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Higher dimensional algebra III: ncategories and the algebra of opetopes
, 1997
"... We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads ..."
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Cited by 74 (6 self)
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We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads over O. Letting I be the initial operad with a oneelement set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘ndimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak ncategory is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ωcategories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘ncoherent Oalgebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal ncategories’, ‘stable ncategories’, ‘virtual nfunctors ’ and ‘representable nprestacks’. We also describe how ncoherent Oalgebra objects may be defined in any (n + 1)coherent Oalgebra.
An introduction to ncategories
 In 7th Conference on Category Theory and Computer Science
, 1997
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Applications of Peiffer pairings in the Moore complex of a simplicial group
, 1998
"... Generalising a result of Brown and Loday, we give for n = 3 and 4, a decomposition of the group, dn NGn ; of boundaries of a simplicial group G as a product of commutator subgroups. Partial results are given for higher dimensions. Applications to 2crossed modules and quadratic modules are discussed ..."
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Cited by 10 (6 self)
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Generalising a result of Brown and Loday, we give for n = 3 and 4, a decomposition of the group, dn NGn ; of boundaries of a simplicial group G as a product of commutator subgroups. Partial results are given for higher dimensions. Applications to 2crossed modules and quadratic modules are discussed. A. M. S. Classication: 18G30, 55U10, 55P10. Introduction Simplicial groups occupy a place somewhere between homological group theory, homotopy theory, algebraic Ktheory and algebraic geometry. In each sector they have played a signicant part in developments over quite a lengthy period of time and there is an extensive literature on their homotopy theory. In homotopy theory itself, they model all connected homotopy types and allow analysis of features of such homotopy types by a combination of group theoretic methods and tools from combinatorial homotopy theory. Simplicial groups have a natural structure of Kan complexes and so are potentially models for weak innity categories. They d...
Freeness Conditions for 2Crossed Modules and Complexes
, 1998
"... Using free simplicial groups, it is shown how to construct a free or totally free 2crossed module on suitable construction data. 2crossed complexes are introduced and similar freeness results for these are discussed. A. M. S. Classication: 18D35 18G30 18G50 18G55. Introduction Crossed modules we ..."
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Cited by 5 (2 self)
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Using free simplicial groups, it is shown how to construct a free or totally free 2crossed module on suitable construction data. 2crossed complexes are introduced and similar freeness results for these are discussed. A. M. S. Classication: 18D35 18G30 18G50 18G55. Introduction Crossed modules were introduced by Whitehead in [23] with a view to capturing the relationship between 1 and 2 of a space. Homotopy systems (which would now be called free crossed complexes [5] or totally free crossed chain complexes (cf. Baues [3, 4]) were introduced, again by Whitehead, to incorporate the action of 1 on the higher relative homotopy groups of a CWcomplex. They consist of a crossed module at the base and a chain complex of modules over 1 further up. Conduche [9] dened 2crossed modules as a model of connected 3types and showed how to obtain a 2crossed module from a simplicial group. A variant of 2crossed modules are the quadratic modules of Baues [3, 4] and he also denes a not...
Categorification
 Contemporary Mathematics 230. American Mathematical Society
, 1997
"... Categorification is the process of finding categorytheoretic analogs of settheoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called ‘c ..."
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Cited by 4 (1 self)
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Categorification is the process of finding categorytheoretic analogs of settheoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called ‘coherence laws’. Iterating this process requires a theory of ‘ncategories’, algebraic structures having objects, morphisms between objects, 2morphisms between morphisms and so on up to nmorphisms. After a brief introduction to ncategories and their relation to homotopy theory, we discuss algebraic structures that can be seen as iterated categorifications of the natural numbers and integers. These include tangle ncategories, cobordism ncategories, and the homotopy ntypes of the loop spaces Ω k S k. We conclude by describing a definition of weak ncategories based on the theory of operads. 1
MartinLöf Complexes
, 2009
"... In this paper we define MartinLöf complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional MartinLöf type theory. We then study the resulting categories of algebras for several theories. Our principal resu ..."
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Cited by 1 (1 self)
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In this paper we define MartinLöf complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional MartinLöf type theory. We then study the resulting categories of algebras for several theories. Our principal result is that there exists a cofibrantly generated Quillen model structure on
MENGER (NÖBELING) MANIFOLDS VERSUS HILBERT CUBE (SPACE) MANIFOLDS – A CATEGORICAL COMPARISON
, 1999
"... Abstract. We show that the nhomotopy category MENGn of connected (n+1)dimensional Menger manifolds is isomorphic to the homotopy category HILBCn of connected Hilbert cube manifolds whose kdimensional homotopy groups are trivial for each k ≥ n + 1. 1. ..."
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Abstract. We show that the nhomotopy category MENGn of connected (n+1)dimensional Menger manifolds is isomorphic to the homotopy category HILBCn of connected Hilbert cube manifolds whose kdimensional homotopy groups are trivial for each k ≥ n + 1. 1.
MATRIX PROBLEMS, TRIANGULATED CATEGORIES AND STABLE HOMOTOPY TYPES
, 903
"... Abstract. We show how the matrix problems can be used in studying triangulated categories. Then we apply the general technique to the classification of stable homotopy types of polyhedra, find out the “representation types ” of such problems and give a description of stable homotopy types in finite ..."
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Abstract. We show how the matrix problems can be used in studying triangulated categories. Then we apply the general technique to the classification of stable homotopy types of polyhedra, find out the “representation types ” of such problems and give a description of stable homotopy types in finite and tame cases. Contents
THREE CROSSED MODULES
, 812
"... We introduce the notion of 3crossed module, which extends the notions of 1crossed module (Whitehead) and 2crossed module (Conduché). ..."
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We introduce the notion of 3crossed module, which extends the notions of 1crossed module (Whitehead) and 2crossed module (Conduché).