, F t ⊗ A x,y 1 ⊗ F x,y / / B F y, F t ⊗ B F x,F y c / / B F x,F t has imag e by Rn that rewrites A z ,t ⊗ A y ,z F 2 + 3 B F y, F t Rn ( c ⊗ (1 ⊗ F x,y )) / / [ A x,y , B F x,F t ] A z ,t ⊗ A y ,z F 2 + 3 B F y, F t B ( F x, − ) / / [ B F x,F y , B F x,F t ] [ F x,y , 1] / / [ B F x,F y , B F x,F t ] The image by Rn o f this la st ar r ow is A z ,t F ′ 2 + 3 [ A y ,z , B F y, F t ] [1 , B ( F x, − )] / / [ A y ,z , [ B F x,F y , B F x,F t ]] [1 , [ F x,y , 1]] / / [ A y ,z , [ B F x,F y , B F x,F t ]] . The 2 -cell ( A z ,t ⊗ A y ,z ) ⊗ A x,y c ⊗ 1 / / A y ,t ⊗ A x,y F 2 x,y,t + 3 B F x,F t 60 has imag e by Rn the 2-cell A z ,t ⊗ A y ,z c / / A y ,t F ′ 2 + 3 [ A x,y , B F x,F t ] which has image b y R n A z ,t A ( y , − ) / / [ A y ,z , A y ,t ] [1 ,F ′ 2 x,y,t ] + 3 [ A y ,z , [ A x,y , B F x,F t ]] . The 2-cell A z ,t ⊗ B F x,F z F z ,t ⊗ 1 / / B F z,F t ⊗ B F x,F z c F x,F z ,F t / / B F x,F t has imag e by Rn the 2-cell A z ,t F z ,t / / B F z,F t B ( F x, − ) / / [ B F x,F z , B F x,F t ] therefore a ccording to Lemma 7.2 1 the 2 -cell ( A z ,t ⊗ A y,z ) ⊗ A x,y A ′ / / A z ,t ⊗ ( A y,z ⊗ A x,y ) 1 ⊗ F 2 x,y,z + 3 A z ,t ⊗ B F x,F z F z ,t ⊗ 1 / / B F z ,F t ⊗ B F x,F z c F x ,F z ,F t / / B F x,F t has imag e by Rn ◦ R n the 2-cell A z ,t F z ,t / / B F z ,F t B ( F x, − ) / / [ B F x,F z , B F x,F t ] [ A x,y , − ] / / [[ A x,y , B F x ,F z ] , [ A x,y , B F x,F t ]] [ F ′ 2 x,y,z , 1] + 3 [ A y,z , [ A x,y , B F x,F t ]] . According to Lemma 7 .21 the 2-cell ( A z ,t ⊗ A y ,z ) ⊗ A x,y A ′ / / A z ,t ⊗ ( A y ,z ⊗ A x,y ) 1 ⊗ c x,y,z / / A z ,t ⊗ A x,z F 2 x,z ,t + 3 B F x,F t has imag e by Rn ◦ R n the 2-cell A z ,t F ′ 2 x,z ,t + 3 [ A x,z , B F x,F t ] [ A x,y , − ] / / [[ A x,y , A x,z ] , [ A x,y B F x,F t ]] [ A ( x, − ) , 1] / / [ A y ,z , [ A x,y , B F x,F t ]] . 7.30 Equivale n c e of Axioms 4.8 and 3.19. PROOF:According to Lemma 7.9, the arr ow A x,y R ′ / / A x,y ⊗ I 1 ⊗ u x / / A x,y ⊗ A x,x F 2 x,x,y + 3 A x,y is e q ual to A x,y F ′ 2 x,x,y + 3 [ A x,x , B F x,F y ] [ u x , 1] / / [ I , B F x,F y ] ev ⋆ / / B F x,F y . Since the image by R n o f A x,y ⊗ B F x,F x F x,y ⊗ 1 / / B F x,F y ⊗ B F x,F x c / / B F x,F y is A x,y F x,y / / B F x,F y B ( F x, − ) / / [ B F x,F x , B F x,F y ] , 61 the ar row A x,y R ′ / / A x,y ⊗ I 1 ⊗ F 0 x + 3 A x,y ⊗ B F x,F x F x,y ⊗ 1 / / B F x,F y ⊗ B F x,F x c / / B F x,F y is e q ual to A x,y F x,y / / B F x,F y B ( F x, − ) / / [ B F x,F x , B F x,F y ] [ F 0 x , 1] + 3 [ I , B F x,F y ] ev ⋆ / / B F x,F y according to Lemma 7.9. 7.31 Equivale n c e of Axioms 4.9 and 3.20. PROOF:The arrow B F y ,F y ⊗ A x,y 1 ⊗ F x,y / / B F y ,F y ⊗ B F x,F y c / / B F x,F y has imag e by Rn B F y ,F y B ( F x, − ) / / [ B F x,F y , B F x,F y ] [ F x,y , 1] / / [ A x,y , B F x,F y ] which has dual A x,y F x,y / / B F x,F y B ( − ,F y ) / / [ B F y ,F y , B F x,F y ] . Therefore according to Lemma 7.8, the 2-cell A x,y L ′ / / I ⊗ A x,y F 0 y ⊗ 1 + 3 B F y ,F y ⊗ A x,y 1 ⊗ F x,y / / B F y ,F y ⊗ B F x,F y c / / B F x,F y is e q ual to A x,y F x,y / / B F x,F y B ( − ,F y ) / / [ B F y ,F y , B F x,F y ] [ F 0 y , 1] + 3 [ I , B F x,F y ] ev ⋆ / / B F x,F y . According to Lemma 7.8 the 2-cell A x,y L ′ / / I ⊗ A x,y u y ⊗ 1 / / A y ,y ⊗ A x,y F 2 x,y,y / / A x,y is e q ual to A x,y ( F ′ 2 x,y,y ) ∗ / / [ A y ,y , A x,y ] [ u y , 1] / / [ I , A x,y ] ev ⋆ / / A x,y Section 5. W e will need the following ch a racteriza tion o f bilinear natural tr a nsformations . Remark 7.32 F or any symmetric monoidal fun ctors F , G : A → [ B , C ] with r esp e ctive underlying functors F ′ , G ′ : A × B → C . Any 2-c el l σ : F → G : A → [ B , C ] in S P C c orr esp onds to a c ol le ction of arr ows σ a,b : F ′ ( a, b ) → G ′ ( a, b ) in C , n atur al in a and b and that satisﬁes the fol lowing two c onditions: - F or al l obje cts a , a ′ , b and b ′ of A the fol lowing diagr am c ommutes F ′ ( a, b ) + F ′ ( a, b ′ ) σ a,b + σ a,b ′   F ∗ 2 b,b ′ a / / F ′ ( a, b + b ′ ) σ a,b + b ′   F ′ ( a ′ , b ) + F ′ ( a ′ , b ′ ) F ∗ 2 b,b ′ ′ a / / F ′ ( a ′ , b + b ′ ); 62 - F or al l obje cts a , a ′ , b and b ′ of A the fol lowing diagr am c ommutes F ′ ( a, b ) + F ′ ( a ′ , b ) σ a,b + σ a ′ ,b   F 2 a,a ′ b / / F ′ ( a + a ′ , b ) σ a + a ′ ,b   F ′ ( a, b ′ ) + F ′ ( a ′ , b ′ ) F 2 a,a ′ b ′ / / F ′ ( a + a ′ , b ′ ) 7.33 Pr o of that 2-rings in the sense of 5.1 ar e exactly one p oint S P C -c ate gories. PROOF:Let us start with a 2-ring A as deﬁned in 5.1. Its cor resp onds to a o ne-p oint S P C -categ ory as follows. Since monoidal functors I → A are in one- to-one corr esp ondence with ob jects of A , the ob ject 1 corresp ond to a strict arrow u : I → A in S P C . That the m ultiplication “ . ”, whic h is a lr eady a functor A × A → A , deﬁnes an arrow ϕ ′ : A → [ A , A ] in S P C corresp onds to the existence of the na tural arrows a b,b ′ and b a,a ′ and the commutation of Diag rams 5.2, 5.3, 5.4, 5 .5 and 5.6. Pr e c isely for any ob ject a of A , the mult iplica tion on the le ft by a , deﬁnes a functor a. − : A → A . That this o ne is monoidal corr esp onds to the existence of the arr ows a b,b ′ natural in b a nd b ′ and such that Diagrams 5.2 commute for all b , b ′ and b ′′ . That the mono idal a. − is symmetric corres po nds to the comm utation of 5.4 for all b and b ′ . Tha t for any arr ow f : a → a ′ , the natural trans formation f . − : a. − → a ′ . − : A → A is monoidal for the structures describ ed previously on a. − a nd a ′ . − corres po nds to the naturality in the argument a o f the maps a b,b ′ . The assignments a 7→ a. − a nd ( f : a → a ′ ) 7→ ( f . − : a. − → a ′ . − ) deﬁne ther efore a functor say ϕ ′ : A → S P C ( A , A ). The ex istence of a natur al transforma tion ( a. − ) + ( a ′ . − ) → ( a + a ′ ) . − corres p o nds to the existence of a rrows b a,a ′ natural in b . This trans fo rmation is mo noidal since Diagrams 5.6 commute. Even tually that the collection of these arr ows for a ll a and a ′ deﬁnes a symmetric mo no idal str uc tur e on the ab ov e functor ϕ ′ results fro m the the natura lity of the collection in the arguments a and a ′ and the commutation Diagra ms 5.3 and 5.5. According to Remar k 7.32, a 2-cell α ′ in S P C as in 3.11 corr esp onds to a na tural collection of ar rows ˜ α a,b,c : a. ( b.c ) → ( a.b ) .c in A such that Diag rams 5.7, 5.8 and 5 .9 commu te. A 2-cell ρ ′ : 1 → ev ⋆ ◦ [ u, 1] ◦ ϕ ′ : A → A in S P C corr e sp onds to a natural c ollection of arrows ˜ ρ a : a. 1 → a such tha t Diagrams 5.10 commute. Similarly a 2-cell λ ′ : 1 → ev ⋆ ◦ [ u, 1] ◦ ϕ ′∗ : A → A amounts to a natural collection o f arr ows ˜ λ a : 1 .a → a such that Diagra ms 5.11 commut e. Then the coherence conditions 3.14 and 3.1 5 for α ′ , ρ ′ and λ ′ ab ov e ar e equiv alent to the co- herence conditions for the asso ciativit y and unit laws of the monoidal categor y ( A , ., I , ˜ α, ˜ ρ, ˜ λ ). 7.34 Pr o of that 2-ring morphisms in the sense of 5.14 ar e exactly S P C -functors. PROOF:According to the tw o o bserv ations below the result b eco mes clear a fter insp e ction o f Ax- ioms 3.1 8, 3.1 9 and 3.2 0. Observe ﬁrst that according to Remark 7.32 a 2-cell A H   ϕ ′ / / [ A , A ] [1 ,H ] # # H H H H H H H H H [ A , B ] B ϕ ′ / / B J                               [ B , B ] [ ϕ ′ , 1] ; ; v v v v v v v v v 63 in S P C is the same thing than a c ollection of arrows Ξ a,b : H a.H b → H ( a + b ) natura l in a and b and satisfying the conditions that fo r any ob jects a , b , c of A , the tw o diag rams in B b elow commute 7.35 H ( a ) .H ( b ) + H ( a ) .H ( c ) H ( a ) H ( b ) ,H ( c ) / / Ξ a,b +Ξ a,c   H ( a ) . ( H ( b ) + H ( c )) H ( a ) .H 2 b,c / / H ( a ) .H ( b + c ) Ξ a,b + c   H ( a.b ) + H ( a.c ) H 2 a.b,a.c / / H ( a.b + a.c ) H ( a b,c ) / / H ( a. ( b + c )) 7.36 H ( a ) .H ( c ) + H ( b ) .H ( c ) H ( c ) H ( a ) ,H ( b ) / / Ξ a,c +Ξ b,c   ( H ( a ) + H ( b )) .H ( c ) H 2 a,b .H ( c ) / / H ( a + b ) .H ( c ) Ξ a + b,c   H ( a.c ) + H ( b.c ) H 2 a.c,b.c / / H ( a.c + b.c ) H ( c a,b ) / / H (( a + b ) .c ) where we write the ϕ ′ as pro ducts. Also according to Remark 7.10 in Appendix an y 2 -cell A H   I u ? ?        u   ? ? ? ? ? ? ? + 3 B in S P C is fully determined by its comp onent a t the genera to r ⋆ which is a n arrow 1 B → H (1 A ) if 1 A and 1 B denote the ima ges u ( ⋆ ), and co nv ersely ev er y such ar row co rresp onds to a 2-cell as ab ov e in this way . 7.37 Pr o of of Pr op osition 5.17. PROOF:That there ar e ident ity 2- cells 3.1 1 amounts to the fact that the diagra m in S P C [ C , D ] [ B , − ] v v n n n n n n n n n n n n n [ A , − ] ) ) S S S S S S S S S S S S S S S [[ B , C ] , [ B , D ]] [1 , [ A , − ]]   [[ A , C ] , [ A , D ]] [[ A , B ] , − ]   [[ B , C ] , [[ A , B ] , [ A , D ]] [[[ A , B ] , [ A , C ]] , [[ A , B ] , [ A , D ]] [[ A , − ] , 1] o o commutes for any ob jects A , B , C and D . Such a diagram inv olves only stric t arrows in S P C and 64 its under lying diag ram in Cat is S P C ( C , D ) P ost u u j j j j j j j j j j j j j j j P ost * * U U U U U U U U U U U U U U U U U S P C ([ B , C ] , [ B , D ]) S P C (1 , [ A , − ])   S P C ([ A , C ] , [ A , D ]) P ost   S P C ([ B , C ] , [[ A , B ] , [ A , D ]]) S P C ([[ A , B ] , [ A , C ]] , [[ A , B ] , [ A , D ]]) S P C ([ A , − ] , 1) o o which commut es accor ding to Lemma [Sch08]-9.10. One has identit y 2 - cells ρ ′ since for any ob jects A a nd B in S P C the comp osite [ A , B ] [ A , − ] / / [[ A , A ] , [ A , B ]] [ v, 1] / / [ I , [ A , B ]] ev ⋆ / / [ A , B ] which is strict and has underlying functor that is an identit y , hence is an iden tity in S P C . One has identit y 2 - cells λ ′ since the comp osite [ A , B ] [ − , B ] / / [[ B , B ] , [ A , B ]] [ v, 1] / / [ I , [ A , B ]] ev ⋆ / / [ A , B ] is the identit y at [ A , B ] as shown b elow. The ar row [ A , B ] [ − , B ] / / [[ B , B ] , [ A , B ]] [ v, 1] / / [ I , [ A , B ]] is v ∗ since it has dual I v / / [ B , B ] [ A , − ] / / [[ A , B ] , [ A , B ]] that is v according to Lemma [Sch08]-18.5. One concludes s ince ev ⋆ ◦ v ∗ = 1. 7.38 Pr o of of Pr op osition 5.19 PROOF:That one has an identit y 2-cell 3.11 amo unts to the commutation of the diagr am I v $ $ H H H H H H H H H v y y s s s s s s s s s s [ I , I ] [ I , − ]   [ I , I ] [1 ,v ]   [[ I , I ] , [ I , I ]] [ v, 1] / / [ I , [ I , I ]] . This dia gram commutes since the arr ow I v / / [ I , I ] [ I , − ] / / [[ I , I ] , [ I , I ]] [ v, 1] / / [ I , [ I , I ]] is I v / / [[ I , I ] , [ I , I ]] [ v, 1] / / [ I , [ I , I ]] according to Lemma [Sch08]-18.5, whic h is e q ual to I v / / [ I , I ] [1 ,v ] / / [ I , [ I , I ]] . 65 according Lemma [Sch08]-18.8. One has an identit y 2-cell ρ ′ since the comp osite I v / / [ I , I ] ev ⋆ / / I is the identit y . One has also an identit y 2-cell λ ′ since the comp osite I v ∗ / / [ I , I ] ev / / I is a n identit y according to Lemma [Sch08]-18.4. 7.39 Pr o of of L emma 5.18. PROOF:According to Lemma [Sch08]-11.4, the diagr am in S P C [ I , [ A , B ]] ev ⋆ / / [ − , [ A , C ]]   [ A , B ] q   [[[ A , B ] , [ A , C ]] , [ I , [ A , C ]] [1 ,ev ⋆ ] / / [[[ A , B ] , [ A , C ]] , [ A , C ]] commutes, ther efore a lso do es the diagram [[ A , B ] , [ A , C ]] [ ˜ F , 1] / / ev ⋆ ' ' O O O O O O O O O O O [ I ′ [ A , C ]] ev ⋆   [ A , C ] . F ro m this and a ccording to Corolla ry [Sc h08]-1 1.6all diag r ams in the pasting b elow [ B , C ] [ F , C ] / / [ A , − ]   [ A , C ] [[ A , B ] , [ A , C ]] [ ˜ F , 1] / / ev F o o o o o 7 7 o o o o o [ I , [ A , C ]] ev ⋆ O O commute and acco rding to Lemma 7.9 the comm utation of the external diagra m ab ove is equiv alent to the commutation of the ﬁrst diagra m of the L e mma . In the pasting [ C , A ] [ C ,F ] / / [ − , B ]   [ C , B ] [[ A , B ] , [ C , B ]] [ ˜ F , 1] / / ev F o o o o o 7 7 o o o o o [ I , [ C , B ]] ev ⋆ O O the top-left diag ram commutes a ccording to Co r ollary [Sch08]-11.8and we hav e already seen that the bottom- left diagram co mm utes. The commutation of the externa l diag ram a bove is equiv alent to the commutation of the second diagram of the Lemma acco rding to Lemma 7.8. Section 6. 66 7.40 Deﬁnition of the 2-c el l θ fr om Axiom 6.22. PROOF:The arrows A ⊗ M L ′ ⊗ 1 / / ( I ⊗ A ) ⊗ M A ′ / / I ⊗ ( A ⊗ M ) is s trict and, as shown b elow, it has the same imag e by Rn a s the ar row L ′ : A ⊗ M → I ⊗ ( A ⊗ M ). The 2 -cell θ corresp o nds then via the adjunction 2.5 to the identit y 2-ce ll. The ima g e by Rn o f L ′ A is A η / / [ M , AM ] [1 ,η ∗ ] / / [ M , [ I , I ( AM )]] [1 ,ev ⋆ ] / / [ M , I ( AM )] . The a rrow A ′ ◦ L ′ ⊗ 1 ab ov e as image by Rn that r ewrites 1. A L ′ / / IA Rn ( A ′ ) / / [ M , I ( AM )] 2. A η ∗ / / [ I , I A ] ev ⋆ / / IA Rn ( A ′ ) / / [ M , I ( AM )] 3. A η ∗ / / [ I , I A ] [1 ,Rn ( A ′ )] / / [ I , [ M , I ( AM )]] ev ⋆ / / [ M , I ( AM )] 4. A Rn ( Rn ( A ′ )) ∗ / / [ I , [ M , I ( AM )]] ev ⋆ / / [ M , I ( AM )] 5. A η / / [ M , AM ] [ − , I ( AM )] / / [[ AM , I ( AM )] , [ M , I ( AM )]] [ η, 1] / / [ I , [ M , I ( AM )]] ev ⋆ / / [ M , I ( AM )] 6. A η / / [ M , AM ] [1 ,η ∗ ] / / [ M , [ I , I ( AM )]] D / / [ I , [ M , I ( AM )]] ev ⋆ / / [ M , I ( AM )] 7. A η / / [ M , AM ] [1 ,η ∗ ] / / [ M , [ I , I ( AM )]] [1 ,ev ⋆ ] / / [ M , I ( AM )] where in the ab ov e deriv ation the equalities b etw een a rrows ho ld for the following rea s ons: - 4 . and 5. b y deﬁnition of A ′ since its image b y Rn ◦ R n is in this case I η / / [ AM , I ( AM )] [ M , − ] / / [[ M , AM ] , [ M , I ( AM )]] [ η, 1] / / [ A , [ M , I ( AM )]] that has dual A η / / [ M , AM ] [ − , I ( AM )] / / [[ AM , I ( AM )] , [ M , I ( AM )]] [ η, 1] / / [ I , [ M , I ( AM )]] ; - 5 . and 6. b y Lemma [Sch08]-10.8; - 6 . and 7. b y Lemma [Sch08]-11.9. 7.41 Pr o of of 6.26. PROOF:The ﬁr st of the 2-cells of Axio m 6.20 can b e decomp osed as the comp osite Ξ 2 ◦ Ξ 1 where Ξ 1 is (( AA ) A ) M A ′ / / ( AA )( AM ) 1 ⊗ ϕ / / ( AA ) M β + 3 M and Ξ 2 is (( AA ) A ) M ( c ⊗ 1) ⊗ 1 / / ( AA ) M β + 3 M . The 2 -cell of 6.26 decomp ose s as Ξ 4 ◦ Ξ 3 where Ξ 3 is ( A ⊗ A ) ⊗ A β ′′ ⊗ 1 + 3 [ M , M ] ⊗ A 1 ⊗ ϕ ′ / / [ M , M ] ⊗ [ M , M ] c / / [ M , M ] and Ξ 4 is ( AA ) A c ⊗ 1 / / AA β ′′ + 3 [ M , M ] . The 2 -cell Ξ 3 has a strict domain and an easy computation gives that its image by R n is A ⊗ A β ′′ + 3 [ M , M ] [ M , − ] / / [[ M , M ] , [ M , M ]]] [ ϕ ′ , 1] / / [ A , [ A , [ M , M ]]] . 67 According to Lemma 7.8 the 2-cell Ξ 1 has an image by Rn with a strict domain and has image b y Rn ◦ R n the 2-cell AA Rn ( β ) + 3 [ M , M ] [ M , − ] / / [[ M , M ] , [ M , M ]] [ ϕ ′ , 1] / / [ A , [ M , M ]] . Therefore the 2- cell Ξ 1 has imag e by Rn the 2-cell Ξ 3 . An eas y computation shows that the 2-cell Ξ 2 has imag e by Rn ( AA ) A c ⊗ 1 / / AA Rn ( β ) + 3 [ M , M ] which is Ξ 4 . 7.42 Pr o of of 6.27. PROOF:The second 2-cells of Axiom 6.2 0 can b e decomp osed as the co mpo site Ξ 3 ◦ Ξ 2 ◦ Ξ 1 where Ξ 1 is (( AA ) A ) M A ′ ⊗ 1 / / ( A ( AA )) M A ′ / / A (( AA ) M ) 1 ⊗ β + 3 AM ϕ / / M Ξ 2 is (( AA ) A ) M A ′ ⊗ 1 / / ( A ( AA )) M (1 ⊗ c ) ⊗ 1 / / ( AA ) M β + 3 M and Ξ 3 is (( AA ) A ) M α ⊗ 1 + 3 AM ϕ / / M . The 2-cell of 6 .27 decomp ose s as Ξ 6 ◦ Ξ 5 ◦ Ξ 4 where Ξ 4 is the 2-c e ll ( AA ) A A ′ / / A ( AA ) 1 ⊗ β ′′ + 3 A ⊗ [ M , M ] ϕ ′ ⊗ 1 / / [ M , M ] ⊗ [ M , M ] c / / [ M , M ] , Ξ 5 is ( AA ) A A ′ / / A ( AA ) 1 ⊗ c / / AA β ′′ + 3 [ M , M ] and Ξ 6 is ( AA ) A α + 3 A ϕ ′ / / [ M , M ] . The 2-cell Ξ 1 has image by Rn the 2- cell ( AA ) A A ′ / / A ( AA ) Rn ( A ′ ) / / [ M , A (( AA ) M )] [1 , 1 ⊗ β ] + 3 [ M , AM ] [1 ,ϕ ] / / [ M , M ] The 2-cell Ξ 7 = A ( AA ) Rn ( A ′ ) / / [ M , A (( AA ) M )] [1 , 1 ⊗ β ] + 3 [ M , AM ] [1 ,ϕ ] / / [ M , M ] has a strict domain and acco rding to Lemma [Sch08]-19.6its image by Rn is Ξ 8 = A ϕ ′ / / [ M , M ] [ M , − ] / / [[ M , M ] , [ M , M ]] [ β ′′ , 1] / / [ AA , [ M , M ]] . The 2 -cell Ξ 9 = A ( AA ) 1 ⊗ β ′′ + 3 A ⊗ [ M , M ] ϕ ′ ⊗ 1 / / [ M , M ] ⊗ [ M , M ] c / / [ M , M ] 68 has also a str ic t domain and its image by R n is Ξ 8 , therefore Ξ 7 = Ξ 9 and Rn (Ξ 1 ) = Ξ 7 ∗ A ′ = Ξ 9 ∗ A ′ = Ξ 4 . The image by Rn o f Ξ 2 is ( AA ) A A ′ / / A ( AA ) 1 ⊗ c / / AA Rn ( β ) / / [ M , M ] which is Ξ 5 . Even tually the image by Rn of Ξ 3 is tr ivially Ξ 6 . T o prov e 6.2 9, we s hall use the following lemma. Lemma 7.43 F or any obje cts A , B , C and D , any 2-c el l C τ + 3 [ D , A ] of S P C and any c ∈ C , the 2-c el ls in S P C [ A , B ] [ D , − ] / / [[ D , A ] , [ D , B ]] [ τ , 1] + 3 [ C , [ D , B ]] ev c / / [ D , B ] and [ A , B ] [ ev c , 1] / / [[ C , A ] , B ] [ τ ∗ , 1] + 3 [ D , B ] ar e e qu al. PROOF:According to Lemma [Sch08]-11.9, The ﬁrst of the 2-c e ll rewrites 1. [ A , B ] [ D , − ] / / [[ D , A ] , [ D , B ]] [ τ , 1] + 3 [ C , [ D , B ]] D / / [ D , [ C , B ]] [1 ,ev c ] / / [ D , B ] according to Lemma [Sch08]-10.9, 2. [ A , B ] [ C , − ] / / [[ C , A ] , [ C , B ]] [ τ ∗ , 1] / / [ D , [ C , B ]] and this las t arrow ha s dual D τ ∗ + 3 [ C , A ] [ − , B ] / / [[ A , B ] , [ C , B ]] [1 ,ev c ] / / [ A , B ] , B ] . One the other ha nd the dual of the second 2-cell of the lemma rewr ites 1. D τ ∗ + 3 [ C , A ] [ ev c , 1] ∗ / / [[ A , B ] , B ] 2. D τ ∗ + 3 [ C , A ] ev c / / A q / / [[ A , B ] , B ] 3. D τ ∗ + 3 [ C , A ] [ − , B ] / / [[ A , B ] , [ C , B ]] [1 ,ev c ] / / [[ A , B ] , B ] where in the ab ov e deriv ation a rrows 2 . and 3. are equal by Corolla ry [Sc h08]-1 1.4. 7.44 Pr o of of 6.29. PROOF:The 2-cell from 6.29 deco mp o ses a s ζ 2 ◦ ζ 1 where ζ 1 = A R ′ / / A ⊗ I 1 ⊗ γ ′ + 3 A ⊗ [ M , M ] ϕ ′ ⊗ 1 / / [ M , M ] ⊗ [ M , M ] c / / [ M , M ] and ζ 2 = A R ′ / / A ⊗ I 1 ⊗ u / / A ⊗ A β ′′ + 3 [ M , M ] . W e show be low that the image by Rn o f the 2-cell Ξ 2 is ζ 1 whereas the image by Rn of Ξ 3 is ζ 2 . The image by Rn o f the 2 -cell Ξ 2 rewrites 69 1. A η / / [ A , AM ] [ γ , 1] + 3 [ M , A ⊗ M ] [1 ,ϕ ] / / [ M , M ] 2. A η / / [ M , AM ] [1 ,ϕ ] / / [ M , M ] [ γ , 1] + 3 [ M , M ] 3. A ϕ ′ / / [ M , M ] [ γ , 1] + 3 [ M , M ] According to Lemma 7 .43 and since the 2-cell γ is ev ⋆ ∗ ( γ ′ ), the 2-cell 3. ab ove is A ϕ ′ / / [ M , M ] [ M , − ] / / [[ M , M ] , [ M , M ]] [ γ ′ , 1] + 3 [ I , [ M , M ]] ev ⋆ / / [ M , M ] . This last 2-cell is actually ζ 1 . T o chec k this use Lemma 7.21 and the fact that the image b y R n of the ar row A ⊗ [ M , M ] ϕ ′ ⊗ 1 / / [ M , M ] ⊗ [ M , M ] c / / [ M , M ] is A ϕ ′ / / [ M , M ] [ M , − ] / / [[ M , M ] , [ M , M ]] . The image by Rn o f the 2 -cell Ξ 3 is A R ′ / / A ⊗ I 1 ⊗ u / / A ⊗ A Rn ( β ) + 3 [ M , M ] which is ζ 2 . 7.45 Pr o of of 6.31. PROOF:The 2-cell Ξ 2 rewrites A ⊗ M ϕ / / M id ϕ ′∗   γ ′′   M [ A , M ] [ u, 1] / / [ I , M ] ev ⋆ O O which image by Rn is the 2-cell A ϕ ′ / / [ M , M ] id [1 ,ϕ ′∗ ]   [1 ,γ ′′ ]   [ M , M ] [ M , [ A , M ]] [1 , [ u, 1]] / / [ M , [ I , M ]] . [1 ,ev ⋆ ] O O On the other hand the 2-cell A L ′ / / I ⊗ A 1 ⊗ ϕ ′ / / I ⊗ [ M , M ] γ ′ ⊗ 1 + 3 [ M , M ] ⊗ [ M , M ] c / / [ M , M ] rewrites 1. A ϕ ′ / / [ M , M ] L ′ / / I ⊗ [ M , M ] γ ′ ⊗ 1 + 3 [ M , M ] ⊗ [ M , M ] c / / [ M , M ] 2. A ϕ ′ / / [ M , M ] [ − , M ] / / [[ M , M ] , [ M , M ]] [ γ ′ , 1] + 3 [ I , [ M , M ]] ev ⋆ / / [ M , M ] 3. A ϕ ′ / / [ M , M ] [1 ,γ ′ ∗ ] + 3 [ M , [ I , M ]] D / / [ I , [ M , M ]] ev ⋆ / / [ M , M ] 4. A ϕ ′ / / [ M , M ] [1 ,γ ′ ∗ ] + 3 [ M , [ I , M ]] [1 ,ev ⋆ ] / / [ M , M ] 5. A ϕ ′ / / [ M , M ] [1 ,γ ′′ ] + 3 [ M , M ] . 70 In the ab ov e der iv atio n the equality betw een arr ows hold for the fo llowing reasons: - 1 . and 2. b y Lemma 7.8, - 2 . and 3. b y Lemma [Sch08]-10.8(whic h can b e impr oved to take 2-cells into account), - 3 . and 4. b y Lemma [Sch08]-11.9. By deﬁnition of the 2-cell θ , the 2 -cell Ξ 3 has image by Rn a n identit y 2- cell. Even tually it is rather s traightforward that 2-c e ll Ξ 4 has imag e by Rn A L ′ / / I ⊗ A u ⊗ 1 / / A ⊗ A β ′′ + 3 [ M , M ] since the 2- cell β ′′ is Rn ( β ). 7.46 PR OO F of 6.32 PROOF:The 2-cell ( AA ) M 1 ⊗ H / / AAN β + 3 N has a strict domain 1 ⊗ H / / A ′ / / 1 ⊗ ψ / / ψ / / and ha s image by Rn AA β ′′ + 3 [ N , N ] [ H, 1] / / [ M , N ] which is image b y Rn is Ξ 1 . The 2- cell ( AA ) M c ⊗ 1 / / AM δ + 3 N has imag e by Rn the 2 - cell AA c / / A δ ′ + 3 [ M , N ] which image by Rn is Ξ 2 . 7.47 Pr o of of 6.33. PROOF:According to Lemma 7.21, the 2- cell ( AA ) M A ′ / / A ( AM ) 1 ⊗ δ + 3 AN ψ / / N has imag e by Rn ◦ R n the 2-ce ll Ξ 4 . According to Lemma 7.21, the 2- cell ( AA ) M A ′ / / A ( AM ) 1 ⊗ ϕ / / AM δ + 3 N has imag e by Rn ◦ R n the 2-ce ll Ξ 6 . Even tually the 2-cell ( AA ) M β + 3 M σ / / N has imag e by Rn ◦ R n the 2-ce ll Ξ 7 . 7.48 Pr o of of 6.34. 71 PROOF:The 2-cell u 2 H : H → [1 , H ] ◦ v : I → [ M , N ] has imag e by ev ⋆ an identit y 2-cell a nd the 2-cell I γ ′ + 3 [ M , M ] [1 ,H ] / / [ M , N ] has imag e by ev ⋆ the 2- cell M γ + 3 M H / / N 7.49 Pr o of of 6.35. PROOF:The arrow H : I → [ M , N ] is str ict has image by the functor ev ⋆ the ar row H : M → N . The 2 -cell I γ ′ + 3 [ N , N ] [ H, 1] / / [ M , N ] has imag e by ev ⋆ the 2- cell M H / / N γ + 3 N . The 2 -cell I u / / A δ ′ + 3 [ M , N ] has imag e by ev ⋆ = S M C (1 , ev ⋆ ) ◦ D the 2-cell M Rn ( δ ) ∗ + 3 [ A , N ] [ u, 1] / / [ I , N ] ev ⋆ / / N which is accor ding to Lemma 7.8 M L ′ / / I ⊗ M u ⊗ 1 / / A ⊗ M δ + 3 M References [Dup08] Ma thieu Dupont , Cat ´ ego ries ab´ eliennes en dimension 2 , Thesis, Universit´ e Ca tho lique de Louv ain, juin 2008. [HyPo02] M.H yland, J. Power , Pseudo-c ommu t ative monads and pseudo-close d 2-c ate gories , Journal of Pure a nd Applied Algebra 175, 20 02, 141 -185. [Lap83] M. Laplaza , Coher enc e for Cate gories with Gr oup Stru ctur e: An alternative appr o ach , Journal of Algebra 8 4, 19 83, 305 -323. [JiPi07] M.Jibladze, T. Pirashvili , Thir d Mac L ane c ohomolo gy via c ate goric al rings , Journal of homotopy an related structures 2, 200 7, 187- 216. 72 [JoSt93] A.Joy al, R.Street , Br aide d tensor c ate gories Adv ances in Ma thematics 10 2, 19 93, 20-78. [Qu87] N.T.Quang , Intr o duction to Ann-c ate gories , T ap chi T oan ho c, 15 , 1987, 14 -24. [Sch08] V. Schmitt , T en sor pr o duct of symmetric monoidal c ate gories , ht tp:// arxiv.o rg/abs /0711.0324 73

Enrichments over symmetric Picard categories

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