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The EckmannHilton argument, higher operads and Enspaces, available at http://www.ics.mq.edu.au
 mbatanin/papers.html of Homotopy and Related Structures
"... The classical EckmannHilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of ..."
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The classical EckmannHilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2category, then its Homset is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category. In this paper we extend this argument to arbitrary dimension. We demonstrate that for an noperad A in the author’s sense there exists a symmetric operad S n (A) called the nfold suspension of A such that the
Iterated wreath product of the simplex category and iterated loop spaces
 Adv. Math
"... Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternat ..."
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Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternative description of the Segal spectrum associated to a Γspace. In particular, each EilenbergMacLane space has a canonical reduced Θnset model. The number of (n + d)dimensional cells of the resulting CWcomplex of type K(Z/2Z, n) is a generalised Fibonacci number.
Generic morphisms, parametric representations and weakly cartesian monads
 THEORY APPL. CATEG
, 2004
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Monads with arities and their associated theories
 J. of Pure and Applied Algebra
"... Abstract. After a review of the concept of “monad with arities ” we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere’s algebraic theories to a general correspondence between mona ..."
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Cited by 4 (1 self)
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Abstract. After a review of the concept of “monad with arities ” we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere’s algebraic theories to a general correspondence between monads and theories for a given category with arities. As application we determine arities for the free groupoid monad on involutive graphs and recover the symmetric simplicial nerve characterisation of groupoids. Introduction. In his seminal work [20] Lawvere constructed for every variety of algebras, defined by finitary operations and relations on sets, an algebraic theory whose nary operations are the elements of the free algebra on n elements. He showed that the variety of algebras is equivalent to the category of models of the associated algebraic
Adding inverses to diagrams II: Invertible homotopy theories are spaces, preprint available at math.AT/0710.2254
"... Abstract. In previous work, we showed that there are appropriate model ..."
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Abstract. In previous work, we showed that there are appropriate model
A PREHISTORY OF nCATEGORICAL PHYSICS
, 2008
"... We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, me ..."
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We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, membranes and spin foams.
Comparing operadic theories of ncategory
, 2008
"... We give a framework for comparing on the one hand theories of ncategories that are weakly enriched operadically, and on the other hand ncategories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (ChengGurski) and examples of ..."
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We give a framework for comparing on the one hand theories of ncategories that are weakly enriched operadically, and on the other hand ncategories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (ChengGurski) and examples of the latter are the definition by Batanin and variants (Leinster). We will show how to take a theory of ncategories of the former kind and produce a globular operad whose algebras are the ncategories we started with. We first provide a generalisation of Trimble’s original theory that allows for the use of other parametrising operads in a very general way, via the notion of categories weakly enriched in V where the weakness is parametrised by an operad P in the category V. We define weak ncategories by iterated weak enrichment using a series of parametrising operads Pi. We then show how to construct from such a theory an ndimensional globular operad for each n ≥ 0 whose algebras
Project Description:
"... d manifolds of a certain dimension and the maps between them are equivalence classes of cobordisms between them, which are manifolds with boundary in the next higher dimension. However, it is in many respects far more natural to deal with an ncobordism "category" constructed from points, edges, surf ..."
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d manifolds of a certain dimension and the maps between them are equivalence classes of cobordisms between them, which are manifolds with boundary in the next higher dimension. However, it is in many respects far more natural to deal with an ncobordism "category" constructed from points, edges, surfaces, and so on through nmanifolds that have boundaries with corners. The structure encodes cobordisms between cobordisms between cobordisms. This is an ncategory with additional structure, and one needs analogously structured linear categories as targets for the appropriate "functors" that define the relevant TQFT's. One could equally well introduce the basic idea in terms of formulations of programming languages that describe processes between processes between processes. A closely analogous idea has long been used in the study of homotopies between homotopies between homotopies in algebraic topology. Analogous structures appear throughout mathematics. In contrast to the original Eilenb
Some properties of the theory of ncategories
, 2001
"... Much interest has recently focused on the problem of comparing different definitions of ncategories. 1 Leinster has made a useful compendium of 10 definitions [8]; May has proposed another definition destined among other things to make comparison easier [9], and he has also led the creation of an u ..."
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Much interest has recently focused on the problem of comparing different definitions of ncategories. 1 Leinster has made a useful compendium of 10 definitions [8]; May has proposed another definition destined among other things to make comparison easier [9], and he has also led the creation of an umbrella research group including comparison as one of the main research topics; Batanin has started a comparison of his definition with Penon’s [2]; and Berger has made a comparison between Batanin’s theory and the homotopy theory of spaces, introducing techniques which should allow for other comparisons starting with Batanin’s theory [3]. The comparison question was first explicitly mentionned by Grothendieck in [7], in a prescient prediction that many different people would come up with different definitions of ncategory; and this theme was again brought up by Baez and Dolan in [1]. The purpose of this short note is to make some observations about properties which one can expect any theory of ncategories to have, and to conjecture that these properties characterize the theory of ncategories. As a small amount of evidence for this conjecture, we show how to go from these properties to the composition law between mapping objects in an ncategory. While we don’t give the proofs here, it is not too hard to see that Tamsamani’s definition of ncategory satisfies the properties listed below. We conjecture that the other definitions satisfy these properties too. This conjecture plus the conjecture of the previous paragraph would give an answer to the comparison question. Rather than giving all of the references for the various different definitions of ncategory, we refer the reader to Leinster’s excellent bibliography [8]. The fundamental tool which we use is the DwyerKan localization [5]. This is a generalization of the classical GabrielZisman localization, which keeps higher homotopy data. Dwyer and Kan obtain a mapping space between two objects, where Gabriel and Zisman obtain only the set of homotopy classes of maps i.e. the π0 of the mapping space. The fundamental observation of Dwyer and Kan is that the mapping spaces (plus their composition and higher homotopy coherence information) are determined by the data of