The periodic table of $n$-categories for low dimensions II: degenerate tricategories

The p erio dic table of n -categories for lo w dimensions I I: degenerate tricategories Eugenia Cheng Departmen t of Mathematics, Univ ersit ´ e de Nice Sophia-An tip olis E-mail: eugenia@math.unice.fr Nic k Gurski Departmen t of Mathematics , Y ale Univ ersit y E-mail: mic haeln.gurski@y ale.edu June 2007 Abstract W e conti nue the pro ject b egun in [5] b y examining d egenerate tricate- gories and comparing them with the structures predicted by the P erio dic table. F or triply dege nerate tricategori es we exhibit a triequiva lence with the partially discrete tricategory of commutativ e monoids. F or the doubly degenerate case we explain ho w to construct a braided monoidal catego ry from a given doubly degenerate category , but show that this does not in- duce a straigh tforward co mparison betw een BrMonCat and T ricat . W e sho w how to alter the natural structure of T ricat in t wo diﬀeren t w ays to pro v ide a comparison, bu t show that only the more bru tal alteration yields an equiv alence. Finally we stud y degenerate t ricategories in order to giv e the ﬁrst fully a lgebraic deﬁnition o f monoidal bicategories and the full tricategory structure M onBicat . Con ten ts In tro duction 2 0.1 T otalities o f str uctures . . . . . . . . . . . . . . . . . . . . . . . . 5 0.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1 T riply dege n e ra te tricategories 9 1.1 Basic res ults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 T ricateg ories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1.3 W eak functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 T ritra nsformations . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 1.5 T rimo diﬁcations and p erturbations . . . . . . . . . . . . . . . . . 1 4 1.6 Overall structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1 2 Doubly de g enerate tricategories 16 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7 2.2 Basic res ults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Braidings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2.4 Overall structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Degenerate tricategorie s 26 3.1 Basic res ults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Overall structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 A App endix: diagrams 43 A.1 Doubly degene r ate tricatego ries . . . . . . . . . . . . . . . . . . . 4 3 A.2 F unctors b et ween doubly degenerate tricatego ries . . . . . . . . . 45 A.3 T rans fo rmations for do ubly degenera te tricateg ories . . . . . . . 46 A.4 Mo diﬁcations for doubly degenera te tricategor ie s . . . . . . . . . 48 A.5 Perturbations for doubly deg e nerate trica tegories . . . . . . . . . 49 In tro duction This work is a contin uation of the work begun in [5], studying the “Perio dic T able” of n -categor ies pro posed by Baez and Dola n [1]. The idea of the Peri- o dic T able is to study “degenerate” n -ca tegories, that is, n -c a tegories in whic h the low est dimensions are tr iv ial. F or s ma ll n this is s upposed to y ie ld well- known a lgebraic structur es such as commut ative mo no ids or braided monoidal categorie s; this he lps us under stand some sp eciﬁc pa rt of the whole n -c a tegory via b etter-known alg ebraic structur e s, a nd a ls o helps us to try to predict what n -categor ies should lo ok like for higher n . More precisely , the idea of deg e ne r acy is as follows. Cons ider an n -categ ory in whic h the lo w est no n- trivial dimens ion is the k th dimension, that is, ther e is o nly one cell o f each dimension low er tha n k . W e call this a “ k -degenerate n -categor y”. W e can then p erform a “dimension shift” and consider the k -cells of the old n - category to b e 0-cells of a new ( n − k )-categ ory , as shown in the schematic diagr am in Figure 1. This yie lds a “ne w ” ( n − k )-categ o ry , but it will always ha ve some s p ecial extra structure: the k -cells of the old n -category ha ve k diﬀerent co mpositions deﬁned on them (along b ounding cells of each lower dimension), so the 0-cells of the “new” ( n − k )-category must have k multiplications deﬁned o n them, int eracting via the in terchange laws from the o ld n -categ ory . Likewise every cell of higher dimension will hav e k “ extra” m ultiplications deﬁned on them as well as comp osition along b ounding cells. In [1], Ba e z and Dola n deﬁne a “ k -tuply monoida l ( n − k )-category” to be a k -degene r ate n -category , but a priori it should be an ( n − k )-catego ry with k -monoidal structures on it, interacting via coherent pseudo- in vertible cells. A direct deﬁnition has not yet been made for gener al n a nd k . (Balteanu e t a l 2 Figure 1: Dimension-shift for k -fo ld degenera te n -categor ies “old” n -category ⊲ “new” ( n − k ) -category 0-cells 1-cells . . . ( k − 1)-cells          trivial k -cells ⊲ 0-cells ( k + 1)-cells ⊲ 1-cells . . . . . . . . . n -cells ⊲ ( n − k )-cells [3] study a lax version o f this, where the monoidal structures in teract via non- inv ertible cells; this g iv es diﬀerent structures not s tudied in the pres en t work.) The Perio dic T able seeks to answer the ques tio n: exactly what sort of ( n − k )- category structure do es the degeneracy pro cess produce? Fig ure 2 sho w s the ﬁrst few columns of the h yp othesised Perio dic T able: the ( n, k )th ent ry predicts what a k -degenerate n -category “is”. (In this table w e follow Ba ez and Dola n and omit the w ord “ w eak ” understanding that all the n -categor ies in consideration are weak.) One consequence of the pre s en t work is that although k -tuply monoidal ( n − k )-ca tegories and k -degenerate n -catego ries are re lated, we see that the relationship is not straightforw ard. So in fact w e ne e d to consider thr e e p ossible structures for eac h n a nd k : • k -degenerate n -categ ories • k -tuply monoida l ( n − k )-categ o ries • the ( n, k )th entry of the P erio dic T able. In [5] we exa mined the top left ha nd co rner o f the table, that is, degen- erate categ ories and deg enerate bicategories . W e fo und that w e had to be careful a bout the exa ct meaning of “is ” . The main problem is the presence of some unw anted extra structure in the “new” ( n − k )-categor ies in the for m of distinguished elemen ts, aris ing from the structure cons tr ain ts in the or iginal n -categor ies — a sp eciﬁed k -cell structure constra int in the “o ld” n -category will appea r as a distinguishe d 0-cell in the “new” ( n − k )-category under the dimension-shift depicted in Figure 1. (F or n = 2 this phenomenon is mentioned by Leinster in [17] and was further described in a talk [18].) This problem b ecomes worse when considering functors, tra nsformations, mo diﬁcations, a nd so on, as we will disc us s in the next se c tion. 3 Figure 2: The h y p othesised Perio dic T able o f n - c ategories set category 2-catego ry 3-catego ry · · · monoid monoidal categ ory monoidal 2- c ategory monoidal 3- c ategory · · · ≡ category with ≡ 2-catego ry with ≡ 3-catego ry with ≡ 4-catego ry with only one ob ject only one ob ject only one ob ject only one ob ject comm utativ e b raided monoida l braided monoida l braided monoida l · · · monoid category 2-categor y 3-catego ry ≡ 2-catego ry with ≡ 3-catego ry with ≡ 4-catego ry with ≡ 5-catego ry with only one ob ject only one ob ject only one ob ject only one ob ject only one 1- cell only one 1-cell only one 1- cell only one 1- cell ′′ symmetric mo noidal sylleptic mono idal sylleptic mono idal · · · category 2-catego ry 3-catego ry ≡ 3-catego ry with ≡ 4-catego ry with ≡ 5-catego ry with ≡ 6-catego ry with only one ob ject only one ob ject only one ob ject only one ob ject only one 1- cell only one 1-cell only one 1- cell only one 1- cell only one 2- cell only one 2-cell only one 2- cell only one 2- cell ′′ ′′ symmetric mo noidal ? · · · 2-category ≡ 4-catego ry with ≡ 5-catego ry with ≡ 6-catego ry with ≡ 7-catego ry with only one ob ject only one ob ject only one ob ject only one ob ject only one 1- cell only one 1-cell only one 1- cell only one 1- cell only one 2- cell only one 2-cell only one 2- cell only one 2- cell only one 3- cell only one 3-cell only one 3- cell only one 3- cell ′′ ′′ ′′ symmetric mo noidal · · · 3-category . . . . . . . . . . . . 4 0.1 T otalities of st ructures Broadly sp eaking we have t wo diﬀerent a ims: 1. Theoretica l: to make pre c ise s tatemen ts about the claims o f the Perio dic T able b y exa mining the totalities of the structures involv ed, that is, not just the degenerate n -catego ries but a lso all the higher morphisms b et ween them. 2. Pra ctical: to ﬁnd the structures predicted by the Perio dic T able in a wa y that somehow naturally arises from degenerate tricateg ories and their morphisms. The p oin t of (2 ) is that in practice w e may simply wan t to know that a given doubly degener ate tricategory is a bra ided monoidal catego ry , or tha t a given functor is a braided monoidal functor, for exa mple, without needing to know if the t he ory of doubly degenerate tricatego ries corresp onds to the theory of bra ided mono idal categ ories. The motiv a ting ex ample dis cussed in [1] is the degenerate n - category of “manifolds with cor ners embedded in n -cube s”; work tow ards constructing s uc h a structure app ears in [2] a nd [6]. In this work we see that although the tricategor ies and functors b eha ve as exp ected, the higher morphisms ar e m uch more genera l than the ones we wan t. Moreo ver, for (1) we see that the ov erall dimensions of the tota lities do not matc h up. On the one hand w e ha v e k -degenerate n -ca tegories, which naturally orga nise themselv es into an ( n + 1)-category —the full sub-( n + 1)- category of nCat ; by contrast, the s tructure pre dic ted by the Perio dic T able is an ( n − k )-c a tegory with extra structure , and these orga nise themse lves into an ( n − k + 1)-catego ry—the full sub-( n − k + 1)-categ ory o f (n-k)Cat . In o rder to compare an ( n + 1)- c a tegory with an ( n − k + 1)-ca tegory we either need to remov e some dimensions fro m the for mer or add some to the la tter. The most obvious thing to do is add dimensions to the la tter in the form of higher identit y c e lls . Ho wev er, we quickly see that this do es no t y ield an equiv alence of ( n + 1)-catego r ies because the ( n + 1)-cells of nCat ar e far fr o m trivial. So instead we tr y to reduce the dimensions o f nCat . W e canno t in gener al apply a simple tr uncation to j -dimensions as this will not re sult in a j -catego ry . Besides, we would also like to restrict our higher mo r phisms in order to achiev e a b etter compar ison with the structures given in the Periodic T able— a priori our higher mor phis ms are to o gener al. So we p erform a construction analo gous to the construction of “ic ons” [16]. The idea o f ico ns is to o r ganise bicateg ories into a bic ate gory ra ther than a tri- category , by disca rding the mo diﬁcations, selecting only tho s e transfor mations that have all their comp onen ts the identit y , and altering their comp osition to ensure closur e. F or the ( n + 1)-ca tegory of n -catego r ies w e ca n try to p erform a n analogous “collapse” to obtain a j -categor y of n - categories, f or any 2 ≤ j ≤ n . W e dis card all cells of nCat of dimensio n greater tha n j , and for dimensions up to j we select 5 only those sp ecial case s where the co mp onents in the lowest ( n − j ) dimensions are the identit y; we then redeﬁne composition to forc e closure. Mor e explicitly , recall that for an m -cell in nCat the data is essen tia lly: • for all 0-cells a n ( m − 1)-c ell • for all 1-cells a n m - cell • for all 2-cells a n ( m + 1)-c ell • for all 3-cells a n ( m + 2)-c ell • . . . • for all ( n − m − 1)-cells an ( n − 2)-cell • for all ( n − m )- c ells an ( n − 1)-c ell • for all ( n − m + 1)-cells an n -cell W e can force the ﬁrst ( n − j + 1 ) of these to be the identit y , so that the ﬁrst non-trivial piece of data is “for a ll ( n − j + 1)-cells a . . . ”. This automatically forces all the morphisms o f dimension higher than j to b e the ident it y , and once we hav e redeﬁned comp osition to force clo sure, we hav e a j -ca tegory o f n -categor ies. Then, using j = n − k + 1 and restricting to the k -degenera te case, we ﬁnd we ha ve a naturally arising ( n − k + 1)-catego ry of k -dege ner ate n -categor ies. So to co mpare with braided mono idal c ategories w e can make a bic ate gory of tricatego r ies [8] with • 0-cells: tricategorie s • 1-cells: functors b et ween them • 2-cells: lax transforma tio ns whose 1- and 2-cell compo nents are iden tities, but 3-cell c o mponents ar e non-triv ia l. T aking the full sub- bica tegory whose 0-cells a re the doubly de g enerate trica t- egories, this do es give rise to braided monoidal categories , braided mono idal functors, and monoidal transformations ; howev er the corr espondence is not at all straig h tforward and more o ver we do not ge t a bie q uiv alenc e of bic a tegories. F or mo no idal bicateg ories we make a tric ate gory o f trica tegories with • 0-cells: tricategorie s • 1-cells: functors b et ween them • 2-cells: lax transformatio ns whose 1-c e ll components are identit ies, but 2- and 3- cell comp onent s are non-trivia l 6 • 3-cells: lax modiﬁcations who se 2- cell comp onen ts are identities, but 3-cell comp onen ts a re non- trivial. T aking the full sub-trica tegory who se 0-cells are the degenerate trica teg ories, this gives rise to monoidal bic a tegories, monoidal functors, monoidal transfor - mations and mo no idal mo diﬁcations; in fact this is how w e deﬁne them, as we will later discuss. Another approach would b e to restr ict the degener ate trica tegories muc h more. W e could adopt the philosophy that for k -degenerate n -ca tegories, when a piece of data says “ for all m - c ells a . . . ”, then if m is o ne of the triv ial dimensions m < k this data should b e trivial as well. This approach seems less natural from the point of vie w of n -categ ories, and might exclude some examples; it produce s an equiv alence of structures for the case of bra ided monoidal categor ies that seems almos t tautological. Essentially we hav e made a functor BrMonCat 4 − → T ricat where BrM o nCat 4 is the partially dis c rete tetraca tegory for med by adding higher identities to BrMo nCat ; we can fairly easily construct such a functor by c ho osing all extra str ucture to be identities. W e migh t try to lo ok for the essential imag e o f this functor, to ﬁnd a sub-tetra category o f T ricat that is equiv alent to BrMo nCat 4 , but it is currently m uch to o har d for us to work with tetraca tegories in this way . Instead, w e can do something very cr ude—ta k e the precise ima ge of this functor (which will b e trivial in dimensions 3 and 4) and somehow force it to b e a bicatego ry . It is not clear wha t this achieves. 0.2 Results The main results of [5] ca n be s ummed up as follows. (Here we write “deg ener- ate” for “1- degenerate”, and “doubly degenera te” for “ 2-degenerate”, a lthough in gener al we also us e “degener ate” for any level of deg e ne r acy .) • Comparing ea ch degene r ate categor y with the monoid for med by its 1- cells, we exhibit a n equiv a lence o f categ ories of these structures, but no t a biequiv alence of bicateg ories. • Comparing each do ubly degener ate bicategor y with the commutativ e monoid formed by its 2 -cells, we exhibit a biequiv a lence of bica tegories o f these structures, but not a n eq uiv a lence of categories or a triequiv alence of tr i- categorie s. • Comparing each degener ate bicategory with the monoidal catego r y for med by its 1 -, 2-, and 3-cells, we exhibit an equiv alence of categories of these structures, but not a biequiv alence of bicatego ries or a triequiv alence of tricategor ies. 7 In the pres en t work w e pr oceed to the next dimension and study degenerate tricategor ies. W e use the fully algebra ic deﬁnition of tr icategory g iv en in [11]; this is based on the deﬁnition g iv en in [9] which is not fully alg ebraic. The results can b e summed up a s follows, but cannot be stated quite so succinctly . • Comparing each tr iply degener ate tricatego ry with the commutativ e mono id formed by its 3 -cells, w e exhibit a tr iequiv a lence o f trica tegories of these structures, but not an eq uiv alence of c a tegories, a biequiv a lence of bicat- egories, or a tetra -equiv alence of tetra-categories . • W e show how doubly deg enerate tricategories give r ise to braided mo noidal categorie s, a nd similar ly functors. F o r compa risons we use the bica te- gory of trica tegories describ ed a bov e . W e exhibit compar ison functors in bo th directions, but no equiv alence except in the “tautolo gical” case (see ab o ve). • A degener ate tricategor y gives, by deﬁnition, a monoidal bicatego ry formed by its 1-cells and 2-cells. The totality of monoida l bica tegories has not pre- viously been understo o d; here w e use the tricategory of bicatego ries de- scrib ed ab ov e, and use this to deﬁne a tricateg ory MonB i ca t of monoidal bicategorie s , in which the higher-dimensional s tr ucture is not directly in- herited fro m T ricat . These results might be thought of as being signiﬁcantly “w orse” than the results for deg enerate categories and bicatego ries, in the sense tha t it is muc h harder to ma ke any precise statements ab o ut the structur es in question b eing equiv alent. Indeed it is a g eneral principle of n -c ategory theory that there is a large (p erhaps dispr opor tio nate) diﬀerence b et ween n = 2 and n = 3: bicate- gories exhibit a level of coherence that do es not gener alise to higher dimensio ns. Two of the main e x amples of this are: 1. Bicatego ries and weak functors form a ca tegory , althoug h “morally” they should o nly be expe cted to form a tricatego ry . Ho wever for n ≥ 3 w eak n -categor ies and weak functors ca nnot form a categor y as comp osition is not strictly asso ciative or unital, at least in the alg ebraic c ase. Note that bicategories, weak functors and weak trans fo rmations do not form a bicategory ; in general n -categories should not form an y co heren t structure in f ewer than ( n + 1)-dimensions without restricting the higher morphisms betw een them and alter ing their comp osition as describ ed ab o ve. 2. Every bica tegory is biequiv alent to a (strict) 2-c a tegory . Ho wev er, it is not the c a se that every tricateg ory is triequiv alent to a strict 3-categor y , and th us the general result for n ≥ 3 is expected to b e more subtle. The ob- struction for tricategor ies is exactly what pro duces a braiding rather than a s ymmetry for doubly deg e nerate tricategor ies (considered as monoidal categorie s), and is also wha t e na bles weak 3 -categories to model homo to p y 3-types where strict 3-ca tegories cannot. A nother wa y of expres s ing this diﬀerence in coher ence is that in a bicatego ry e v ery diagram of c o nstrain ts 8 commutes, whereas in a tricatego ry there are diagrams of co ns train ts that might not co mm ute. The second rema rk above shows the imp ortance of understanding the r e - lationship betw een doubly degenerate tricategories and braided monoidal cat- egories. W e will show that although every doubly degenerate tricategor y do es give rise to a braided monoida l categor y of its 2-cells and 3 -cells, the relationship is no t straig h tforward. The pro cess of pro ducing the braiding is complicated, and there is a grea t deal o f “extra structure” on the r esulting braided monoidal category . The disparity is even greater for functors, trans fo rmations and mo di- ﬁcations, meaning that it is not clear in what sense br aided monoida l catego ries and doubly degenerate tricateg ories should b e cons idered equiv alent. In fact the sour ce of all this diﬃcult y can b e seen in the ca se of doubly deg enerate bicategorie s , although in that cas e ma n y of the diﬃculties can res olv ed to pro- duce a strict Ec kmann-Hilton argument. When categorifying this to the case of doubly degenera te tricatego ries, we might ho pe to pro duce a categoriﬁe d Eckmann-Hilton argument yielding a categoriﬁed commutativit y , i.e. a braid- ing. How ever, the diﬃculties inv o lv ed a re also “categor iﬁed”, as w e will further discuss in Se c tio n 2.1. The orga nisation of the pa per is a s follo ws; it is worth noting that ea c h sec- tion is signiﬁca n t for diﬀerent r e asons, as we will p oin t out. In Section 1 we examine triply degener ate tricategories ; the signiﬁcance o f this section is that this is a “stable” case, and the results therefore hav e implications for the Sta- bilisation Hypo thesis. In Sectio n 2 w e examine doubly degener ate trica tegories, whose signiﬁcance we ha v e discussed ab o ve; we dr a w the r eader’s a tten tion to the extensive informal o verview given in Section 2.1 as the severe technical de- tails are in danger of obscuring the imp ortant principles involv ed. In Section 3 we examine degenera te tricategories (i.e. 1-degenerate tricat- egories); the main pur p ose of this section is to give the ﬁrst full deﬁnition of algebraic monoidal bicatego ries, together with their functors, trans formations and mo diﬁcations. Some of the large technical diag rams are deferred to the Appendix. Finally w e no te that the present pap er is necessarily mo st concer ned with studying pr e cisely what str uctures do a rise a s deg enerate tricategor ies, since go od comparisons do not na turally ar is e. F o r the purp oses o f corr ectly inter- preting the P erio dic T able, it seems lik ely that a direct deﬁnition of “ k -tuply monoidal” higher ca teg ory will b e more fruitful. The case of doubly degener- ate tricategorie s shows us tha t a k -degenera te n -category does not give rise to k monoidal struc tur es on t he as sociated ( n − k )-category in a straightforward way; how ever it is poss ible that k -tuply monoidal ( n − k )-categories deﬁned directly could more na tur ally yield the desired entries in the Perio dic T a ble. 1 T riply degenerate tricategories In this se c tio n, we will study triply degenera te trica tegories and the higher morphisms b et ween them—functors , tr ansformations, mo diﬁcations and p er- 9 turbations. By the Perio dic T a ble, triply degenerate tricatego ries are expected to be commut ative monoids; by results of [5] we now expect them to b e co mm u- tative monoids e qu ipp e d with some distinguishe d invertible elements aris ing from the structur e constraints in the trica tegory . The pro cess of ﬁnding ho w many such elemen ts there a re is highly technical and not particularly enlightening; we simply examine the data a nd axioms for a tr ic ategory and calculate which constraints determine the other s in the triply deg enerate case. The imp ortance of these r esults is not in the ex act num b er of distinguis hed inv ertible elements, but rather in the fact that there are any at a ll, and more than in the bicategory case. W e exp ect n -degenerate n -categories to have increasing num b ers of dis- tinguished inv ertible e lemen ts as n increases, a nd thus for the pr ecise alg ebraic situation to b ecome more a nd more intractible in a somewhat uninteresting wa y . The other impo rtan t part of this result is to ex amine whether the hig her morphisms b et ween triply degener ate tricateg ories rectify the s ituation—if a n y higher mor phisms e s sen tially ignore the distinguished inv ertible e lemen ts al- ready sp eciﬁed, then we can still hav e a s tructure eq uiv alent to commutativ e monoids. F or do ubly degenerate bicatego ries, this happ ened at the transforma - tion level; for triply deg enerate tr icategories, this happ ens at the mo diﬁcation level. As ex p ected from r esults of [5], the top level mo rphisms, tha t is the per turbations, destroy the p ossibilit y of an equiv alence on the level of tetr acat- egories. Throughout this sectio n we use results of [5] to characterise the (sing le) doubly degenera te hom-bicateg ory of a tr iply degener a te tricategor y . 1.1 Basic r esults The ov erall r esults for triply deg enerate tr icategories a re as follows; w e will discuss the calculations that lead to these results in the fo llowing sections . W e should also p oint o ut that the results in this se c tio n show tha t the higher- dimensional hypotheses w e ma de in [5 ] a r e incorrect. Theorem 1.1 . 1. A triply de gener ate tric ate gory T is pr e cisely a c ommutative monoid X T to gether with eight distinguishe d invertible elements d, m, a, l , r, u, π, µ . 2. Extending the ab ove c orr esp ondenc e, a we ak functor S → T is pr e cisely a monoid homomorphism F : S → T to gether with four distinguishe d invertible elements m F , χ, ι, γ . 3. Extending the ab ove c orr esp ondenc e, a tritr ansformation α : F → G is pr e cisely t he assertion t ha t ( F, m F ) = ( G, m G ) to gether with distinguishe d invertible elements Π and α T . 4. Extending the ab ove c orr esp ondenc e, a trimo diﬁc ation m : α ⇒ β is pr e- cisely the assertion that α and β ar e p ar al lel. 5. Extending the ab ove c orr esp ondenc e, a p erturb ation σ : m ⇛ n is pr e cisely an element σ in T . 10 1.2 T ricategories In this sec tio n we p erform the calculations for the tr iply degener ate tricatego ries themselves. First we pr ove a useful lemma concerning adjoint equiv alences. The data for a tricategory in v olves the speciﬁc a tion o f v ar ious adjoint equiv alences whose comp onents are themselv es a djoin t equiv a lences in the doubly-degener a te hom-bicategor ies. W e are thus in teres ted in adjoint equiv ale nc e s in doubly de- generate bicatego ries. Lemma 1 . 2. L et B b e a doubly de gener ate bic ate gory. Then an adjoint e quiv- alenc e ( f , g , η , ε ) in B c onsists of an invertible element η ∈ X B with ε = η − 1 . Pr o of. The triangle identities yield the following equation in an y bicategory . η ∗ 1 g = a − 1 ◦ 1 g ∗ ε − 1 ◦ r − 1 g ◦ l g Using the fact that B is doubly deg enerate, we see that in the commutativ e monoid X B (with unit written as 1) a = 1 , 1 g = 1 , and r = l . W e als o note that ∗ = ◦ , so the above equation reduces to the f act that η a nd ε ar e inv er se to each other. A priori , a triply deg enerate tricategor y T co nsists of the following data, which we will nee d to try to “ reduce”: • a single ob ject ⋆ ; • a doubly deg enerate bicategory T ( ⋆, ⋆ ), whic h will b e co nsidered as a commutativ e mono id with distinguis hed in v ertible element, ( T , d T ); • a w eak functor T ( ⋆, ⋆ ) × T ( ⋆, ⋆ ) → T ( ⋆, ⋆ ), which will be considered a s a monoid homomorphism together with a distinguished in vertible elemen t, ( ⊗ , m T ); • a weak functor I : 1 → T ( ⋆, ⋆ ), which will b e considered as the unique monoid homomor phism 1 → T tog ether with a disting uis hed inv er tible element u T ; • an adjoint equiv alence a : ⊗ ◦ ⊗ × 1 ⇒ ⊗ ◦ 1 × ⊗ , which is the a ssertion that ⊗ is strictly as sociative as a binary operatio n on T together with a distinguished inv ertible element a T ; • adjoint equiv alences l : ⊗ ◦ I × 1 ⇒ 1 , r : ⊗ ◦ 1 × I ⇒ 1, which is the assertion that 1 is a unit for ⊗ a s a binary operatio n on T , to g ether with distinguished inv ertible elements l T , r T ; • and fo ur distinguished in vertible ele ments π T , µ T , λ T , ρ T . Thu s we have a commutativ e mo noid T , a monoid homomorphis m ⊗ : T × T → T , 11 and distinguis hed invertible elements d T , m T , u T , a T , l T , r T , π T , µ T , λ T , ρ T . The fact that ⊗ is a monoid homomor phism is expressed in the follo wing equa tion, where we have written the mo no id structure on T as concatenatio n. ( ab ) ⊗ ( cd ) = ( a ⊗ c )( b ⊗ d ) The adjoin t equiv alence s l , r ea c h imply that 1 is a unit for ⊗ . Using this a nd the e q uation ab ov e, the E ckmann-Hilton arg umen t immedia tely implies that a ⊗ b = ab . W e will la ter need to us e the natura lit y iso morphisms; it is simple to compute that that the naturality isomorphism for the transformatio n a is 1, and the naturality isomorphis ms for l and r are b oth m T . There are three tricateg ory axioms that we must now c heck to ﬁnd the depe ndence b et ween distinguis hed in vertible e le men ts. Using the a bov e , it is straightforward to check that the ﬁrst trica tegory axiom is v a c uous, the second gives the equa tion λπ = d 2 m 4 T , and the third giv es the equation ρπ = d 2 m 4 T . Since λ, ρ, π , and d are inv ertible, λ = ρ = π − 1 d 2 m 4 T . Thu s λ and ρ are determined by the remaining data, hence w e hav e the result as summaris e d above. 1.3 W eak functors In this section we characteris e weak functors betw een triply deg enerate tri- categorie s. A priori a weak functor F : S → T betw een triply dege ne r ate tricategor ies consists of the following data, which we will try to simplify: • a weak functor F ⋆,⋆ : S ( ⋆, ⋆ ) → T ( ⋆, ⋆ ), which by the results of [5] is a monoid homomo rphism F : S → T together with a distinguished inv e r tible element m F ∈ T ; • an adjo int equiv alence χ : ⊗ ′ ◦ ( F × F ) ⇒ F ◦⊗ , which is the trivial asse r tion that F ( a ⊗ b ) = F a ⊗ ′ F b together with a distinguished inv e rtible e lemen t χ ∈ T ; • an adjoint eq uiv ale nce ι : I ′ ⋆ ⇒ F ◦ I ⋆ , which is the trivial asser tio n that F 1 = 1 tog ether with a distinguished invertible element ι ∈ T ; • and invertible mo diﬁcations ω , γ , and δ . 12 Thu s we have a monoid homo morphism F a nd six distinguished inv e r tible ele - men ts m F , χ, ι, ω , γ , and δ . It is stra ig h tforward to co mput e that the naturality isomorphism for χ is given by the invertible element F m S · ( m T m F ) − 1 and the naturality isomorphis m for ι is given by m F . There ar e tw o a xioms for weak functors for trica teg ories. In the case o f tr iply degenerate tricateg ories, the ﬁr st axio m reduces to the equa tion ω · π T · F m 2 S · m − 2 T · F d 2 S · d − 2 T = F π S th us by inv er tibilit y ω is determined by the re s t of the da ta. The second axiom reduces to the equation ω · δ · γ · µ T · F m 2 S · m − 2 T · F d 2 S · d − 2 T = F µ S . By the previous equation and the in vertibilit y of all terms inv olved, δ and γ determine each o ther o nce the r est of the da ta is ﬁxed, hence we have the result as summaris e d above. 1.4 T ritr an sformations In this section w e characterise tritransformatio ns fo r triply degenerate tricate- gories. First we need the follo wing lemma, which is a simple ca lc ula tion. Lemma 1.3. L et T b e a triply de gener ate tric ate gory. Then the functor T (1 , I ⋆ ) = I ⋆ ◦ − : T ( ⋆, ⋆ ) → T ( ⋆ , ⋆ ) is given by the identity homomorphism t o gether with the distinguishe d invertible element d − 1 m . Ad ditional ly, T (1 , I ⋆ ) = T ( I ⋆ , 1 ) . A priori , the data for a tritransfor mation α : F → G of triply degenerate tricategor ies consists of: • an adjoint equiv alence α : T (1 , I ⋆ ) ◦ F ⇒ T ( I ⋆ , 1 ) ◦ G , which co nsists of the a ssertion that F = G as mono id homo morphisms together with a distinguished inv ertible element α T ; and • distinguished in vertible elements Π and M . It is ea sy to compute that the natura lit y iso morphism fo r the tr ansformation α is m − 1 F m G . The ﬁrst tra nsformation axio m reduces to the eq ua tion m G = m F , the second a xiom reduces to the equation Π µ T l T γ F = M m 4 T d 2 T a − 1 T γ G , and the third to the equation Π δ F = a − 1 T l − 1 T d 2 T m 4 T µ − 1 M δ G . Thu s we s ee that Π determines M , and that the second and third a xioms com- bine to yield no ne w informa tion. So we ha ve remaining distinguished in vertible elements Π and α T , giving the results as summarise d abov e. 13 1.5 T rimo diﬁcations and p erturbations The data for a trimo diﬁcation m : α ⇒ β consists of a s ingle inv ertible element m in T , a nd there are tw o axioms. The ﬁrst is the eq ua tion m 2 · Π · Gd S = Π · F d S · m which reduces to m = 1 since F = G as monoid homomorphisms. The second axiom als o r educes to m = 1, thus there is a unique trimo diﬁcation be tw een any t wo parallel trans formations. Note that this mea ns that any diagra m of trimo diﬁcations in this se tting comm utes, a fac t tha t will be useful la ter. The data for a per turbation σ : m ⇛ n consists of a n element σ in T . The single axio m is v acuous so a perturbatio n is precisely an ele men t σ ∈ T . 1.6 Ov erall structure W e now compare the to ta lities of, on the one hand triply degener ate trica te- gories, and on the other hand comm utativ e monoids. Recall that for the case of doubly degenerate bicategories we w ere able to attempt comparis ons at the level of categor ies, bicategor ie s and tricategor ie s of suc h, simply b y trunca ting the full s ub-tricategory of Bicat to the required dimension. How ever, for triply degenerate tricategories w e show that truncating the full sub-tetraca tegory of T ricat do es no t yield a category o r a bicategor y; trunca tion do es yield a tri- category , and this is the only level that yields an equiv alence with commutative monoids. As in [5] we compare with the discrete j -categ ories of comm uta tiv e monoids obtained by adding higher iden tity cells to CMon . Note that we do not actually prov e that we have a tetraca tegory of triply degenerate tricategories; for the compariso n, we simply prov e that the natura l putative functor is not full and faithful therefore ca nno t b e an equiv alence. W e have a 4- dimensional structure with • 0-cells: triply degener ate tricatego ries • 1-cells: w eak functors b et w een them • 2-cells: tritransformatio ns betw een those • 3-cells: trimo diﬁcations b et w een those • 4-cells: per turbations b et ween those. W e write T ricat (3) j for the truncation of this structure to a j -dimensional structure, and CM o n j for the j -categor y of commutativ e monoids and their morphisms (and higher iden tities where necessary ). There are ob vio us assignments triply deg enerate tricateg o ry 7→ underly ing commutativ e mo noid weak functor 7→ underly ing homomo r phism of monoids 14 which, together w ith the unique ma ps on hig her cells , form the underlying morphism on j -globular sets for putativ e functors ξ j : T ricat (3) j → CMon j . Theorem 1.4. 1. T ri ca t (3) 1 is not a c ate gory. 2. T ri ca t (3) 2 is not a bic ate gory. 3. T ri ca t (3) 3 is a t ric ate gory, and ξ 3 deﬁnes a functor which is a t rie quiva- lenc e. 4. ξ 4 do es not give a tetr a-e quivalenc e of t etr a-c ate gories. The rest of this section will cons titut e a g radual pro of o f the v arious parts o f this theorem. W e b egin b y constr ucting the hom-bicategories for a tricategory structure on T ricat (3) 3 . Prop osition 1.5. L et X , Y b e t rip ly de gener ate tric ate gories. Then t he r e is a bic ate gory T ricat (3) 3 ( X, Y ) with 0-c el ls we ak functors F : X → Y , 1-c el ls tritr ansformations α : F ⇒ G , and 2-c el ls trimo diﬁc ations m : α ⇛ β . Pr o of. T o give the bica tegory structur e , w e need only provide unit 1-cells and 1-cell comp osition since there is a unique trimo diﬁcation b etw een every pair o f parallel tritransforma tions. It is s imple to re a d oﬀ the requir e d distinguished inv ertible elements fr om the co rresp o nding form ulae for composites of tritra ns- formations and from the data for the unit tritransformation. Remark 1.6. Note that comp osition of 1-ce lls in T ricat (3) 3 ( X, Y ) is strictly asso ciative, but is no t str ictly unital. In particular, this shows that T ricat (3) 2 is not a bicategory , proving Theo rem 1.4, part 2 . W e now construct the compositio n functor ⊗ : T ricat (3) 3 ( Y , Z ) × T ricat (3 ) 3 ( X, Y ) → T ricat (3) 3 ( X, Z ) . for a n y triply degener ate tricatego ries X , Y , Z . W e de ﬁne the co mposite GF of functors F : X → Y , G : Y → Z by the follo wing formulae which ca n b e read oﬀ dire ctly from the for mulae giving the c o mposite of functors b et ween tricategor ies. m GF = m G Gm F χ GF = χ G G ( χ F d Y ) d − 2 Z ι GF = ι G G ( ι F d Y ) d − 2 Z γ G = d − 2 Z m 2 Z m 2 G γ G G ( γ F d Y m Y ) The formulae for the comp osite β ⊗ α of tw o transformations ar e derived simi- larly , and thus we hav e a weak functor ⊗ for comp osition as requir ed. Similarly , ther e is a unit functor I X : 1 → T ricat (3) 3 ( X, X ) whose v alue o n the unique 0-cell is the identit y functor on X . 15 Remark 1.7. The formulae ab ov e make it obvious that ⊗ is not strictly as so- ciative o n 0-cells , and that the identit y functor is not a strict unit for ⊗ . This shows that T ricat (3) 1 is not a category , proving Theore m 1 .4, par t 1. Next we need to sp ecify the required co ns train t adjoint equiv alences . It is straightforward to ﬁnd adjoin t eq uiv a lences A : ⊗ ◦ ⊗ × 1 ⇒ ⊗ ◦ 1 × ⊗ L : ⊗ ◦ I × 1 ⇒ 1 R : ⊗ ◦ 1 × I ⇒ 1 in the appro priate functor bicateg ories; the actua l c hoice of adjoint equiv alence is ir relev an t, s ince there is a unique mo diﬁcation b et ween any pair of pa rallel transformatio ns. Finally , to ﬁnish c onstructing the tr icategory T ricat (3) 3 we must deﬁne in- vertible mo diﬁcations π , µ, λ, ρ and chec k three axioms. Howev er since there a re unique trimodiﬁcations b et ween parallel tritra nsformations, these mo diﬁcations are uniquely deter mined and the axioms automatica lly hold. W e now e x amine the morphism ξ 3 of 3-globular sets and sho w that it deﬁnes a functor T ricat (3 ) 3 − → CMon 3 ; in fact functorialit y is trivia l as CMon 3 has discrete hom-2-catego ries. F urther - more we s how it is an e q uiv alenc e a s follo ws. The functor is clearly sur jectiv e on ob jects, and the functor on hom-bica tegories T ricat (3) 3 ( X, Y ) → CMon 3 ( ξ 3 X , ξ 3 Y ) is easily seen to b e surjective on ob jects as well. This functor on hom-bicategories is also a lo cal equiv alence since CMon 3 is discrete at dimensions tw o and three and T ricat (3) 3 has unique 3- cells b e t ween parallel 2-cells. This ﬁnishes the pro of of Theo rem 1.4, part 3 . F or pa rt 4, we observe that the mor phism ξ 4 of 4-glo bular sets is clea rly not lo cally faithful on 4-cells. This ﬁnishes the proo f o f Theor em 1.4. 2 Doubly degenerate tricategories W e now co mpare doubly degenera te tr icategories with braided mono idal cat- egories. As des cribed informally in the Introduction the co mpa rison is no t straightforward. Therefore we b egin by directly listing the structure that we get on the monoidal ca tegory g iven b y the (unique) degenerate hom-bicategor y; this is s imply a matter o f writing out the deﬁnitions as nothing simpliﬁes in this case. Afterwards, we show how to extra ct a br a ided monoidal category from this structure. Essentially , all of the data listed in Section 2.2 can b e thought of as “extra structure” that arises on the bra ided monoidal categ ory we will construct. Finally , we construc t s ome compar ison functors. W e will begin with an informal o verview o f this whole section as w e feel that for many r eaders the ideas will be a t least as impor tan t as the technical details. 16 2.1 Ov erview It is widely accepted that a doubly degenerate bicategor y “is” a commutativ e monoid, a nd that a doubly degenerate tricateg ory “is” a braided monoida l cat- egory . Moreov er, it is widely accepted that the pro of of the bicategor y case is “simply” a question o f applying the Eckmann-Hilton argument to the multipli- cations given b y horizo n tal and vertical compo sition, and that the tricategor y result is prov ed by do ing this pro cess up to isomorphism. In this sectio n we give an infor mal overview o f the extent to which this is a nd is not the cas e. W e believe that this is impor ta n t b ecause the disparity will increase as dimen- sions increase, and beca use th is issue seems to lie at the heart of v arious critical phenomena in hig her-dimensional ca tegory theory , such as: 1. why we do not exp ect every w eak n - c ategory to be equiv alent to a strict one 2. why weak n -c a tegories a re exp ected to mode l homoto p y n -types while strict o ne s are known not to do so [10, 1, 22] 3. why some diagr ams of co nstrain ts in a tr ic ategory do not in genera l com- m ute, and wh y these do not arise in free tricategor ies [11] 4. why s trict computads do not form a presheaf categ ory [19] 5. why the existing deﬁnitions of n -categories based o n r eﬂexiv e globular sets fail to b e fully w eak [7 ] 6. why a no tio n o f semistrict n -ca tegory with weak units but stric t inter- change may b e weak enoug h to model homotopy n - t yp es and giv e c o her- ence re s ults [21, 1 5, 13] 7. why we need weak n - categories at all, a nd not just strict ones. A do ubly deg enerate bicatego ry B has only one 0-cell ⋆ and only one 1-cell I ⋆ . T o show that the 2-cells for m a commut ative mo no id w e ﬁrst use the fact that they are the m orphisms of the single hom-category B ( ⋆, ⋆ ); since this hom- category has only one ob ject I ⋆ we know it is a mo noid, with multip lication given by v ertica l co mposition o f 2 -cells. T o show that it is a c ommutative monoid, we apply the Eckmann-Hilton argument to the t wo multiplications deﬁned o n the set of 2 - cells: vertical c omposition a nd hor izon tal comp osition. Recall tha t the E ckmann-Hilton ar gumen t says: Let A b e a set with tw o binary op erations ∗ and ◦ such that 1. ∗ and ◦ a re unital with the same unit 2. ∗ and ◦ distr ibute over each other i.e. ∀ a, b, c, d ∈ A ( a ∗ b ) ◦ ( c ∗ d ) = ( a ◦ c ) ∗ ( b ◦ d ) . 17 Then ∗ and ◦ are in fact equa l a nd this op eration is comm uta tiv e. Note that the t wo binary op erations are usually called products (with implied as sociativity) but in fac t asso ciativit y is irr elev an t to the a rgumen t. How ever, in our case a diﬃculty ar ises b e cause hor izon ta l compo sition in a bicategory is not strictly unital. The situa tion is r escued by the fact that l I = r I in any bicateg ory . This, together with the naturality o f l and r , enables us to prov e, albeit lab oriously , that horizo n tal co mposition is strictly unital for 2-cells in a doubly de gener ate bic a tegory , and moreover that the vertical 2- cell iden tity also acts as a horizo n tal identit y . Thus we can in f act apply the Ec kmann-Hilton argument. Generalising this a rgumen t to doubly degenerate tricategor ies dir ectly is tricky . There are v ario us candidates for a “catego riﬁed Eckmann-Hilton ar- gument” pro vided by Joyal and Street [14, 4]. The idea is to r eplace all the equalities in the argument b y isomo rphisms, but as usual we need to take so me care over sp e cifying these isomor phisms rather than merely asserting their ex is- tence; see Deﬁnition 2. How ever, when we try and apply this result to a doubly degenera te tric a t- egory we hav e so me further diﬃculties: comp osition along bounding 0- c ells is diﬃcult to manipula te a s a m ultiplication, b ecause we cannot us e co herence results fo r trica tegories. Coherence for tr icategories [12] tells us that “ ev er y diagram of constraints in a fre e tricategor y (on a categ ory-enriched 2- graph) commutes”. In par ticular this means that if we need to use cells that do not arise in a free tricatego ry , then w e cannot use coherence r esults to check axioms. This is the case if w e attempt to built a m ultiplication out of comp osition a long 0-cells; we hav e to use the fact tha t we o nly hav e one 1-cell in our tricategory , and there fo re that v arious comp osites of 1- cells are all “accidentally” the s ame. This comes do w n to the fact that the free tricatego ry on a doubly deg enerate tricategor y is not itself doubly deg enerate; it is not clear how to constr uct a “free doubly deg enerate tricateg ory”. How ever, to re c tify this situa tio n we can lo ok at an alternative w ay of proving the result for degenerate bicategories, that does not make suc h ident iﬁcations. W e still use the Eckmann-Hilton argument but instead of attempting to apply it us ing horizo n tal compos ition of 2-cells, w e deﬁne a new binar y o peration on 2-cells that is derived from hor izon tal comp osition as follows: β ⊙ α = r ◦ ( β ∗ α ) ◦ l − 1 (Essentially this is what we use to prove that horizo n tal co mp osition is str ictly unital in the previous argument.) Unlik e horizontal composition, this op eration do es “c a tegorify co rrectly”, that is, given a doubly degenerate trica tegory we can deﬁne a multiplication on its a ssocia ted monoidal ca tegory by using the ab o ve formula (this is the c o n tent of Theorem 2 .8 ), and w e can manipulate it using cohere nc e for tricategories. T o extract a br aiding from this we then hav e to follow the s teps of the Eckmann-Hilton a r gumen t a nd k eep track of a ll the isomorphisms used; this is Prop osition 2 .7. 18 W e s e e that we use insta nces of the following cells, in a lengthy comp osite: • naturality c onstrain ts for l I and r I • constra in ts for weak int erchange of 2-cells • isomor phisms showing that l I ∼ = r I This shows v er y clearly wh y a theor y with w e a k unit s but stric t in terchange is enoug h to pro duce braiding s – the braiding is built from all o f the ab ov e structure con traints, so if any one of th em is w eak then braidings will still arise. Thu s if w e hav e stric t units then we need weak interchange, but if we main tain weak units we can hav e str ict in terchange and still get a br aided monoidal category . As men tioned above w e do, howev er , get a certa in amount of extra structure on the braided monoida l categor y that aris e s, and there do es not seem to b e a s traigh tforward way of organising it, or of descr ibing coherently the tricateg orical situation in whic h this extra structure is trivial. 2.2 Basic r esults The results in this s ection ar e all obtaine d b y simply r ewriting the appropriate deﬁnitions using the results of [5]. Many o f the diagrams needed in the theorems below are e xcessiv ely lar ge, and hav e be e n relegated to the Appendix. Just as w e bega n the previous section by characterising adjoin ts in doubly degenerate bicatego ries, we b egin this s ection by reca lling the deﬁnition of “dual pair” of ob jects in a monoidal catego ry , since this c ha racterises adjoints for 1- cells in degenerate bicatego ries; even tually we will of co urse b e interested in adjoint equiv alences, no t just adjoin ts. Deﬁnition 1. Le t M b e a monoidal catego ry . Then a dual p air in M consists of a pa ir of o b jects X , X · together with morphisms ε : X ⊗ X · → I , η : I → X · ⊗ X satisfying the tw o eq ua tions below, where all unmarked isomorphisms are given by coherence isomo rphisms. X X I ∼ = / / X I X ( X · X ) 1 η / / X ( X · X ) ( X X · ) X ∼ = / / ( X X · ) X I X ε 1 / / I X X ∼ =   X X 1 + + W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W X · I X · ∼ = / / I X · ( X · X ) X · η 1 / / ( X · X ) X · X · ( X X · ) ∼ = / / X · ( X X · ) X · I 1 ε / / X · I X · ∼ =   X · X · 1 + + W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W 19 Theorem 2.1 . A doubly de gener ate tric ate gory B is pr e cisely • a monoidal c ate gory ( B , ⊗ , U , a, l, r ) given by the single de gener ate hom- bic ate gory; • a monoidal functor ⊠ : B × B → B fr om c omp osition; • a monoid I in B and an isomorphism I ∼ = U as monoids in B ; this c omes fr om the functor fo r units I − → B ( ⋆, ⋆ ) • a dual p air ( A, A · , ε A , η A ) with ε A , η A b oth invertible, and natur al isomor- phisms A ⊗  ( X ⊠ Y ) ⊠ Z )  ∼ =  X ⊠ ( Y ⊠ Z )  ⊗ A A · ⊗  X ⊠ ( Y ⊠ Z )  ∼ =  ( X ⊠ Y ) ⊠ Z )  ⊗ A · ; subje ct to diagr ams given in the App endix, • a dual p air ( L, L · , ε L , η L ) with with ε L , η L b oth invertible, and natur al isomorphi sms L ⊗ ( I ⊠ X ) ∼ = X ⊗ L L · ⊗ X ∼ = ( I ⊠ X ) ⊗ L · subje ct to diagr ams given in the App endix, • a dual p air ( R, R · , ε R , η R ) with with ε L , η L b oth invertible, and natur al isomorphi sms R ⊗ ( X ⊠ I ) ∼ = X ⊗ R R · ⊗ X ∼ = ( X ⊠ I ) ⊗ R · ; subje ct to diagr ams given in the App endix, • and isomorphisms ( U ⊠ A ) ⊗  A ⊗ ( A ⊠ U )  ! π ∼ = A ⊗ A ( U ⊠ L ) ⊗  A ⊗ ( R · ⊠ U )  ! µ ∼ = U L ⊠ U λ ∼ = L ⊗ A U ⊠ R · ρ ∼ = A ⊗ R · ; al l subje ct to thr e e axioms which app e ar in the App endix. Remark 2. 2. It is important to note tha t ⊠ does not a priori g iv e a monoidal structure on the categor y B ; the obstruction is that lax transfo r mations b e- t ween weak functor s of degenera te tricategories are mor e gener al than monoidal transformatio ns b et ween the asso ciated m onoidal functors (see [5]). As noted in 20 Section 2.1 it may b e possible to pro ve that ⊠ is a v alid monoidal structure, but since we cannot us e cohere nce for tric ategories to help us, the pro of is no t very evident. Thus to extract a braiding from all this structure, we will not simply apply an Eckmann-Hilton-style arg umen t to ⊗ a nd ⊠ (see Section 2.3). W e now describ e f unctors, transforma tions, mo diﬁcations and p erturbations in a simila r spirit. Theorem 2 .3. A we ak funct or F : B → B ′ b etwe en doubly de gener ate tric ate- gories is pr e cisely • a monoidal functor F : B → B ′ ; • a dual p air ( χ, χ · , ε χ , η χ ) in B ′ with ε χ , η χ b oth invertible, and n atur al isomorphi sms χ ⊗ ′ ( F X ⊠ ′ F Y ) ∼ = F ( X ⊠ Y ) ⊗ ′ χ χ · ⊗ ′ F ( X ⊠ Y ) ∼ = ( F X ⊠ ′ F Y ) ⊗ ′ χ · subje ct to diagr ams given in the App endix, • a dual p air ( ι, ι · , ε ι , η ι ) with ε ι , η ι b oth invertible, and natura l isomorphi sms ι ⊗ ′ I ′ ∼ = F I ⊗ ′ ι ι · ⊗ ′ F I ∼ = I ′ ⊗ ′ ι · subje ct to diagr ams given in the App endix, • and isomorphisms F A ⊗ ′  χ ⊗ ′ ( χ ⊠ ′ U ′ )  ω ∼ = χ ⊗ ′  ( U ′ ⊠ ′ χ ) ⊗ ′ A ′  F L ⊗ ′  χ ⊗ ′ ( ι ⊠ ′ U ′ )  γ ∼ = L ′ F R · δ ∼ = χ ⊗ ′  ( U ′ ⊠ ′ ι ) ⊗ ′ ( R ′ ) ·  ; al l subje ct to axioms given in the App endix. Theorem 2.4. A we ak tr ansformation α : F → G in t he ab ove setting is pr e cisely • a dual p air ( α, α · , ε α , η α ) with ε α , η α b oth invertible, and natur al isomor- phisms α ⊗ ′ ( U ′ ⊠ ′ F X ) ∼ = ( GX ⊠ ′ U ′ ) ⊗ ′ α α · ⊗ ′ ( GX ⊠ ′ U ′ ) ∼ = ( U ′ ⊠ ′ F X ) ⊗ ′ α · subje ct to diagr ams given in the App endix, • and isomorphisms ( χ G ⊗ ′ U ′ ) ⊗ ′ ( A ′ ) · ⊗ ′  ( U ′ ⊠ ′ α ) ⊗ ′  A ′ ⊗ ′ ( α ⊠ ′ U ′ )   ! Π ∼ = α ⊗ ′  ( U ′ ⊠ ′ χ F ) ⊗ ′ A ′  α ⊗ ′  ( U ′ ⊠ ′ ι F ) ⊗ ′ ( R ′ ) ·  M ∼ = ( ι G ⊠ ′ U ′ ) ⊗ ′ ( L ′ ) · ; 21 al l subje ct to thr e e axioms given in the App endix. The analogous result for la x transforma tions s hould b e obvious, with dual pair replaced by dis tinguished ob ject since in the lax case we have a noninv er tible morphism instead o f an a djoin t equiv a lence. Theorem 2.5 . A mo diﬁc ation m : α ⇒ β is pr e cisely • an obje ct m ∈ B ′ and • an isomorphi sm ( U ′ ⊠ m ) ⊗ ′ α ∼ = β ⊗ ′ ( m ⊠ ′ U ′ ) subje ct to two axioms given in the App endix. Theorem 2.6. A p ertu rb ation σ : m ⇛ n is pr e cisely a morphism σ : m → n in B ′ satisfying the single axiom in the App endix. 2.3 Braidings In this se c tio n we show that the underlying monoida l categ ory o f a do ubly degenerate tricategor y do es ha ve a bra iding o n it. T o show this, we use the fact that to g iv e a braiding for a monoidal s tr ucture, it suﬃces to give the structure of a m ultiplicatio n on the monoidal category in question. W e give the relev ant deﬁnitions b elo w; for additional details , see [14]. Deﬁnition 2. Let M b e a monoidal categ o ry , and equip M × M with the comp onen twise monoidal structure. Then a multiplic ation ϕ on M consis ts o f a monoidal functor ϕ : M × M → M and inv er tible mono idal trans formations ρ : ϕ ◦ (id × I ) ⇒ id , λ : ϕ ◦ ( I × id) ⇒ id where I : 1 → M is the canonica l monoidal functor whose v alue on the sing le ob ject is the unit of M and whose structure constr ain ts are giv en by unique co herence isomor phisms. The following result, due to Joy al and Street [14], says that a m ultiplication naturally gives r ise to a braiding. Prop osition 2.7. L et M b e a monoidal c ate gory with multiplic ation ϕ . Then M is br aide d with br aiding giv en by the c omp osite b elow. ab λ − 1 ρ − 1 − → ϕ ( I , a ) ϕ ( b, I ) ∼ = − → ϕ ( I b, aI ) ϕ ( l,r ) − → ϕ ( b, a ) ϕ ( r − 1 ,l − 1 ) − → ϕ ( bI , I a ) ∼ = − → ϕ ( b, I ) ϕ ( I , a ) ρλ − → ba W e will use this constr uction to pro vide a braiding for the monoidal c a tegory asso ciated to a doubly degenera te trica tegory . As ca n be seen from the ab ove formula, this bra iding is “natural” but not ex a ctly “simple” . Theorem 2.8 . Le t B b e a doubly de gener ate tric ate gory, and also denote by B the monoidal c ate gory asso ciate d to t he single (de gener ate) hom-bic ate gory. Then ther e is a multiplic ation ϕ on B with ϕ ( X, Y ) = R ⊗  ( X ⊠ Y ) ⊗ L ·  . 22 This result is a length y calcula tion tha t req uires re peated use o f the coher- ence theor m for tric a tegories as well as coher ence for bicatego ries and functors. W e thus o mit it, and only reco rd the following cr ucial cor ollary . Corollary 2.9. L et B b e a doubly de gener ate tric ate gory, and also denote by B the monoidal c ate gory asso ciate d to t he single (de gener ate) hom-bic ate gory. Then B is a br aide d m ono idal c ate gory. The situatio n for functors is similar , with braided monoida l functors arising from “multiplicativ e ” functors a s follows. Deﬁnition 3. Let ( M , ϕ ) and ( N , ψ ) b e mo noidal catego ries eq uipp ed with m ultiplications. A mult ip lic ative funct or F : ( M , ϕ ) → ( N , ψ ) c onsists of a monoidal functor F : M → N and an in vertible monoidal transformation χ : ψ ◦ ( F × F ) ⇒ F ◦ φ , satisfying unit ax ioms. Prop osition 2.10. L et ( M , ϕ ) and ( N , ψ ) b e monoidal c ate gories e quipp e d with multiplic ations, and let F : ( M , ϕ ) → ( N , ψ ) b e a mu ltipl ic ative functor b etwe en them. Then the underlying monoidal functor F is br aide d when M and N ar e e quipp e d with the br aidings induc e d by their r esp e ctive multiplic ations. The following theorem s a ys that functors betw een do ubly degenera te tri- categorie s do give r ise to m ultiplicative functor s, a nd as a co rollary , braided monoidal functors. The pro of of the theorem is another long calcula tio n inv olv- ing coherence . Theorem 2.11. L et B and B ′ b e doubly de gener ate tric ate gories, and let F : B → B ′ b e a functor b etwe en t he m. Then the monoidal functor F b etwe en the m onoidal c ate gories B and B ′ c an b e give n the st ructur e of a multiplic ative functor when we e quip B and B ′ with the m u ltipli c ations of The or em 2.8. Corollary 2. 12. L et B and B ′ b e doubly de gener ate t ric ate gories, and let F : B → B ′ b e a functor b etwe en them. Then the monoidal functor F is br aide d with r esp e ct to the br aide d monoidal c ate gories B and B ′ as in Co r ol lary 2.9. The situation for tr ansformations do es not lend itself to the same sort of analysis: a tra nsformation of doubly degener ate tricategor ies is r ather diﬀerent from a monoidal transforma tion. This also o ccurs in the study o f degener a te bicategorie s , wher e transformations of degener ate bicategories are ra ther diﬀ er- ent from mo no idal trans formations. Thus, a s discussed in the Introduction, we mo dify T ricat so that the 2-ce lls b et w een doubly degener ate tric a tegories do give rise to monoidal transforma tions [8]. W e co nstruct a bicategory ^ T ricat with • 0-cells: tricategorie s , • 1-cells: functors, and • 2-cells: “lo cally ico nic lax tra nsformations”, tha t is, trans formations whos e 1- and 2-cell comp onents are identities. 23 Recall that an icon is an oplax transforma tion of bicategor ies, all of whose com- po nen t 1- cells are identities. Specifying a 2 - cell o f ^ T ricat thus in volv es sp eci- fying, for every 2- cell in the source , a 3- cell comp onent in the target, tog e ther with 3-ce ll constraint data Π a nd M (as exhibited in Theor em 3.3 for example) satisfying the relev an t axio ms; there is eﬀectiv ely no low er -dimensional data. Since icons b et ween dege ne r ate bicategor ies yield mono idal transforma tions (see [5]), the follo wing r esult is immediate. Theorem 2.13. L et α : F ⇒ G b e a 2-c el l in ^ T ricat whose sour c e and tar get tric ate gories ar e doubly de gener ate. Then α gives rise to a monoidal tr ansfor- mation b etwe en the br aide d monoidal functors c orr esp onding to F and G . This gives us a slightly better compariso n with braided monoidal categor ies— we can at least ha ve functors in b oth directions—but in the next section w e will compare the overall structures and see that we still do not get a biequiv alence of bicategor ies. 2.4 Ov erall structure In this section we attempt to compare the tota lities of structur es inv olved. That is, on the one hand w e hav e doubly degener ate trica tegories, and on the other hand we have braided monoida l categories as predicted by the Perio dic T a ble . First observe tha t the full 4-dimensional structure of tricateg ories do es not yield an equiv alence. W e ca n add higher iden tit y cells to the bicatego ry BrMonCat of braided monoida l ca tegories to form a discrete tetracateg ory , but it is clear that this cannot b e equiv alent to the tetracateg ory T ricat (2 ) of do ubly deg en- erate trica tegories, which ha s too ma ny non-trivial 4-c ells. Instead we can examine ^ T ricat (2), the full sub-bicategory of ^ T ricat whose 0-cells are t he doubly degenerate tricategories. W e will no w s ho w that there are naturally arising compar ison functors to and from BrMonCat , but these do not exhibit a biequiv alence. F rom the r esults of the previous s ection we hav e a mo rphism of underlying globular sets U : ^ T ricat (2 ) − → BrMonCat , assigning • to e a c h 0 - cell asso ciated braided mono idal category as in Co rollary 2.9, • to each 1-cell the asso ciated braided mono idal functor as in Corollary 2.12, and • to each 2-cell the as sociated monoidal transfor ma tion as in Theorem 2.13. Prop osition 2.14. The morphism U deﬁnes a strict functor of bic ate gories. Pr o of. It is trivia l that U str ictly pr eserves compo sition and units; the functor axioms fo llow by noting that U s e nds the constraint isomorphisms a, l, r to ident ities. 24 There is also a comparison functor in the opp osite direction F : BrMonCat − → ^ T ricat (2 ) . This is simply a matter o f choosing all the extr a structur e to be given by iden- tities where this makes sense, a nd us ing isomorphis ms given canonica lly by coherence constra in ts elsewhere. W e can then chec k the follo wing theorem. Theorem 2. 15. The c omp osite functor U F : BrMonCat → BrMonCat is the identity 2-fun ctor. Pr o of. It is obvious that U F ( X ) has the same monoida l structure as X , and following the deﬁnition o f the bra iding carefully and using coherenc e gives that the braided structures a re the same a s well. It follows immediately that U F is the identit y on 1 - and 2-cells. It remains to check that the constraints for this comp osite are also identities; this follows from the deﬁnition of the constra in ts for F and the deﬁnitio n of U . W e can now sho w that these comparison functors do not exhibit an eq uiv - alence. The pro blem is at the level of 2-cells; the 2-cells of ^ T ricat (2) hav e an extra c hoice of the structure constra in ts Π and M , a nd these are forgotten b y the functor U . Theorem 2 .16. The functor U : ^ T ricat (2 ) − → BrMonCat is lo c al ly ful l but not lo c al ly faithful. Pr o of. The ﬁr st statemen t follows from Theor em 2.15. F or the seco nd, let Z denote the symmetr ic monoidal catego ry with only one ob ject x and Z ( x, x ) = Z / 2, with the comp osition and monoidal s tructure given on morphisms by addi- tion and all coherenc e isomor phisms the identit y . The iden tit y functor 1 : Z → Z is a strict braided mono idal functor. There a re tw o natural transformatio ns 1 ⇒ 1 corres p onding to the t wo diﬀerent group elements, but only the identit y is monoidal. T o show tha t U is not locally faithful, we w ill prove that there is more than a s ingle 2- c e ll F (1) ⇒ F (1) in ^ T ricat (2); howev er, there is o nly one 2-cell U F (1) = 1 ⇒ 1 = U F (1). A 2-cell F (1) ⇒ F (1) in ^ T ricat (2) consists of a mo noidal transformatio n and t wo group elemen ts satisfying fo ur axio ms. W e show that these axioms allow for t wo diﬀerent 2-cells. The ﬁrst axiom r educes to the equation Π = Π since all the other cells inv olved are iden tities. The second and thir d a x ioms b oth r educe to Π + M = 0. The fourth axiom is then the equation that Π + Π = Π + Π. Thu s there are tw o diﬀerent 2-cells F (1) ⇒ F (1) in ^ T ricat (2 ) corresp onding to the tw o diﬀeren t choices of Π. Remark 2.17. The same pro of shows that the stric t functor F is not lo cally full. Finally no te that we could restrict ^ T ricat (2 ) further just for the purp oses of getting a bie q uiv alenc e , a s describ ed in th e following theorem. W e wr ite I for the dual pair ( I , I , l , l − 1 ); this is v a lid in an y monoidal categ ory . 25 Theorem 2.1 8 . Ther e is a 2-c ate gory T ricat (2) ′ 2 with • 0-c el ls those doubly de gener ate tric ate gories with ⊠ = ⊗ , monoid I with isomorphi sm I ∼ = I the identity, al l dual p airs I , and al l isomorphisms given by unique c oher enc e isomorphisms; • 1-c el ls those functors with al l dual p airs I and al l isomorphi sms given by unique c oher enc e isomorphisms; and • 2-c el ls those lax tr ansformations with distinguishe d obje ct I and al l c on- str aints given by unique c oher enc e isomorphi sms. The functors F and U then r estrict to co mpa rison functors to and from this 2- category , and this does pro duce a biequiv alence, alb eit a somewha t tautolo gical one. 3 Degenerate tricategories W e now study degenera te tricategor ies, and use them to make a deﬁnition of monoidal bicategory . This deﬁnition diﬀers from existing deﬁnitions [9, 20] only in that it is fully algebra ic. The diﬀerence b et ween these structures b ecomes more sig niﬁcan t at the level of transformation. In Sec tio n 3.2 w e will e x plore these diﬀere nces in the pro cess of deﬁning a tric ategory of mo no idal bicategories. Since we will deﬁne monoidal bicategories to b e degener ate tricategor ies, a pro cess o f “compar ison” might seem r a ther cir c ula r. In eﬀect we do little more than observe tha t our deﬁnition of transformation is s igniﬁcan tly diﬀerent from that inherited from T ricat , just as in the case of tr ansformations betw een degenerate bicatego ries [5]. 3.1 Basic r esults The results in this s ection ar e all obtaine d b y simply r ewriting the appropriate deﬁnitions using the results o f [5]. First w e characterise degener ate trica tegories and functors b et ween them; this is straightforw ard. Theorem 3.1 . A de gener ate tric ate gory B is pr e cisely • a single hom-bic ate gory which we will also c al l B ; • a functor ⊗ : B × B → B ; • a functor I : 1 → B ; • adjoint e quivalenc e a , l , and r as in the deﬁnition of a tric ate gory; and • invertible mo diﬁc ations π , µ, λ , and ρ as in the deﬁnition of a tric ate gory al l subje ct to the tr ic ate gory axioms. 26 Theorem 3.2 . A we ak functor F : B → B ′ b etwe en de gener ate tric ate gories is pr e cisely • a we ak functor F : B → B ′ ; • adjoint e quivalenc es χ and ι as in the deﬁnition of we ak fun ctor b etwe en tric ate gories; and • invertible mo diﬁc ations ω , δ , and γ as in the deﬁnition of we ak functor, as shown b elow al l subje ct to axioms which ar e identic al t o the functor axioms aside fr om sou rc e and tar get c onsider ations. W e now characterise weak tra nsformations, mo diﬁcations and p erturbations. Here we actually include all the diagr ams in the deﬁnition, b ecause in Sectio n 3.2 we will mo dify these deﬁnitions in order to co nstruct a tricateg ory of monoidal bicategorie s . Theorem 3.3 . A we ak tr ansformation α : F → G b etwe en we ak funct ors of de gener ate tric ate gories is pr e cisely • an obje ct α in the tar get bic ate gory B ′ , c orr esp onding to t he c omp onent α ⋆ of the tr ansformation; • an adjoint e quivalenc e as disp laye d b elow; B B ′ G   B B ′ F / / B ′ B ′ α ⊗ ′ −   B ′ B ′ −⊗ ′ α / / α s { o o o o o o o o o o o o o o o o • and invertible mo diﬁc ations as displaye d b elow, wher e we write [ a, b ] for Hom( a, b ) and [ b, c ; a, b ] fo r Hom( b, c ) × Hom( a, b ) ; 27 [ b, c ; a, b ] [ a, c ] ⊗   [ a, c ] [ Ga, Gc ] G ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q [ Ga, Gc ] [ F a, Gc ] T ′ ( α a , 1) ; ; w w w w w w w w w w w w w w w w w w w w w w w w [ b, c ; a, b ] [ F b, F c ; F a, F b ] F × F / / [ F b, F c ; F a, F b ] [ F b, Gc ; F a, F b ] T ′ (1 ,α c ) × 1   [ F b, Gc ; F a, F b ] [ F a, Gc ] ⊗   [ b, c ; a, b ] [ Gb, Gc ; F a, F b ] G × F   [ Gb, Gc ; F a, F b ] [ Gb, Gc ; F a, Gb ] 1 × T ′ (1 ,α b )   [ Gb, Gc ; F a, Gb ] [ F a, Gc ] ⊗ / / [ Gb, Gc ; F a, F b ] [ F b, Gc ; F a, F b ] / / T ′ ( α b , 1 ) × 1 [ b, c ; a, b ] [ Gb, Gc ; Ga, Gb ] G × G } } { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { [ Gb, Gc ; Ga, Gb ] [ Gb, Gc ; F a, Gb ] 1 × T ′ ( α a , 1) / / [ Gb, Gc ; Ga, Gb ] [ Ga, Gc ] ⊗ $ $ J J J J J J J J J J J J J J J J J J J J J J J J J Π    [ b, c ; a, b ] [ a, c ] ⊗   [ a, c ] [ Ga, Gc ] G ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q [ Ga, Gc ] [ F a, Gc ] T ′ ( α a , 1) ; ; w w w w w w w w w w w w w w w w w w w w w w w w [ b, c ; a, b ] [ F b, F c ; F a, F b ] F × F / / [ F b, F c ; F a, F b ] [ F b, Gc ; F a, F b ] T ′ (1 ,α c ) × 1   [ F b, Gc ; F a, F b ] [ F a, Gc ] ⊗   [ b, c ; a, b ] [ F b, F c ; F a, F b ] F × F   [ F b, F c ; F a, F b ] [ F a, F c ] ⊗   [ F a, F c ] [ F a, Gc ] T ′ (1 ,α c ) / / [ a, c ] [ F a, F c ] F / / ⇓ α × 1 ⇓ a 1 × α ⇐ ⇓ a · χ ⇐ ⇓ a χ ⇐ ⇓ α 28 1 T ( a, a ) I a   T ( a, a ) T ′ ( Ga, Ga ) G   T ′ ( Ga, Ga ) T ′ ( F a , Ga ) T ′ ( α a , 1) / / 1 T ′ ( F a , Ga ) α a   1 T ′ ( F a , F a ) I F a   T ′ ( F a , F a ) T ′ ( F a , Ga ) T ′ (1 ,α a ) " " E E E E E E E E E E E E E E T ( a, a ) T ′ ( F a , F a ) F / / M _ * 4 1 T ( a, a ) I a y y r r r r r r r r r r r r r T ( a, a ) T ′ ( Ga, Ga ) G   T ′ ( Ga, Ga ) T ′ ( F a , Ga ) T ′ ( α a , 1) / / 1 T ′ ( F a , Ga ) α a   1 T ′ ( Ga, Ga ) I Ga { { ι ⇐ ⇓ α r · ⇐ ι ⇐ l · ⇐ 29 al l subje ct to the fol lowing axioms. (( αF f ) F g ) F h (( Gf α ) F g ) F h ( α 1)1 O O (( Gf α ) F g ) F h ( Gf ( αF g )) F h a 1 K K          ( Gf ( αF g )) F h ( Gf ( Gg α )) F h (1 α )1 K K          ( Gf ( Gg α )) F h Gf (( Gg α ) F h ) a G G           Gf (( Gg α ) F h ) Gf ( Gg ( αF h )) 1 a ? ?             Gf ( Gg ( αF h )) Gf ( Gg ( Ghα )) 1(1 α ) / / Gf ( Gg ( Ghα )) ( Gf Gg )( Ghα ) a ·   7 7 7 7 7 7 7 7 7 7 7 ( Gf Gg )( Ghα ) G ( f g )( Ghα ) χ 1   ? ? ? ? ? ? ? ? ? ? ? ? G ( f g )( Ghα ) ( G ( f g ) Gh ) α a ·   ( G ( f g ) Gh ) α G (( f g ) h ) α χ 1   ' ' ' ' ' ' ' ' ' G (( f g ) h ) α G ( f ( g h )) α Ga 1   (( αF f ) F g ) F h ( αF f )( F g F h ) a   7 7 7 7 7 7 7 7 7 7 7 ( αF f )( F g F h ) ( αF f ) F ( g h ) 1 χ   : : : : : : : : : : : : : : : : : : : ( αF f ) F ( g h ) α ( F f F ( g h )) a / / α ( F f F ( g h )) αF ( f ( g h )) 1 χ B B                αF ( f ( g h )) G ( f ( g h )) α α B B                (( αF f ) F g ) F h ( α ( F f F g )) F h a 1 9 9 s s s s s s s s s s s ( α ( F f F g )) F h ( αF ( f g )) F h (1 χ )1 G G        ( αF ( f g )) F h ( G ( f g ) α ) F h α 1 G G        ( Gf ( Gg α )) F h (( Gf Gg ) α ) F h a · 1 5 5 l l l l l l l (( Gf Gg ) α ) F h ( G ( f g ) α ) F h ( χ 1)1   ( ( ( ( ( ( ( ( ( ( (( Gf Gg ) α ) F h ( Gf Gg )( αF h ) a ; ; w w w w w w w w w Gf ( Gg ( αF h )) ( Gf Gg )( αF h ) a ·   ? ? ? ? ? ? ? ? ( Gf Gg )( αF h ) ( Gf Gg )( Ghα ) 1 α - - Z Z Z Z Z Z Z ( G ( f g ) α ) F h G ( f g )( αF h ) a 6 6 m m m m m m m m m m m m m m ( Gf Gg )( αF h ) G ( f g )( αF h ) χ 1   < < < < < < < < < < < < < G ( f g )( αF h ) G ( f g )( Ghα ) 1 α . . \ \ \ \ \ \ \ \ ( αF ( f g )) F h α ( F ( f g ) F h ) a & & M M M M M M M M M M M M α ( F ( f g ) F h ) αF (( f g ) h ) 1 χ & & L L L L L L L L L L L L αF (( f g ) h ) G (( f g ) h ) α α ; ; w w w w w w w w w w w w w w αF (( f g ) h ) αF ( f ( g h )) 1 F a   ( ( ( ( ( ( ( ( ( ( ( ( ( α ( F f F g )) F h α (( F f F g ) F h ) a B B B B B B B B B B B B B α (( F f F g ) F h ) α ( F ( f g ) F h ) 1( χ 1) E E            α (( F f F g ) F h ) α ( F f ( F g F h )) 1 a   2 2 2 2 2 2 2 2 2 2 α ( F f ( F g F h )) α ( F f F ( g h )) 1(1 χ )   2 2 2 2 2 2 2 2 2 2 2 2 2 ( αF f )( F g F h ) α ( F f ( F g F h )) a - - [ [ [ [ [ [ [ [ [ [ [ [ [ ⇓ Π1 ⇓ π ∼ = ∼ = ∼ = ⇓ Π ∼ = ⇓ π ∼ = ⇓ 1 ω ∼ = 30 (( αF f ) F g ) F h (( Gf α ) F g ) F h ( α 1)1 O O (( Gf α ) F g ) F h ( Gf ( αF g )) F h a 1 K K          ( Gf ( αF g )) F h ( Gf ( Gg α )) F h (1 α )1 K K          ( Gf ( Gg α )) F h Gf (( Gg α ) F h ) a G G           Gf (( Gg α ) F h ) Gf ( Gg ( αF h )) 1 a ? ?             Gf ( Gg ( αF h )) Gf ( Gg ( Ghα )) 1(1 α ) / / Gf ( Gg ( Ghα )) ( Gf Gg )( Ghα ) a ·   7 7 7 7 7 7 7 7 7 7 7 ( Gf Gg )( Ghα ) G ( f g )( Ghα ) χ 1   ? ? ? ? ? ? ? ? ? ? ? ? G ( f g )( Ghα ) ( G ( f g ) Gh ) α a ·   ( G ( f g ) Gh ) α G (( f g ) h ) α χ 1   ' ' ' ' ' ' ' ' ' G (( f g ) h ) α G ( f ( g h )) α Ga 1   (( αF f ) F g ) F h ( αF f )( F g F h ) a   7 7 7 7 7 7 7 7 7 7 7 ( αF f )( F g F h ) ( αF f ) F ( g h ) 1 χ   : : : : : : : : : : : : : : : : : : : ( αF f ) F ( g h ) α ( F f F ( g h )) a / / α ( F f F ( g h )) αF ( f ( g h )) 1 χ B B                αF ( f ( g h )) G ( f ( g h )) α α B B                ( αF f ) F ( g h ) ( Gf α ) F ( g h ) α 1 L L             ( αF f )( F g F h ) ( Gf α )( F g F h ) α 1 J J              ( Gf α )( F g F h ) ( Gf α ) F ( g h ) 1 χ   5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 (( Gf α ) F g ) F h ( Gf α )( F g F h ) a & & M M M M M M M M M ( Gf ( αF g )) F h Gf (( αF g ) F h ) a / / Gf (( αF g ) F h ) Gf ( α ( F g F h )) 1 a   ' ' ' ' ' ' ' ' ' ( Gf α )( F g F h ) Gf ( α ( F g F h )) a ; ; w w w w w w w Gf ( α ( F g F h )) Gf ( αF ( g h )) 1(1 χ )   ? ? ? ? ? ? ? ? ? ? ? ? ( Gf α ) F ( g h ) Gf ( αF ( g h )) a E E              Gf ( αF ( g h )) Gf ( G ( g h ) α ) 1 α H H            Gf ( G ( g h ) α ) ( Gf G ( g h )) α a · % % J J J J J J J J ( Gf G ( g h )) α G ( f ( g h )) α χ 1 ) ) R R R R R R R Gf ( Gg ( Ghα )) Gf (( Gg Gh ) α ) 1 a ·                          Gf (( Gg Gh ) α ) Gf ( G ( g h ) α ) 1( χ 1)   , , , , , , , , , , , , , , , , , , , ( Gf Gg )( Ghα ) (( Gf Gg ) Gh ) α a ·                (( Gf Gg ) Gh ) α ( Gf ( Gg Gh )) α a 1   ' ' ' ' ' ' ' ' ( Gf ( Gg Gh )) α ( Gf G ( g h )) α (1 χ )1   Gf (( Gg Gh ) α ) ( Gf ( Gg Gh )) α a · # # G G G G G G G G G G G G G G G G G G G G (( Gf Gg ) Gh ) α ( G ( f g ) Gh ) α ( χ 1)1 ' ' P P P P P P P P P Gf (( αF g ) F h ) Gf (( Gg α ) F h ) 1( α 1) U U + + + + + + + + + + + + + + + + + + + + + ∼ = ⇓ π ∼ = ⇓ 1Π ∼ = ∼ = π ⇐ ∼ = ∼ = ⇓ ω 1 ⇓ Π 31 αF f ( αI ) F f r · 1 ? ?         ( αI ) F f ( αF I ) F f (1 ι )1 ? ?         ( αF I ) F f ( GI α ) F f α 1 < < y y y y y y ( GI α ) F f GI ( αF f ) a ; ; v v v v v v GI ( αF f ) GI ( Gf α ) 1 α # # H H H H H H GI ( Gf α ) ( GI Gf ) α a · " " E E E E E E ( GI Gf ) α G ( I f ) α χ 1   ? ? ? ? ? ? ? ? G ( I f ) α Gf α Gl 1   ? ? ? ? ? ? ? ? αF f ( I α ) F f l · 1 $ $ J J J J J J J J J J J J J J J ( I α ) F f I ( αF f ) a * * T T T T T T T T T T I ( αF f ) I ( Gf α ) 1 α 4 4 j j j j j j j j j j I ( Gf α ) Gf α l : : t t t t t t t t t t t t t t t ( αF I ) F f α ( F I F f ) a , , Y Y Y Y Y α ( F I F f ) αF ( I f ) 1 χ , , Y Y Y Y Y Y αF ( I f ) G ( I f ) α α , , Y Y Y Y Y Y Y ( αI ) F f α ( I F f ) a + + X X X X X X X X X X X α ( I F f ) αF f 1 l + + X X X X X X X X X X X X X αF f Gf α α / / α ( I F f ) α ( F I F f ) 1( ι 1) H H         αF ( I f ) αF f 1 F l   αF f αF f 1 / / I ( αF f ) αF f l D D                  ⇓ Π ∼ = ⇓ µ ⇓ 1 γ ∼ = ∼ = ⇓ λ αF f ( αI ) F f r · 1 ? ?         ( αI ) F f ( αF I ) F f (1 ι )1 ? ?         ( αF I ) F f ( GI α ) F f α 1 < < y y y y y y ( GI α ) F f GI ( αF f ) a ; ; v v v v v v GI ( αF f ) GI ( Gf α ) 1 α # # H H H H H H GI ( Gf α ) ( GI Gf ) α a · " " E E E E E E ( GI Gf ) α G ( I f ) α χ 1   ? ? ? ? ? ? ? ? G ( I f ) α Gf α Gl 1   ? ? ? ? ? ? ? ? αF f ( I α ) F f l · 1 $ $ J J J J J J J J J J J J J J J ( I α ) F f I ( αF f ) a * * T T T T T T T T T T I ( αF f ) I ( Gf α ) 1 α 4 4 j j j j j j j j j j I ( Gf α ) Gf α l : : t t t t t t t t t t t t t t t ( I α ) F f ( GI α ) F f ( ι 1)1 K K                                   I ( αF f ) GI ( αF f ) ι 1 O O I ( Gf α ) GI ( Gf α ) ι 1 O O I ( Gf α ) ( I Gf ) α a · O O ( I Gf ) α ( GI Gf ) α ( ι 1)1 O O ( I Gf ) α Gf α l 1 , , Z Z Z Z Z Z Z Z Z Z Z ⇓ M 1 ∼ = ∼ = ⇓ γ 1 ⇓ λ ∼ = 32 αF f Gf α α F F           Gf α Gf ( αI ) 1 r · @ @             Gf ( αI ) Gf ( αF I ) 1(1 ι ) : : t t t t t t t t t t t t t t t Gf ( αF I ) Gf ( GI α ) 1 α   ? ? ? ? ? ? ? ? ? ? ? ? Gf ( GI α ) ( Gf GI ) α a ·   ? ? ? ? ? ? ? ? ? ? ? ? ( Gf GI ) α G ( f I ) α χ 1   ? ? ? ? ? ? ? ? ? ? ? ? αF f αF f 1 / / αF f Gf α α 5 5 l l l l l l l l l l Gf α G ( f I ) α Gr · 1 < < y y y y y y y y y y y y y y αF f ( αF f ) I r · 1 1 c c c c c c c c c c c c c c c c ( αF f ) I ( Gf α ) I α 1 V V - - - - - - - - Gf α ( Gf α ) I r · 2 2 d d d d d ( Gf α ) I Gf ( αI ) a O O ( αF f ) I ( αF f ) F I 1 ι 7 7 n n n n n n n n n n ( αF f ) F I ( Gf α ) F I α 1 X X 2 2 2 2 2 2 2 2 ( Gf α ) F I Gf ( αF I ) a L L             ( Gf α ) I ( Gf α ) F I 1 ι 7 7 n n n n n n n n n ( αF f ) I α ( F f I ) a   = = = = = = = = = = α ( F f I ) α ( F f F I ) 1(1 ι ) = = | | | | | | | | | | | ( αF f ) F I α ( F f F I ) a   8 8 8 8 8 8 α ( F f I ) αF f 1 r   ; ; ; ; ; ; ; ; ; αF f αF ( f I ) 1 F r · ? ?                 α ( F f F I ) αF ( f I ) 1 χ ' ' O O O O O O αF ( f I ) G ( f I ) α α 2 2 e e e e e e e ∼ = ⇓ ρ ∼ = ∼ = ∼ = ⇓ 1 δ ∼ = ⇓ ρ ⇓ Π αF f Gf α α F F           Gf α Gf ( αI ) 1 r · @ @             Gf ( αI ) Gf ( αF I ) 1(1 ι ) : : t t t t t t t t t t t t t t t Gf ( αF I ) Gf ( GI α ) 1 α   ? ? ? ? ? ? ? ? ? ? ? ? Gf ( GI α ) ( Gf GI ) α a ·   ? ? ? ? ? ? ? ? ? ? ? ? ( Gf GI ) α G ( f I ) α χ 1   ? ? ? ? ? ? ? ? ? ? ? ? αF f αF f 1 / / αF f Gf α α 5 5 l l l l l l l l l l Gf α G ( f I ) α Gr · 1 < < y y y y y y y y y y y y y y Gf α Gf ( I α ) 1 l · 2 2 e e e e e e e e e e e e e e e e e e e Gf ( I α ) Gf ( GI α ) 1( ι 1) 7 7 n n n n n n n n n Gf α Gf α 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 P P P P P P P P P P P P P P P P P P Gf ( I α ) ( Gf I ) α a · ! ! B B B B B B B B B B B ( Gf I ) α Gf α r 1   4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 ( Gf I ) α ( Gf GI ) α (1 ι )1 5 5 l l l l l l l ⇓ 1 M ∼ = ⇓ µ ∼ = ⇓ δ 1 33 Theorem 3 . 4. L et α and β b e tr ansformations F ⇒ G of de gener ate t ric ate- gories. Then a mo diﬁc ation m : α ⇒ β is pr e cisely • a 1-c el l m : α → β in B ′ and • invertible mo diﬁc ations as shown b elow, T ( a, b ) T ′ ( Ga, Gb ) G   T ( a, b ) T ′ ( F a , F b ) F / / T ′ ( F a , F b ) T ′ ( F a , Gb ) T ′ (1 ,α b )   T ′ ( Ga, Gb ) T ′ ( F a , Gb ) T ′ ( α a , 1) / / α r z n n n n n n n n n n n n n n n n n n T ′ ( Ga, Gb ) T ′ ( F a , Gb ) T ′ ( β a , 1) B B ( m a ) ∗   m _ * 4 T ( a, b ) T ′ ( F a , F b ) F / / T ( a, b ) T ′ ( Ga, Gb ) G   T ′ ( F a , F b ) T ′ ( F a , Gb ) T ′ (1 ,β b )   T ′ ( Ga, Gb ) T ′ ( F a , Gb ) T ′ ( β a , 1) / / T ′ ( F a , F b ) T ′ ( F a , Gb ) T ′ (1 ,α b ) z z ( m b ) ∗ k s β }                            al l su bje ct to the following two axioms (u nmarke d isomorphi sms ar e natur ality isomorphi sms). ( αF f ) F g ( Gf α ) F g α 1 / / ( Gf α ) F g Gf ( αF g ) a / / Gf ( αF g ) Gf ( Gg α ) 1 α / / Gf ( Gg α ) ( Gf Gg ) α a · / / ( Gf Gg ) α G ( f g ) α χ 1 / / G ( f g ) α G ( f g ) β G 1 m   ( αF f ) F g ( β F f ) F g ( mF 1) F 1   ( β F f ) F g β ( F f F g ) a ' ' P P P P P P P P P P P P P P P P P P P β ( F f F g ) β F ( f g ) 1 χ / / β F ( f g ) G ( f g ) β β 7 7 n n n n n n n n n n n n n n n n n n n ( β F f ) F g ( Gf β ) F g β 1 / / ( Gf β ) F g Gf ( β F g ) a / / Gf ( β F g ) Gf ( Gg β ) 1 β / / Gf ( Gg β ) ( Gf Gg ) β a · / / ( Gf Gg ) β G ( f g ) β χ 1 / / ( Gf α ) F g ( Gf β ) F g ( G 1 m ) F 1   Gf ( αF g ) Gf ( β F g ) G 1( mF 1)   Gf ( Gg α ) Gf ( Gg β ) G 1( G 1 m )   ( Gf Gg ) α ( Gf Gg ) β ( G 1 G 1) m   ( αF f ) F g ( Gf α ) F g α 1 / / ( Gf α ) F g Gf ( αF g ) a / / Gf ( αF g ) Gf ( Gg α ) 1 α / / Gf ( Gg α ) ( Gf Gg ) α a · / / ( Gf Gg ) α G ( f g ) α χ 1 / / G ( f g ) α G ( f g ) β G 1 m   ( αF f ) F g ( β F f ) F g ( mF 1) F 1   ( β F f ) F g β ( F f F g ) a ' ' P P P P P P P P P P P P P P P P P P P β ( F f F g ) β F ( f g ) 1 χ / / β F ( f g ) G ( f g ) β β 7 7 n n n n n n n n n n n n n n n n n n n ( αF f ) F g α ( F f F g ) a ' ' P P P P P P P P P P P P P P P P P P P α ( F f F g ) αF ( f g ) 1 χ / / αF ( f g ) G ( f g ) α α 7 7 n n n n n n n n n n n n n n n n n n n α ( F f F g ) β ( F f F g ) m ( F 1 F 1)   αF ( f g ) β F ( f g ) mF (11)   ⇓ m 1 ∼ = 1 m ⇓ ∼ = ∼ = ⇓ Π ⇓ Π ∼ = ⇓ m ∼ = 34 α αI r · / / αI αF I 1 ι / / αF I GI α α / / GI α GI β 1 m   α β m   β I β l · ) ) S S S S S S S S S S S S S S S S S S S S S S S S S S S S S I β GI β ι 1 5 5 k k k k k k k k k k k k k k k k k k k k k k k k k k k k β β I r · / / β I 1 ι 1 ι / / 1 ι GI β β / / αI β I m 1   αF I 1 ι m 1   α αI r · / / αI αF I 1 ι / / αF I GI α α / / GI α GI β 1 m   α β m   β I β l · ) ) S S S S S S S S S S S S S S S S S S S S S S S S S S S S S I β GI β ι 1 5 5 k k k k k k k k k k k k k k k k k k k k k k k k k k k k α I α l · ) ) T T T T T T T T T T T T T T T T T T T T T T T T T T T T T I α GI α ι 1 4 4 j j j j j j j j j j j j j j j j j j j j j j j j j j j j I α I β 1 m   ∼ = ∼ = ⇓ m ⇓ M ⇓ M ∼ = ∼ = Theorem 3.5. A p erturb ation σ : m ⇛ n of d e gener ate tric ate gories is pr e cisely a 2-c el l σ : m ⇒ n in B ′ such that t he fol lowing axiom holds. α ⊗ F f Gf ⊗ α α / / Gf ⊗ α Gf ⊗ β 1 ⊗ m   α ⊗ F f β ⊗ F f m ⊗ 1   β ⊗ F f Gf ⊗ β β / / α ⊗ F f β ⊗ F f n ⊗ 1 & & α ⊗ F f Gf ⊗ α α / / Gf ⊗ α Gf ⊗ β 1 ⊗ n   Gf ⊗ α Gf ⊗ β 1 ⊗ m x x α ⊗ F f β ⊗ F f n ⊗ 1   β ⊗ F f Gf ⊗ β β / / σ ⊗ 1 ⇐ 1 ⊗ σ ⇐ m s { 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 s { 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 3.2 Ov erall structure In this section, we will construct a tricatego ry of monoidal bicatego ries. The ob jects and 1 -cells will be given b y degenerate tricatego ries and functor s b et w een them, but the higher cells will b e given only by sp ecial tra nsformations a nd mo diﬁcations which ha ve their comp onen ts at the low es t dimension chosen to be the identit y , as discussed in the Introduction. This is similar to the c a se of doubly dege nerate tricateg ories ab o v e, and is in direct analogy with the 2 - dimensional v ersion in which the bicategory of mono idal categor ies, mono idal 35 functors a nd monoidal trans formations can b e found as a full s ub- bicategory of the bicatego ry of icons. Deﬁnition 4. 1. A monoidal bic ate gory is a degene r ate tricatego ry . 2. A we ak monoidal functor , whic h we no w shorten to monoidal functor, is a weak functor betw een the cor responding degener ate tricatego ries. W e now deﬁne monoidal transformations as a sp ecial cas e of lax transforma- tions wher e the s ing le o b ject comp onen t is the identit y , the la x tr ansformation α is a ctually weak, and where the t w o mo diﬁcations Π and M ar e inv ertible. The da ta and axioms presented here use collapsed versions of the transfo r mation diagrams, making use o f the left and right unit adjoint equiv alences to simplify the diagr ams inv olved. Deﬁnition 5 . Let B , B ′ be monoidal bicateg ories and F, G : B → B ′ be monoidal functors b et ween them. A monoidal tr ansformation α : F ⇒ G consists of • a weak transforma tion α : F ⇒ G b et w een the underlying w eak functors, • an in vertible mo diﬁcation as displa yed b elow, B × B B ′ × B ′ F × F % % B × B B ′ × B ′ G × G : : ⇓ α × α B × B B ⊗   B B ′ G : : B ′ × B ′ B ′ ⊗ ′   χ G u } r r r r r r r r r r r r r r B × B B ′ × B ′ F × F % % B × B B ⊗   B B ′ F $ $ B B ′ G : : ⇓ α B ′ × B ′ B ′ ⊗ ′   ; χ F v ~ u u u u u u u u u u u u u u ; Π _ * 4 • and a n inv er tible mo diﬁcation as display ed be low, 1 B ′ I ′ / / 1 B I " " E E E E E E E E B B ′ F < < y y y y y y y y B B ′ G N N ι F   ⇓ α 1 B ′ I ′ / / 1 B I " " E E E E E E E E B B ′ F < < y y y y y y y y ι G   M _ * 4 36 all sub ject to the follo wing three axio ms. ( F xF y ) F z ( GxGy ) F z ( αα )1 ; ; w w w w w w w w w ( GxGy ) F z ( GxGy ) G z (11) α 7 7 o o o o o o o o o o o o ( GxGy ) G z G ( xy ) Gz χ 1 / / G ( xy ) Gz G  ( xy ) z  χ ' ' O O O O O O O O O O O O G  ( xy ) z  G  x ( y z )  GA # # G G G G G G G G G ( F xF y ) F z F x ( F y F z ) A & & L L L L L L L L L L L L L L L F x ( F y F z ) F xF ( y z ) 1 χ ) ) S S S S S S S S S S S F xF ( y z ) F  x ( y z )  χ 5 5 k k k k k k k k k k F  x ( y z )  G  x ( y z )  α 8 8 r r r r r r r r r r r r r r ( F xF y ) F z F ( xy ) F z χ 1 / / ( GxGy ) F z G ( xy ) F z χ 1 / / F ( xy ) F z G ( xy ) F z α 1 9 9 r r r r r r r r r r r G ( xy ) F z G ( xy ) Gz 1 α 9 9 r r r r r r r r r r r F ( xy ) F z G ( xy ) Gz αα D D F ( xy ) F z F  ( xy ) z  χ / / F  ( xy ) z  F  x ( y z )  F A & & M M M M M M M M M M M M M M F  ( xy ) z  G  ( xy ) z  α 5 5 j j j j j j j j j j j j j j j j ∼ = ⇓ Π1 ∼ = ⇓ Π ∼ = ⇓ ω F ( F xF y ) F z ( GxGy ) F z ( αα )1 ; ; w w w w w w w w w ( GxGy ) F z ( GxGy ) G z (11) α 7 7 o o o o o o o o o o o o ( GxGy ) G z G ( xy ) Gz χ 1 / / G ( xy ) Gz G  ( xy ) z  χ ' ' O O O O O O O O O O O O G  ( xy ) z  G  x ( y z )  GA # # G G G G G G G G G ( F xF y ) F z F x ( F y F z ) A & & L L L L L L L L L L L L L L L F x ( F y F z ) F xF ( y z ) 1 χ ) ) S S S S S S S S S S S F xF ( y z ) F  x ( y z )  χ 5 5 k k k k k k k k k k F  x ( y z )  G  x ( y z )  α 8 8 r r r r r r r r r r r r r r ( GxGy ) F z Gx ( Gy F z ) A   6 6 6 6 6 6 6 F x ( F y F z ) Gx ( Gy F z ) α ( α 1) O O ( GxGy ) G z Gx ( Gy Gz ) A         Gx ( Gy F z ) Gx ( Gy Gz ) 1(1 α ) @ @         Gx ( Gy Gz ) GxG ( y z ) 1 χ , , Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y GxG ( y z ) G  x ( y z )  χ , , Y Y Y Y Y Y F xF ( y z ) GxG ( y z ) αα > > } } } } } } } } } } } } } } } } } } } } } } } } } F x ( F y F z ) Gx ( F y F z ) α (11) > > } } } } } } } Gx ( F y F z ) Gx ( Gy Gz ) 1( αα ) O O Gx ( F y F z ) GxF ( y z ) 1 χ 2 2 e e e e e GxF ( y z ) GxG ( y z ) 1 α 2 2 d d d d d d d d d F xF ( y z ) GxF ( y z ) α 1 K K                ∼ = ∼ = ∼ = ⇓ ω G ⇓ 1Π ∼ = ∼ = ⇓ Π 37 I ′ F x F I F x ι 1 D D         F I F x GI F x α 1 7 7 o o o o o o o GI F x GI Gx 1 α / / GI Gx G ( I x ) χ ' ' O O O O O O O G ( I x ) Gx Gl   4 4 4 4 4 4 4 I ′ F x I ′ Gx 1 α + + W W W W W W W W W W W W W W W W W W I ′ Gx Gx l ′ 3 3 g g g g g g g g g g g g g g g g g g g I ′ F x GI F x ι 1 I I I ′ Gx GI Gx ι 1 J J                   ⇓ M 1 ∼ = ⇓ γ G I ′ F x F I F x ι 1 D D         F I F x GI F x α 1 7 7 o o o o o o o GI F x GI Gx 1 α / / GI Gx G ( I x ) χ ' ' O O O O O O O G ( I x ) Gx Gl   4 4 4 4 4 4 4 I ′ F x I ′ Gx 1 α + + W W W W W W W W W W W W W W W W W W I ′ Gx Gx l ′ 3 3 g g g g g g g g g g g g g g g g g g g F I F x GI Gx αα 7 7 F I F x F ( I x ) χ , , Z Z Z Z Z Z Z Z Z Z Z F ( I x ) G ( I x ) α 2 2 d d d d d d d d d d d I ′ F x F x l ′ / / ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ F x Gx α / / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` F ( I x ) F x F l   ∼ = ⇓ Π ⇓ γ F ∼ = ∼ = F xI ′ F xF I 1 ι D D         F xF I F xGI 1 α 7 7 o o o o o o o F xGI GxGI α 1 / / GxGI G ( xI ) χ ' ' O O O O O O O G ( xI ) Gx Gr   4 4 4 4 4 4 4 F xI ′ GxI ′ α 1 + + W W W W W W W W W W W W W W W W W W GxI ′ Gx r ′ 3 3 g g g g g g g g g g g g g g g g g g g F xI ′ F xGI 1 ι I I GxI ′ GxGI 1 ι J J                   ⇓ 1 M ∼ = ⇓ δ G F xI ′ F xF I 1 ι D D         F xF I F xGI 1 α 7 7 o o o o o o o F xGI GxGI α 1 / / GxGI G ( xI ) χ ' ' O O O O O O O G ( xI ) Gx Gr   4 4 4 4 4 4 4 F xI ′ GxI ′ α 1 + + W W W W W W W W W W W W W W W W W W GxI ′ Gx r ′ 3 3 g g g g g g g g g g g g g g g g g g g F xF I GxGI αα 7 7 F xF I F ( xI ) χ , , Z Z Z Z Z Z Z Z Z Z Z F ( xI ) G ( xI ) α 2 2 d d d d d d d d d d d F xI ′ F x r ′ / / ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ F x Gx α / / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` F ( xI ) F x F r   ∼ = ⇓ Π ⇓ δ F ∼ = ∼ = Note that in the previous dia gram we have wr itten δ F and δ G when in fact their mates are us ed. 38 W e now deﬁne mono idal mo diﬁcations b et ween mono idal bicategories in a similar fashion, as a special case of lax modiﬁcations with the component at th e single ob ject b eing given by an iden tity . Using the le ft and r ig h t unit adjoint equiv alences, we a re then able to s implif y the diagrams to those given b elow. Deﬁnition 6. Let α, β : F ⇒ G b e mono ida l trans formations b et ween monoidal functors. A monoidal mo diﬁc ation m : α ⇛ β consists of a mo diﬁcation m : α ⇛ β b et ween the underlying tr ansformations s uc h that the following tw o axioms hold. F xF y GxF y α 1 2 2 GxF y GxGy 1 α ( ( GxGy G ( xy ) χ % % L L L L L L L L L L L F xF y F ( xy ) χ + + V V V V V V V V V V V V V V V V F ( xy ) G ( xy ) β 3 3 h h h h h h h h h h h h h h h h F xF y GxF y β 1 @ @ GxF y GxGy 1 β 3 3 F xF y GxGy β β < < ⇓ m 1 ⇓ 1 m ∼ = ⇓ Π β F xF y GxF y α 1 9 9 r r r r r r r r r r r GxF y GxGy 1 α / / GxGy G ( xy ) χ % % L L L L L L L L L L L F xF y F ( xy ) χ - - [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ F ( xy ) G ( xy ) β 1 1 c c c c c c c c c c c c c c c F xF y GxGy αα 9 9 F ( xy ) G ( xy ) α , , ∼ = ⇓ Π α ⇓ m I ′ F I ι ; ; w w w w w w w w w w w F I GI α   F I GI β # # G G G G G G G G G G I ′ GI ι / / ⇓ M β ⇓ m I ′ F I ι ; ; w w w w w w w w w w w F I GI α # # G G G G G G G G G G I ′ GI ι / / ⇓ M α The re s t of this s e c tion will be devoted to deﬁning the structure of the tricat- egory M onBicat whose 0-cells a r e monoidal bicategories, 1-c ells a re monoidal functors, 2- c e lls a re monoidal transformations, and 3-c e lls a re monoidal modi- ﬁcations. W e b egin b y deﬁning the hom-bicategor ies fo r this tricategory; note that comp osition is not inherited directly from T ricat but can be thought of as a “hybrid” of the resp ectiv e structures of T ri cat and Bicat . F or 1-cell co mposition, consider mono idal transforma tions α : F ⇒ G and β : G ⇒ H . W e deﬁne a mo noidal tra nsformation β α as fo llo ws: • its under lying tra ns formation is the comp osite β α , • the inv er tible mo diﬁcation Π β α has co mponent at ( X , Y ) given by the 39 diagram b elo w, F X ⊗ F Y HX ⊗ H Y ( β α ) ⊗ ( β α ) / / H X ⊗ H Y H ( X ⊗ Y ) χ H   F X ⊗ F Y F ( X ⊗ Y ) χ F   F ( X ⊗ Y ) G ( X ⊗ Y ) α / / G ( X ⊗ Y ) H ( X ⊗ Y ) β / / F X ⊗ F Y GX ⊗ GY α ⊗ α # # G G G G G G G G G G G GX ⊗ GY H X ⊗ H Y β ⊗ β ; ; w w w w w w w w w w w GX ⊗ GY G ( X ⊗ Y ) χ G   ∼ = ⇓ Π α ⇓ Π β • and the in vertible mo diﬁcation M β α is given by the dia g ram b elo w. I ′ F I ι F ? ?          F I GI α / / GI H I β   ? ? ? ? ? ? ? ? ? I ′ GI ι G < < I ′ H I ι H / / ⇓ M α ⇓ M β The three a xioms are e a sily chec ked b y a simple diagra m c hase. F or identit y 1-cells , conside r a monoidal functor F . Then the iden tit y tra ns- formation u : F ⇒ F ca n b e equipp ed with th e struc tur e of a monoidal tr a nsfor- mation with b oth Π u and M u being given by unique coherence isomorphisms. The axioms follow immediately from the coherence theor em for trica tegories. F or vertical 2-cell compo sition, consider monoidal modiﬁca tions m : α ⇛ β and n : β ⇛ γ . Then w e can c heck that the comp osite nm : α ⇛ γ in B i ca t is in fact monoida l, and lik ewis e the iden tity . F or ho rizon tal 2-c ell comp osition, cons ider monoidal mo diﬁcations as dis- play ed b elo w. X Y F   X Y G / / X Y H G G α % % β y y γ % % δ y y m _ * 4 n _ * 4 Then we can chec k that the comp osite n ∗ m : γ α ⇛ δ β in Bi cat is in fact monoidal, and that this comp osition is functorial. 40 F or coher ence isomor phis ms in the hom- bicategories, consider monoidal trans- formations α : F ⇒ G , β : G ⇒ H , and γ : H ⇒ J . • Let r : αu F ⇛ α b e the mo diﬁcation with comp onen t a t X the rig ht unit isomorphism r α X . It follows from coherence for tricategor ies that r a nd r − 1 are monoidal. • Let l : u G α ⇛ α b e the mo diﬁcation with comp onent a t X the left unit isomorphism l α X . Observe as ab ov e that this mo diﬁcation and its inverse l − 1 are monoidal. • Let a : ( γ β ) α ⇛ γ ( β α ) b e the mo diﬁcation with comp onen t a t X the asso ciativity isomorphism a γ X β X α X is monoidal. Observe as ab ov e that this mo diﬁcation and its in verse a − 1 are monoidal. Theorem 3.6 . The ab ove struct ur e deﬁnes a bic ate gory MonBicat ( X, Y ) . Pr o of. The axioms follow from the bicategory axioms in Y . W e next deﬁne comp osition along 0 -cells for the tricatego r y MonBicat , which we will denote ⊠ ; w e simply extend the deﬁnition of comp osition in the tricategor y Bicat whic h w e no w reca ll. Consider functor s, transfo r mations, and mo diﬁcations a s b elo w. X Y F   X Y F ′ E E α " " α ′ | | Γ _ * 4 Y Z G   Y Z G ′ E E β " " β ′ | | ∆ _ * 4 Then we have the following fo rm ulae. G ⊗ F := GF β ⊗ α := ( G ′ ∗ α ) ◦ ( β ∗ F ) (∆ ⊗ Γ) x := G ′ Γ x ∗ ∆ F x Now supp ose a ll of the ab ov e data a r e monoida l. 1. The comp osite G ⊠ F is the compo site o f the functors of the underlying degenerate tr ic ategories. 2. The comp osite β ⊠ α has underly ing transformation β ⊗ α as ab o ve together with 41 • inv er tible mo diﬁcation Π giv e n by the diag ram b elo w, and GF X ⊗ GF Y G ′ F X ⊗ G ′ F Y β ⊗ β / / G ′ F X ⊗ G ′ F Y G ′ F ′ X ⊗ G ′ F ′ Y G ′ α ⊗ G ′ α / / GF X ⊗ GF Y G ′ F ′ X ⊗ G ′ F ′ Y ( β ⊠ α ) ⊗ ( β ⊠ α ) ) ) GF X ⊗ GF Y G ( F X ⊗ F Y ) χ G   G ( F X ⊗ F Y ) GF ( X ⊗ Y ) Gχ F   G ( F X ⊗ F Y ) G ′ ( F X ⊗ F Y ) β / / GF ( X ⊗ Y ) G ′ F ( X ⊗ Y ) β / / G ′ F X ⊗ G ′ F Y G ′ ( F X ⊗ F Y ) χ G ′   G ′ ( F X ⊗ F Y ) G ′ F ( X ⊗ Y ) G ′ χ F   G ′ ( F X ⊗ F Y ) G ′ ( F ′ X ⊗ F ′ Y ) G ′ ( α ⊗ α ) / / G ′ F ( X ⊗ Y ) G ′ F ′ ( X ⊗ Y ) G ′ α / / G ′ F ′ X ⊗ G ′ F ′ Y G ′ ( F ′ X ⊗ F ′ Y ) χ G ′   G ′ ( F ′ X ⊗ F ′ Y ) G ′ F ′ ( X ⊗ Y ) G ′ χ F ′   ∼ = ⇓ Π β ∼ = ∼ = ⇓ G ′ Π α • inv er tible mo diﬁcation M g iv en by the dia gram b elo w. I ′′ GI ′ ι G / / GI ′ GF I G ι F / / GF I G ′ F I β F I / / G ′ F I G ′ F ′ I G ′ α I / / GI ′ G ′ I ′ β I ′ & & L L L L L L L L L L G ′ I ′ G ′ F I Gι F 8 8 r r r r r r r r r r I ′′ G ′ I ′ ι G ′ / / G ′ I ′ G ′ F ′ I G ′ ι F : : ∼ = ⇓ M β ⇓ G ′ M α 3. The modiﬁcation ∆ ⊗ Γ is a monoidal mo diﬁcation, so we can put ∆ ⊠ Γ = ∆ ⊗ Γ. Theorem 3.7 . The assignments ab ove extend to a functor ⊠ : MonBi ca t ( Y , Z ) × MonBi ca t ( X, Y ) → MonBi cat ( X , Z ) . Pr o of. The co nstrain t mo diﬁcations are the same as thos e given in m y [12]; we need only chec k that t hey are monoida l modiﬁca tio ns, which is accomplished b y a lengthy , but r outine, diag ram chase. The functor a xioms follow fr om coherence and the tra nsformation axio ms. W e now deﬁne units for the compo sition ⊠ . Prop osition 3. 8. L et X b e a monoidal bic ate gory. Ther e is a functor I X : 1 → MonBicat ( X , X ) whose value on the single obje ct is the identity m ono idal functor and whose value on the single 1-c el l is the identity monoida l tr ansfor- mation. Pr o of. F unctoriality determines that the v alue on the single 2- c e ll is the identit y . The unit constraint is the identit y , and the compo s ition co nstrain t is given b y the left (or right) unit isomo rphism in X , which we hav e alr eady determined is a monoidal mo diﬁcation. The axioms then follow from coherence. W e now deﬁne the adjoin t equiv a lences a : ⊠ ◦ ( ⊠ × 1) ⇒ ⊠ ◦ (1 × ⊠ ) l : ⊠ ◦ ( I X × 1) ⇒ 1 r : ⊠ ◦ (1 × I X ) ⇒ 1 . 42 The underlying adjoint equiv alences of tra nsformations are all the same a s the relev ant adjoin t equiv alences in Bicat . It remains to pr o vide the comp onen t mo diﬁcations, chec k that these choices g iv e monoidal transformatio ns, check that the unit and counit mo diﬁcations ar e mo no idal, and check the tria ngle ident ities. All the cells involv ed are coherence cells, and we c a n use co herence for tricateg ories to chec k that all necessar y diagrams commute. Theorem 3.9 . Ther e is a tric ate gory MonBicat with • 0-c el ls monoidal bic ate gories; • hom-bic ate gories give n by the bic ate gories MonBicat ( X, Y ) deﬁne d ab ove; • c omp osition functor given by ⊠ ; • unit given by the fun ctor I X : 1 → MonBicat ( X, X ) ; • adjoint e quivalenc es a , l , r as ab ove; and • invertible m o diﬁc ations π , λ, ρ, µ with e ach mo diﬁc ation having c omp onent s given by unique c oher enc e c el ls in the tar get bic ate gory. F u rthermor e, the obvious for getful fun ctor MonBicat → Bicat is a strict func- tor b etwe en tric ate gories. Pr o of. The tr ic ategory a xioms follow from coherence for bicategories. The fact that the mo diﬁcations above are monoidal follows fro m coherenc e for trica te- gories. A App endix: diagrams Here we include all the diagrams tha t were omitted fro m the text itself. A.1 Doubly degenerate tricategories W e will write the mono ida l structure ⊗ as concatenation, inv oking coherence in order to ignore asso ciation. The monoidal functor ⊠ will be written a s · to sav e space, and we will enclose terms inv olving ⊠ with squa re bracket s when necessary to av o id an exce s s o f paren theses. Iso morphisms will remain larg ely unmarked as there will alwa ys b e a n o bvious choice. The dual pair A and its attenda nt natural isomorphisms must satisfy the 43 following axioms. ( A [( X 1 · Y 1 ) · Z 1 ])[( X 2 · Y 2 ) · Z 2 ] ([ X 1 · ( Y 1 · Z 1 )] A )[( X 2 · Y 2 ) · Z 2 ] / / ([ X 1 · ( Y 1 · Z 1 )] A )[( X 2 · Y 2 ) · Z 2 ] [ X 1 · ( Y 1 · Z 1 )]( A [( X 2 · Y 2 ) · Z 2 ]) / / [ X 1 · ( Y 1 · Z 1 )]( A [( X 2 · Y 2 ) · Z 2 ]) [ X 1 · ( Y 1 · Z 1 )]([ X 2 · ( Y 2 · Z 2 )] A )   [ X 1 · ( Y 1 · Z 1 )]([ X 2 · ( Y 2 · Z 2 )] A ) ([ X 1 · ( Y 1 · Z 1 )][ X 2 · ( Y 2 · Z 2 )]) A   ([ X 1 · ( Y 1 · Z 1 )][ X 2 · ( Y 2 · Z 2 )]) A [( X 1 X 2 ) ·  ( Y 1 Y 2 ) · ( Z 1 Z 2 )  ] A   ( A [( X 1 · Y 1 ) · Z 1 ])[( X 2 · Y 2 ) · Z 2 ] A  [( X 1 · Y 1 ) · Z 1 ][( X 2 · Y 2 ) · Z 2 ]    A  [( X 1 · Y 1 ) · Z 1 ][( X 2 · Y 2 ) · Z 2 ]  A [  ( X 1 X 2 ) · ( Y 1 Y 2 )  · ( Z 1 Z 2 )]   A [  ( X 1 X 2 ) · ( Y 1 Y 2 )  · ( Z 1 Z 2 )] [( X 1 X 2 ) ·  ( Y 1 Y 2 ) · ( Z 1 Z 2 )  ] A / / A AU / / AU A [( U · U ) · U ] / / A [( U · U ) · U ] [ U · ( U · U )] A   A U A   U A [ U · ( U · U )] A / / Similar axio ms hold for A · , and the dual pair s L a nd R . The three tr icategory a xioms now take the following form, where the lab els π , µ, λ, ρ indicate that the arr ow is built up from that constraint using ⊗ and ⊠ ; unmarked arrows are given b y naturality or unique coherence isomorphisms. [ U · ( U · A )][ U · A ] A [( U · A ) · U ][ A · U ][( A · U ) · U ] [ U · ( U · A )][ U · A ][ U · ( A · U )] A [ A · U ][( A · U ) · U ] / / [ U · ( U · A )][ U · A ][ U · ( A · U )] A [ A · U ][( A · U ) · U ] [ U · A ][ U · A ] A [ A · U ][ ( A · U ) · U ] π   [ U · A ][ U · A ] A [ A · U ][ ( A · U ) · U ] [ U · A ] AA [( A · U ) · U ] π   [ U · A ] AA [( A · U ) · U ] [ U · A ] A [ A · ( U · U )] A   [ U · A ] A [ A · ( U · U )] A [ U · A ] A [ A · U ] A   [ U · A ] A [ A · U ] A AAA π   [ U · ( U · A )][ U · A ] A [( U · A ) · U ][ A · U ][( A · U ) · U ] [ U · ( U · A )][ U · A ] A [ A · U ][ A · U ] π   [ U · ( U · A )][ U · A ] A [ A · U ][ A · U ] [ U · ( U · A )] AA [ A · U ] π   [ U · ( U · A )] AA [ A · U ] A [( U · U ) · A ] A [ A · U ]   A [( U · U ) · A ] A [ A · U ] A [ U · A ] A [ A · U ]   A [ U · A ] A [ A · U ] AAA π / / 44 A [( U · L ) · U ] [ A · U ][( R · · U ) · U ] [ U · ( L · U )] A [ A · U ][( R · · U ) · U ] D D      [ U · ( L · U )] A [ A · U ][( R · · U ) · U ] [ U · ( LA )] A [ A · U ][( R · · U ) · U ] λ D D      [ U · ( LA )] A [ A · U ][( R · · U ) · U ] [ U · L ][ U · A ] A [ A · U ][( R · · U ) · U ] / / [ U · L ][ U · A ] A [ A · U ][( R · · U ) · U ] [ U · L ] AA [ ( R · · U ) · U ] π   5 5 5 5 5 [ U · L ] AA [ ( R · · U ) · U ] [ U · L ] A [ R · · ( U · U )] A   5 5 5 5 5 [ U · L ] A [ R · · ( U · U )] A [ U · L ] A [ R · · U ] A   [ U · L ] A [ R · · U ] A U A µ   U A A   A [( U · L ) · U ] [ A · U ][( R · · U ) · U ] A [  ( U · L ) A ( R · · U )  · ( U U U )]   A [  ( U · L ) A ( R · · U )  · ( U U U )] A [  ( U · L ) A ( R · · U )  · U ]   A [  ( U · L ) A ( R · · U )  · U ] A [ U · U ] µ   A [ U · U ] AU / / AU A / / [ U · ( U · L )][ U · A ][ U · ( R · · U )] A [ U · ( U · L )][ U · A ] A [( U · R · ) · U ] D D      [ U · ( U · L )][ U · A ] A [( U · R · ) · U ] [ U · ( U · L )][ U · A ] A [( AR · ) · U ] ρ D D      [ U · ( U · L )][ U · A ] A [( AR · ) · U ] [ U · ( U · L )][ U · A ] A [ A · U ][ R · · U ] / / [ U · ( U · L )][ U · A ] A [ A · U ][ R · · U ] [ U · ( U · L )] AA [ R · · U ] π   5 5 5 5 5 [ U · ( U · L )] AA [ R · · U ] A [( U · U ) · L ] A [ R · · U ]   5 5 5 5 5 A [( U · U ) · L ] A [ R · · U ] A [ U · L ] A [ R · · U ]   A [ U · L ] A [ R · · U ] AU µ   AU A   [ U · ( U · L )][ U · A ][ U · ( R · · U )] A [( U U U ) ·  ( U · L ) A ( R · · U )  ] A   [( U U U ) ·  ( U · L ) A ( R · · U )  ] A [ U ·  ( U · L ) A ( R · · U )  ] A   [ U ·  ( U · L ) A ( R · · U )  ] A [ U · U ] A µ   [ U · U ] A U A / / U A A / / A.2 F unctors b et wee n doubly degenerate tricategories Here we provide the diagrams omitted in the c ha racterisation of weak functor s betw een doubly degenerate tricategories. First note that the dual pairs χ a nd ι must s atisfy axio ms simila r to those for A , L , and R . Then, the t wo functor 45 axioms b ecome the tw o diagrams b elow. [ F ( U · A )] F A [ F ( A · U )] χ [ χ · U ][( χ · U ) · U ] [ F ( U · A )] F Aχ [ F A · F U ][ χ · U ][( χ · U ) · U ] : : t t t t t t t [ F ( U · A )] F Aχ [ F A · F U ][ χ · U ][( χ · U ) · U ] [ F ( U · A )] F Aχ [ F A · U ][ χ · U ][( χ · U ) · U ] 5 5 l l l l l l l l l l [ F ( U · A )] F Aχ [ F A · U ][ χ · U ][( χ · U ) · U ] [ F ( U · A )] F Aχ [  F Aχ ( χ · U )  · ( U U U )] ) ) R R R R R R R R [ F ( U · A )] F Aχ [  F Aχ ( χ · U )  · ( U U U )] [ F ( U · A )] F Aχ [  F Aχ ( χ · U )  · U ] $ $ J J J J [ F ( U · A )] F Aχ [  F Aχ ( χ · U )  · U ] [ F ( U · A )] F Aχ [  χ ( U · χ ) A  · U ] ω   [ F ( U · A )] F Aχ [  χ ( U · χ ) A  · U ] [ F ( U · A )] F Aχ [ χ · U ][( U · χ ) · U ][ A · U ]   [ F ( U · A )] F Aχ [ χ · U ][( U · χ ) · U ][ A · U ] [ F ( U · A )] χ [ U · χ ] A [( U · χ ) · U ][ A · U ] ω   [ F ( U · A )] χ [ U · χ ] A [( U · χ ) · U ][ A · U ] χ [ U · F A ][ U · χ ][ U · ( χ · U )] A [ A · U ]   χ [ U · F A ][ U · χ ][ U · ( χ · U )] A [ A · U ] χ [ U · χ ][ U · ( U · χ )][ U · A ] A [ A · U ] ω   χ [ U · χ ][ U · ( U · χ )][ U · A ] A [ A · U ] χ [ U · χ ][ U · ( U · χ )] AA π   [ F ( U · A )] F A [ F ( A · U )] χ [ χ · U ][( χ · U ) · U ] F A F Aχ [ χ · U ][( χ · U ) · U ] F π   F A F Aχ [ χ · U ][( χ · U ) · U ] F Aχ [ U · χ ] A [ ( χ · U ) · U ] ω   F Aχ [ U · χ ] A [ ( χ · U ) · U ] F Aχ [ U · χ ][ χ · ( U · U )] A   F Aχ [ U · χ ][ χ · ( U · U )] A F Aχ [ U · χ ][ χ · U ] A   F Aχ [ U · χ ][ χ · U ] A F Aχ [ χ · U ][ U · χ ] A   F Aχ [ χ · U ][ U · χ ] A χ [ U · χ ] A [ U · χ ] A ω   χ [ U · χ ] A [ U · χ ] A χ [ U · χ ] A [ ( U · U ) · χ ] A / / χ [ U · χ ] A [ ( U · U ) · χ ] A χ [ U · χ ][ U · ( U · χ )] AA / / [ F ( U · L )] F A [ F ( R · · U )] χ [ F ( U · L )] F Aχ [ F R · · U ] 7 7 o o o o o o o o [ F ( U · L )] F Aχ [ F R · · U ] [ F ( U · L )] F Aχ [  χ ( U · ι ) R ·  · U ] δ 7 7 o o o o o o o [ F ( U · L )] F Aχ [  χ ( U · ι ) R ·  · U ] [ F ( U · L )] F Aχ [ χ · U ][( U · ι ) · U ] [ R · · U ] ' ' O O O O O O O [ F ( U · L )] F Aχ [ χ · U ][( U · ι ) · U ] [ R · · U ] [ F ( U · L )] χ [ U · χ ] A [( U · ι ) · U ][ R · · U ] ω ' ' O O O O O O O O [ F ( U · L )] χ [ U · χ ] A [( U · ι ) · U ][ R · · U ] χ [ U · F L ] [ U · χ ][ U · ( ι · U )] A [ R · · U ]   χ [ U · F L ] [ U · χ ][ U · ( ι · U )] A [ R · · U ] χ [ U · L ] A [ R · · U ] γ   χ [ U · L ] A [ R · · U ] χU µ   [ F ( U · L )] F A [ F ( R · · U )] χ F U χ F µ   F U χ U χ   U χ χ / / χ χU / / A.3 T ransformations for doubly degenerate tricategories Now w e provide the diagra ms omitted from the characteris ation of weak trans- formations in the context of doubly degener ate tricategor ie s. As b efore, the dua l pair α must satisfy tw o axioms similar to those for A , L , and R . The three 46 transformatio ns axioms b ecome the diag rams b elow. Since w e need lax trans- formations as well as weak ones, it should b e noted that the diag rams below do not c hange in the lax case, except tha t we ha ve a distinguis hed ob ject α ins tead of the dual pair, and thus α satisﬁes axioms similar to those for A, L , and R . [ GA · U ][ χ G · U ] A · [ χ G · U ] A · [ U · ( U · α )][ U · A ] A [( U · α ) · U ][ A · U ][( α · U ) · U ] [ GA · U ][ χ G · U ] A · [ χ G · U ][ U · α ] A · [ U · A ] A [( U · α ) · U ][ A · U ][ ( α · U ) · U ] , , Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z [ GA · U ][ χ G · U ] A · [ χ G · U ][ U · α ] A · [ U · A ] A [( U · α ) · U ][ A · U ][ ( α · U ) · U ] [ GA · U ][ χ G · U ] A · [ U · α ][ χ G · U ] A · [ U · A ] A [( U · α ) · U ][ A · U ][ ( α · U ) · U ]   [ GA · U ][ χ G · U ] A · [ U · α ][ χ G · U ] A · [ U · A ] A [( U · α ) · U ][ A · U ][ ( α · U ) · U ] [ GA · U ][ χ G · U ] A · [ U · α ][ χ G · U ] A [ A · · U ][( U · α ) · U ][ A · U ][( α · U ) · U ] π   [ GA · U ][ χ G · U ] A · [ U · α ][ χ G · U ] A [ A · · U ][( U · α ) · U ][ A · U ][( α · U ) · U ] [ GA · U ][ χ G · U ] A · [ U · α ] A [( χ G · U ) · U ][ A · · U ][( U · α ) · U ][ A · U ][( α · U ) · U ]   [ GA · U ][ χ G · U ] A · [ U · α ] A [( χ G · U ) · U ][ A · · U ][( U · α ) · U ][ A · U ][( α · U ) · U ] [ GA · U ][ χ G · U ] A · [ U · α ] A [ α · U ][( U · χ F ) · U ][ A · U ] Π   [ GA · U ][ χ G · U ] A · [ U · α ] A [ α · U ][( U · χ F ) · U ][ A · U ] [ GA · U ] α [ U · χ F ] A [( U · χ F ) · U ][ A · U ] Π   [ GA · U ] α [ U · χ F ] A [( U · χ F ) · U ][ A · U ] α [ U · F A ][ U · χ F ][ U · ( χ F · U )] A [ A · U ]   α [ U · F A ][ U · χ F ][ U · ( χ F · U )] A [ A · U ] α [ U · χ F ][ U · ( U · χ F )][ U · A ] A [ A · U ] ω   α [ U · χ F ][ U · ( U · χ F )][ U · A ] A [ A · U ] α [ U · χ F ][ U · ( U · χ F )] AA π   α [ U · χ F ][ U · ( U · χ F )] AA α [ U · χ F ] A [( U · U ) · χ F ] A   [ GA · U ][ χ G · U ] A · [ χ G · U ] A · [ U · ( U · α )][ U · A ] A [( U · α ) · U ][ A · U ][( α · U ) · U ] [ GA · U ][ χ G · U ][( χ G · U ) · U ] A · A · [ U · ( U · α )][ U · A ][ U · ( α · U )] A [ A · U ][( α · U ) · U ]   [ GA · U ][ χ G · U ][( χ G · U ) · U ] A · A · [ U · ( U · α )][ U · A ][ U · ( α · U )] A [ A · U ][( α · U ) · U ] [ χ G · U ][( U · χ G ) · U ][ A · U ] A · A · [ U · ( U · α )][ U · A ][ U · ( α · U )] A [ A · U ][( α · U ) · U ] ω   [ χ G · U ][( U · χ G ) · U ][ A · U ] A · A · [ U · ( U · α )][ U · A ][ U · ( α · U )] A [ A · U ][( α · U ) · U ] [ χ G · U ][( U · χ G ) · U ] A · [ U · A · ][ U · ( U · α )][ U · A ][ U · ( α · U )] A [ A · U ][( α · U ) · U ] π   [ χ G · U ][( U · χ G ) · U ] A · [ U · A · ][ U · ( U · α )][ U · A ][ U · ( α · U )] A [ A · U ][( α · U ) · U ] [ χ G · U ] A · [ U · ( χ G · U )][ U · A · ][ U · ( U · α )][ U · A ][ U · ( α · U )] A [ A · U ][( α · U ) · U ]   [ χ G · U ] A · [ U · ( χ G · U )][ U · A · ][ U · ( U · α )][ U · A ][ U · ( α · U )] A [ A · U ][( α · U ) · U ] [ χ G · U ] A · [ U · α ][ U · ( U · χ F )][ U · A ] A [ A · U ][( A · U ) · U ] Π   [ χ G · U ] A · [ U · α ][ U · ( U · χ F )][ U · A ] A [ A · U ][( A · U ) · U ] [ χ G · U ] A · [ U · α ][ U · ( U · χ F )] AA [( A · U ) · U ] π   [ χ G · U ] A · [ U · α ][ U · ( U · χ F )] AA [( A · U ) · U ] [ χ G · U ] A · [ U · α ] A [ U · χ F ][ α · U ] A   [ χ G · U ] A · [ U · α ] A [ U · χ F ][ α · U ] A [ χ G · U ] A · [ U · α ] A [ α · U ][ U · χ F ] A ' ' O O O O O O O O [ χ G · U ] A · [ U · α ] A [ α · U ][ U · χ F ] A α [ U · χ F ] A [( U · U ) · χ F ] A / / 47 [ GL · U ][ χ G · U ] A · [ U · α ] A [ α · U ][( U · ι F ) · U ][ R · · U ] [ GL · U ][ χ G · U ] A · [ U · α ] A [( ι G · U ) · U ][ L · · U ] M / / [ GL · U ][ χ G · U ] A · [ U · α ] A [( ι G · U ) · U ][ L · · U ] [ GL · U ][ χ G · U ] A · [ U · α ][ ι G · U ] A [ L · · U ]   [ GL · U ][ χ G · U ] A · [ U · α ][ ι G · U ] A [ L · · U ] [ GL · U ][ χ G · U ] A · [ ι G · U ][ U · α ] A [ L · · U ]   [ GL · U ][ χ G · U ] A · [ ι G · U ][ U · α ] A [ L · · U ] [ GL · U ][ χ G · U ][( ι G · U ) · U ] A · [ U · α ] A [ L · · U ]   [ GL · U ][ χ G · U ][( ι G · U ) · U ] A · [ U · α ] A [ L · · U ] [ L · U ] A · [ U · α ] A [ L · · U ] γ G   [ L · U ] A · [ U · α ] A [ L · · U ] L [ U · α ] A [ L · · U ] λ   [ GL · U ][ χ G · U ] A · [ U · α ] A [ α · U ][( U · ι F ) · U ][ R · · U ] [ GL · U ] α [ U · χ F ] A [( U · ι F ) · U ][ R · · U ] Π   [ GL · U ] α [ U · χ F ] A [( U · ι F ) · U ][ R · · U ] [ GL · U ] α [ U · χ F ][ U · ( ι F · U )] A [ R · · U ]   [ GL · U ] α [ U · χ F ][ U · ( ι F · U )] A [ R · · U ] α [ U · F L ] [ U · χ F ][ U · ( ι F · U )] A [ R · · U ]   α [ U · F L ] [ U · χ F ][ U · ( ι F · U )] A [ R · · U ] α [ U · L ] A [ R · · U ] γ F   α [ U · L ] A [ R · · U ] αU µ / / αU αLA [ L · · U ] λ / / αLA [ L · · U ] L [ U · α ] A [ L · · U ] / / W e will no w drop the subs c ripts for χ and ι as they are determined b y the rest of the diagram. [ χ · U ] A · [ U · α ][ U · ( U · ι )][ U · R · ] α [ χ · U ] A · [ U · α ][ U · ( U · ι )] AR · α ρ 7 7 o o o o o o o o [ χ · U ] A · [ U · α ][ U · ( U · ι )] AR · α [ χ · U ] A · [ U · α ] A [ α · U ][ U · ι ] R · 7 7 o o o o o o o o [ χ · U ] A · [ U · α ] A [ α · U ][ U · ι ] R · α [ U · χ ] A [ U · ι ] R · Π ' ' O O O O O O O O α [ U · χ ] A [ U · ι ] R · α [ U · χ ][ U · ( U · ι )] AR · ' ' O O O O O O O O α [ U · χ ][ U · ( U · ι )] AR · α [ U · F R · ][ U · R ] AR · δ   α [ U · F R · ][ U · R ] AR · α [ U · F R · ] U ρ   α [ U · F R · ] U [ GR · · U ] α   [ χ · U ] A · [ U · α ][ U · ( U · ι )][ U · R · ] α [ χ · U ] A · [ U · ( ι · U )][ U · L · ] α M   [ χ · U ] A · [ U · ( ι · U )][ U · L · ] α [ χ · U ][( U · ι ) · U ] A · [ U · L · ] α   [ χ · U ][( U · ι ) · U ] A · [ U · L · ] α [ GR · · U ][ R · U ] A · [ U · L · ] α δ   [ GR · · U ][ R · U ] A · [ U · L · ] α [ GR · · U ] U α µ / / [ GR · · U ] U α [ GR · · U ] α / / A.4 Mo diﬁcations for doubly degenerate tr ic at egor ies Here we give the t w o a xioms fo r a mo diﬁcation m : α ⇒ β in the context o f doubly degenerate trica tegories. Once aga in we w ill only mark some of the maps, a nd the maps mark ed by m a re obtained fro m the sing le isomorphism in 48 the deﬁnition of the mo diﬁcation. [ GU · m ][ χ · U ] A · [ U · α ] A [ α · U ] [ χ · U ] [( GU · GU ) · m ] A · [ U · α ] A [ α · U ] / / [ χ · U ] [( GU · GU ) · m ] A · [ U · α ] A [ α · U ] [ χ · U ] A · [ GU · ( GU · m )][ U · α ] A [ α · U ]   [ χ · U ] A · [ GU · ( GU · m )][ U · α ] A [ α · U ] [ χ · U ] A · [ U · β ][ GU · ( m · F U )] A [ α · U ] m   [ χ · U ] A · [ U · β ][ GU · ( m · F U )] A [ α · U ] [ χ · U ] A · [ U · β ] A [( GU · m ) · F U ][ α · U ]   [ χ · U ] A · [ U · β ] A [( GU · m ) · F U ][ α · U ] [ χ · U ] A · [ U · β ] A [ β · U ][( m · F U ) · F U ] m   [ χ · U ] A · [ U · β ] A [ β · U ][( m · F U ) · F U ] β [ U · χ ] A [ ( m · F U ) · F U ] Π   [ GU · m ][ χ · U ] A · [ U · α ] A [ α · U ] [ GU · m ] α [ U · χ ] A Π   [ GU · m ] α [ U · χ ] A β [ m · F ( U · U )][ U · χ ] A m   β [ m · F ( U · U )][ U · χ ] A β [ U · χ ][ m · ( F U · F U )] A   β [ U · χ ][ m · ( F U · F U )] A β [ U · χ ] A [( m · F U ) · F U ] / / [ U · m ] α [ U · ι ] R · β [ m · U ][ U · ι ] R · m / / β [ m · U ][ U · ι ] R · β [ U · ι ][ m · U ] R ·   β [ U · ι ][ m · U ] R · β [ U · ι ] R · m   β [ U · ι ] R · m [ ι · U ] L · m M   [ U · m ] α [ U · ι ] R · [ U · m ][ ι · U ] L · M   [ U · m ][ ι · U ] L · [ ι · U ][ U · m ] L ·   [ ι · U ][ U · m ] L · [ ι · U ] L · m / / A.5 P erturbations for doubly degenerate tricategories Here we give the single p erturbation axiom in the context of doubly deg enerate tricategor ies. 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The periodic table of $n$-categories for low dimensions II: degenerate tricategories

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