Claudio Hermida, Michael Makkai, and John Power. On weak higher-dimensional categories I -- 2. Journal of Pure and Applied Algebra, 157:247--277, 2001.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... mathematical structures may be generalized quite dramatically, so as to include geometric structures like topological spaces and the much lesser known approach spaces (see [21] but also Lambek s [19] multicategories which enjoy renewed interest in higher dimensional category theory (see [13] [14]) Indeed, it is well known that a topological space may be completely described by a convergence relation, i.e. by a function 2, where UX is the set of ultrafilters on X satisfying the two basic axioms . m(X) z) Here x x means that the ultrafilter x on X converges to ....

C. Hermida, M. Makkai and J. Power, On weak higher-dimensional categories I - 2, J. Pure Appl. Algebra 157 (2001) 247-277.

....is conceptually analogous to Street s definition, except that it works one n at a time. Their n opetopic sets are given by a presheaf category, and their n categories are suitably restricted n opetopic sets. Variants of their definition are given and studied by Hermida, Makkai, and Power [70, 71, 72, 73, 73, 75], Makkai and Zawadowski [106, 108, 156] Leinster [94, 95] and Cheng, who also gives comparisons among definitions within this family [32, 33, 34, 35] Another conceptually similar definition, using an alternative diagram scheme, has recently been given by Higuchi, Miyada and 15 Tsujishita [76, ....

Claudio Hermida, Michael Makkai, John Power, On weak higher dimensional categories I, part 1, Journal of Pure and Applied Algebra 154 (2000), no. 1-3, 221--246.

....is conceptually analogous to Street s definition, except that it works one n at a time. Their n opetopic sets are given by a presheaf category, and their n categories are suitably restricted n opetopic sets. Variants of their definition are given and studied by Hermida, Makkai, and Power [70, 71, 72, 73, 73, 75], Makkai and Zawadowski [106, 108, 156] Leinster [94, 95] and Cheng, who also gives comparisons among definitions within this family [32, 33, 34, 35] Another conceptually similar definition, using an alternative diagram scheme, has recently been given by Higuchi, Miyada and 15 Tsujishita [76, ....

Claudio Hermida, Michael Makkai, John Power, On weak higher dimensional categories, available via http://fcs.math.sci.hokudai.ac.jp/doc/info/ncat.html, 1997, 104 pages.

....is conceptually analogous to Street s definition, except that it works one n at a time. Their n opetopic sets are given by a presheaf category, and their n categories are suitably restricted n opetopic sets. Variants of their definition are given and studied by Hermida, Makkai, and Power [66, 67, 68, 69, 69, 71], Makkai and Zawadowski [102, 104, 148] Leinster [89, 90] and Cheng, who also gives comparisons among definitions within this family [28, 29, 30, 31] Another conceptually similar definition, using an alternative diagram scheme, has recently been given by Higuchi, Miyada and Tsujishita [72, 114] ....

Claudio Hermida, Michael Makkai, John Power, On weak higher dimensional categories I, part 1, Journal of Pure and Applied Algebra 154 (2000), no. 1-3, 221--246.

....is conceptually analogous to Street s definition, except that it works one n at a time. Their n opetopic sets are given by a presheaf category, and their n categories are suitably restricted n opetopic sets. Variants of their definition are given and studied by Hermida, Makkai, and Power [66, 67, 68, 69, 69, 71], Makkai and Zawadowski [102, 104, 148] Leinster [89, 90] and Cheng, who also gives comparisons among definitions within this family [28, 29, 30, 31] Another conceptually similar definition, using an alternative diagram scheme, has recently been given by Higuchi, Miyada and Tsujishita [72, 114] ....

Claudio Hermida, Michael Makkai, John Power, On weak higher dimensional categories, available via http://fcs.math.sci.hokudai.ac.jp/doc/info/ncat.html, 1997, 104 pages.

....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 I ....

....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. Opetopes seem to have been defined first in [BD] they are also explained in [Baez] and used in the [HMP] approach to lax n categories. My understanding of the latter is based on [Hy] Various ideas in this chapter also appear in [Her] Categories of trees are employed in [Bor] Sny] KM1] KM2] and [Soi] 3 Other. The Tamsamani approach is laid out in [Tam] and explored further in Simpson s ....

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C. Hermida, M. Makkai, J. Power, On weak higher dimensional categories (1997). Available via http://triples.math.mcgill.ca.

....all instances of the interchange axiom are replaced by dimension raising compositions C p C n C q C p q n 1 , that have to satisfy naturality, functoriality, associativity, and more axioms. It must be remarked that there is also the notion of weak n category, in many different ( incarnations [1, 2, 12], which are very much in vogue, but these are not of concern for this paper. Just like the notion of strict n category, the notion of nD tas is dimension invariant: for an nD tas C and for each pair of m arrows c and c 0 in C with common (m 1) source and target, the collection of elements C ....

C. Hermida, M. Makkai, and J. Power, On weak higher dimensional categories, J. Pure Appl. Algebra, to appear.

....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 ....

C. Hermida, M. Makkai, and J. Power, On weak higher dimensional categories, J. Pure Appl. Algebra, to appear.

....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 ....

C. Hermida, M. Makkai, and J. Power, On weak higher dimensional categories, preprint available at http://hypatia.dcs.qmw.ac.uk/authors/M/ MakkaiM/papers/multitopicsets/

....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: ....

C. Hermida, M. Makkai, and J. Power, On weak higher dimensional categories, J. Pure Appl. Algebra (to appear).

....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 ....

C. Hermida, M. Makkai, and J. Power, On weak higher dimensional categories I, J. Pure Appl. Algebra (to appear).

....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 ....

.... w groupoid was already used for cubical sets with extra groupoid structures [13] Then there are weak n groupoids , referred to above, which weaken the strictness of the groupoid condition [38] Another use of weak occurs in weak n categories , which have weakened axioms for composition [32], and which some people prefer to call n categories [3, 49] And then there is the term Graycategory , which doesn t give any indication that it is 3 dimensional, and for which the boldface font is a bit tiresome. The reason for all these problems is basically that categorical terminology was ....

C. Hermida, M. Makkai, and J. Power, On weak higher dimensional categories, J. Pure Appl. Algebra (to appear).

....are interpreted in W as face and degeneracy operators, by extending the terminology used for simplicial sets. In [BD] the opetopic weak n categories (for finite n) are in fact defined without first describing the shape category, but the latter is implicit: it is the category of opetopes. In [HMP], the shape category is made explicit: it is the category of multitopes. In [MM2] where the definition of multitopic # category is completed, and, also, the ambient structure comprising all multitopic # categories is clarified, the shape category (the category of multitopes) plays an active role. ....

C. Hermida, M. Makkai, J. Power, On weak higher dimensional categories, I. Part 1. J. Pure and Applied Alg. 153 (2000), pp 221-246. Parts 2 and 3: to appear in the same Journal.

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Claudio Hermida, Michael Makkai, and John Power. On weak higher-dimensional categories I -- 2. Journal of Pure and Applied Algebra, 157:247--277, 2001.

No context found.

Claudio Hermida, Michael Makkai, and John Power. On weak higher dimensional categories, 1997. Available via http://triples.math.mcgill.ca.

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C. Hermida, M. Makkai, J. Power, On weak higher dimensional categories I.1, I.2, I.3. Journal of Pure and Applied Algebra 154 (

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Claudio Hermida, Michael Makkai, John Power, On weak higher-dimensional categories I, part 3, to appear in Journal of Pure and Applied Algebra.

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Claudio Hermida, Michael Makkai, John Power, On weak higher-dimensional categories I, part 3, to appear in Journal of Pure and Applied Algebra.

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