R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Memoirs Amer. Math. Soc. 117 (1995) Number 558.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....proof objects for equality. Discussions with John Power revealed that the extracted proof is essentially a logical version of the proof of coherence for monoidal categories obtained as a special case of the proof of coherence for bicategories given in the recent paper by Gordon, Power, and Street [9]. Their proof relies on Street s bicategorical Yoneda lemma. The interest of our proof is that it is a nice application of the Curry Howard interpretation. It was discovered independently and uses only some elementary category theory and the elementary reasoning needed for a proof of normalization ....

R. Gordon, A. J. Power, and R. Street. Coherence for tricategories. In Memoirs of the American Mathematical Society. To appear.

.... of restricting attention to the linear counterparts of homomorphisms ( pseudo functors) between bicategories and pseudo natural transformations (with isomorphisms as 2 cell components) in order to secure this composition of the transformations between linear morphisms, as in the tricategory Bicat [11], seemed like a very unattractive option. After all, the lax ness of the components of a linear functor is an essential feature. This provided us with considerable motivation to develop the notion of a poly module which could support two compositions in order to subsume linear natural ....

....GA and to every pair of objects B# a natural transformation B#A, B# #A,B# G#A,B# ##A,B# C#FA,FB# C#FA,#B# C#GA,GB# C##A,GB# ## C#FA,GB# (44) Its value at f B#A, B# is a 2 cell Ff #B ## #A#Gf , which for brevity we denote by #f . The 2 cell orientation is that used in [11]; it reflects the orientation of # from F to G also at the level of 2 cells. Moreover, it parallels the usage of monoidal transformations in [8] The presence of tensors in both domain and codomain implies that in a multi or poly bicategory this notion cannot be interpreted unless is ....

Gordon, R., Power, A. J., and Street, R. Coherence for Tricategories, vol. 558 of Memoirs of the AMS. AMS, Providence, RI, 1995.

....with feedback for studying concurrency. The reader is referred to [KSW] for an overview of this work. Most of the technical definitions in this paper are taken from [DS] and as is done there, we first deal with Gray monoids (which are strict monoidal bicategories) using the coherence theorem of [GPS] to transfer our definitions and results. 2 Preliminaries In order to facilitate calculations in particular, the passage between algebraic and diagrammatic representations of expressions horizontal and vertical composition will be written in diagrammatic order: if f : A B and f 0 : B ....

....than Gray monoids. Of course, a Gray monoid is a type of monoidal bicategory. It is the case, however, that for any monoidal bicategory B there exists a Gray monoid M and strong monoidal biequivalence (that is, a triequivalence) E : B M. This follows from the coherence theorem for tricategories ([GPS]) In this section we extend the results of the previous section to the setting of monoidal bicategories. Strong monoidal biequivalences between monoidal bicategories preserve adjoints and right biduals. If B is a monoidal bicategory then we can define the monoidal bicategory Sq(B) of squares in B ....

Gordon R, Power J and Street R, Coherence for tricategories, Mem. Am. Math. Soc., 117 (1995), 558.

....for the lax. For us, an operad is a multicategory with just one object; this conflicts with [BD] as explained after Definition I.2.2. Related Work Preliminaries on Bicategories. The basic definitions are taken from [B en] and [Gray] and the outline of the coherence theorem from [St2] and [GPS] The same material is covered in more detail in [Lei2] see also [Lack] for another summary. I have not seen the ideas on bias elsewhere; however, Tamsamani appears to establish that his definition of n category is equivalent to the usual one in the case n = 2 ( Tam] and this must require ....

....of lax n category, which appears in [Bat] and is summarized in [St3] The operads used in this chapter are the same as Batanin s, but our notions of contractibility differ. The final section, on cubical structures, is to my knowledge original. III: Gray categories. These were defined in [GPS] following [Gray] We use an equivalent definition, given in [Bat] Sections III.3 and III.4 are based on these sources, with what appears to be a new emphasis (on which processes are canonical) Sections III.6 and III.7, on Cayley representation, seem to be new. IV: The Opetopic Approach. ....

[Article contains additional citation context not shown here]

R. Gordon, A. J. Power, Ross Street, Coherence for tricategories (1995). Memoirs of the AMS, vol. 117, no. 558.

.... (weak) n categories as coecient objects [19, 18] Part of this development will be by clarifying the notion of n stack [12, 14, 3, 4] In 1995, Gordon, Power and Street made signi cant progress by proving a coherence theorem for tricategories, showing that they are triequivalent to Gray categories [13]. For higher dimensions, I introduced teisi and I conjectured that weak 4 categories are weak equivalent to 4 dimensional teisi [10] Teisi also come into the picture on the homotopical side. In 1977 Brown and Higgins introduced crossed complexes, generalizing crossed modules, and proved a ....

R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), no. 558.

....of ve di erent Zamolodchikov equations in braided monoidal 2 categories. Hence higher dimensional Mac Lane s pentagon expresses the relations between these proofs concisely. 1 Introduction The coherence theorem for tricategories states that every tricategory is triequivalent to a Gray category [6]. But there is also another coherence theorem for tricategories, stating that tricategories are (algebras for a) contractible (operad) 1] which roughly says that all diagrams in a tricategory commute . In the basic reference situation of monoidal categories [13, 14] the two coherence theorems ....

R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), no. 558.

.... (weak) n categories as coecient objects [19, 18] Part of this development will be by clarifying the notion of n stack [12, 14, 3, 4] In 1995, Gordon, Power and Street made signi cant progress by proving a coherence theorem for tricategories, showing that they are triequivalent to Gray categories [13]. For higher dimensions, I introduced teisi and I conjectured that weak 4 categories are weak equivalent to 4 dimensional teisi [11] Teisi also come into the picture on the homotopical side. In 1977 Brown and Higgins introduced crossed complexes, generalizing crossed modules, and proved a ....

R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), no. 558.

....1] f g, where ( 2 P (n) 2 32 A. Mutlu, T. Porter The categorical theory of tricategories is still relatively undeveloped and so we have not attempted to identify relationships between the above elements, the kernel kernel commutators and the complicated conditions for a tricategory, cf. [13]. ....

R. Gordon, A.J.Power, and R.Street, Coherence for tricategories, Memoirs A.M.S. 117, (1995), 558 pages

.... of restricting attention to the linear counterparts of homomorphisms ( pseudo functors) between bicategories and pseudo natural transformations (with isomorphisms as 2 cell components) in order to secure this composition of the transformations between linear morphisms, as in the tricategory Bicat [9], seemed like a very unattractive option. After all, the monoidalness of the components of a linear functor is an essential feature. This provided us with considerable motivation to extend the notion of poly module, which does not su er from the whiskering problem, as far as possible in order to ....

....A AG and to every pair of objects hA; Bi a natural transformation BhA;Bi F hA;Bi GhA;Bi hA;Bi C hAF; BF i ChAF;B i C hAG; BGi ChA ;BGi C hAF; BGi (5 00) Its value at f 2 BhA; Bi is a 2 cell fF B A fG, which for brevity we denote by f . We prefer the 2 cell orientation used in [9], since it re ects the orientation of from F to G also at the level of 2 cells. Moreover, it parallels the usage of monoidal transformations in [7] The presence of tensors in both domain and codomain implies that in a multi or polybicategory this notion cannot be interpreted unless is ....

Gordon, R., Power, A. J., and Street, R. Coherence for Tricategories, vol. 558 of Memoirs of the AMS. AMS, Providence, RI, 1995.

....no longer explicitly mention the associativity isos. In some sense it is taken care of by the geometry of the diagrams, this applies to pasting diagrams as well as to string diagrams. More formally, we utilize the fact that every bicategory is biequivalent to a 2 category, e.g. Theorem 1. 4 in [4]. 3. Interpolads and monads In general, endo 1 cells, modules and modisms do not form a bicategory, since identity modules may not exist. Identity modules must at least be idempotent under module composition. The only reasonable candidate for an identity module on A a A is a , equipped with ....

Gordon, R., Power, A. J., and Street, R. Coherence for Tricategories. No. 558 in Memoirs of the AMS. AMS, Providence, RI, 1995.

....will no longer explicitly mention the associativity isos. In some sense it is taken care of by the geometry of the diagrams, this applies to pasting diagrams as well as to string diagrams. More formally, we utilize the fact that every bicategory is biequivalent to a 2 category, cf. Theorem 1. 4 in [4]. 11 3 Interpolads and monads In general, endo 1 cells, modules and modisms do not form a bicategory, since identity modules may not exist. Identity modules must at least be idempotent under module composition. The only reasonable candidate for an identity module on A a A is a , equipped ....

Gordon, R., Power, A. J., and Street, R. Coherence for Tricategories. No. 558 in Memoirs of the AMS. AMS, Providence, RI, 1995.

....monoidal 2 category e C di erent from st(C ) a reliance on invertibility of the braiding, and an ignoring of J issues. Braided monoidal 2 categories are 4 dimensional teisi with one object and one arrow. It is known that every tricategory is triequivalent to a 3dimensional tas ( Gray category) [5]. The calculations in the proof of theorem 3 establish that many (3 dimensional) diagrams in braided monoidal 2 categories commute. This can be interpreted as a very rudimentary and restricted 4 dimensional coherence theorem. It is known that permutations are the objects of a (strict) n ....

R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), no. 558.

....subjects respectively. Familiarity with 2dimensional algebra is assumed and the reader may wish to consult [B en67] or [KS74] for general theory on this subject; in particular we shall use the theory of mates [KS74] We use the string calculus of [JS93] and the definition of a monoidal bicategory [GPS95]. The approach taken, and terminology used in this article follows [DS97] and [McC99b] the latter being heavily drawn upon. The author thanks R. Street for his mathematical advice and inspiration, and thanks S. Lack for his patience and advice. Thanks is also due to C. Butz, G. Katis, C. Hermida ....

.... and let Gray denote the symmetric monoidal closed category whose underlying category is Gray 0 and whose tensor product is the Gray tensor product ; see for example [DS97] A Gray category is a Gray enriched category in the sense of [Kel82] and may be considered to be a semi strict tricategory [GPS95]. In particular, every 3 category may be considered to be a Gray category. There is a factorization system (E; M) on the category Gray 0 . The class E consists of those 2functors that are bijective on objects and bijective on arrows, and the class M consists of those 2 functors that are locally ....

R. Gordon, A.J. Power, and R. Street. Coherence for tricategories, Mem. Amer. Math. Soc., 558, 1995.

....on categories with pullbacks it seems natural to take as domain of variation the 2 category Pbk of categories with pullbacks, pullback preserving functors and cartesian transformations. Let Bicat denote the tricategory of bicategories, homomorphisms, pseudo natural transformations and modi cations [GPS95]. A.3. Proposition. The bicategory of spans construction extends to a trihomomorphism Spn( Pbk Bicat Proof. Given a pullback preserving functor F : B C , the homomorphism Spn(F) Spn(B ) Spn(C ) acts as follows R d c = 7 FR Fd x x x x x F c F F F F ....

R. Gordon, A. J. Power, and R. Street. Coherence for tricategories. Memoirs of the AMS, 117(558), 1995.

....categories with pullbacks it seems natural to take as domain of variation the 2 category Pbk of categories with pullbacks, pullback preserving functors and cartesian transformations. Let Bicat denote the tricategory of bicategories, homomorphisms, pseudo natural transformations and modifications [GPS95]. A.3. Proposition. The bicategory of spans construction extends to a trihomomorphism Spn( Pbk Bicat Proof. Given a pullback preserving functor F : B C , the homomorphism Spn(F ) Spn(B ) Spn(C ) acts as follows R d Gamma Gamma Gamma Gamma Gamma Gamma Gamma c OEOE = ....

R. Gordon, A. J. Power, and R. Street. Coherence for tricategories. Memoirs of the AMS, 117(558), 1995.

....Theory and Applications of Categories, Vol. 4, No. 7 172 The categorical theory of tricategories is still relatively undeveloped and so we have not attempted to identify relationships between the above elements, the kernel kernel commutators and the complicated conditions for a tricategory, cf. [13]. ....

R. Gordon, A.J.Power, and R.Street, Coherence for tricategories, Memoirs A.M.S. 117, (1995), 558 pages

....satisfy a coherence law, the pentagon identity, which says that the following diagram commutes: fg)h)i (fg) hi) f(g(hi) f(gh) i f( gh)i) 6 where all the arrows are 2 morphisms built using the associator. Weak 3 categories or tricategories were defined by Gordon, Power and Street [18] in a paper that appeared in 1995. In a tricategory, the pentagon identity holds only up to an invertible 3morphism, which satisfies a further coherence law of its own. When one explicitly lists the coherence laws this way, the definition of weak n category tends to grow ever more complicated ....

R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Memoirs Amer. Math. Soc. 117 (1995) Number 558.

....proof objects for equality. Discussions with John Power revealed that the extracted proof is essentially a logical version of the proof of coherence for monoidal categories obtained as a special case of the proof of coherence for bicategories given in the recent paper by Gordon, Power, and Street [9]. Their proof relies on Street s bicategorical Yoneda lemma. The interest of our proof is that it is a nice application of the Curry Howard interpretation. It was discovered independently and uses only some elementary category theory and the elementary reasoning needed for a proof of normalization ....

R. Gordon, A. J. Power, and R. Street. Coherence for tricategories. In Memoirs of the American Mathematical Society. To appear.

.... the tensor product on the collection of categories, which is defined by Steiner [16] which extends Gray s tensor product on 2 categories [6] and which is closely related to Brown Higgins s tensor product on groupoids [2] This description will lead to a better understanding of Gray categories [5], so that hopefully methods of chapter 1 can be used to show the homotopical relevance of Graygroupoids [9] and their higher dimensional generalizations, still to be defined. Another application of pasting presentations may be possible in connection with ideas of Tonks [22] 2 Pasting schemes ....

R. Gordon, A. J. Power, and R. Street. Coherence for tricategories. unpublished, 1993.

....a monoidal 2 category with the above tensor product as part of the monoidal structure. Kapranov and Voevodsky [17] have defined the notion of a weak monoidal structure on a strict 2 category, which should be sufficient for the purpose at hand. On the other hand the work of Gordon, Power and Street [14] gives a fully general notion of weak monoidal 2 category, namely a 1 object tricategory. This should also be suitable for studying the tensor product on 2Hilb, though it might be considered overkill. Both these sorts of monoidal 2 category involve various extra structures besides the tensor ....

R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Memoirs Amer. Math. Soc. 117 (1995) Number 558.

....which gives yet another way of interpreting braidings, namely as coming from 0 composition of 2 arrows in a tricategory. One dimension up, there have been several attempts at defining braided monoidal bicategories. The general strategy here is to invoke the coherence theorem for tricategories [15], which implies that it is sufficient to define braidings on semistrict monoidal 2 categories. The first attempt was by Kapranov and Voevodsky [24, 23] who gave a long list of data and axioms. However, their definition contains several inaccuracies and errors, which was noted by Baez and Neuchl ....

....which should be a special case of a coherence theorem for tetracategories. Obviously, such a theorem will not involve (strict) 4 categories, as even in dimension 3 not every tricategory is triequivalent to a 3 category. But the fact that every tricategory is triequivalent to a Gray category [15] gives strong evidence that the semistrict 4 dimensional categorical structures to which tetracategories will be tetraequivalent will have compositions that are dimension raising. Exactly with the application of defining semistrict 4 categories in mind, I introduced a tensor product for ....

[Article contains additional citation context not shown here]

R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), no. 558.

....with feedback for studying concurrency. The reader is referred to [KSW] for an overview of this work. Most of the technical definitions in this paper are taken from [DS] and as is done there, we first deal with Gray monoids (which are strict monoidal bicategories) using the coherence theorem of [GPS] to transfer our defintions and results. 2 Preliminaries 2.1 Gray monoids To begin with, we recall the data for a Gray monoid M from [DS] It is a 2 category equipped with the following: a) an object I ; b) for all objects A, two 2 functors LA ; RA : M M satisfying conditions LA (B) RB (A) ....

....Gray monoids. Of course, a Gray monoid is a type of monoidal bicategory. It is the case, however, that for any monoidal bicategory B there exists a Gray monoid M and strong monoidal biequivalence (that is, a triequivalence) E : B M. This follows from the coherence theorem for tricategories ([GPS]) In this section we extend the results of the previous section to the setting of monoidal bicategories. We assume that the reader is familiar with the notion of an adjoint in a bicategory as well as a right bidual in a monoidal bicategory. Note that strong monoidal biequivalences preserve ....

Gordon R, Power J and Street R, Coherence for tricategories, Mem. Am. Math. Soc., 117 (1995), 558.

.... kind arise as algebraic 3 types of arc connected, simply connected spaces, and from higher dimensional category theory, where braided monoidal categories arise as one object monoidal bicategories [16] These motivations have subsequently been brought together by the definition of tricategories [13] of which Joyal and Tierney s Gray groupoids, which are algebraic 3 types [17] are a special instance. But they have been joined by many further applications of braided monoidal categories, for example in the theory of knots and braids [12] and in relation to quantum groups [21] These ....

....is a 3 category except that horizontal composition of 2arrows results in a 3 isomorphism between the two possible ways of composing these 2arrows vertically, rather than these two vertical composites being equal. More formally, a Gray category is a category enriched in the monoidal category Gray [14, 13]. A braided (strict) monoidal category can be reconsidered as a Gray category which has only one object and one arrow. This reconsideration has the form of a reindexing : for a braided (strict) monoidal category C , the corresponding Gray category, which I will denote by Sigma 2 (C ) has (one ....

[Article contains additional citation context not shown here]

R. Gordon, A. J. Power, and R. Street. Coherence for tricategories. Memoirs Amer. Math. Soc., 117(558), 1995.

....so on: true equational laws are only to be required at the level of n morphisms. Unfortunately, determining the correct coherence laws is a rather tricky business, so that weak n categories have been defined so far only for n 3. They are usually called bicategories [2] for n = 2 and tricategories [17] for n = 3. A major challenge for higher dimensional algebra is to find a good theory of weak n categories for all n. In any event, one expects quite generally that in either the strict or the weak context an (n 1) category C with only one object can be regarded as an n category C by ....

....condition is not an equation but an isomorphism. In other words, a weak 3 category with only one object and one 1 morphism can be thought of as a weak braided monoidal category: one equipped with a natural isomorphism R x;y : x Omega y y Omega x satisfying certain coherence laws [17, 19]. More generally, we may define a braided monoidal n category to be an (n 2) category with one object and one 1 morphism. More generally still, a (n k) category with only one j morphism for each j k can be regarded as a special sort of n category, a k tuply monoidal n category . These play ....

[Article contains additional citation context not shown here]

R. Gordon, A. Powers, and R. Street, Coherence for tricategories, Memoirs Amer. Math. Soc. 117 (1995) Number 558.

.... displayed on the screen) 6 Related Work Discussions with Martin Hyland and John Power revealed that the extracted proof is essentially a logical version of the proof of coherence for bicategories (in the special case of monoidal categories) given in the recent paper by Gordon, Power, and Street [12]. Their proof relies on Street s bicategorical Yoneda lemma. In our case a proof with similar structure was instead discovered by using the Curry Howard interpretation which makes explicit the connection between the formal proof of normalization and the proof of coherence. The present work can be ....

R. Gordon, A. J. Power, and R. Street. Coherence for tricategories. In Memoirs of the American Mathematical Society. To appear.

....A slightly different tensor product is obtained if one requires f Omega g to be an isomorphism. The corresponding internal hom Pseud(C ; D ) has as arrows pseudo natural transformations. 2 Cat with this monoidal closed structure, which is in fact symmetric, is, following Gordon, Power and Street [15], denoted by Gray, and this way Gray itself becomes a Gray category. Gray introduced his tensor product of 2 categories in order to describe lax natural transformations and their composition properly. Since then, Gray s tensor product has gained wider significance, particularly in the form of ....

....to classify homotopy 3 types [6] while Gray groupoids do suffice [21, 5] for this. The second instance is in category theory, where there is a coherence theorem stating that every tricategory is equivalent, in some precise sense, to a Gray category, but not necessarily to a 3 category [15]. It should be noted here that the pseudo version of the tensor product seems to be more important than the lax version, at least from the evidence from tricategories, as the proof of the coherence theorem does not work for lax tricategories. The difference between the cartesian product and Gray s ....

[Article contains additional citation context not shown here]

R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Memoirs Amer. Math. Soc. 117, No. 558 (1995).

....target for every 2 cell in P . In this case the pasting composite of the diagram is well defined. So we have rediscovered a result of D.Verity [19] Example 6.4. Another useful monad considered in [3] is the monad on Glob 3 generated by the 3 operad G which has as algebras the Gray categories [6]. The 2 dimensional skeleton of G is the monad Ssq whose algebras are sesquicategories (i.e. 2 categories without interchange law [14] The 1 skeleton of G coincides with (D s ) 1 . Hence, a G computad consists of a 3 globular set C whose 2 skeleton is isomorphic to a free sesquicategory ....

Gordon R., Power A.J., Street R., Coherence for Tricategories. Memoirs of the AMS, v.117, n.558, 1995.

....no longer explicitly mention the associativity isos. In some sense it is taken care of by the geometry of the diagrams, this applies to pasting diagrams as well as to string diagrams. More formally, we utilize the fact that every bicategory is biequivalent to a 2 category, e.g. Theorem 1. 4 in [4]. 3. Interpolads and monads In general, endo 1 cells, modules and modisms do not form a bicategory, since identity modules may not exist. Identity modules must at least be idempotent under module composition. The only reasonable candidate for an identity module on A F NaN F NaN a A is a , ....

Gordon, R., Power, A. J., and Street, R. Coherence for Tricategories. No. 558 in Memoirs of the AMS. AMS, Providence, RI, 1995.

....Weak 2 categories are usually known as bicategories [12] but there is a strictification theorem saying that all of these are equivalent (or more precisely, biequivalent) to strict 2 categories. Weak 3 categories, or tricategories , have recently been developed by Gordon, Power, and Street [37]. These are not all triequivalent to strict 3 categories, but there is a strictification theorem saying they are triequivalent to semistrict 3 categories . These are categories enriched over 2Cat thought of as a monoidal category not with its Cartesian product, but with a weakened product similar ....

....with a single object, we obtain a strict (resp. weak) monoidal category Similarly, starting with strict, semistrict, or weak 3 categories with only one object, we obtain corresponding sorts of monoidal 2 categories, i.e. 2 categories having tensor products of objects, morphisms, and 2 morphisms [37, 46]. We can iterate this process, and construct from an (n k) category C with only one object, one morphism, and so on up to one (k Gamma 1) morphism, an n category C whose j morphisms are the (j k) morphisms of C. In doing so we obtain a particular sort of n category with extra structure ....

[Article contains additional citation context not shown here]

R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Memoirs Amer. Math. Soc. 117 (1995) Number 558.

....do classify homotopy 3 types [37] This suggests that Gray categories are the right 3 dimensional categorical structures to consider. Further evidence for this comes from the theory of braids [35, 36] braided (strict) monoidal categories are precisely one object, one arrow Gray categories [28]. w w w Cat) Omega categories (the explanation of the name will follow later) 17, Section 3 12] are higher dimensional categorical structures generalizing Gray categories. They are like w categories except that 0 composition of a p arrow and a q arrow results in a (p q Gamma 1) arrow whose ....

....lower dimensional composites [21, p. 8] There are two main tests for the above conjecture. The first one is that such structures in which moreover all elements are invertible should classify all homotopy types. The second one is that, just as every tricategory is triequivalent to a Gray category [28], these structures should feature in a coherence theorem for weak n categories [6, 4, 32] Even the failure of these test cases would be interesting, as that would give an abstract homotopy theory which is richer than for w groupoids but still not as rich as for topological spaces, and it would ....

R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), no. 558.

....categories [14] In 1967, B enabou invented bicategories [4] and he proved that every bicategory is biequivalent to a 2 category. In 1993, Gordon, Power and Street introduced tricategories, and they proved that every tricategory is triequivalent not to a 3 category, but to a Gray category [19]. In algebraic topology, some weakening is needed in dimension 3 as well: homotopy 2 types are classified by 2 groupoids [30, 15, 28] but homotopy 3 types are classified not by 3 groupoids, but by Gray groupoids [24, 23, 5] In order to find out what happens in higher dimensions, and to avoid ....

....9 Lax q transformations and quasi functors of more variables Analogous to the quasi natural transformations of [20] and to the m fold homotopies of [9] I introduce the notion of lax q transformation. This notion unifies the pseudo natural transformations, modifications and perturbations of [19], and makes the terminology ready for higher dimensions. It answers a suggestion of [2] that cubes can be used as domains for higher homotopies of categories, negatively: it is the globes that are used as such. Analogous to the quasi functors of two and of n variables of [20] and to the ....

R. Gordon, A. J. Power, and R. Street. Coherence for tricategories. unpublished, 1993.

....and 2 functors has a symmetric monoidal closed structure for which the closed structure is given by the 2 category P s(C ; D ) of 2 functors from C to D and pseudo natural transformations between them. The category 2 Cat together with this symmetric monoidal closed structure is called Gray [5]. De nition 21 A Gray monoid is a monoid in the symmetric monoidal closed category Gray, i.e. a 2 category K together with a 2 functor : K K K ; subject to the monoid laws. Spelling this out, we have Proposition 22 A Gray monoid consists of a 2 category K, for each object X of K, ....

....is strict symmetric monoidal. If K is symmetric monoidal, we cannot obtain a Gray monoid structure on Kl(T ) because the monoidal structure is not strict on the objects of Kl(T ) The most natural structure to consider in this regard is that of a monoidal bicategory, or a one object tricategory [5]. In loc. cit. there is a coherence theorem that says every monoidal bicategory is equivalent to a Gray monoid. But although the structure on Kl(T ) is more general than that of a Gray monoid, it is not as complicated as the structure of a monoidal bicategory, as the monoidal structure of Kl(T ) ....

R. Gordon, A.J. Power, and R. Street, Coherence for tricategories, Memoirs Amer. Math. Soc. 558 (1995).

No context found.

R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Memoirs Amer. Math. Soc. 117 (1995) Number 558.

No context found.

R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Memoirs Amer. Math. Soc. 117 (1995) Number 558.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC