John Baez and James Dolan. Higher-dimensional algebra III: n-categories and the algebra of opetopes. Adv. Math., 135(2):145--206, 1998. Also available via http://math.ucr.edu/home/baez.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is to (n k) categories with only one object, one 1 morphism, and so on up to one (k 1) morphism for some k. There is a reindexing process that allows one to transform such an (n k) category to an n category with additional structure, called a k tuply monoidal n category. Baez and Dolan [8, 9] have formulated very interesting conjectures as to the structure of such special kinds of n categories. As yet unknown higher dimensional analogues of Gray categories are relevant here since these speculations are best understood in semi strict situations. For example, as a starting point, ....

....and Fiedorowicz [56] used to compare the operadic and # space approaches to 1 fold loop space theory. With considerable input from Simpson and Toen, May has recently made progress on this comparison, although further model theoretic work is necessary to pin down all of the details. Baez and Dolan [5, 8, 9] gave a remarkable definition designed to allow very general diagram shapes, with what they call opetopic sets replacing globular sets. With n opetopic sets replacing simplicial sets, their definition is conceptually analogous to Street s definition, except that it works one n at a time. Their ....

John C. Baez, James Dolan, Higher-dimensional algebra III: n-categories and the algebra of opetopes, Advances in Mathematics 135 (1998), no. 2, 145--206.

....is to (n k) categories with only one object, one 1 morphism, and so on up to one (k 1) morphism for some k. There is a reindexing process that allows one to transform 6 such an (n k) category to an n category with additional structure, called a k tuply monoidal n category. Baez and Dolan [7, 8] have formulated very interesting conjectures as to the structure of such special kinds of n categories. As yet unknown higher dimensional analogues of Gray categories are relevant here since these speculations are best understood in semi strict situations. To give the idea, when n = 0, we obtain ....

....definition can be expressed recursively and it seems probable that it can be compared to May s definition by recursive use of arguments analogous to those that Thomason [141] and Fiedorowicz [49] used to compare the operadic and # space approaches to 1 fold loop space theory. Baez and Dolan [5, 7, 8] gave a remarkable definition designed to allow very general diagram shapes, with what they call opetopic sets replacing globular 13 sets. With n opetopic sets replacing simplicial sets, their definition is conceptually analogous to Street s definition, except that it works one n at a time. ....

John C. Baez, James Dolan, Higher-dimensional algebra III: n-categories and the algebra of opetopes, Advances in Mathematics 135 (1998), no. 2, 145--206.

....which generalizes the correspondence between a category and its nerve Delta op Set ( Tam] we say nothing about this, but for some speculation on page 28. Baez and Dolan presented their definition in terms of opetopes, and this inspired a related description by Hermida, Makkai and Power ( BD] HMP] Finally, Batanin defined lax n and categories using globular structures ( Bat] On the second and third approaches we have much more to say. The contents of this essay are as follows. A preliminary chapter reviews some basic bicategory theory, which we shall call upon later. Chapter ....

....and 2 categories the strict. Our policy is always to say lax or strict except in dimensions 2 and 3, and there use 2 category and 3 category for the strict versions and bicategory and tricategory 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 ....

[Article contains additional citation context not shown here]

J. Baez, J. Dolan, Higher-dimensional algebra III: n-categories and the algebra of opetopes (1998). Adv. Math. 135, pp. 145--206. Also available via http://math.ucr.edu/home/baez.

.... 2 category of module categories (or better 2 Hilbert spaces) in the next step, i.e. weak module 2 categories (For the notions of higher category theory see e.g. 24] The notion of a weak n category for n 4 has long not been precise but there are now different approaches available, see [21] [23] and [25] 27] and there is hope that they can be proved to be equivalent. We therefore feel free to proceed as if there were a single coherent concept. In general, sets of the next higher level correspond to the module structures of the foregoing one, i.e. the finite part of the quantum von ....

J. C. Baez, J. Dolan, Higher-dimensional algebra III: n-categories and the algebra of opetopes, preprint, q-alg/9702014.

....if s(x) s(y) then J between x and y there are no elements of lower dimension, so one of x and y must be a target. 2 6 Extensions Weak n or categories, of whatever kind, are not pre teisi as they do not have strict identities. For kinds of weak n categories which are not de ned algebraically [1, 15] it even makes no sense to ask for strict identities; for Batanin s weak categories, which are de ned using operads [2] it does. Say that an element of a diagrammatic globular set A is high if y is not a source nor a target: there is no z with y = s(z) or y = t(z) Say it is low if it is both ....

J. C. Baez and J. Dolan, Higher dimensional algebra III: n-categories and the algebra of opetopes, Adv. Math. 135 (1998), 145-206.

....that it is no longer visible. In other words the algebra does not seem to re ect the geometry in 1 2 A. Mutlu, T. Porter any simple way. Some interesting recent work in modelling geometry by algebra has tended towards the explicit use of weak in nity categories (Batanin, 2, 3] Baez and Dolan, [1], Leinster, 18] Tamsamani, 23] These use globular, multisimplicial or operad algebra models, but not simplicial groups or groupoids as such, yet some of the structure of weak n groupoids is already apparent in the related simplicial group models. For the transfer of simplicial homotopic ....

J. Baez and J. Dolan, Higher-dimensional algebra III: n-categories and the algebra of opetopes (1997) to appear in Adv. In Math. Also available via http://math.ucr.edu/home.baez/README.html or q-alg 9702

....is also due to C. Butz, G. Katis, C. Hermida and M. Weber. 2. Pseudomonoids In this section the definition of a pseudomonoid in a monoidal 2 category is provided. This definition mildly generalizes that of a pseudomonoid in a Gray monoid [DS97, Section 3] and may be considered a categorification [BD98] of the definition of a monoid in a monoidal category. Motivating examples are provided. Recall that a monoidal bicategory is a tricategory [GPS95, Section 2.2] with exactly one object and that a monoidal 2 category is a monoidal bicategory whose underlying bicategory is a 2 category [GPS95, ....

J. Baez and J. Dolan. Higher dimensional algebra III: n-categories and the algebra of opetopes, Adv. Math., 135 (1998), 145--206.

....[Bat98b] based on homotopical algebra, more speci cally on the theory of operads. He considers contractible operads to organise the spaces of coherent operations, so that weak n categories amount to algebras for such operads. The approach most relevant to our present concerns is that of [BD98]. Its most prominent feature is that the operations of composition are not given explicitly as structure, but rather speci ed by a universal property . This is, in 3 principle, the most attractive approach from a category theoretic perspective. Of course, to specify something universally there ....

....(or rather, their higher dimensional extension which we will take up in a subsequent paper) is to have a solid basis for such a universal approach to weak n categories. Notice that a major di erence between our proposal (here examined thoroughly in its lower dimensional instance) and that of [BD98] is our consideration of a multicategory structure on the cellular structures we consider, in order to have a proper framework for the notion of universality, viz. our representability condition. The present paper may be considered then as the first instalment of our approach to higher dimensional ....

J. Baez and J. Dolan. Higher-dimensional algebra III: n-categories and the algebra of opetopes. Advances in Mathematics, 135:145-206, 1998.

....[Bat98b] based on homotopical algebra, more specifically on the theory of operads . He considers contractible operads to organise the spaces of coherent operations, so that weak n categories amount to algebras for 3 such operads. The approach most relevant to our present concerns is that of [BD98]. Its most prominent feature is that the operations of composition are not given explicitly as structure, but rather specified by a universal property . This is, in principle, the most attractive approach from a category theoretic perspective. Of course, to specify something universally there must ....

....(or rather, their higher dimensional extension which we will take up in a subsequent paper) is to have a solid basis for such a universal approach to weak n categories. Notice that a major difference between our proposal (here examined thoroughly in its lower dimensional instance) and that of [BD98] is our consideration of a multicategory structure on the cellular structures we consider, in order to have a proper framework for the notion of universality, viz. our representability condition. The present paper may be considered then as the first instalment of our approach to ....

J. Baez and J. Dolan. Higher-dimensional algebra III: n-categories and the algebra of opetopes. Advances in Mathematics, 135:145--206, 1998.

....are in typing relation to terms. The notion of higher dimensional syntax characterises these hierarchies of typing relation. A syntactic item of dimension n will be in typed by an item of of next lower dimension n Gamma 1. 1 The construction First recall that John Baez microcosm principle [BD98] tells that in order to characterise categorically the notion of a monoid we need the structure of a monoidal category. So in a monoidal category (C; ffl; I) we get a notion of a monoidal object M with morphisms m : M ffl M M and u : I M satisfying the obvious commuting diagrams. Also recall in ....

....n 2 f A . Given two objects g 2 A Theta B and h 2 f A we get a new object by use of indexed product g c i Theta h 2 f A so we define 13 g c Theta h : g c i Theta h; p 1 ; g d 2 A Theta B. Putting things together we get a definition of combing on top of an operad (see Baez [BD98]) For object g 2 A Theta B we define : FunCmb g : X m2N X n2N X f2FinSet(n;m) g c Theta (A f ; in n ) 2 A Theta B Note that we used copairing [f; g] C C 0 A in category C to get a indexed coproduct f g 2 e A in the comma category. Further we used cocompleteness of C ....

[Article contains additional citation context not shown here]

J. Baez and J. Dolan. Higher dimensional algebra III: n-categories and the algebra of opetopes. Adv. Math., 135:145--206, 1998.

....so thinly around that it is no longer visible. In other words the algebra does not seem to reflect the geometry in any simple way. Some interesting recent work in modelling geometry by algebra has tended towards the explicit use of weak infinity categories (Batanin, 2, 3] Baez and Dolan, [1], Leinster, 18] Tamsamani, 23] These use globular, multisimplicial or operad algebra models, but not simplicial groups or groupoids as such, yet some of the structure of weak n groupoids is already apparent in the related simplicial group models. For the transfer of simplicial homotopic ....

J. Baez and J. Dolan, Higher-dimensional algebra III: n-categories and the algebra of opetopes (1997) to appear in Adv. In Math. Also available via http://math.ucr.edu/home.baez/README.html or q-alg 9702

....better understood for the case n = k = 2. A weak n category is the most general n category, in the sense that there are relations specified only between n morphisms. Baez and Dolan give a definition of weak n category for all n, from which one may define weak k tuply monoidal (n k) categories [3]. In practice, however, it can be quite difficult to work with the full generality of weak monoidal n categories, so formulations have been given of semistrict monoidal and braided monoidal categories for n 2. The idea motivating these semistrict categories is to give definitions which are easier ....

.... Crane and Frenkel give a method for constructing Hopf categories from Lusztig s canonical bases for quantum groups [5] Another possibility for finding appropriate 2 categories is homotopy 2 types of double loop spaces, which Baez and Dolan suggest are likely to be braided monoidal 2 categories [2, 3]. Further, Baez suggests that braided monoidal 3 Hilbert spaces are likely to be braided monoidal 2 categories with duals [1] Finally, there are several examples of braided monoidal 2 categories which have been constructed from solutions of the Zamolodchikov tetrahedron equations [11, 7] and it ....

J. Baez and J. Dolan, Higher-dimensional algebra III: n-Categories and the algebra of opetopes, to appear in Adv. Math., preprint available as q-alg/9702014 and at http://math.ucr.edu/home/baez/

....and Voevodsky motivated their definition by referring to their MAIN PRINCIPLE OF CATEGORY THEORY [24, p. 179] that in any category it is unnatural and undesirable to speak about equality of two objects. This PRINCIPLE also motivated more recent developments in the theory of weak n categories [6, 3, 17], which have now made it possible to define braided monoidal bicategories more conceptually, as tetracategories [32] with one object and one arrow. In practical situations, having the coherence data around all the time is often undesirable. This is already clear from braided monoidal categories: ....

J. C. Baez and J. Dolan, Higher dimensional algebra III: n-categories and the algebra of opetopes, Adv. Math. 135 (1998), 145--206.

....example in the theory of knots and braids [12] and in relation to quantum groups [21] These motivations extend to higher dimensional generalizations of braided monoidal categories. These are expected to arise as algebraic homotopy types of particular connected spaces, and as weak n categories [2, 4] which have only one element in low dimensions. And there should be applications to higher dimensional TQFTs and to n tangles, see Baez et al. [1, 3] Day and Street [10] and Crane and Yetter [7, 8] Rather than dealing with weak higher dimensional categories in their full generality, I will ....

J. C. Baez and J. Dolan. Higher dimensional algebra III: n-categories and the algebra of opetopes. available as http://math.ucr.edu/home/baez/op.ps, 1997.

....before, I take the dimension raising principle as basic. So for Gray categories C and D , their tensor product C Omega D has as generators expressions c Omega d of dimension p q for c 2 C p and d 2 D q , for p q 3. The faces of such a generator c Omega d are 1 Because weak n categories [1, 3, 18] more or less correspond to Grothendieck s stacks [17] I have started to refer to hypothetical higher dimensional categorical structures with dimension raising horizontal composition as teisi (Tas, plural teisi (pronounced TAY see) is Welsh for stack . 11] Theory and Applications ....

....of Gray categories I describe is the lax one; the pseudo version is an easy modification. Taking this viewpoint back to Gray s tensor product, the 3 dimensional generators here are the non identity version of Gray s naturality axioms. This is reminiscent of Baez and Dolan s plus construction [1], which also replaces relations ( reduction laws ) by generators ( operations ) and rewrites between relations by new relations. Their construction is a universal one, though, whereas here only a few of the rewrites are considered. The next basic thing is the behaviour of the tensor product with ....

J. C. Baez and J. Dolan, Higher dimensional algebra III: n-categories and the algebra of opetopes, Adv. Math. 135 (1998), 145--206.

....dimensional category theory. One of the main ingredients of any proposed definition of weak n category is the shape of diagrams (pasting scheme) we accept to be composable. In a globular approach [3] each k cell has a source and target (k Gamma1) cell. In the opetopic approach of Baez and Dolan [1] and the multitopic approach of Hermida, Makkai and Power [7] each k cell has a unique (k Gamma 1) cell as target and a whole (k Gamma 1) dimensional pasting diagram as source. In the theory of strict n categories both source and target may be a general pasting diagram [9, 14, 15] The globular ....

Baez J., Dolan J., Higher DimensionalAlgebra III: n-Categories and the Algebra of Opetopes, Adv. Math. 135 (1998), pp.145-206.

....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 give a basis for the study of the weaker structures that then are weakly equivalent to ....

J. C. Baez and J. Dolan, Higher dimensional algebra III: n-categories and the algebra of opetopes, Adv. Math. 135 (1998), 145--206.

....the conditions R (AjA;A) 1 and R (AjA;A) 1 which previously appeared in the definition of an unframed self dual object A. We refer to the paper in which the tangle hypothesis was first stated as HDA0 [3] and refer to the earlier papers in this series as HDA1 [7] HDA2 [1] and HDA3 [4]. 2 A Topological Description of 2 Tangles In this section we describe the 2 category T of 2 tangles using the language of differential topology, and prove that T is a braided monoidal 2 category with duals. First we carefully describe the objects, 1 morphisms, and 2 morphisms of T , and show ....

J. Baez and J. Dolan, Higher-dimensional algebra III: n-Categories and the algebra of opetopes, Adv. Math. 135 (1998), 145-206.

....field theory and more traditional approaches to algebraic topology. The present paper covers some aspects of this program in more detail, taking advantage of work that has been done in the meantime. Various other aspects are treated in a series of papers entitled Higher Dimensional Algebra [2, 5, 6, 8]. 2 n Categories One philosophical reason for categorification is that it refines our concept of sameness by allowing us to distinguish between isomorphism and equality. In a set, two elements are either the same or different. In a category, two objects can be the same in a way while still ....

....satisfy various laws. As with 2 categories, we can try to impose these laws either strictly or weakly. Strict n categories have been understood for quite some time now [23, 28] but more interesting for us are the weak ones. Various definitions of weak n category are currently under active study [5, 10, 36, 57, 58, 61, 62, 63], and we discuss our own in Section 5. Here, however, we wish to sketch the main challenges any theory of weak n categories must face, and some of the richness inherent in the notion of weak n category. Nota bene: Throughout the rest of this paper, n category will mean weak n category unless ....

[Article contains additional citation context not shown here]

J. Baez and J. Dolan, Higher-dimensional algebra III: n-Categories and the algebra of opetopes, to appear in Adv. Math., preprint available as qalg /9702014.

....as homotopy n types. 6 In short, by iterated categori cation, the whole of homotopy theory should spring forth naturally from pure algebra This dream is well on its way to being realized, but it still holds many challenges for mathematics. Various de nitions of n category have been proposed [5, 9, 23, 28], and for some of these the notion of fundamental n groupoid has already been explored. However, nobody has yet shown that these di erent de nitions are equivalent. Finding a clear treatment of n categories is a major task for the century to come. 3 Natural Numbers and Finite Sets Starting from ....

....rig is not Nfxg, since the isomorphism class of a structure type F contains more information than the numbers jF n j. However, there is a homomorphism from the decategori cation of FinSetfxg onto Nfxg. In fact, one can make the analogy between power series and structure types very precise [5]. But instead of doing this here, let us give an illustration of just how far the analogy goes. Suppose F is a structure type and X is a nite set. De ne the groupoid F (X) as follows: F (X) 1 X n=0 (F n X n ) n : Here n stands for the group of permutations of the set n. This group ....

J. Baez and J. Dolan, Higher-dimensional algebra III: n-Categories and the algebra of opetopes, Adv. Math. 135 (1998), 145-206.

....we can actually find invariants this way in practice: are there any interesting examples of braided monoidal 2 categories with duals We believe there are many examples and that the problem is mainly a matter of developing the machinery to get our hands on them. First, there is plenty of evidence [2, 3] suggesting that we can obtain braided monoidal 2 categories from the homotopy 2 types of double loop spaces. Second, Neuchl and the first author have shown how to obtain braided monoidal 2 categories from monoidal 2 categories by a quantum double construction [5] It seems plausible that ....

J. Baez and J. Dolan, Higher-dimensional algebra III: n-Categories and the algebra of opetopes, to appear in Adv. Math., preprint available as q-alg/9702014 and at http://math.ucr.edu/home/baez/

No context found.

John Baez and James Dolan. Higher-dimensional algebra III: n-categories and the algebra of opetopes. Adv. Math., 135(2):145--206, 1998. Also available via http://math.ucr.edu/home/baez.

No context found.

J. Baez, J. Dolan, Higher-dimensional algebra III: n-categories and the algebra of opetopes (

No context found.

J. C. Baez, J. Dolan, Higher-dimensional algebra III: n-categories and the algebra of opetopes, preprint, q-alg/9702014.

No context found.

J. Baez and J. Dolan, Higher-dimensional algebra III: n-Categories and the algebra of opetopes, Adv. Math. 135 (1998), 145-206.

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