Length 3 Complexes of Abelian Sheaves and Picard 2-Stacks

Length 3 Complexes of Ab elian Shea v es and Picard 2-Stac ks A. Emin T atar Departmen t of Mathematics, Florida State Univ ersit y T allahassee, FL 32306 -4510, USA atatar@m ath.fsu.edu Abstract W e deﬁn e a tricateg ory T [ − 2 , 0] of length 3 complexes of ab elian shea v es, whose hom- bigroup oids consist of weak morph isms of suc h complexes. W e also deﬁne a 3-categ ory 2Pic ( S ) of Picard 2-st ac ks, whose hom-2-group oids consist of add itiv e 2-functors. W e pro v e that these categories are triequiv alen t as tricategories. As a consequence w e ob- tain a generalization of Deligne’s analogous result ab out Picard stac ks in SGA4, Exp os ´ e XVI I I(Del igne (1973) [11 ]). Con t e n ts In t roduction 2 1 Preliminaries 4 1.1 Butterﬂies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 ( A, B )-t orsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 ( A , B )-torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Ab elian Sheav es a nd Picard Stacks . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 T ricategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Picard 2-S tac ks as T orsors 7 2.1 2-Stac ks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Picard 2-Stack Asso ciated to a Complex . . . . . . . . . . . . . . . . . . . . . 12 2.3 Homotop y Exact Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 The 3-categor y of Picard 2 -Stac ks . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 The 3-category of Complexes of Ab elian Shea v es 14 3.1 Deﬁnition of C [ − 2 , 0] ( S ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Ab elian Sheav es a nd Picard 2- Stac ks . . . . . . . . . . . . . . . . . . . . . . . 15 4 W eak Morphisms of Complexes of Ab elian Shea v es 17 4.1 Deﬁnition of Fr ac ( A • , B • ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7 4.2 F rac ( A • , B • ) is a bigroup oid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1 5 Biequiv al ence of F rac ( A • , B • ) a nd Hom ( A • , B • ) 21 5.1 Morphisms of Picard 2-Stac ks a s F ractions . . . . . . . . . . . . . . . . . . . . 21 5.2 Hom-categories of F rac ( A • , B • ) a nd Hom ( A • , B • ) . . . . . . . . . . . . . . . . 25 6 The T ricategory of Complexes of Ab elian Shea v es 30 6.1 Deﬁnition of T [ − 2 , 0] ( S ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.2 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 7 Stac kiﬁcation 34 A App endix 34 A.1 Deﬁnition of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 A.2 Monoidal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 A.3 Extending the Additiv e Structure to F ree Ab elian Gro up . . . . . . . . . . . . 42 In tro duction Let D [ − 1 , 0] ( S ) b e t he subcategory of the derive d cat ego ry of category of complexes of ab elian shea v es A • o ver a site S with H − i ( A • ) 6 = 0 only for i = 0 , 1 . Let Pic ♭ ( S ) denote the category of Picard stacks o ver S with 1-morphisms isomorphism classes of additiv e func tors. In SGA4 Exp o s ´ e XVI I I, Deligne sho ws the following. Prop osition. [11, Prop osition 1.4.15] The functor ℘ ♭ : D [ − 1 , 0] ( S ) / / Pic ♭ ( S ) giv en by sending a length 2 complex of a belian shea v es, A • : A − 1 → A 0 o ver S to its asso ciated Picard stac k [ A − 1 → A 0 ] ∼ , a n isomorphism class of fractions from A • to B • to an isomorphism class of morphisms of asso ciated Picard stac ks is an equiv alence. The purp ose of this pa per is to generalize the ab o v e res ult to Picard 2-stac ks o v er S . Let 2Pic ♭♭ ( S ) denote the category of Picard 2-stac ks, whose mor phisms are equiv alence classes o f additiv e 2- f unctors. Let D [ − 2 , 0] ( S ) be the subcategory of the deriv ed category of category of complexes of ab elian shea v es A • o ver S with H − i ( A • ) 6 = 0 fo r i = 0 , 1 , 2. Theorem I. The functor 2 ℘ ♭♭ : D [ − 2 , 0] ( S ) / / 2Pic ♭♭ ( S ) giv en b y sending a length 3 complex of abelian she a v es, A • : A − 2 → A − 1 → A 0 o ver S to its asso ciated Picard 2- stac k [ A − 2 → A − 1 → A 0 ] ∼ , an eq uiv alence class of fractions from A • to B • to an equiv alence class of mor phism s of asso ciated Picard 2 -stac ks is an equiv alence. Basically , it giv es a geometric description o f the deriv ed catego r y of length 3 complexes of ab elian shea v es. It states that an y Picard 2- stac k ov er a site S is biequiv alen t to a Picard 2-stac k asso ciated to a length 3 complex of ab elian sheav es and that any morphism of Picard 2-stac ks comes from a f raction o f su c h complexes . A complex of ab elian shea v es, whose only non-zero cohomology groups are placed at degrees -2,-1, and 0 can b e t ho ug h t as a length 3 2 complex of ab elian shea v es, and therefore a morphism in D [ − 2 , 0] ( S ) b et w een an y tw o complexes A • and B • is giv en b y an equiv alence class of f raction ( q , M • , p ) : A • M • p / / q o o B • with q being a quasi-isomorphism. Ho wev er, w e prov e a muc h strong er stateme n t, so that the latter theorem b ecomes an imme- diate consequence o f it. Let 2Pic ( S ) b e the 3-category of Picard 2-stacks where 1-morphisms are a dditiv e 2-functors, 2-morphisms are natural 2-transforma t io ns, and 3-morphisms are mod- iﬁcations. Length 3 complex es of ab elian shea v es ov er S placed in degrees [ − 2 , 0 ] form a 3- category C [ − 2 , 0] ( S ) by adding to the regular morphisms of complexes, the degree -1 a nd -2 morphisms. Then w e easily construct an explicit trihomomorphism 2 ℘ : C [ − 2 , 0] ( S ) / / 2Pic ( S ) , that is a 3-functor b et we en 3-categories. Under this construction, le ngth 3 c omplexes of abelian shea v es correspo nd t o Picard 2-stacks . Although morphisms of suc h complexes induce mor- phisms betw een ass o ciated Picard 2-stac ks, not all of them are obtained in this w ay . In this sense, the 1-morphisms of C [ − 2 , 0] ( S ) are not geometric and the reason is their strictness. W e resolv e this problem b y w eak ening C [ − 2 , 0] ( S ) a s follows : W e intro duce a tricategory T [ − 2 , 0] ( S ) (a tricategory is a w eak vers ion of a 3-category in the sense of [14]) with same ob jects as C [ − 2 , 0] ( S ). F or a ny t w o complexes o f ab elian shea v es A • and B • , morphisms b et w een A • and B • in T [ − 2 , 0] ( S ) is the bigroup oid F rac ( A • , B • ), whose main prop ert y is that it satisﬁes π 0 ( F rac ( A • , B • )) ≃ Hom D [ − 2 , 0] ( S ) ( A • , B • ), where π 0 denotes the isomorphism classes of o b jects. Roughly speaking, ob jects of Frac ( A • , B • ) are fractions from A • to B • in t he ordinary sens e and its 2-morphisms a re certain comm uta tiv e diagrams (4.2) called “ diamo n ds ”. Then we pro ve: Theorem I I. The triho momorphism 2 ℘ : T [ − 2 , 0] ( S ) / / 2Pic ( S ) deﬁned by sending A • a length 3 complex of ab elian sheav es to its asso ciated Picard 2-stac k is a triequiv alence. Since in particular a triequiv alence is essen tially surjectiv e, every Picard 2-stac k is biequiv- alen t to a Picard 2-stac k asso ciated to a complex of ab elian shea v es. Then by ignoring the 3-morphisms and passing to t he equiv alence class of morphisms in the triequiv alence of Theo- rem I I, w e deduce Theorem I. Organization of the p ap er This pap er is orga nized a s follo ws: In Section 1, w e recall imp ortan t p oin ts of butterﬂies in ab elian context, ( A, B )-tor sors, where A a nd B are a belian sheav es, and ( A , B )-torsors, where A and B are Picard stac ks. W e also remind the reader some imp ortant results f r o m [3] that we will refer con tinuously . In Section 2, w e explain brieﬂy the basics on 2-stack s with structures, a nd exact sequences of Picard 2-stack s. W e also giv e an example of Picard 2- stac k, namely Tors ( A , A 0 ), where A is a Picard stack and A 0 is an ab elian sheaf. This example will b e of great imp ortance for the 3 rest since it will b e the Picard 2- stack asso ciated to A • a length 3 complex of ab elian shea ve s. W e deﬁne at the end of the section the 3-category 2Pic ( S ) of Picard 2-stac ks, a s w ell. In Section 3 , w e ﬁrst deﬁne the 3 -category C [ − 2 , 0] ( S ) of length 3 complexes of ab elian shea v es. W e construct an explicit trihomomorphism from this 3-category to the 3-categor y of Picard 2- stac ks. In Section 4, for any t w o length 3 complexes of a b elian shea v es A • and B • , w e deﬁne a bigroup oid Frac ( A • , B • ). It is a w eakene d ve rsion o f the hom-2 -category Ho m C [ − 2 , 0] ( A • , B • ) in the sense that π 0 ( F rac ( A • , B • )) ≃ Hom D [ − 2 , 0] ( S ) ( A • , B • ). In Section 5, w e sho w that fo r an y tw o length 3 complexes o f ab elian sheav es A • and B • , there exis ts a biequiv alence as bigro up oids b et w een F rac ( A • , B • ) and the 2-catego r y Hom ( A • , B • ) o f morphisms of Picard 2-stack s asso ciated to A • and B • . In Section 6, w e deﬁne the tricategory T [ − 2 , 0] ( S ). It consis ts of same ob jects as C [ − 2 , 0] ( S ) and for an y t wo length 3 complexes A • and B • of ab elian s hea ves , F rac ( A • , B • ) as the hom- bigroup oid. W e extend the trihomomorphism constructed in Section 3 to a trihomomorphism on T [ − 2 , 0] ( S ). W e pro v e Theorem I I whic h sa ys that the latter trihomomorphism is a triequiv- alence and fr o m whic h Theorem I follows. In Section 7, w e informally discuss the stack v ersions of what has b een done in the previous sections. Ac kno wledgemen ts I would lik e to ex press m y profound gratitude to m y advisor, Ettore Aldrov andi, for helping me at all stages of the pa per, whic h is pa rt of m y Ph.D. thesis. I w ould lik e to thank Behrang No ohi for helpful conv ersations. I also thank Chris P ortw o o d for pro ofreading. 1 Preliminaries The metho d tha t we ar e going to adopt to prov e our results is g o ing to use mostly the language and tec hniques dev elop ed in the papers of Aldrov a ndi and No ohi suc h as butterﬂies, tor sors, etc. So it is w ort hwhile to men tion here some of their work. W e ﬁnish with a few w ords a bout bicategories and tricategories. Before, let us ﬁx our conv en tions and no t ations. Throughout the pap er, w e will work with shea v es, stac ks, etc. deﬁned ov er a site S . F o r simplicit y , w e will assume that S has ﬁb ered products. Fib ered 2-categories, 2 -functors, and natural 2-transformations will be used in the sense of Hakim [16]. A comple x of ab elian s hea ve s will mean a length 3 complex of ab elian shea v es ov er t he site S unless otherwise stated. It will b e denoted as A • : A − 2 δ A / / A − 1 λ A / / A 0 . F or any complex of ab elian sheav es A • , A • < 0 denotes the complex A • < 0 : A − 2 δ A / / A − 1 / / 0 and therefore f • < 0 : A • < 0 → B • < 0 a morphism of complexes b et w een A • < 0 and B • < 0 . 4 1.1 Butterﬂies The reader can refer to [21] and [22 ] for details of butterﬂies ov er a p oin t or to [3] for a treatmen t o ver a s ite. Here, w e will remind the basic deﬁnitions follo wing the latter point of view in an ab elian con text. Deﬁnition 1.1. Let A • : A − 1 → A 0 and B • : B − 1 → B 0 b e t wo length 2 complexes of ab elian shea v es. A butterﬂy from A • to B • is a commutativ e diag ram of ab elian sheaf morphisms of the form A − 1   κ ! ! C C C C C C C C B − 1   ı } } { { { { { { { { E ρ } } { { { { { { { {  ! ! C C C C C C C C A 0 B 0 (1.1) where E is an ab elian sheaf, the NW-SE sequence is a complex, and the NE-SW sequence is an extension. [ A • , E , B • ] will denote a butterﬂy from A • to B • . A morphism of butterﬂies ϕ : [ A • , E , B • ] → [ A • , E ′ , B • ] is an ab elian sheaf isomorphism E → E ′ satisfying certain comm utative diagrams. Tw o such morphisms comp ose in an obv ious w ay . Therefore butterﬂies from A • to B • form a group oid denoted b y B ( A • , B • ). A butterﬂy is ﬂipp able o r r eversible if b oth dia g onals of (1.1) are extensions. F or more ab out crossed mo dules and butterﬂies in the ab elian con text, w e refer the reader to [21, § 12] and [3, § 8]. 1.2 ( A, B ) -torsors Let A → B b e a morphism of, not ne cessarily ab elian, shea ve s. An ( A, B )-torsor is a pair ( L, x ), where L is an A - torsor and x : L → B is an A -equiv arian t morphis m of sheav es (see [1 2]). A morphism b et w een tw o pairs ( L, x ) and ( K , y ) is a morphism of shea ve s F : L → K compatible with the a ction of A suc h that the diagram L x   1 1 1 1 1 1 1 1 1 1 1 1 1 F / / K y                B comm utes. ( A, B )- torsors form a cat ego ry denoted by Tors ( A, B ). 1.3 ( A , B ) -torsors Let A b e a gr - stac k, not necessarily Picard. A stack P in g r o upoids is a ( righ t) A -torsor if there exists a morphism of stac ks m : P × A / / P 5 compatible with the group law s in A , and the morphism (pr , m) : P × A / / P × P is an equiv alence, and for all U ∈ S , P U is not empt y . [6, § 6.1 ] Let A → B b e a morphism of gr-stacks . An ( A , B )-t orsor is a pair ( L , x ), where L is an A -to r sor, and x : L → B is an A -equiv arian t morphis m o f stac ks [1, § 6.1], [3, § 6.3.4]. A 1-morphism o f ( A , B )- torsors is a pair ( F , µ ) : ( L , x ) / / ( K , y ) , where F : L → K is a morphism o f stac ks suc h that L F / / x   3 3 3 3 3 3 3 3 3 3 3 3 3     ~  σ F K y                B and µ is a natural transformation of stacks L × A   F × 1 / /       µ K × A   L F / / K expressing t he compatibilit y of F with the torsor structure. A 2- morphism o f ( A , B )- torsors ( F , µ ) ⇒ ( G, ν ) is giv en b y a nat ur a l transformation φ : F ⇒ G satisfyin g the conditions giv en in [3, § 6.3.4 ]. ( A , B )-torsors form a 2- stac k denoted by Tors ( A , B ). 1.4 Ab elian Shea ve s and Picard S tac ks W e recall Deligne’s w ork ab out ab elian s hea ve s and Picard stac ks fro m [11, § 1.4]. They a re going to b e referred sans c esse thro ughout the pap er. These results are also revisited b y Aldro v andi and No ohi in [3]. In order to be consisten t with the rest of the pap er, w e recall them as they a r e announced in [3 ]. Theorem. [3, The or em 8.3.1] F or any two length 2 c omplexes of ab elia n she aves A • and B • , ther e is an e quivalenc e of gr oup oids Hom ( A • , B • ) ∼ / / B ( A • , B • ) , wher e Hom ( A • , B • ) i s the gr oup oid of add itive functors b etwe en the Pic ar d stacks asso cia te d to A • and B • . Prop osition. [3 , Pr op osition 8.3.2] L et A b e a Pic ar d stack. Then ther e exists a length 2 c omplex of ab eli an she aves A • : A − 1 → A 0 such that A is e quivalent to Pic ar d stack [ A − 1 → A 0 ] ∼ . 6 Let C [ − 1 , 0] ( S ) denote the bicategory of mo r phisms of ab elian sheav es o v er S with com- m utative squares as 1-mor phisms and homotopies a s 2-morphisms. Let Pic ( S ) denote the 2-category of Picard stack s o v er S with 1-mo r phisms b eing additiv e functors and 2- morphisms b eing na tural 2 -transformations. Putting the ab o v e results together, Deligne pro v es: Theorem. [3, Pr op osition 8.4.3] The functor C [ − 1 , 0] ( S ) / / Pic ( S ) deﬁne d by se nding a morphism of ab elian she a v e s A • : [ A 1 → A 2 ] to its a sso ciate d Pic ar d stack [ A 1 → A 2 ] ∼ is a bie quivalenc e of bic ate gories. R emark 1.2 . In the same pap er, the authors also pro v e these facts in the non-ab elian con t ext b y not assuming that stac ks and shea ve s are necessarily Picard or ab elian. 1.5 T ricat egories Ev en though the la nguage of bicategories and tricategories is go ing to b e extensiv ely used, w e are not g oing to remind here in full detail bicategories or tricategories. Just fo r mot iv a tion, a 3-category can be though t as t he category of 2- categories with 2-functors or w eak 2- functors in the sense of B ´ enab ou [5] and a tricategory as a w eake ned v ersion of a 3-categor y . How ev er, w e w ant to recall the triequiv alence since the pro of of Theorem 6 .4 will follo w its deﬁnition. F or more a b out bicategories and tricategories, we refer the reader to [14, 15, 20, 5]. Deﬁnition 1.3. [20] A trihomomorphism of tricategories T : C → D is called a tr ieq uiv alence if it induces biequiv alence s T X,Y : C ( X , Y ) → D ( T X , T Y ) of hom-bicategories for all o b jects X , Y in C ( T is lo cally a biequiv alence), and eve ry ob ject in D is biequiv alen t in D to an ob ject o f the form T X where X is an ob ject in C . 2 Picard 2-Stacks as T orsors In this section, o ur goal is to giv e some of the fundamen tal facts abo ut 2-stack s and torsors that will b e nee ded throughout the pap er. Our main references for 2-stac ks with structures suc h as monoidal, gro up-lik e, braided, Picard are [7, 8] and fo r torsors [1, 6] 2.1 2-Stac ks Deﬁnition 2.1. [9, Deﬁnition 6.2] A 2- stack P is a ﬁb ered 2-category in 2-gr o upoids suc h that • f or all X , Y ob jects in S U , Hom P U ( X , Y ) is a stac k ov er S /U ; • 2 -desce n t data is eﬀectiv e f o r ev ery ob ject in P . In the ab o v e deﬁnition 2-gr oupoids are considered in the sense of Breen [8], that is, 1- morphisms ar e we akly in v ertible. Deﬁnition 2.2. [8, Deﬁnition 8 .4 ] A gr- 2-stac k P is a 2- stac k with a mor phism ⊗ : P × P → P of 2-stack s, an asso ciativit y constrain t a compatible with ⊗ , a left unit l and a r ig h t unit r constrain ts compatible with a , and an inv erse constrain t i with resp ect to ⊗ compatible with units. 7 A more detailed deﬁnition of gr-2-stac k can b e found in [7]. Next, fo llo wing [8, § 8.4] w e add to gr-2 -stac ks comm utat ivity constraints with a n increasing lev el of strictness. Deﬁnition 2.3. A gr-2-stack P is said to b e: • b r aide d , if there exists a functoria l natura l tra nsfor mat ion R X,Y : X ⊗ Y / / Y ⊗ X that satisfy the 2-bra iding axioms of Kapranov and V o ev o dsky [18] t o gether with the additional condition that, in their terminology , the pair of 2 -morphisms deﬁning the induced Z -systems coincide. The corrected and full 2-bra iding axioms can b e found in [4]. • s tr o n gly br aide d , if it is braided and for any X , Y tw o ob jects, there exists a functorial 2-morphism X ⊗ Y 1 X ⊗ Y * * R Y ,X R X,Y 4 4       S X,Y X ⊗ Y . (2.1) suc h that the tw o compatibility conditions giv en in [10, Deﬁnition 15 ] are satisﬁed. • s ymmetric , if it is strongly braided and the follo wing whisk erings coincide: X ⊗ Y 1 X ⊗ Y * * R Y ,X R X,Y 4 4       S X,Y X ⊗ Y R X,Y / / Y ⊗ X , (2.2) X ⊗ Y R X,Y / / Y ⊗ X 1 Y ,X * * R X,Y R Y ,X 4 4       S Y ,X Y ⊗ X . (2.3) • Pi c ar d , if it is symmetric and f or any ob ject X , t here exists a functorial 2 -morphism X ⊗ X 1 X,X , , R X,X 2 2       S X X ⊗ X (2.4) additiv e in X (i.e. there is a relation b et w een S X ⊗ Y , S X , and S Y ) suc h that S X,X = S X ∗ S X . F urther do wn in the pap er, w e will be talking ab out the 3-category of Picard 2-stac ks whic h requires the concept o f morphism of Picard 2- stac ks. F ollo wing Breen [8], we will call suc h a morphism additive 2-functor. It will be a cartes ian 2- functor b et w een the underlying ﬁbered 2-categories compatible with the monoidal, braided, and Picard structures carried b y the 2- categories. The compatibilit y with monoidal structure is already know n. In Gordon, P o wer, Street [14], a monoidal 2-category is deﬁned as a one-o b ject tricat ego ry . More in detail, one can t hink of a monoidal 2-category as the hom- 2-category of a one-ob ject tricatego r y , whose asso ciativit y and unit constrain ts hold up t o 2-isomorphisms and whose mo diﬁcations are in ve rtible. Then the trihomomorphism [14 , Deﬁnition 3.1] b et w een suc h tricategories will be the right deﬁn ition of morphism b et w een monoidal 2-catego ries. F or the compatibilit y with the rest of the structures, w e refer the reader to t he author’s thesis [26]. Here is a tec hnical result that w e will use sev eral times in our pro ofs. 8 Lemma 2.4. L et P b e a Pic ar d 2-stack and A, B b e two ab elian she aves with additive 2-functors φ : A / / P and ψ : B / / P . Then A × P B is a Pic ar d stack. Pr o of. The ﬁb ered catego ry A × P B with ﬁb ers ( A × P B ) | U consisting of • o b jects ( a, f , b ), where a ∈ A ( U ), b ∈ B ( U ), and f : φ ( a ) → ψ ( b ) is a 1-morphism in P U ; • mo r phisms ( a, f , α, g , b ), where φ ( a ) f + + g 3 3       α ψ ( b ) is a 2-morphism in P U ; is a prestac k since for any U ∈ S , 1-morphisms of P f orm a stac k o v er S /U . It is in fact a stack . Let (( U i → U ) , ( a i , f i , b i ) , α i,j ) i,j ∈ I b e a descen t datum with ( U i → U ) i ∈ I a co v ering of U , ( a i , f i , b i ) an ob ject in ( A × P B ) U i and α i,j a 1- mo r phism in ( A × P B ) U ij b et w een ( a j , f j , b j ) | U ij and ( a i , f i , b i ) | U ij . Since a i | U ij = a j | U ij , b i | U ij = b j | U ij and A and B are sheav es, there exist a ∈ A ( U ) and b ∈ B ( U ) such that a | U i = a i and b | U i = b i . Then the collection (( U i → U ) , f i , α i,j ) i,j ∈ I satisﬁes the descen t in Hom( φ ( a ) , ψ ( b )), whic h is eﬀectiv e since P is a Picard 2-stac k. That is, there exis ts f ∈ Hom( φ ( a ) , ψ ( b )) and β i : f | U i ⇒ f i compatible with α i,j suc h that for all i ∈ I , ( a i , f | U i , β i , f i , b i ) is a morphism fro m ( a, f , b ) | U i to ( a i , f i , b i ). Th us, the descen t (( U i → U ) , ( a i , f i , b i ) , α i,j ) i,j ∈ I is eﬀectiv e. Next, w e show tha t A × P B is Picard. First, let us recall the notation fro m Deﬁnition 2.2. ⊗ P is the monoidal op eration, a , l , r , i , R − , − , S − , − , and S − are resp ectiv ely asso ciativit y , left unit, right unit, in v erse, braiding, symmetry , and Picard constrain ts. The unnamed a r r o ws in the diagrams b elo w are structural equiv alence s resulting from additive 2- functors φ and ψ . monoidal structur e : The multiplic ation is deﬁned as ( a 1 , f 1 , b 1 ) ⊗ ( a 2 , f 2 , b 2 ) := ( a 1 + a 2 , f 1 f 2 , b 1 + b 2 ) , where f 1 f 2 is the morphism tha t mak es the diagram φ ( a 1 ) ⊗ P φ ( a 2 )   f 1 ⊗ P f 2 / / ψ ( b 1 ) ⊗ P ψ ( b 2 )   φ ( a 1 + a 2 ) f 1 f 2 / / ψ ( b 1 + b 2 )     C K N m comm ute up to a 2 - isomorphism N m . F or an y three ob jects ( a i , f i , b i ) for i = 1 , 2 , 3, the asso ciator is giv en b y the morphism ( a 1 + a 2 + a 3 , f 1 ( f 2 f 3 ) , α f 1 ,f 2 ,f 3 , ( f 1 f 2 ) f 3 , b 1 + b 2 + b 3 ), where α f 1 ,f 2 ,f 3 is deﬁned as the 2- isomorphism of the b ottom fa ce that makes the follow ing cub e comm utative ( we ignored ⊗ P for compactness). 9 φ ( a 1 )( φ ( a 2 ) φ ( a 3 ))   a { { w w w w w w w w w w w w w w w w w w w w w w f 1 ⊗ P ( f 2 ⊗ P f 3 ) / / ψ ( b 1 )( ψ ( b 2 ) ψ ( b 3 ))   a { { w w w w w w w w w w w w w w w w w w w w w w φ ( a 1 ) φ ( a 2 + a 3 )   ψ ( b 1 ) ψ ( b 2 + b 3 )   ( φ ( a 1 ) φ ( a 2 )) φ ( a 3 )   ( f 1 ⊗ P f 2 ) ⊗ P f 3 / / ( ψ ( b 1 ) ψ ( b 2 )) ψ ( b 3 )   φ ( a 1 + a 2 ) φ ( a 3 )   ψ ( b 1 + b 2 ) ψ ( b 3 )   φ ( a 1 + a 2 + a 3 ) f 1 ( f 2 f 3 ) / / = y y t t t t t t t t t t t t t t t t t t t t       α f 1 ,f 2 ,f 3 ψ ( b 1 + b 2 + b 3 ) = y y t t t t t t t t t t t t t t t t t t t t φ ( a 1 + a 2 + a 3 ) ( f 1 f 2 ) f 3 / / ψ ( b 1 + b 2 + b 3 ) The other 2- isomorphisms of the cub e are, t he left and right 2-isomorphisms represen t the compatibility o f the additive 2-functors ψ and φ with the asso ciativit y constrain t (see Dat a HTD5 in [14]), the bac k a nd front ones are of the form N m , the top one is giv en b y the asso ciativit y constrain t a of P on t he 1-mor phisms. The o b ject I := (0 A , e, 0 B ), where 0 A (resp. 0 B ) is the unit ob ject in A (r es p. in B ) and e is deﬁned by t he 2-comm utat iv e diagra m 1 P = / /         N u 1 P   φ (0 A ) e / / ψ (0 B ) is the unit in the ﬁb ered pro duct A × P B . I comes with the functorial morphisms l ( a,f ,b ) := ( a, ef , L f , f , b ) and r ( a,f ,b ) := ( a, f e, R f , f , b ), where L f is deﬁned as the 2 - isomorphism of t he front face that mak es the diagram comm ute (similar dia gram for R f ). φ (0 A ) ⊗ P φ ( a ) e ⊗ P f / / w w o o o o o o o o o o o o ψ (0 B ) ⊗ P ψ ( b ) w w n n n n n n n n n n n φ (0 A + a ) =   ef / /       L f ψ (0 B + b ) =   1 P ⊗ P φ ( a ) O O 1 P ⊗ P f / / w w o o o o o o o o o o o o 1 P ⊗ P ψ ( b ) O O w w n n n n n n n n n n n n φ ( a ) f / / ψ ( b ) 10 The other 2- isomorphisms of the cub e are, t he left and right 2-isomorphisms represen t the compatibilit y of the additiv e 2-functors ψ and φ with the unit constraint (see D ata HTD6 in [14]), the top and b ottom ones ar e of the form N m , a nd the back one is of the form N u . br aiding : The morphism betw een ( a 1 , f 1 , b 1 ) ⊗ ( a 2 , f 2 , b 2 ) and ( a 2 , f 2 , b 2 ) ⊗ ( a 1 , f 1 , b 1 ) is giv en b y ( a 1 + a 2 , f 1 f 2 , β f 1 ,f 2 , f 2 f 1 , b 1 + b 2 ), where β f 1 ,f 2 is the 2-isomorphism of t he b ottom face of the comm utativ e cub e. φ ( a 1 ) ⊗ P φ ( a 2 ) R v v m m m m m m m m m m m m m f 1 ⊗ P f 2 / /   ψ ( b 1 ) ⊗ P ψ ( b 2 ) R v v m m m m m m m m m m m m m   φ ( a 2 ) ⊗ P φ ( a 1 ) f 2 ⊗ P f 1 / /   ψ ( b 2 ) ⊗ P ψ ( b 1 )   φ ( a 1 + a 2 )       β f 1 ,f 2 f 1 f 2 / / = v v m m m m m m m m m m m m m ψ ( b 1 + b 2 ) = v v m m m m m m m m m m m m m φ ( a 1 + a 2 ) f 2 f 1 / / ψ ( b 1 + b 2 ) (2.5) The other 2- isomorphisms of the cub e are, t he left and right 2-isomorphisms represen t the compatibility o f the additiv e 2-functors ψ and φ with the bra iding structure [26], the fron t a nd bac k ones are of the form N m , a nd the top one represen t s the compatibilit y of R − , − with P . gr oup like : In v erse of an ob ject ( a, f , b ) is deﬁned as ( − a, g , − b ), where there exists a 2-isomorphism γ : f g ⇒ e deﬁned as the 2- morphism o f the fron t face that makes the c ub e comm utative. φ ( a ) ⊗ P φ ( − a ) v v n n n n n n n n n n n n f ⊗ P g / /   ψ ( b ) ⊗ P ψ ( − b ) v v n n n n n n n n n n n n   φ ( a + ( − a )) f g / / =         γ ψ ( b + ( − b )) =   1 P = / / v v m m m m m m m m m m m m m m 1 P v v m m m m m m m m m m m m m m φ (0 A ) e / / ψ (0 B ) The other 2- isomorphisms of the cub e are, t he left and right 2-isomorphisms represen t the compatibility of the additive 2-functors ψ and φ with the in v erse ob ject constrain t [26], the top (resp. b ottom) one is of the form N m (resp. N u ), the bac k one is the in ve rse ob ject constrain t i . symmetry : W e ha v e to v erify that the 2-morphism of the b ottom face of the diagram obtained b y concatenation of the appropriate tw o cubes of the form (2 .5) is iden tit y . This follo ws from the fact tha t , 2-morphism o f the top fa ce of the concatenated cub e pastes to iden tit y with the help of the structural 2- mo r phism s of type (2 .1 ). Pic ar d: The m orphism from ( a, f , b ) ⊗ ( a, f , b ) to itself is iden tit y because the 2-morphism of the top fa ce of the diagr am (2.5) b ecomes iden t it y when it is pasted with (2.4). 11 The compatibility conditions for eac h structure are trivially satisﬁed. 2.2 Picard 2-Stac k Asso ciated to a Complex An immediate example of a Picard 2- stack is the Picard 2-stack a ss o ciated to a complex of ab elian shea v es whic h is in a sense the only example (see Lemma 6 .3 ). It is already explained in [21] and in [3] how to associate a 2-gr o upoid to a length 3 complex. How ev er, this 2-group oid is not a 2-stack . It is no t ev en a 2-prestac k (i.e. 1-morphisms only form a prestac k but not a stac k and 2- descen t data are not eﬀectiv e). Therefore to obtain a 2-stack one has to a pply the stac kiﬁcation t wice. Instead, w e are going to use a torsor mo del for asso ciated stac ks. It is more geometric, in tuitiv e, and can b e found in [1] for the ab elian case, and in [3] for the non-ab elian case. Consider A • a complex of a belian she a v es. Let A b e the associated Picard stac k, that is [ A − 2 → A − 1 ] ∼ ≃ Tors ( A − 2 , A − 1 ) a nd let Λ A : A → A 0 b e a n additiv e functor of Picard stacks , where A 0 is considered as a discrete stac k (no non-trivial morphisms). It a sso ciates to an ob ject ( L, s ) in Tors ( A − 2 , A − 1 ) a n elemen t λ A ( s ) in A 0 . W e consider Tors ( A , A 0 ) consisting of pairs ( L , s ), where L is an A -torsor and s : L → A 0 is an A -equiv arian t map with res p ect to Λ A . A morphism b et wee n an y t w o pairs is giv en b y another pair ( F , γ ) ( F , γ ) : ( L 1 , s 1 ) / / ( L 2 , s 2 ) , where F is an A - torsor morphism compatible with the torsor structure up to γ . F also ﬁts in to the comm uta t iv e diagram. L 1 F / / s 1   3 3 3 3 3 3 3 3 3 3 3 3 3 L 2 s 2                A 0 A 2-morphism ( L 1 , s 1 ) ( F, γ ) , , ( G,δ ) 2 2       θ ( L 2 , s 2 ) is give n b y a natura l tra nsformation θ : F ⇒ G t ha t makes the dia g ram L 1 × A F × 1 , , G × 1 2 2   L 2 × A   L 1 F + + g d b _ \ Z W G 3 3 L 2 γ   δ n v θ × 1   θ   comm ute. It is an immediate result of the follo wing prop osition that t he 2-stac k To rs ( A , A 0 ), whic h w e ha ve just constructed is Picard. 12 Prop osition 2.5. F or any A → B m orphism o f Pic ar d stacks, Tors ( A , B ) is a Pic ar d 2- stack. Pr o of. F rom [3, § 6.3.4], it fo llows that Tors ( A , B ) is a 2-stac k. Its group- like structure is deﬁned in [6, § 4.5] a nd Pic ar dness is relativ ely easy to ve rify . Deﬁnition 2.6. F or any complex of ab elian shea ves A • , we deﬁne the Picard 2-stac k asso ciated to A • as Tors ( A , A 0 ). 2.3 Homotop y Exact Sequence Let Tors ( A , A 0 ) b e the associated Picard 2-stac k to A • , the n there is a sequence of Picard 2-stac ks A Λ A / / A 0 π A / / Tors ( A , A 0 ) , (2.6) where A 0 is considered as discrete Picard 2-stac k (no non-trivial 1 - morphisms and 2-morphisms). The morphism π A assigns to an elemen t a of A 0 ( U ) the pair ( A , a ), where a is identiﬁed with the morphism A → A 0 sending 1 A = ( A − 2 , δ A ) t o a . (2 .6) is homotopy exact in the sense that A satisﬁes the pullback diagr a m. A / / Λ A   0   A 0 π A / / Tors ( A , A 0 )     @ H (2.7) Since A is the Picard stack asso ciated to the morphism of ab elian shea v es δ A : A − 2 → A − 1 , it ﬁts into the comm uta tiv e pullbac k square o f Picard stac ks (see the pro of of non-ab elian v ersion of Prop osition 8.3.2 in [3]) . A − 2 / / δ A   0   A − 1 π A / / A     = E (2.8) Then pasting the diagrams 2 .7 and 2.8 a t A , we obta in 13 A − 2 / / δ A   0   A − 1 π A / / λ A % % 8 ; > A E H J A Λ A   / /     = E 0   A 0 π A / / Tors ( A , A 0 )     @ H (2.9) 2.4 The 3-category of Picard 2-S tac ks Picard 2-stac ks ov er S form an ob vious 3-category whic h w e denote by 2Pic ( S ). 2Pic ( S ) has a hom-2-group oid consisting of a dditive 2-functors, we akly inv ertible natural 2-transfor ma t io ns, and strict mo diﬁcations. F o r a n y tw o Picard 2 -stac ks P and Q , it is denoted by Hom ( P , Q ). If P and Q are Picard 2-stacks asso ciated to complexes of ab elian shea v es A • and B • , then the hom-2-group oid will b e denoted as Hom ( A • , B • ). 3 The 3-cate g ory o f Complexes of Ab elian Shea v e s W e start with a deﬁnition of a 3-categor y C [ − 2 , 0] ( S ) of complexes of a b elian shea v es ov er S . W e end with a n explicit construction of a trihomomorphism 2 ℘ b et w een C [ − 2 , 0] ( S ) and the 3-category 2Pic ( S ) of Picard 2-stack s o v er S . 3.1 Deﬁnition of C [ − 2 , 0] ( S ) Although the 3 - category of complexes is very we ll kno wn, in o rder to setup our notatio n and terminology , w e will describ e it explicitly . Its ob jects are length 3 complexes of ab elian shea v es placed in degrees [ − 2 , 0]. F or a pair of ob jects A • , B • , the hom-2-group oid Hom C [ − 2 , 0] ( S ) ( A • , B • ) is deﬁned a s follo ws: • A 1-morphism f • : A • → B • is a degree 0 map giv en b y strictly comm utative squares. A − 2 δ A / / f − 2   A − 1 λ A / / f − 1   A 0 f 0   B − 2 δ B / / B − 1 λ B / / B 0 (3.1) 14 • A 2-morphism s • : f • ⇒ g • is a homotop y map give n by the diagram A − 2 δ A / / g − 2   f − 2   A − 1 λ A / / g − 1   f − 1   s − 1 { { { { { { { { } } { { { { { { { { A 0 g 0   f 0   s 0 | | | | | | | | } } | | | | | | | | B − 2 δ B / / B − 1 λ B / / B 0 (3.2) satisfying the relatio ns g 0 − f 0 = λ B ◦ s 0 , g − 1 − f − 1 = δ B ◦ s − 1 + s 0 ◦ λ A , g − 2 − f − 2 = s − 1 ◦ δ A . • A 3-morphism v • : s • ⇛ t • is a ho motop y map betw een ho mo t o pies s • and t • giv en b y the diagram A − 2 δ A / / g − 2   f − 2   A − 1 λ A / /     s − 1   t − 1 y y A 0 g 0   f 0   s 0   t 0 y y v v v m m m m m m m m m m m m m m m B − 2 δ B / / B − 1 λ B / / B 0 (3.3) satisfying the relatio ns s 0 − t 0 = δ B ◦ v , s − 1 − t − 1 = − v ◦ λ A . R emark 3.1 . In f act, the hom-2-gro up oid Hom C [ − 2 , 0] ( S ) ( A • , B • ) is the 2-gro upoid asso ciated to τ ≤ 0 (Hom • ( A • , B • )), t he smo oth truncation of the hom complex Hom • ( A • , B • ), that is to the complex Hom − 2 ( A • , B • ) / / Hom − 1 ( A • , B • ) / / Z 0 (Hom 0 ( A • , B • )) of ab elian g r oups, where for i = 1 , 2 the elemen ts of Hom − i ( A • , B • ) are morphisms of complexes from A • to B • of degree − i , and where Z 0 (Hom 0 ( A • , B • )) is the ab elian g roup of co cycles. 3.2 Ab elian Shea ve s and Picard 2-S tac ks Lemma 3.2. Th e r e is a trihomomorph ism 2 ℘ : C [ − 2 , 0] ( S ) / / 2Pic ( S ) (3.4) b etwe en the 3-c ate gory C [ − 2 , 0] ( S ) of c om p lexes of ab elian she aves a n d the 3-c ate gory 2Pic ( S ) of Pic ar d 2-stacks. Pr o of. W e will giv e a step b y step construction of the trihomomorphism and lea v e t he v eriﬁ- cation of the axioms to the reader. • Using the notations in section 2.2, giv en a complex A • , w e deﬁne 2 ℘ ( A • ) as the asso ciated Picard 2- stac k, that is 2 ℘ ( A • ) := Tors ( A , A 0 ). 15 • F or any morphism f • : A • → B • of complexes (see diagr a m (3.1)), there exists a commu- tativ e square of Picard stac ks A Λ A / / F   A 0 f 0   B Λ B / / B 0 (3.5) where F is induced by f • < 0 : A • < 0 → B • < 0 . F rom the square (3.5), w e construct a 1- morphism 2 ℘ ( f • ) in 2Pic ( S ) 2 ℘ ( f • ) : Tors ( A , A 0 ) / / Tors ( B , B 0 ) that sends an ( A , A 0 )-torsor ( L , x ) t o ( L ∧ A F B , f 0 ◦ x + Λ B ) where L ∧ A F B denotes the con tracted pro duct of the A - torsors L a nd B suc h that the A -torsor structure of B is induc ed b y the morphis m F . F or details, the reader can refer to [6, § 6.7] and [1, § 5.1, § 6.1]. • F or any 2-morphism s • : f • ⇒ g • of complexes (see diagram (3.2)), there exists a diagram of Picard stack s A F   G   Λ A / / A 0 f 0   g 0   ˆ s 0 } } } } } } } ~ ~ } } } } } } } } B Λ B / / B 0 (3.6) suc h that for any ( L, a ) in A , we hav e the relation G ( L, a ) − F ( L, a ) = ˆ s 0 ◦ Λ A ( L, a ) with ˆ s 0 ( a ) = ( B − 2 , s 0 ( a )) . F rom the relation, w e construct a natural 2- transformation θ : Tors ( A , A 0 ) 2 ℘ ( f • ) - - 2 ℘ ( g • ) 1 1 Tors ( B , B 0 ) θ   in 2Pic ( S ) tha t a ss igns to any ob ject ( L , x ) in Tors ( A , A 0 ) a 1-morphism θ ( L , x ) θ ( L , x ) : ( L ∧ A F B , x F ) / / ( L ∧ A G B , x G ) (3.7) in Tors ( B , B 0 ), where x F = f 0 ◦ x + Λ B and x G = g 0 ◦ x + Λ B . The morphism (3.7) is deﬁned b y sending ( l, b ) to ( l, b − s 0 ◦ x ( l )). 16 • F or an y 3-morphism v • : s • ⇛ t • of complexes (see dia g ram 3.3 ), there exists a mo diﬁ- cation Γ: Tors ( A , A 0 ) 2 ℘ ( f • ) + + 2 ℘ ( g • ) 3 3 θ ⇓ ⇛ Γ ⇓ φ Tors ( B , B 0 ) in 2Pic ( S ) that assigns to any ( L , x ) ob ject of Tors ( A , A 0 ) a natural 2-transformation Γ ( L , x ) , ( L ∧ A F B , x F ) θ ( L ,x ) + + φ ( L ,x ) 3 3 ( L ∧ A G B , x G ) ⇓ Γ ( L ,x ) in Tors ( B , B 0 ), where θ ( L ,x ) , φ ( L ,x ) are of the form (3 .7). The natural 2-transforma t io n Γ ( L , x ) is deﬁned b y assigning t o any ob ject ( l , b ) in ( L ∧ A F B , x F ) a morphism Γ ( L , x ) ( l , b ) : ( l , b − s 0 ◦ x ( l )) / / ( l , b − t 0 ◦ x ( l )) in ( L ∧ A G B , x G ) giv en by the triple ( i d l , 1 A , β ) with β being the isomorphism b − s 0 ◦ x ( l ) / / b − s 0 ◦ x ( l ) + δ B ◦ v ◦ x ( l ) , and id l the iden tity of l in L , and 1 A the unit elemen t in A . 4 W eak Morph i sms o f Complexes of Ab elian Shea v es W e ﬁx t w o complexes of abelian shea ves A • and B • . W e deﬁne Frac ( A • , B • ) a weak ened analog of the hom-2-g roupo id Hom C [ − 2 , 0] ( S ) ( A • , B • ). W e also prov e that Frac ( A • , B • ) is a bigroup oid. 4.1 Deﬁnition of F rac ( A • , B • ) F rac ( A • , B • ) is a consists of ob jects, 1 -morphisms, and 2-morphisms suc h that: • An ob ject is an ordered triple ( q , M • , p ), called fraction ( q , M • , p ) : A • M • q o o p / / B • with M • a complex of ab elian shea ve s, p a morphism of complexes, and q a quasi- isomorphism. • A 1-mor phism from the fraction ( q 1 , M • 1 , p 1 ) to the f raction ( q 2 , M • 2 , p 2 ) is an ordered triple ( r , K • , s ) with K • a complex of ab elian shea ve s, r and s quasi-isomorphisms making the diagram 17 M • 1 p 1 ' ' N N N N N N N N N N N N N q 1 w w p p p p p p p p p p p p p A • K • p / / q o o s O O    r      B • M • 2 p 2 7 7 p p p p p p p p p p p p p q 2 g g N N N N N N N N N N N N N (4.1) comm utative. • A 2-morphism f r om the 1- morphism ( r 1 , K • 1 , s 1 ) to the 1-morphism ( r 2 , K • 2 , s 2 ) is an isomorphism t • : K • 1 → K • 2 of complexes of ab elian sheav es suc h that the diagram that w e will call “ diamo nd ” A • K • 1 v v m m m m m m m m m m m m m m m m m , , X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X r 1 + + + + + + + + + + +   + + + + + + + + + + + s 1 { { { = = { { { t • C C C C C ! ! C C C C C M • 2 q 2 L L L L L L L L L L L L L f f L L L L L L L L L L L L L L p 2 r r r r r r r r r r r r r 8 8 r r r r r r r r r r r r r r M • 1 q 1 r r r r r r r r r r r r r x x r r r r r r r r r r r r r r p 1 L L L L L L L L L L L L L & & L L L L L L L L L L L L L L K • 2 l l X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 6 6 m m m m m m m m m m m m m m m m m r 2 { { { } } { { { s 2 + + + + + + + + + U U + + + + + + + + + + + + + + B • (4.2) comm utes. R emark 4.1 . F or reasons of clarit y , w e w ill repre sen t the ab o v e 2- morphism b y the following planar comm utativ e diag ram M • 1 p 1 & & L L L L L L L L L L L L L L L L L L L L L L L L q 1 x x r r r r r r r r r r r r r r r r r r r r r r r r A • K • 1 t • / / _ _ _ _ _ _ _ _ s 1       E E       r 1 4 4 4 4 4 4   4 4 4 4 4 4 K • 2 s 2 4 4 4 4 4 4 Y Y 4 4 4 4 4 4 r 2               B • M • 2 p 2 8 8 r r r r r r r r r r r r r r r r r r r r r r r r q 2 f f L L L L L L L L L L L L L L L L L L L L L L L L where w e ha v e ignored the maps f r om K • ’s to A • and B • . R emark 4 .2 . F r o m the deﬁnition of 2-mo r phisms, it is immediate that all 2-morphisms are isomorphisms. 18 4.2 F rac ( A • , B • ) is a bigroup oid Prop osition 4.3. L et A • and B • b e t wo c omplexes of ab elian she aves. Then F rac ( A • , B • ) is a bigr oup oid. Pr o of. W e will describ e the necessary data to deﬁne the bigroup oid without v erifying that they satisfy the r eq uired axioms. • F or an y tw o comp osable morphisms ( r 1 , K • 1 , s 1 ) : ( q 1 , M • 1 , p 1 ) → ( q 2 , M • 2 , p 2 ) and ( r 2 , K • 2 , s 2 ) : ( q 2 , M • 2 , p 2 ) → ( q 3 , M • 3 , p 3 ) sho wn by the diagram M • 1 p 1 # # F F F F F F F F F F F F F F F F F F F F F F q 1 { { x x x x x x x x x x x x x x x x x x x x x x K • 1 q ′ S S S S S S S S S ) ) S S S S S S S S S p ′ l l l l l l l l l u u l l l l l l l l l s 1 O O    r 1      A • M • 2 p 2 / / q 2 o o B • K • 2 q ′′ k k k k k k k k k 5 5 k k k k k k k k k p ′′ S S S S S S S S i i S S S S S S S S S s 2 O O    r 2      M • 3 p 3 ; ; x x x x x x x x x x x x x x x x x x x x x x q 3 c c F F F F F F F F F F F F F F F F F F F F F F the comp osition is deﬁned b y the pullbac k diagram. K • 1 × M • 2 K • 2 pr 2 % % K K K K K K K K K K pr 1 y y s s s s s s s s s s K • 1 r 1 % % L L L L L L L L L L L s 1 } } { { { { { { { { = K • 2 r 2 ! ! C C C C C C C C s 2 y y r r r r r r r r r r r M • 1 M • 2 M • 3 That is the comp osition is the triple ( r 2 ◦ pr 2 , K • 1 × M • 2 K • 2 , s 1 ◦ pr 1 ). • F or tw o 2-morphisms t • 1 : ( r 1 , K • 1 , s 1 ) ⇒ ( r 2 , K • 2 , s 2 ) a nd t • 2 : ( r 2 , K • 2 , s 2 ) ⇒ ( r 3 , K • 3 , s 3 ) sho wn b y the diagram M • 1 ( ( P P P P P P P P P P P P P P P P P P P P P P P P P P P P P w w 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 n n n A • K • 1 t • 1 / / _ _ _ _ _ _ s 1 } } } } } } } } > > } } } } } } } } r 1 A A A A A A A A A A A A A A A A K • 2 s 2 O O r 2   t • 2 / / _ _ _ _ _ _ K • 3 s 3 A A A A A A A A ` ` A A A A A A A A r 3 } } } } } } } } ~ ~ } } } } } } } } B • M • 2 7 7 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 n n n g g P P P P P P P P P P P P P P P P P P P P P P P P P P P P P 19 the v ertical comp osition is deﬁned by t • 2 ◦ t • 1 . • F or tw o 2-morphisms t • : ( r 1 , K • 1 , s 1 ) ⇒ ( r 2 , K • 2 , s 2 ) a nd u • : ( r ′ 1 , L • 1 , s ′ 1 ) ⇒ ( r ′ 2 , L • 2 , s ′ 2 ) shown b y the diagram M • 1 & & 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 N N N N x x p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p K • 1 r 1 C C C ! ! C C C s 1 { { { = = { { { t • / / _ _ _ _ _ _ _ _ K • 2 s 2 C C C a a C C C r 2 { { { } } { { { A • M • 2 / / o o B • L • 1 s ′ 1 { { { = = { { { r ′ 1 C C C ! ! C C C u • / / _ _ _ _ _ _ _ _ L • 2 r ′ 2 { { { } } { { { s ′ 2 C C C a a C C C M • 3 8 8 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p f f 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 N N N N the horizontal composition is give n by the natural morphism K • 1 × M • 2 L • 1 → K • 2 × M • 2 L • 2 b et w een the pullbac ks of pairs ( r 1 , s ′ 1 ) and ( r 2 , s ′ 2 ) ov er M • 2 . An y three comp osable 1 -morphisms ( r 1 , K • 1 , s 1 ), ( r 2 , K • 2 , s 2 ), and ( r 3 , K • 3 , s 3 ) can b e pic- tured as a seque nce of thr ee f ractions K • 1 r 1 ! ! C C C C C C C C s 1 } } { { { { { { { { K • 2 r 2 ! ! C C C C C C C C s 2 } } { { { { { { { { K • 3 r 3 ! ! C C C C C C C C s 3 } } { { { { { { { { M • 1 M • 2 M • 3 M • 4 simply by igno ring the ma ps to A • and B • . They can b e comp osed in tw o diﬀeren t w a ys, either ﬁrst b y pulling bac k ov er M • 2 then ov er M • 3 or vice v ersa. The resulting fractions will b e ( r , ( K • 1 × M • 2 K • 2 ) × M • 3 K • 3 , s ) and ( r ′ , K • 1 × M • 2 ( K • 2 × M • 3 K • 3 ) , s ′ ), resp ectiv ely , where r and r ′ (resp. s and s ′ ) are equal to r 3 (resp. s 1 ) comp osed with appropriate pro jection maps. The 2-isomorphism b et w een thes e fractions is g iven by the na t ural isomorphism b etw een the pullbac ks. Th us, the asso ciativit y of comp osition of 1-mo r phism s is weak . W e also observ e that 1-morphisms ar e w eakly in vertible . Let ( r, K • , s ) b e a 1 - morphism from ( q 1 , M • 1 , p 1 ) to ( q 2 , M • 2 , p 2 ), then ( s, K • , r ) is a weak in ve rse of ( r , K • , s ) in the sense that the comp o sition ( r ◦ pr , K • × M • 2 K • , r ◦ pr) is equiv alen t to the iden tit y , that is there is a natural 2-transformation θ : r ◦ pr ⇒ id ◦ ( r ◦ pr) as sho wn in the b elo w diagra m. 20 M • 1 p 1 & & M M M M M M M M M M M M M M M M M M M M M M M M q 1 x x r r r r r r r r r r r r r r r r r r r r r r r r A • K • r ◦ pr / / _ _ _ _ _ _ _ _ r ◦ pr        E E       r ◦ pr 4 4 4 4 4 4 4   4 4 4 4 4 4 _ _ _ _ + 3 θ M • 1 id 4 4 4 4 4 4 Z Z 4 4 4 4 4 4 id               B • M • 1 p 1 8 8 r r r r r r r r r r r r r r r r r r r r r r r r q 1 f f L L L L L L L L L L L L L L L L L L L L L L L L _ _ _ _ + 3 θ Th us, F rac ( A • , B • ) is a bigroup oid. R emark 4.4 . In the terminology of [2], what w e ha v e called fra ctions are called in the non- ab elian context w eak mo r phism s of 2-crossed mo dules o r butterﬂies of gr-stacks or bats of shea v es. 5 Biequiv alenc e of F rac ( A • , B • ) and Hom ( A • , B • ) Fix again t wo complexes of ab elian shea v es A • and B • . In this section, w e prov e that the bigroup oid F rac ( A • , B • ) of fractions deﬁned in Section 4 is biequiv alen t to the 2- group oid Hom ( A • , B • ) o f additiv e 2- functors from 2 ℘ ( A • ) to 2 ℘ ( A • ) deﬁned in Section 2 .4. 5.1 Morphisms of Picard 2-Stac ks as F racti ons Lemma 5.1. A morphism f : A • → B • is a quasi-isomorphism if and only if 2 ℘ ( f ) : 2 ℘ ( A • ) / / 2 ℘ ( B • ) is a bie quivalenc e. Pr o of. Giv en f : A • → B • a morphism of complexes, w e kno w ho w to induce a mo r phism of Picard 2-stac ks (see construction of trihomomor phism 2 ℘ ( f )). It is also known that a 2-stack (not necessarily Picard) can b e seen as a 2-g erb e ov er its o wn π 0 b ounded by the stack A ut (I) of automorphisms of iden tity [8, § 8.1]. In particular, the Picard 2 - stac ks Tors ( A , A 0 ) and Tors ( B , B 0 ) are 2-gerb es ov er their own π 0 b ounded by A ut (I 2 ℘ ( A • ) ) ≃ [ A − 2 → ker( δ A )] ∼ and A ut (I 2 ℘ ( B • ) ) ≃ [ B − 2 → ker( δ B )] ∼ , resp ectiv ely . F urthermore, if f is a quasi-isomorphism, then H − i ( A • ) ≃ H − i ( B • ) for i = 0 , 1 , 2 and th us, π i (2 ℘ ( A • )) ≃ π i (2 ℘ ( B • )) for i = 0 , 1 , 2. So Tors ( A , A 0 ) and Tors ( B , B 0 ) are 2- gerbes with equiv alen t bands. Therefore they are equiv alen t. Giv en an additiv e 2-functor F in Hom ( A • , B • ), we will sho w in the next lemma that there is a corresp onding ob ject in F rac ( A • , B • ). Lemma 5.2. F or any additive 2-functor F : 2 ℘ ( A • ) → 2 ℘ ( B • ) , ther e exists a fr action ( q , M • , p ) such that F ◦ 2 ℘ ( q ) ≃ 2 ℘ ( p ) . 21 Pr o of. F rom the sequences A Λ A / / A 0 π A / / 2 ℘ ( A • ) and B Λ B / / B 0 π B / / 2 ℘ ( B • ) , w e can construct the comm utativ e diag r a m A × B µ F   & & M M M M M M M M M M M x x r r r r r r r r r r r A ν F & & L L L L L L L L L L L L Λ A   B Λ B   ξ F x x r r r r r r r r r r r r E F pr 2 & & L L L L L L L L L L L L pr 1 x x r r r r r r r r r r r r A 0 π A   F ◦ π A * * U U U U U U U U U U U U U U U U U U U U U U B 0 π B   2 ℘ ( A • ) F / / 2 ℘ ( B • ) (5.1) where E F := A 0 × F ,B B 0 . It follo ws from the comm utativit y of the ab o v e diagram that µ F = (Λ A , Λ B ). The sequen ce B ξ F / / E F pr 1 / / A 0 (5.2) is homotop y exact since it is the pullbac k of the exact sequence B → B 0 → 2 ℘ ( B • ). F rom Lemma 2.4, it follo ws that E F is a Picard stac k. Therefore by [3, Prop osition 8.3.2], there exists a length 2 complex E • = [ δ E : E − 1 F → E 0 F ] of ab elian shea ve s suc h tha t the asso ciated Picard stac k Tors ( E − 1 F , E 0 F ) is equiv alen t to E F . Then by [3, The orem 8.3.1], there exists a butterﬂy represen ting µ F : A − 2 × B − 2 δ A × δ B   κ % % K K K K K K K K K K E − 1 F δ E   ı } } { { { { { { { { P F  ! ! C C C C C C C C ρ y y s s s s s s s s s s A − 1 × B − 1 π A × π B   E 0 F π E F   A × B µ F / / E F (5.3) with P F ≃ ( A − 1 × B − 1 ) × E F E 0 F . F rom a diﬀeren t p ersp ectiv e, this butterﬂy can b e seen as 0 / /   E − 1 F δ E / / ı   E 0 F id   A − 2 × B − 2 κ / / id   P F  / / ρ   E 0 F   A − 2 × B − 2 δ A × δ B / / A − 1 × B − 1 / / 0 (5.4) 22 where eac h column is an exact s equence of ab elian sheav es. The only non- t rivial sequenc e is the second column and its exactness follows from the deﬁnition of a butterﬂy (1.1). So w e ha ve a short exact sequence of complexes of ab elian shea ve s 0 / / E • F / / M • F / / A • < 0 × B • < 0 / / 0 , (5.5) where M • F := A − 2 × B − 2 / / P F / / E 0 F , (5.6) E • F := 0 / / E − 1 F / / E 0 F , A • < 0 × B • < 0 := A − 2 × B − 2 / / A − 1 × B − 1 / / 0 . F rom the low er part of the diagram (5 .4 ) and the deﬁnition o f P F , w e deduce that there are morphisms of complexes A − 2 × B − 2 κ   pr 1 w w o o o o o o o o o o o o pr 2 ' ' O O O O O O O O O O O O A − 2 δ A   B − 2 δ B   P F    pr 2 ◦ ρ w w o o o o o o o o o o o o o pr 1 ◦ ρ ' ' O O O O O O O O O O O O O A − 1 λ A   B − 1 λ B   E 0 F w w 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 O O A 0 B 0 M • F q w w o o o o o o o o o o o o o p ' ' O O O O O O O O O O O O O A • B • (5.7) W e claim that q is a quasi-isomorphism, that is H − 2 ( M • F ) ≃ k er ( δ A ) , H − 1 ( M • F ) ≃ k er( λ A ) / im( δ A ) , H 0 ( M • F ) ≃ cok er( λ A ) . Indeed, fr o m the exact sequence (5.5 ), we obta in the lo ng exact sequence of homology shea ves 0 / / H − 2 ( M • F ) / / H − 2 ( A • < 0 ) × H − 2 ( B • < 0 ) / / H − 1 ( E • F ) ED BC GF ∂   H − 1 ( M • F ) / / H − 1 ( A • < 0 ) × H − 1 ( B • < 0 ) / / H 0 ( E • F ) / / H 0 ( M • F ) / / 0 . (5.8) 23 On the other hand, by [3, Prop osition 6.2 .6 ] applied to the exact sequenc e (5.2), w e get a long exact sequence of homotop y groups 0 / / π 1 ( B ) / / π 1 ( E F ) / / π 1 ( A 0 ) / / π 0 ( B ) / / π 0 ( E F ) / / π 0 ( A 0 ) / / 0 . (5.9) Since π 1 ( A 0 ) = H − 1 ( A 0 ) = 0 a nd π 0 ( A 0 ) = H 0 ( A 0 ) = A 0 , it follow s from (5.9 ) that w e ha ve a n isomorphism H − 2 ( B • < 0 ) ≃ / / H − 1 ( E • F ) (5.10) and an exact sequence 0 / / H − 1 ( B • < 0 ) / / H 0 ( E • F ) / / A 0 / / 0 . (5.11) (5.10) implies that ∂ = 0 in (5.8). Therefore from ( 5.8) again, w e obta in a short exact sequence 0 / / H − 2 ( M • F ) / / H − 2 ( A • < 0 ) × H − 2 ( B • < 0 ) / / H − 1 ( E • F ) / / 0 from whic h w e deduce tha t H − 2 ( M • F ) ≃ H − 2 ( A • < 0 ) = k er( δ A ). No w, apply the snak e lemma to the short exact sequence (5.1 1) and to 0 / / H − 1 ( B • < 0 ) / / H − 1 ( A • < 0 ) × H − 1 ( B • < 0 ) / / H − 1 ( A • < 0 ) / / 0 in order to get the dashed exact sequence 0   / / H − 1 ( M • F )   / / k er ( λ A ) / im( δ A )   ED BC          GF _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ @A          / / _ _ _ _ _ 0 / / H − 1 ( B • < 0 ) / /   H − 1 ( A • < 0 ) × H − 1 ( B • < 0 ) / /   H − 1 ( A • < 0 ) / /   0 0 / / H − 1 ( B • < 0 ) / /   H 0 ( E • F ) / /   A 0 / /   0 0 / / H 0 ( M • F ) / / cok er( λ A ) from whic h it follows H − 1 ( M • F ) ≃ k er ( λ A ) / im( δ A ), and H 0 ( M • F ) ≃ cok er ( A 0 ) a s w an t ed. W e end this pro of b y sho wing that F ◦ 2 ℘ ( q ) ≃ 2 ℘ ( p ). (5.7) induces a diagram of Picard 2-stac ks 24 2 ℘ ( M • F ) 2 ℘ ( p ) % % K K K K K K K K K K 2 ℘ ( q ) y y s s s s s s s s s s 2 ℘ ( A • ) F / / 2 ℘ ( B • ) . (5.12) W e claim that (5.12) comm utes up t o a natural 2- transformation. T o show that, it is enough to lo ok at 2 ℘ ( M • F ) lo cally . Give n U ∈ S , 2 ℘ ( M • F ) U is the 2- group oid associated to the complex of ab elian groups (for the deﬁnition of the 2- group oid asso ciated to a complex see [3 ] or [21]) A − 2 ( U ) × B − 2 ( U ) δ / / P F ( U ) λ / / E 0 F ( U ) Then, a n o b ject o f 2 ℘ ( M • F ) U is an elemen t e o f E 0 F ( U ). Since E F := A 0 × F ,B B 0 ≃ Tors ( E − 1 F , E 0 F ), e can b e tak en as ( a, f , b ), where a ∈ A 0 ( U ), b ∈ B 0 ( U ), and f : F ( a ) → b is a 1-morphism in 2 ℘ ( B • ) U . A 1-morphism of 2 ℘ ( M • F ) U from e 1 to e 2 is giv en b y an e lemen t p of P F ( U ) suc h that λ ( p ) + e 1 = e 2 in E 0 F ( U ). W e can a gain ta k e λ ( p ), e 1 , and e 2 as ( a, f , b ), ( a 1 , f 1 , b 1 ), and ( a 2 , f 2 , b 2 ), resp ectiv ely . Therefore, the addition in E 0 F ( U ) should b e replaced by the monoida l op eration on E F b et w een the triples , that is ( a, f , b ) ⊗ E F ( a 1 , f 1 , b 1 ) = ( a 2 , f 2 , b 2 ). This monoidal op eration is describ ed in the pro of o f the tech nical Lemma 2.4. It creates a diagram comm utative up to a 2-isomorphism in 2Pic ( B • ) U that deﬁnes f 2 . F ( a 2 ) ≃   f 2 / / b 2 ≃   F ( a ) ⊗ B F ( a 1 ) f ⊗ B f 1 / / b ⊗ B b 1     A I θ The collection ( f , θ ) giv es the natura l 2- transformation b etw een 2 ℘ ( q ) ◦ F and 2 ℘ ( p ) . R emark 5.3 . Since q is a quasi-isomorphism in C [ − 2 , 0] ( S ), the tec hnical lemma 5.1 implies that 2 ℘ ( q ) is a biequiv alence in 2Pic ( S ). Therefore, b y choosing an in vers e of 2 ℘ ( q ) up to a natural 2-transformation we can write F a s F ≃ 2 ℘ ( p ) ◦ 2 ℘ ( q ) − 1 . 5.2 Hom-categories of F rac ( A • , B • ) and Hom ( A • , B • ) In the next tw o lemmas, w e are going to explore the relatio n b et wee n 1-morphisms (resp. 2-morphisms) o f F rac ( A • , B • ) and natura l 2-transformatio ns (res p. mo diﬁcations) of Picard 2-stac ks. Supp ose we hav e a natura l 2-transformatio n θ : 2 ℘ ( A • ) F + + G 3 3       θ 2 ℘ ( B • ) (5.13) b et w een the t wo additiv e 2 -functors F, G : 2 ℘ ( A • ) → 2 ℘ ( B • ). By Lemma 5 .2, w e know that there are fractions ( q F , M • F , p F ) a nd ( q G , M • G , p G ) asso ciated to F and G . 25 Lemma 5.4. F or any natur al 2-tr ansformation θ a s in (5.13), ther e is a 1-morphism in F rac ( A • , B • ) b etwe en the fr actions ( q F , M • F , p F ) and ( q G , M • G , p G ) . Pr o of. F or F and G , w e hav e the f o llo wing diagrams similar to (5.1) A × B µ F   & & M M M M M M M M M M M x x r r r r r r r r r r r A ν F & & L L L L L L L L L L L L Λ A   B Λ B   ξ F x x r r r r r r r r r r r r E F & & L L L L L L L L L L L L x x r r r r r r r r r r r r A 0 π A   F ◦ π A * * U U U U U U U U U U U U U U U U U U U U U U B 0 π B   2 ℘ ( A • ) F / / 2 ℘ ( B • ) A × B µ G   & & M M M M M M M M M M M x x r r r r r r r r r r r A ν G & & L L L L L L L L L L L L Λ A   B Λ B   ξ G x x r r r r r r r r r r r r E G & & L L L L L L L L L L L L x x r r r r r r r r r r r r A 0 π A   G ◦ π A * * U U U U U U U U U U U U U U U U U U U U U U B 0 π B   2 ℘ ( A • ) G / / 2 ℘ ( B • ) where E F := A 0 × F ,B B 0 and E G := A 0 × G,B B 0 are Picard stac ks b y Lemma 2.4. Therefore b y [3, Prop osition 8.3.2 ], there exist E − 1 F → E 0 F and E − 1 G → E 0 G morphisms of a belian shea v es suc h that the Picard stac k asso ciat ed to them are resp ectiv ely E F and E G . The natural 2-transfor ma t io n θ : F ⇒ G induces an equiv alence H : E G → E F of Picard stack s deﬁned as follo ws: • F or an y ( a, g , b ) ob ject of ( E G ) U , H (( a, g , b )) := ( a, f , b ), where f ﬁts in to the comm utative diagram F ( a ) θ a   f / / = b ∼   G ( a ) g / / b • F or an y ( a, g , σ , g ′ , b ) morphism of ( E G ) U , H (( a, g , σ , g ′ , b )) := ( a, f , τ , f ′ , b ), where τ is deﬁned b y the fo llo wing whisk ering. F ( a ) θ a / / G ( a ) g ) ) g ′ 5 5       σ b By [3, Theorem 8.3.1], H corresp onds t o a butterﬂy [ E • G , N , E • F ]. Since H is an equiv alence , this butterﬂy is ﬂippable. W e comp ose H and µ G b y comp osing their corresp onding butterﬂies 26 A − 2 × B − 2 δ A × δ B   κ ′ & & M M M M M M M M M M M E − 1 F δ E   ı ′ z z u u u u u u u u u P G × E − 1 G E 0 G N  ′ $ $ I I I I I I I I I I ρ ′ x x p p p p p p p p p p p A − 1 × B − 1 π A × π B   E 0 F π E F   A × B H ◦ µ G / / E F where P G × E − 1 G E 0 G N is pull-out/ pull-bac k construction as deﬁned in [3, § 5.1]. There is also a direct morphism µ F from A × B to E F . µ F is equiv alen t to H ◦ µ G since they b oth map an ob ject of A × B to an ob ject in E F whic h is isomorphic to the unit ob ject in 2 ℘ ( B • ). Then b y [3, Theorem 8.3.1], there exists an isomorphism k b et w een the corresp onding butterﬂies of µ F and H ◦ µ G , that is the do t t ed arrow in the diagr a m b elo w suc h that all regions comm ute. A − 2 × B − 2 δ A × δ B   κ ) ) R R R R R R R R R R R R R R R R R R R κ ′ / / P G × E − 1 G E 0 G N C C C C C  ′ ! ! C C C C C C C C C C C C C C w w w w w ρ ′ { { w w w w w w w w w w w w w w w w k      E − 1 F δ E   ı v v n n n n n n n n n n n n n n n n n ı ′ o o P F  ( ( R R R R R R R R R R R R R R R R R ρ u u j j j j j j j j j j j j j j j j j j A − 1 × B − 1 π A × π B   E 0 F π E F   A × B H ◦ µ G - - µ F 1 1 E F (5.14) Let M • F : A − 2 × B − 2 → P F → E 0 F and M • G : A − 2 × B − 2 → P G → E 0 G . W e claim that, there exists a complex K • with quasi-isomorphisms r F and r G suc h that all regions in the diag r am M • F p F ' ' N N N N N N N N N N N N N q F w w p p p p p p p p p p p p p A • K • p / / q o o r F O O r G   B • M • G p G 7 7 p p p p p p p p p p p p p q G g g N N N N N N N N N N N N N (5.15) comm ute. Pr o of of the claim : Let K • : A − 2 × B − 2 → P G × E 0 G N → N and deﬁne r F b y the comp osition 27 K • r F   A − 2 × B − 2 / / P G × E 0 G N / / quotien t   N quotien t   A − 2 × B − 2 / / P G × E − 1 G E 0 G N / /   N/ E − 1 G   M • F A − 2 × B − 2 / / P F / / E 0 F (5.16) and r G b y the diagram K • r G   A − 2 × B − 2 / / P G × E 0 G N   / / N   M • G A − 2 × B − 2 / / P G / / E 0 G (5.17) The comm utat ivit y o f the diag r am (5.1 6 ) follows from comp osition of butterﬂies. Since P G × E − 1 G E 0 G N ≃ P F and the butterﬂy [ E • G , N , E • F ] is ﬂippable, r F is a quasi-isomorphism. The diagram (5.1 7) comm utes b ecause its left square is a pullback . This implies that r G is a quasi-isomorphism. It remains to sho w that q F ◦ r F = q G ◦ r G , that is in the dia g ram b elo w eac h column closes to a comm utativ e square. A • A − 2 / / A − 1 / / A 0 M • F q F O O A − 2 × B − 2 O O / / P F / / O O E 0 F O O K • r F O O r G   A − 2 × B − 2 / / P G × E 0 G N O O   / / N O O   M • G q G   A − 2 × B − 2 / /   P G   / / E 0 G   A • A − 2 / / A − 1 / / A 0 It is obv ious fo r the ﬁrst column. The comm utativity of the tria ngles P G × E − 1 G E 0 G N k / / ρ ′ % % J J J J J J J J J J J J J J J J J J J J P F ρ   A − 1 × B − 1 E G H / / pr 1   3 3 3 3 3 3 3 3 3 3 3 3 3 E F pr 2                A 0 imply that the middle and last columns close to a commutativ e square, resp ective ly (the ﬁrst triangle is extracted from diagram (5.14)). 28 In the same w a y , we also show that p F ◦ r F = p G ◦ r G . No w, suppose w e ha v e a mo diﬁcation Γ: 2 ℘ ( A • ) F ( ( G 6 6 θ ⇓ ⇛ Γ ⇓ φ 2 ℘ ( B • ) (5.18) b et w een t w o natural 2-transformations θ , φ : F ⇒ G . W e ha v e prov ed in Lemmas 5.2 and 5.4 that b oth θ and φ corresp ond to a 1-morphism in Frac ( A • , B • ). Lemma 5.5. Given a m o diﬁc ation Γ a s in (5.18), ther e exists a 2-morphism b etwe en the two 1-morphisms c orr esp onding to θ and φ . Pr o of. Using t he same notatio ns as in Lemma 5.4, w e construct a diagra m of Picard stac ks E G H θ ) ) H φ 5 5       T E F , where T is a nat ura l transformation. F or an y ob j ect ( a, g , b ) in E G , T ( a,g, b ) is a morphism in E F deﬁned b y F ( a ) f θ $ $ f φ : :       1 g ∗ Γ a b , where F ( a ) θ a ( ( φ a 6 6       Γ a G ( a ) , and H θ ( a, g , b ) = ( a, f θ , b ), H φ ( a, g , b ) = ( a, f φ , b ) . By [3 , Theorem 5 .3.6], the natural trans- formation T corresp onds to an isomorphism t b et w een the cen ters of the butterﬂies asso ciated to H θ and H φ . E 0 G δ E G   κ φ & & M M M M M M M M M M M M M κ θ / / N θ < < < < <  θ   < < < < < < < < < < < <      ρ θ               E − 1 F δ E F   ı φ x x p p p p p p p p p p p p p ı θ o o N φ  φ & & N N N N N N N N N N N N N ρ φ x x q q q q q q q q q q q q q t O O    E 0 G π E G   E 0 F π E F   E G H θ - - H φ 1 1 E F ⇓ T (5.19) t induces a n isomorphism of complexes t • . 29 K • φ t •   A − 2 × B − 2 / / P G × E 0 G N φ id × t   / / N φ t   K • θ A − 2 × B − 2 / / P G × E 0 G N θ / / N θ The pro of ﬁnishe s by s ho wing that all the regions in the diagram (4.2) comm ute. The only regions, whose comm utativity ar e non-trivial, are the triangles in the middle sharing an edge mark ed b y the isomorphism t • . They commute a s w ell since in t he diagram b elo w M • G A − 2 × B − 2 / / P G / / E 0 G K • φ r G,φ O O t •   A − 2 × B − 2 / / P G × E 0 G N φ pr 1 O O id × t   / / N φ ρ φ O O t   K • θ r G,θ   A − 2 × B − 2 / / P G × E 0 G N θ pr 1   / / N θ ρ θ   M • G A − 2 × B − 2 / / P G / / E 0 G eac h column closes to a comm utative triangle. This is immediate for the ﬁrst tw o columns. The triangle formed b y the last column comm utes as we ll, since it is a piece of the comm utativ e diagram (5.1 9). F or an y tw o complexes of ab elian she a v es A • and B • , the pro ofs of Lemmas 5.2 and 5 .4 deﬁne us a 2 -functor 2 ℘ ( A • ,B • ) : F rac ( A • , B • ) / / Hom ( A • , B • ) (5.20) b et w een the bigroup oid F rac ( A • , B • ) and the 2- g roupo id Hom ( A • , B • ) of a dditive 2-functors b et w een 2 ℘ ( A • ) a nd 2 ℘ ( B • ) considered as a bigroup oid. In fact, we hav e prov ed: Theorem 5.6. F or an y two c omplexes of a b elian she aves A • and B • , 2 ℘ ( A • ,B • ) is a bie quiva - lenc e of bigr oup oids. 6 The T ricate g ory o f Complexes of Ab eli an Shea v es After proving in Section 5 that for an y tw o complexes of ab elian shea v es A • and B • , Frac ( A • , B • ) is biequiv alen t as a bigroup oid to Hom ( A • , B • ), it is clear that the triho mo mo r phism 2 ℘ (3.4) deﬁned in Section 3.2 cannot b e a t r ieq uiv alence. T o attain the triequiv alence , w e need to consider a t least a tricategory with same ob jects as C [ − 2 , 0] ( S ) and with hom-bicategories of the form F rac ( A • , B • ). F urthermore, there is the question of essen tial surjectivit y whic h w e deal with in this section. 6.1 Deﬁnition of T [ − 2 , 0] ( S ) W e deﬁne the tricategory T [ − 2 , 0] ( S ) promised at t he b eginning of the section. 30 Deﬁnition-Prop osition 6.1. T [ − 2 , 0] ( S ) with obje cts c omplexes of ab elian she aves, an d hom - bigr oup oids F rac ( A • , B • ) , for any two c omplex es of ab elian she aves A • and B • , is a tric ate gory. Pr o of. W e ha v e to v erify that T [ − 2 , 0] ( S ) has the dat a giv en in [15 , Deﬁnition 3.3.1]. • O b jects are complexes of ab elian shea ve s. • F or an y t w o complexes of ab elian she a ves A • and B • , F ra c ( A • , B • ) is the hom- bicategory . • F or an y three complexes of a b elian shea ves A • , B • , a nd C • , the comp osition is giv en b y the w eak functor ⊗ T : F rac ( A • , B • ) × F ra c ( B • , C • ) / / F rac ( A • , C • ) , whic h is deﬁned o n 1. ob jects, by M • 1 q 1           p 1   ? ? ? ? ? ? ? ? M • 2 q 2           p 2   ? ? ? ? ? ? ? ? M • 1 × B • M • 2 q 1 ◦ pr 1 w w o o o o o o o o o o o o o p 2 ◦ pr 2 ' ' O O O O O O O O O O O O O A • B • B • C • A • C • = ⊗ T 2. 1-morphisms, by M • 1 q 1           p 1   ? ? ? ? ? ? ? ? M • 2 q 2           p 2   ? ? ? ? ? ? ? ? M • 1 × B • M • 2 q 1 ◦ pr 1 w w o o o o o o o o o o o o o p 2 ◦ pr 2 ' ' O O O O O O O O O O O O O A • K • s 1 O O r 1   y 1 / / x 1 o o B • ⊗ T B • L • s 2 O O r 2   y 2 / / x 2 o o C • = A • K • × B • L • s 1 × s 2 O O r 1 × r 2   y 1 ◦ pr 2 / / x 1 ◦ pr 1 o o C • N • 1 q ′ 1 _ _ ? ? ? ? ? ? ? ? p ′ 1 ? ?         N • 2 q ′ 2 _ _ ? ? ? ? ? ? ? ? p ′ 2 ? ?         N • 1 × B • N • 2 q ′ 1 ◦ pr 1 g g O O O O O O O O O O O O O p ′ 2 ◦ pr 2 7 7 o o o o o o o o o o o o o 3. 2-morphisms, by M • 1 q 1           p 1   ? ? ? ? ? ? ? ? M • 2 q 2           p 2   ? ? ? ? ? ? ? ? M • 1 × B • M • 2 q 1 ◦ pr 1 w w o o o o o o o o o o o o o p 2 ◦ pr 2 ' ' O O O O O O O O O O O O O A • K • 1 → K • 2 B • ⊗ T B • L • 1 → L • 2 C • = A • K • 1 × B • L • 1 → K • 2 × B • L • 2 C • N • 1 q ′ 1 _ _ ? ? ? ? ? ? ? ? p ′ 1 ? ?         N • 2 q ′ 2 _ _ ? ? ? ? ? ? ? ? p ′ 2 ? ?         N • 1 × B • N • 2 q ′ 1 ◦ pr 1 g g O O O O O O O O O O O O O p ′ 2 ◦ pr 2 7 7 o o o o o o o o o o o o o H H       V V - - - - - -   - - - - - -         H H       V V - - - - - -   - - - - - -         D D        Z Z 4 4 4 4 4 4 4   4 4 4 4 4 4 4          W e lea ve deﬁning the r es t of the data as well as v erifying that they satisfy the axioms to the reader. The t r ihomomorphism (3.4) extends to a t r iho momorphism 2 ℘ : T [ − 2 , 0] ( S ) / / 2Pic ( S ) (6.1) 31 on the tricat ego ry T [ − 2 , 0] ( S ) a s follows 1 : On ob jects, it is deﬁned as explained in Section 3 .2. On 1-, 2-, 3-morphisms , b y the biequiv alence 2 ℘ ( A • ,B • ) , whe re A • and B • are any tw o complexes of ab elian shea ve s. Theorem 5.6 implies that (6.1) is a lready fully faithful in t he a ppropriate sense. In order to prov e the triequiv alence, one needs to sho w that it is essen tially surjectiv e, as w ell. The esse n tia l surjectivit y dep ends on the fo llo wing tec hnical lemma, whic h is similar to Lemme 1.4.3 in [1 1]. W e give its pr o of in the App endix ( A). Prop osition 6.2. F or any se t E , deno te by Z ( E ) the fr e e a b elian gr oup gener ate d by E . L et C b e a Pic ar d 2-c ate gory an d F 0 : E → C b e a set map. Then F 0 extends to an additive 2-functor F : Z ( E ) → C wher e Z ( E ) is c onsider e d as a 2-c ate gory (trivial ly Pic a r d). Lemma 6.3. L et P b e a Pic ar d 2-stack, then ther e exists a c omp l e x of ab eli an sh e av es A • such that 2 ℘ ( A • ) is bie quivalent to P . Pr o of. There is a construction analogo us t o the sk eleton o f categories. F or any 2 -category P , w e construct 2sk( P ) a 2-categor y tha t has one ob ject p er equiv alence class in P . W e observ e that 2sk( P ) is a full sub 2-cat ego ry of P , that is the inclusion 2sk( P ) → P is a biequiv alenc e. Let P b e a Picard 2-stac k. W e note that Ob 2sk( P ) : U → Ob(2sk( P U )) is a presheaf of sets. W e consider A 0 the a b elian sheaf ov er S asso ciated t o the presheaf { U → Z (Ob(2sk( P U ))) } where Z (Ob(2sk( P U ))) is the free ab elian group asso ciated t o Ob(2sk( P U )). By Prop osition 6.2 , the inclusion i : Ob 2 sk ( P ) → P extends to π P : A 0 / / P an essen tially surjectiv e additive 2- functor on A 0 . Deﬁne A b y the pullbac k diagram A / / Λ A   0   A 0 π P / / P     < D (6.2) of morphisms of Picard 2-stac ks, which is similar to (2.7). Then, the sequence of Picard 2-stac ks A / / A 0 / / P is exact sequence in the sense of Section 2.3 . On t he other hand, from Lemma 2 .4 , it follows that A is a Picard stac k. Therefore by [3, Prop osition 8.3.2], there exists a morphism of ab elian shea v es δ A : A − 2 → A − 1 , where A − 2 is deﬁned b y the pullback diag ram A − 2 / / δ A   0   A − 1 π A / / A     = E (6.3) 1 W e commit an abuse of no tation b y ca lling both functor s (3.4) and (6.1) by 2 ℘ . 32 and A := Tors ( A − 2 , A − 1 ). No w putting the diagrams ( 6 .2) and (6.3) tog ether, A − 2 / / δ A   0   A − 1 π A / / λ A % % 8 ; ? B E H K A Λ A   / /     = E 0   A 0 π P / / P     ; C (6.4) w e ha ve a diagram o f Picard 2-stac ks. It implies that A • : A − 2 δ A / / A − 1 λ A / / A 0 is a complex. The Picard 2-stac k asso ciated t o A • , that is 2 ℘ ( A • ) := Tors ( A , A 0 ), v eriﬁes b y deﬁnition the ab o ve diagram (see 2.9). The biequiv alenc e 2 ℘ ( A • ) ≃ P is almost immediate. Essen t ia l surjectivit y follows from the deﬁnition of π P and equiv alence of hom-categories f rom the fact that A 0 and 0 pull bac k to A o ver 2 ℘ ( A • ) and ov er P . 6.2 Main Theorem Considering 2Pic ( S ) as a tricategory , our main resu lt follows from Theorem 5.6 a nd Lemma 6.3. Theorem 6.4. Th e trihomomorphism (6.1) is a trie quivalenc e. An immediate consequence of Theorem 6.4, whic h w as also t he mo t iv a tion for this pap er, is the fo llo wing. Let 2Pic ♭♭ ( S ) denote t he categor y of Picard 2-stac ks obtained from 2Pic ( S ) b y ignoring the mo diﬁcations and taking as morphisms the equiv alence classes of additiv e 2- functors. Let D [ − 2 , 0] ( S ) b e the sub category of the deriv ed category of category of complexes of ab elian shea ves A • o ver S with H − i ( A • ) 6 = 0 f or i = 0 , 1 , 2 . W e deduce from Theorem 6.4 the following, whic h generalizes D eligne’s result [11, Prop osition 1.4.15] fr o m Picard stac ks to Picard 2-stac ks. Corollary 6.5. The functor (6.1) induc es an e quivalenc e 2 ℘ ♭♭ : D [ − 2 , 0] ( S ) / / 2Pic ♭♭ ( S ) (6.5) of c ate gories. Pr o of. It is enough to o bse rv e from the calculations in Section 4 that π 0 ( F rac ( A • , B • )) ≃ Hom D [ − 2 , 0] ( S ) ( A • , B • ). Since the ob jects of D [ − 2 , 0] ( S ) a re same as the ob jects of T [ − 2 , 0] ( S ), the essen tial surjectivit y follows from Lemma 6.3. 33 7 Stac kiﬁcati on W e wan t to conclude with an info r mal discussion of stack versions of some of o ur results. W e will assum e that all structures are strict unless otherwise stated. Througho ut the pap er, w e dealt with 2- and 3- categories and their we ak ened v ersions bi- and tr icat ego ries. They can b e stac kiﬁed. 2-stac ks ov er a site are w ell kno wn [8]. The collection of 2-stac ks o v er S , denoted by 2 St a ck ( S ), comprise a 3 -category structure. W e can consider the ﬁb ered 3-category 2 S t a ck ( S ), whose ﬁb er ov er U is the 3- category 2 St ack ( S /U ) of 2-stack s o v er S /U . In [8, Remark 1.12], Breen claims that 2 S t ack ( S ) is a 3-stac k. Hirsc howitz and Simps on in [17], generalize this result to w eak n -stacks . Theorem. [17, Th´ eor ` eme 20.5] The we ak ( n + 1) -p r estack of we ak n -stacks nW S t a ck ( S ) is a we ak ( n + 1) -stack over S . W e can use t he ab ov e fa cts to deduce that the 3-prestac k of Picard 2-stack s 2 P ic ( S ) with ﬁb ers 2Pic ( S /U ) o ve r U is a 3-stack. Claim. H om ( A • , B • ) ﬁb er e d over S in 2-gr oup oids is a 2-stack wher e for an y U ∈ S , the 2- gr oup oid Hom ( A • | U , B • | U ) of additive 2-functors fr om 2 ℘ ( A • ) | U to 2 ℘ ( B • ) | U deﬁnes the ﬁb er over U . W e ha v e also ﬁb ered analogs for eac h hom-bicategory F ra c ( A • , B • ) and for T [ − 2 , 0] ( S ). It follo ws from the ab o v e claim and T heorem 5.6 that t he bi-prestac k F rac ( A • , B • ) of fractions from A • to B • with ﬁb ers deﬁned by F rac ( A • | U , B • | U ) is a bistac k. Then, once an appropriate notion of 3-descen t has b een sp eciﬁed and all descen t data are sho wn to b e eﬀectiv e, w e conclude b y the ch aracterization pro position [17, Prop osition 10.2 ] fo r n -stac ks that the tri- prestac k of complexes T [ − 2 , 0] ( S ) with ﬁb ers T [ − 2 , 0] ( S /U ) is a tristac k. The c haracterization prop osition cited ab ov e brieﬂy say s that P is an n -stac k ov er S if and only if all descen t data are eﬀectiv e and for any X , Y ob jects of P U , Hom P U ( X , Y ) is an n − 1 stack ov er S /U . R emark 7.1 . The c haracterization prop osition in [1 7, Prop osition 10.2] is originally enounced for Segal n - categories, n -prestacks , and n -stack s. But again in the same pap er, it has b een remark ed t ha t the pro position holds for no n-Segal structures [17, § 20] where in this case, the w eak structure is assumed to b e the one deﬁned by T amsamani. Its deﬁnition can b e found in [24] and [25]. Ho wev er, we are b eing v ery infor mal and not discussing here the connection of the w eak structure of our categories, pre-stac ks and, stac ks with the ones men tioned ab o ve. Finally , we deﬁne the t r iho momorphism o f tristac ks b y lo calizing the triequiv alence (6.1 ). T [ − 2 , 0] ( S ) / / 2 P ic ( S ) , (7.1) where 2 P ic ( S ) is considered naturally a s a tristac k. W e deduce then its stack ana lo g Theorem 7.2. (7. 1 ) is a trie quivalenc e of tristacks. A App end i x W e giv e the pro of o f Prop osition (6.2). W e assume that the set E is w ell ordered and denote the order on E b y  . In what follo ws, w e deﬁne 34 1. a 2-functor F : Z ( E ) → C , 2. for any t wo words w 1 and w 2 in Z ( E ), a functorial 1-morphism λ w 1 ,w 2 λ w 1 ,w 2 : F ( w 1 ) ⊗ F ( w 2 ) / / F ( w 1 + w 2 ) , 3. for any three w ords w 1 , w 2 , a nd w 3 in Z ( E ), a 2-morphism ψ w 1 ,w 2 ,w 3 (A.8), 4. for any t wo words w 1 and w 2 in Z ( E ), a 2-morphism φ w 1 ,w 2 (A.10). A.1 Deﬁnition of F W e construct t he 2-f unctor F : Z ( E ) → C as f o llo ws: • F or any generator a ∈ E , F a := F 0 a , • F or any generator a ∈ E , F ( − a ) := ( F a ) ∗ , where ( F a ) ∗ is in vers e of F a in C , • F (0) is t he unit elemen t in C , where 0 denotes the unit elemen t in Z ( E ). • F or any w ord w in Z ( E ), w e – simplify w so that there a re no cancelations and denote the simpliﬁed w ord b y w s , – order the letters of w s from least to greatest and denote the simpliﬁed and ordered w ord b y w s,o . F ( w ) is deﬁned b y multiplyin g the letters of w s,o from left to right. F or instance le t w = 2 a + b − c − a − 2 b . After cancelations and ordering the letters w s,o = a − b − c and F ( w ) = F ( w s,o ) = (( F a ⊗ ( F b ) ∗ ) ⊗ F c ) . The order on the set E is needed since without the order tw o words that diﬀer by t he p osition of letters would map to diﬀeren t ob jects in C although they are the same word in Z ( E ). F or the reasons o f compactness, w e use juxtap osition for the group op eration ⊗ on the 2-category C . A.2 Monoidal Case The items (2- ( 4) describ es the additiv e structure of the 2 -functor F . W e ﬁrst deﬁne them on the w ords tha t do not ha v e letters with negative co eﬃcien ts. That is, they are constructed ﬁrst on the f r ee ab elian monoid N ( E ). In App endix A.3, w e extend their deﬁnitions to the free ab elian group Z ( E ). W e leav e the v eriﬁcation o f their compatibilit y with the Picard structure to the autho r ’s thesis [26]. 35 Deﬁnition of λ w 1 ,w 2 : Let w 1 = a 1 + . . . + a m and w 2 = b 1 + . . . + b n b e tw o w ords in N ( E ). The w ord w 1 + w 2 is deﬁned b y concatenation of w 1 and w 2 and then b y an ( m, n )-sh uﬄe so that the letters o f w 1 and w 2 are ordered from least to greatest. W e denote w 1 + w 2 b y c 1 + . . . + c m + n . F rom the deﬁnition of F , F ( w 1 ) ⊗ F ( w 2 ) = ( . . . (( F a 1 F a 2 ) F a 3 ) . . . F a m ) ⊗ ( . . . (( F b 1 F b 2 ) F b 3 ) . . . F b n ) (A.1) F ( w 1 + w 2 ) = ( . . . (( F c 1 F c 2 ) F c 3 ) . . . F c m + n ) (A.2) W e deﬁne the functorial morphism λ w 1 + w 2 : F ( w 1 ) ⊗ F ( w 2 ) → F ( w 1 + w 2 ) in t w o ste ps as follo ws: Step 1: Correct Brac ke ting In this step, we deﬁne the morphism ( . . . (( F a 1 F a 2 ) F a 3 ) . . . F a m ) ⊗ ( . . . (( F b 1 F b 2 ) F b 3 ) . . . F b n ) → (((( . . . ( ( F a 1 F a 2 ) F a 3 ) . . . F a m ) F b 1 ) F b 2 ) . . . F b n ) , (A.3) whic h mov es the pairs of parenthe sis of F ( w 2 ) one b y one to the left from the o uter most to the inner most without changing the place of paren thesis o f F ( w 1 ). (A.3 ) is comp osition of n − 1 many morphisms of the form ( . . . (( F ( w 1 )( F ( w ′ 2 ) F b i )) F b i +1 ) . . . F b n ) → ( . . . ((( F ( w 1 ) F ( w ′ 2 )) F b i ) F b i +1 ) . . . F b n ) , (A.4) where w ′ 2 is a sub word of w 2 . Step 2: Ordering Letters Once the morphism (A.3) is applied, the letters of w 1 and w 2 are paren thesized from left. Next, w e deﬁne the morphism (((( . . . ( ( F a 1 F a 2 ) F a 3 ) . . . F a m ) F b 1 ) F b 2 ) . . . F b n ) → ( . . . (( F c 1 F c 2 ) F c 3 ) . . . F c m + n ) , (A.5) that sh uﬄes the letters of w 1 and w 2 to order them from least to greatest, that is c 1  c 2  . . .  c m + n . The rule is, ﬁnd the smallest letter of w 2 in w 1 + w 2 suc h that it has a letter o f w 1 on its left that is greater, c hange their places. Dep ending on the p osition of the letters, there are tw o cases. Either the letters are in the same paren thesis , then (A.5) simply p erm utes them ( . . . (( F c 1 F c 2 ) F c 3 ) . . . F c m + n ) → ( . . . (( F c 2 F c 1 ) F c 3 ) . . . F c m + n ) , (A.6) or they ar e in diﬀeren t pairs of paren thesis and (A.5) ﬁrst groups them together b y mo ving the appropriate pair of pa ren thesis to the rig h t, then p ermu tes the letters , and mo ves the pair of parenthe sis mov ed to the righ t to the left, that is (( . . . ((( ( . . . ( F c 1 F c 2 ) . . . ) F c k − 1 ) F c k +1 ) F c k ) . . . ) F c m + n ) → (( . . . ((( . . . ( F c 1 F c 2 ) . . . ) F c k − 1 )( F c k +1 F c k )) . . . ) F c m + n ) → (( . . . ((( . . . ( F c 1 F c 2 ) . . . ) F c k − 1 )( F c k F c k +1 )) . . . ) F c m + n ) → (( . . . ((( ( . . . ( F c 1 F c 2 ) . . . ) F c k − 1 ) F c k ) F c k +1 ) . . . ) F c m + n ) (A.7) 36 where c k is a letter o f w 2 in w 1 + w 2 with 1 < k < m + n and c k − 1 is a letter o f w 1 suc h that c k ≺ c k − 1 . W e rep eat the a bov e pro cess to ev ery letter of w 2 in w 1 + w 2 . W e deﬁne the morphism (A.5) as comp osition of t he morphisms of the form (A.6) or (A.7). W e can illustrate the map (A.5) by the lattice paths [13, Chapter 7.3 D ]. It is clear tha t there is a 1-1 corresp ondence b et w een the lattice paths from (0 , 0) to ( m, n ) and the ( m, n )-sh uﬄes. (A.2) can b e seen as the lattice path cor r esp onding to the ( m, n )-shuﬄ e of the w ords w 1 , w 2 that deﬁnes w 1 + w 2 and (A.1) a s the lattice path corresp onding to the concatenation of the w ords w 1 and w 2 (i.e. the empty ( m, n )-sh uﬄe). W e denote these paths by L w 1 + w 2 and L w 1 ,w 2 , respectiv ely . F rom this p erspectiv e, the map (A.5) can b e thought as applying an ( m, n )-sh uﬄe to the concatenation of the w ords w 1 and w 2 . (0 , 0) ( m, n ) • • • • • • • • • • • • • • • • • • • • a 1 . . . a m b 1 . . . . . . b n Lattice P ath L w 1 ,w 2 ✲ (0 , 0) ( m, n ) • • • • • • • • • • • • • • • • • • • • a 1 . . . a m Lattice P ath L w 1 + w 2 b 1 . . . . . . b n The morphisms (A.6) and (A.7) describe the basic mov emen t. They substitute the p oin t ( i, j ) on the lattice path with t he p oin t ( i − 1 , j + 1) as show n in the picture b elo w. ✲ ( i − 1 , j + 1) ( i − 1 , j ) ( i, j + 1) ( i, j ) ( i − 1 , j + 1) ( i − 1 , j ) ( i, j + 1) ( i, j ) • • • • • • • • The o ve rall mo v emen t is describ ed b y the morphism (A.5) where each step is a basic mo vem en t. W e deﬁne the follow ing sp ecial p oint on the lattice path in order to explain the mec hanism of the mov emen ts. W e call the p oin t ( i, j ) o n the lattice path t he c orner p oint if the p oin t s ( i − 1 , j ) and ( i, j + 1) are o n the lattice path, as w ell. The morphism (A.5) pic ks at ev ery step the corner p oin t ( i, j ) with the least y -co ordinate that is not on the latt ice path L w 1 + w 2 and substitutes it with ( i − 1 , j + 1 ). W e sho w in the picture b elo w the transformation of the latt ice path L w 1 ,w 2 to the lattice path L w 1 + w 2 . 37 • • • • • • • • • • • • • • • • • • • • a 1 . . . a m b 1 . . . . . . b n ✲ • • • • • • • • • • • • • • • • • • • • a 1 . . . a m b 1 . . . . . . b n ✲ • • • • • • • • • • • • • • • • • • • • a 1 . . . a m b 1 . . . . . . b n ❄ • • • • • • • • • • • • • • • • • • • • a 1 . . . a m b 1 . . . . . . b n ✛ • • • • • • • • • • • • • • • • • • • • a 1 . . . a m b 1 . . . . . . b n ✛ • • • • • • • • • • • • • • • • • • • • a 1 . . . a m b 1 . . . . . . b n The morphism (A.3) obtained in t he ﬁrst step follow ed by the morphism (A.5) constructed in the second step deﬁnes λ w 1 ,w 2 . W e remark that if a ll the letters o f w 1 are less t han all the letters of w 2 , the n w 1 + w 2 is obtained by concatenating the words w 1 and w 2 without the sh uﬄe. That is L w 1 + w 2 coincides with L w 1 ,w 2 . In this case λ w 1 ,w 2 is of the form (A.3). W e also observ e that the morphism λ w 1 ,w 2 is a path in the 1 - sk eleton of p erm uto- associahedron K Π m + n − 1 where m and n are lengths of the w ords w 1 and w 2 , resp ectiv ely . K Π m + n − 1 is a p olytop e whose vertices are all p ossible orderings and groupings o f strings o f length m + n and whose edges are all p ossible adjacen t p erm utations and all p ossible paren thesis mo v emen ts. F or more details ab out p erm uto-asso ciahedron, w e refer to [1 9] a nd [27 ]. Deﬁnition of ψ w 1 ,w 2 ,w 3 : F or an y three words w 1 , w 2 , w 3 in N ( E ), w e deﬁne the 2-morphism ψ w 1 ,w 2 ,w 3 (( F ( w 1 ) F ( w 2 )) F ( w 3 )) a   λ w 1 ,w 2 / / F ( w 1 + w 2 ) F ( w 3 ) λ w 1 + w 2 ,w 3 / /       ψ w 1 ,w 2 ,w 3 F ( w 1 + w 2 + w 3 ) ( F ( w 1 )( F ( w 2 ) F ( w 3 ))) λ w 2 ,w 3 / / F ( w 1 ) F ( w 2 + w 3 ) λ w 1 ,w 2 + w 3 / / F ( w 1 + w 2 + w 3 ) (A.8) 38 b et w een the 1-morphisms λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 ◦ a and λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 from (( F ( w 1 ) F ( w 2 )) F ( w 3 )) to F ( w 1 + w 2 + w 3 ) 1 . These 1-morphisms are paths in the 1-sk eleton of K Π m + n + p − 1 where n , m , and p are the lengths of the w ords w 1 , w 2 , and w 3 , resp ectiv ely . This follo ws from the fact that ev ery ma p in the diagram (A.8) is in the 1-sk eleton of K Π m + n + p − 1 . In order to b etter understand these paths, w e in terpret them in terms of 3- dimensional lattice paths. Assume that the letters of the words w 1 , w 2 , and w 3 represen t resp ectiv ely the unit interv a ls on the x -, y -, a nd z -axis. F ( w 1 + w 2 + w 3 ) can b e represen ted b y the 3 -dimensional lattice path corresponding to the ( m, n, p )-shu ﬄe of the w ords w 1 , w 2 , w 3 that deﬁnes w 1 + w 2 + w 3 and (( F ( w 1 )( F ( w 2 )) F ( w 3 )) b y the 3- dimens ional lattice path corresp onding to the empt y sh uﬄe of the w ords w 1 , w 2 , w 3 . Therefore, the paths λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 ◦ a and λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 can b e though t as tw o diﬀeren t w ay s of shuﬄin g w 1 , w 2 , w 3 to obtain w 1 + w 2 + w 3 . The path λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 ◦ a ﬁrst do es the ( n, p )-sh uﬄe then the ( m, n )-sh uﬄe. On the other hand the path λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 do es the ( m, n )-shuﬄ e ﬁrst, then the ( n, p )-sh uﬄe. In this sense the 2-morphism ψ w 1 ,w 2 ,w 3 can b e seen as the connection b et w een the tw o diﬀeren t w ay s of doing the ( m, n, p )-sh uﬄe. T o deﬁne the 2- morphism ψ w 1 ,w 2 ,w 3 , we need the following lemmas . Lemma A.1. L et w 1 and w 2 b e two elem ents of N ( E ) . λ w 2 ,w 3 = c and λ w 1 ,w 2 = id if and only if λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 ◦ a = λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 Pr o of. W e ﬁrst remark that λ w 2 ,w 3 = c and λ w 1 ,w 2 = id is equiv alent to ass uming w 2 and w 3 are letters suc h that w 2 is greater than w 3 and w 2 is greater than or equal to all letters of w 1 . These facts imply that the map λ w 1 + w 2 ,w 3 ﬁrst p erm utes F ( w 2 ) and F ( w 3 ) then shuﬄe s F ( w 1 ) and F ( w 3 ) without c hanging the p osition of F ( w 2 ). Th us λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 ◦ a = λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 . In the other direction, w e observ e that the morphism a can b e only part of t he morphism λ w 1 ,w 2 + w 3 whic h means λ w 1 ,w 2 = id. This requires w 2 to be a letter greater than or equal to all letters o f w 1 and λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 ◦ a = λ w 1 + w 2 ,w 3 . W e also observ e that a paren thesis mo vem en t caused b y λ w 2 ,w 3 eﬀects only t he pla ces of the paren t hesis ar o und the letters of w 2 and w 3 and suc h a mo v emen t cannot b e caused by λ w 1 + w 2 ,w 3 . This means λ w 2 ,w 3 do es not cause any parenthes is mo ve men ts. Hence, w e deduce that w 3 is also a letter. If w 2  w 3 then λ w 2 ,w 3 and λ w 1 + w 2 ,w 3 b ecome iden tit y morphisms and w e o btain λ w 1 ,w 2 + w 3 ◦ a = id whic h is not p ossible. Therefore λ w 2 ,w 3 should consist of a single p erm utation. Lemma A.2. L et w 1 , w 2 , an d w 3 b e thr e e elements o f N ( E ) . T h en the f o l lowings ar e e quiva- lent. 1. The p ath λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 is strictly include d in λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 ◦ a . That is V ( w 1 ,w 2 | w 3 ) the ve rtex set of the p ath λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 is strictly include d in V ( w 1 | w 2 ,w 3 ) the vertex s e t of the p ath λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 ◦ a . 2. λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 = λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 ◦ a − 1 . 3. λ w 2 ,w 3 = id . Pr o of. It is clear that (2) implies (1). (3) ⇒ (2): λ w 2 ,w 3 = id is equiv alen t to assuming that b oth w 2 and w 3 are letters and w 2 ≺ w 3 . This requires F ( w 1 ) F ( w 2 + w 3 ) to b e o f the form F ( w 1 )( F ( w 2 ) F ( w 3 )). Since all 1 W e co mmit an abus e of nota tion in diag ram (A.8). By λ w 1 ,w 2 and λ w 2 ,w 3 we mean λ w 1 ,w 2 ⊗ id w 3 and id w 1 ⊗ λ w 2 ,w 3 , resp ectively . 39 the morphisms λ ’s start with mo ving paren thesis to the left, λ w 1 ,w 2 + w 3 starts exactly with a − 1 . Therefore λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 = λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 ◦ a − 1 . (1) ⇒ (3): In all the ve rtices that λ w 2 ,w 3 pass through, F ( w 1 ) is group ed separately fr om F ( w 2 ) and F ( w 3 ). Therefore an y parenthes is mo ve men t or p erm utatio n that is part of λ w 2 ,w 3 do es not c hange the paren thesis a round F ( w 1 ). How ev er, on the path λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 the same mo vem en ts that describ e λ w 2 ,w 3 are part o f the morphism λ w 1 + w 2 ,w 3 . Since this path passes through the v ertices that group F ( w 1 ) and F ( w 2 ), the paren thesis mo v emen ts and p erm uta - tions change the parenthes is around F ( w 1 ). This con tradicts to the fact that λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 is included in λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 ◦ a . W e remark that Lemma A.2 can b e also expressed as λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 is strictly included in λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 ◦ a if and only if V ( w 1 | w 2 ,w 3 ) = V ( w 1 ,w 2 | w 3 ) ∪ { ( F ( w 1 )( F ( w 2 ) F ( w 3 ))) } . W e can return to t he deﬁnition of the 2 - morphism ψ w 1 ,w 2 ,w 3 . By Lemmas A.1 and A.2, the paths λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 ◦ a and λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 are g o ing to satisfy o ne of the following three cases. 1. The pa ths may b e the same. In this case, t he 2-morphism ψ w 1 ,w 2 ,w 3 is iden tity . 2. The path λ w 1 + w 2 ,w 3 ◦ λ w 1 ,w 2 is strictly included in λ w 1 ,w 2 + w 3 ◦ λ w 2 ,w 3 ◦ a . In t his case, b y Lemma A.2, the 2- morphism ψ w 1 ,w 2 ,w 3 is aa − 1 ⇒ id. 3. The paths may enclose a 2-cell. This 2-cell is a tiling of p entagonal and rectangular 2 - cells. The p en tagonal 2-cells are either MacLane P en tag o ns or their deriv ativ es obtained b y in verting the direction of an edge. The rectangular 2-cells are of the f o rm • a 1 / / a 2   • a 2   • a 1 / / • • a 1 / / c 1   • c 1   • a 1 / / • • c 1 / / c 2   • c 2   • c 1 / / • (A.9) where a 1 , a 2 are either leftw ard or right w ard paren thesis mov emen ts and c 1 , c 2 p erm ute adjacen t o b jects. R ectangular 2-cells can b e also derive d from (A.9) b y in v erting the direction of an edge. These 2-cells comm ute up t o structural 2-morphisms deﬁned by the Picard structure of the 2-category C . Theorem 3.3 in [2 3] implies that these 2- morphisms comp ose in a unique wa y . W e let ψ w 1 ,w 2 ,w 3 b e this comp osition. Deﬁnition of φ w 1 ,w 2 : The last piece of the additive structure o f F is the 2-morphism φ w 1 ,w 2 F ( w 1 ) F ( w 2 ) c   λ w 1 ,w 2 / /       φ w 1 ,w 2 F ( w 1 + w 2 ) F ( w 2 ) F ( w 1 ) λ w 2 ,w 1 / / F ( w 2 + w 1 ) (A.10) b et w een the 1-morphisms λ w 2 ,w 1 ◦ c and λ w 1 ,w 2 from F ( w 1 ) F ( w 2 ) to F ( w 1 + w 2 ) where w 1 and w 2 are an y t wo w ords in N ( E ). W e notice tha t the path λ w 2 ,w 1 ◦ c is not necess arily in the 1-sk eleton of K Π m + n − 1 . The reason is that t he braiding c is not an adjacen t p erm utation unless w 1 and w 2 are letters. In the case where the words w 1 and w 2 are letters, φ w 1 ,w 2 is deﬁned by the ta ble 40 w 1 w 2 φ w 1 ,w 2 a a id a b id ⇒ c 2 b a id where id ⇒ c 2 is giv en b y the Picard structure of the 2-category C . No w, w e assume that w 1 and w 2 are tw o w ords suc h that their sum of lengths is m + n ≥ 3. The 2-morphism φ w 1 ,w 2 is deﬁned in the follow ing w a y . W e ﬁrst transform the pat h λ w 2 ,w 1 ◦ c to a path in the 1 -sk eleton of K Π m + n − 1 . Second w e apply the pro cess that deﬁnes ψ w 1 ,w 2 ,w 3 to the new path and the path λ w 1 ,w 2 . φ w 1 ,w 2 is then deﬁned as the appro priate comp o sition of the 2-morphisms o bta ined a t the ﬁrst and the second step. Therefore to deﬁne φ w 1 ,w 2 , it suﬃces to describ e how w e transform the path λ w 2 ,w 1 ◦ c into a pa t h in the 1-ske leton of K Π m + n − 1 . The main idea is to substitute the edge c that is not in the 1-sk eleton b y a sequence of ﬁv e ot her edges. This sequence is an a lternating collection of three left w ard or right w ard paren thesis mo v emen t s and tw o bra idings. The paren thesis mov emen ts are certainly in the 1-sk eleton; how ev er the braidings ma y not b e. If they are not, then we substitute eac h of those braidings b y a sequence of ﬁve ot her edges as ab o v e. W e ke ep substituting until all the braidings b ecome p erm utations of adjoint ob jects, therefore part of the 1-sk eleton. W e kno w that the substitution pro cess is going to terminate b ecause after each substitution braidings p erm ute pa ren thesized ob jects with shorter length. W e describe this pro cess on the sample w 1 = b + e and w 2 = a + c + d . The bra iding c p er- m utes F ( w 1 ) a nd F ( w 2 ). First, w e substitute c b y the braidings c ( a,c,d | e ) and c ( a,c,d | b ) . c ( a,c,d | e ) p erm utes the parenthe sized ob ject (( F aF c ) F d ) with F e and c ( a,c,d | b ) p erm utes (( F aF c ) F d ) with F b . They are going to b e substituted b y c ( d | e ) and c ( a,c | e ) and b y c ( a,c | b ) and c ( d | b ) , re- sp ectiv ely . Since c ( d | e ) p erm utes F d and F e and c ( d | b ) p erm utes F d and F b , t hey ar e edges in the 1-sk eleton and therefore cannot b e subs tituted. In the diagram b elo w, w e illustrat e the complete pro cess of substituting c by adjacen t p erm utations c ( a | b ) , c ( c | b ) , c ( d | b ) , c ( a | e ) , c ( c | e ) , and c ( d | e ) using lattice paths. 41 • • • • • • • • • • • • ⑦ c b e a c d ✲ c ( a,c,d | e ) ❄ c ( a | e ) ❅ ❅ ❅ ❅ ❅ ❘ c ( a,c | e ) • • • • • • • • • • • • b e a c d ✲ c ( a,c,d | b ) ❄ c ( a | b ) ❅ ❅ ❅ ❅ ❅ ❘ c ( a,c | b )      ✒ c ( d | e )      ✒ c ( d | b ) • • • • • • • • • • • • b e a c d • • • • • • • • • • • • b e a c d • • • • • • • • • • • • b e a c d • • • • • • • • • • • • b e a c d      ✒ c ( c | e ) • • • • • • • • • • • • b e a c d      ✒ c ( c | b ) This pro cess deﬁnes a 2- morphism as follows . Substituting a braiding b y an alternat- ing s equence of three leftw ard or righ tw ard paren thesis mo v emen ts and tw o braidings means substituting an edge in a hexagonal 2- cell b y the other ﬁve edges. Such hexagonal 2-cells com- m ute up to a 2 - morphism giv en b y the Picard structure of the 2-category C . The a ppropriate comp osition of these 2 - morphisms deﬁnes the 2- morphism of the ﬁrst step. A.3 Extending the Additiv e S tru c ture to F ree Ab elian Group Here w e extend the deﬁnition of the 2 -functor F so that it transforms the trivial Picard structure of the free ab elian group Z ( E ) generated b y the set E to the Picard structure of the 2-category C . 42 Extending λ w 1 ,w 2 : The extension of λ w 1 ,w 2 , denoted by e λ w 1 ,w 2 , to the w or ds in Z ( E ) should tak e in to cons ideration the cance lations that might occur in w 1 + w 2 . If w 2 do es no t ha v e a letter that app ears with an opp osite sign in w 1 then there aren’t a ny cancelations in w 1 + w 2 and e λ w 1 ,w 2 = λ w 1 ,w 2 . Otherwise, e λ w 1 ,w 2 orders the letters of w 1 and w 2 from least to greatest, left par enthes izes, and do es the cancelations starting with the image of the least letter. That is e λ w 1 ,w 2 is equal to p ost comp osition of λ w 1 ,w 2 with the morphisms of the form ( . . . ((( F ( w ) F c i )( F c i ) ∗ ) F c i +1 ) . . . F c n + m ) / / ( . . . (( F ( w )( F c i ( F c i ) ∗ )) F c i +1 ) . . . F c n + m ) ED BC GF inv F c i @A / / ( . . . (( F ( w ) I ) F c i +1 ) . . . F c n + m ) r F ( w ) / / ( . . . ( F ( w ) F c i +1 ) . . . F c n + m ) (A.11) for ev ery cancelation. In (A.11) w is a sub w o r d of w 1 + w 2 , I is a unit elemen t in the Picard 2-category and inv F c i and r F ( w ) are s tructural morphisms due to the Picard structure of the 2-category . By the Picard structure, w e can a lso assume for simplicit y that when e λ w 1 ,w 2 orders letters from least to greatest the in vers e of a n ob ject is alw a ys a dj acen t to the ob ject and it is on its left. W e note that us ing λ w 1 ,w 2 for the morphism that orders the letters of w 1 and w 2 from least to g reatest and left paren thesizes them is an abus e of notation. Here λ w 1 ,w 2 do es not ma p to t he ob ject F ( w 1 + w 2 ) but to a n ob ject that we denote F ( w 1 , 2 ). F ( w 1 , 2 ) is pro duct of the images of all letters in w 1 and w 2 paren thesized from the left, o r dered from least to greatest, and if there exists in v erse of an ob ject is placed on its left. F o r instance, if w 1 = b + c and w 2 = a − b , then λ w 1 ,w 2 : ( F bF c )( F a ( F b ) ∗ ) / / ((( F aF b ) F b ) ∗ ) F c ) , where F ( w 1 , 2 ) = (( F aF b )( F b ) ∗ ) F c ). Th us e λ w 1 ,w 2 can b e expressed as comp osition of F ( w 1 ) F ( w 2 ) λ w 1 ,w 2 / / F ( w 1 , 2 ) τ w 1 ,w 2 / / F ( w 1 + w 2 ) , (A.12) where τ w 1 ,w 2 is comp osition of morphisms of the fo r m (A.11) for ev ery cancelation. W e re- mark t hat λ w 1 ,w 2 as in the mo no idal case deﬁnes a path in the 1 -sk eleton of the p erm uto - asso ciahedron K Π m + n − 1 . How ev er if there a re cancelations, e λ w 1 ,w 2 is not a path in the 1- sk eleton of K Π m + n − 1 . Extending ψ w 1 ,w 2 ,w 3 : The extension of ψ w 1 ,w 2 ,w 3 , denoted b y e ψ w 1 ,w 2 ,w 3 , to t he w ords w 1 , w 2 , w 3 in Z ( E ) is a 2- morphism (( F ( w 1 ) F ( w 2 )) F ( w 3 )) a   e λ w 1 ,w 2 / / F ( w 1 + w 2 ) F ( w 3 ) e λ w 1 + w 2 ,w 3 / /       e ψ w 1 ,w 2 ,w 3 F ( w 1 + w 2 + w 3 ) ( F ( w 1 )( F ( w 2 ) F ( w 3 ))) e λ w 2 ,w 3 / / F ( w 1 ) F ( w 2 + w 3 ) e λ w 1 ,w 2 + w 3 / / F ( w 1 + w 2 + w 3 ) (A.13) b et w een the 1-morphisms e λ w 1 ,w 2 + w 3 ◦ e λ w 2 ,w 3 ◦ a and e λ w 1 + w 2 ,w 3 ◦ e λ w 1 ,w 2 . As noticed, these paths ma y not b e in t he 1-sk eleton of K Π m + n + p − 1 . How ev er, there exists a ve rtex V 0 of the 43 p erm uto-asso ciahedron K Π m + n + p − 1 that b oth paths e λ w 1 + w 2 ,w 3 ◦ e λ w 1 ,w 2 and e λ w 1 ,w 2 + w 3 ◦ e λ w 2 ,w 3 pass through. Therefore the diagram (A.13) can b e rewritten a s: (( F ( w 1 ) F ( w 2 )) F ( w 3 )) a   / /       ψ ′ w 1 ,w 2 ,w 3 V 0 / / F ( w 1 + w 2 ) F ( w 3 ) e λ w 1 + w 2 ,w 3 / /       ρ w 1 ,w 2 ,w 3 F ( w 1 + w 2 + w 3 ) ( F ( w 1 )( F ( w 2 ) F ( w 3 ))) / / V 0 / / F ( w 1 ) F ( w 2 + w 3 ) e λ w 1 ,w 2 + w 3 / / F ( w 1 + w 2 + w 3 ) (A.14) where b oth horizontal morphisms to V 0 are paths on K Π m + n + p − 1 . So w e compute ψ ′ w 1 ,w 2 ,w 3 in the same w a y as ψ of the mono ida l case. After the vertex V 0 , the morphisms on the diagra m (A.14) are not any more in the 1-sk eleton of K Π m + n + p − 1 b ecause of the cancelations. The region b et w een the t w o paths fro m V 0 to F ( w 1 + w 2 + w 3 ) can b e ﬁlled with the structural 2-morphisms o f the Picard structure in particular by the ones in v olving the inv erse and unit ob jects. The 2-morphism ρ w 1 ,w 2 ,w 3 is then the unique pasting of those structural 2- morphisms . Hence, w e deﬁne e ψ w 1 ,w 2 ,w 3 as pasting of ψ ′ w 1 ,w 2 ,w 3 and ρ w 1 ,w 2 ,w 3 . Extending φ w 1 ,w 2 : The extension of φ w 1 ,w 2 , denoted b y e φ w 1 ,w 2 is a 2-morphism F ( w 1 ) F ( w 2 ) c   e λ w 1 ,w 2 / /       e φ w 1 ,w 2 F ( w 1 + w 2 ) F ( w 2 ) F ( w 1 ) e λ w 2 ,w 1 / / F ( w 2 + w 1 ) (A.15) b et w een the 1-morphisms e λ w 2 ,w 1 ◦ c and e λ w 1 ,w 2 from F ( w 1 ) F ( w 2 ) to F ( w 1 + w 2 ) where w 1 and w 2 are a ny t w o w ords in Z ( E ). W e rewrite the diagram (A.15) b y expressing e λ w 1 ,w 2 and e λ w 2 ,w 1 as comp ositions using (A.12) . F ( w 1 ) F ( w 2 ) c   λ w 1 ,w 2 / /       φ ′ w 1 ,w 2 F ( w 1 , 2 ) τ w 1 ,w 2 / / F ( w 1 + w 2 ) F ( w 2 ) F ( w 1 ) λ w 2 ,w 1 / / F ( w 2 , 1 ) τ w 2 ,w 1 / / / / F ( w 2 + w 1 ) (A.16) The square on t he left commu tes up to the 2-morphism φ ′ w 1 ,w 2 deﬁned in the same wa y as φ of the monoidal case. The square on the righ t commu tes since F ( w 1 , 2 ) = F ( w 2 , 1 ) and therefore τ w 1 ,w 2 = τ w 2 ,w 1 . Hence, e φ w 1 ,w 2 is the whisk ering φ ′ w 1 ,w 2 ∗ τ w 1 ,w 2 . References [1] Ettore Aldro v andi. 2-gerb es b ound by complexes of g r -stac ks, and cohomology . J. Pur e Appl. Algebr a , 212(5):99 4–1038, 2008. [2] Ettore Aldrov andi and Behrang Noohi. Butt erﬂies I II: Higher butterﬂies a nd higher gr- stac ks. I n pr ep ar ation . 44 [3] Ettore Aldrov andi and Behrang No ohi. Butterﬂies I: Morphisms of 2-group stac ks. A d- vanc es in Mathem a tics , 22 1 (3):687 – 773, 20 09. [4] John C. Ba ez and Martin Neuc hl. Higher- dimens ional alg ebra. I. Braided monoidal 2 - categories. A dv. Math. , 121(2):196– 244, 1996. [5] Jean B ´ enab ou. Introduction t o bicategor ies. In R ep orts of the Midwest Cate gory Semina r , pages 1–77. Springer, Berlin, 1967. [6] La wrence Breen. Bitorseurs et cohomo lo gie no n ab ´ elienne. In The Gr othendie ck Festschrift, Vol. I , v olume 86 of Pr o gr. Math. , pages 401–476. Birkh¨ auser Boston, Boston, MA, 1990. [7] La wrence Br een. Th ´ eorie de Sc hreier sup ´ erieure. A nn. Sci. ´ Ec ole Norm. Sup. (4) , 25(5):465– 514, 1992. [8] La wrence Breen. On the classiﬁcation of 2- gerbes and 2-stac ks. Ast ´ e ri s q ue , (2 2 5):160, 1994. [9] La wrence Breen. No tes on 1- and 2-gerb es. In T owar ds Higher Cate gories, J.C. Baez and J.P. May (e ds.) , volume 152 of The IMA V olumes in Mathematics and its Applic ations , pages 193–23 5 . Springer, New Y or k, 2010. [10] Brian Day and Ro ss Street. Monoidal bicategories and Hopf alg ebroids. A dv. Math. , 129(1):99– 157, 1997. [11] Pierre Deligne. La fo r mule de dualit´ e globale, 1973. SG A 4 I I I, Exp os ´ e XVI I I. [12] Pierre Deligne. V ari ´ et ´ es de Shim ura: in terpr´ etatio n mo dulaire, et tec hniques de con- struction de mo d` eles canoniques. In A utomorph i c f o rms, r epr esentations and L -functions (Pr o c. Symp os. Pur e Math., Or e gon State Univ., Corval lis, Or e., 197 7), Part 2 , Pro c. Symp os. Pure Math., XXXII I, pages 247– 289. Amer. Math. So c., Pro vidence, R.I., 1979 . [13] I. M. Gelfand, M. M. Kapra nov, a nd A. V. Z elev insky . D iscriminants, r esultants and multidimensional determinants . Mo dern Birkh¨ auser Classics. Birkh¨ auser Boston Inc., Boston, MA, 2008 . Reprint of the 199 4 edition. [14] R. Gordon, A. J. P ow er, and Ross Street. Coherence for tricatego ries. Mem. Amer. Math. So c. , 117(558) :vi+81 , 1995. [15] Nic k Gurski. An algebraic theory o f tricategories. PhD Thesis , 200 6 . [16] Monique Ha kim. T op os annel´ es et sch´ em a s r elatifs . Springer-V erlag, Berlin, 197 2 . Ergeb- nisse der Mathematik und ihrer Grenzgebiete, Band 64. [17] Andre Hirscho witz and Carlos Simpson. D esce n te po ur les n-champs (descen t f o r n-stac ks), 1998. [18] M. M. Kaprano v and V. A. V oevodsky . 2- categories and Zamolo dc hik ov tetrahedra equa- tions. In A l g ebr aic gr oups and their gen er alizations: quantum and inﬁnite-dim e nsional metho ds (Unive rs i ty Park, P A, 199 1 ) , v olume 56 of Pr o c. Symp os. Pur e Math. , pa g es 177–259. Amer. Math. So c., Provide nce, RI, 1994. 45 [19] Mikhail M. Kaprano v. The p erm utoasso ciahedron, Mac L a ne’s coherence theorem and asymptotic zones f o r the KZ equation. J. Pur e Appl. Algebr a , 85(2):11 9–142, 1993. [20] Stephen Lack. Bicat is no t tr ieq uiv alen t to G ra y . The ory Appl. Cate g. , 18:No. 1, 1– 3 (electronic), 2007 . [21] Behrang No ohi. On weak ma ps b et we en 2- g roups, 2005. [22] Behrang No ohi. Notes on 2- g roupo ids, 2-groups and crossed mo dules. Homol o gy, Homo- topy Appl. , 9(1):75 –106 (electronic), 2007. [23] A. J. P ow er. A 2 - categorical pasting t heorem. J. Algebr a , 129(2):4 3 9–445, 1990. [24] Carlos Simpson. A closed mo del structure for n -catego r ies, in ternal hom , n -stack s and generalized seifert-v an k amp en, 1997. [25] Zouhair T amsamani. Sur de s notions de n -cat ´ egorie et n -group o ¨ ıde non strictes via des ensem bles m ulti- simplic iaux. K -The ory , 16(1) :5 1–99, 1999. [26] Ahmet E. T atar. On the picard 2-stac ks. PhD Thesis, In pr ep ar ation . [27] G ¨ un t er M. Ziegler. L e ctur es o n p olytop es , v olume 15 2 of Gr aduate T exts in Mathem a tics . Springer-V erlag, New Y ork, 1995 . 46

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