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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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Cited by 25 (5 self)
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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.
Algebras of higher operads as enriched categories II
 In preparation
"... Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the ..."
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Cited by 13 (8 self)
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Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the category of nglobular sets from any normalised (n + 1)operad A, in such a way that the algebras for A may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)category is something like a category enriched in weak ncategories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.
Iterated bar complexes of Einfinity algebras and homology theories
, 2008
"... We proved in a previous article that the bar complex of an E ∞algebra inherits a natural E ∞algebra structure. As a consequence, a welldefined iterated bar construction B n (A) can be associated to any algebra over an E ∞operad. In the case of a commutative algebra A, our iterated bar constructi ..."
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We proved in a previous article that the bar complex of an E ∞algebra inherits a natural E ∞algebra structure. As a consequence, a welldefined iterated bar construction B n (A) can be associated to any algebra over an E ∞operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E ∞algebras. We use this effective definition to prove that the nfold bar construction admits an extension to categories of algebras over Enoperads. Then we prove that the nfold bar complex determines the homology theory associated to the category of algebras over an Enoperad. In the case n = ∞, we obtain an isomorphism between the homology of an infinite bar construction and the usual Γhomology with trivial coefficients.
A∞ FUNCTORS FOR LAGRANGIAN CORRESPONDENCES
"... Abstract. We describe a construction of A ∞ functors associated to monotone Lagrangian correspondences, and a proof that the composition of A∞ functors is homotopic to the functor for the composition, in the case that the composition is smooth and embedded. 1. ..."
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Abstract. We describe a construction of A ∞ functors associated to monotone Lagrangian correspondences, and a proof that the composition of A∞ functors is homotopic to the functor for the composition, in the case that the composition is smooth and embedded. 1.
The EckmanHilton argument and higher operads, preprint
, 2006
"... To the memory of my father. 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 ..."
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To the memory of my father. 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 begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the EckmanHilton type argument in terms of the calculation of left Kan extensions in an appropriate 2category. Then we apply this scheme to the case of noperads in the author’s sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation Symn from a certain subcategory of noperads to the
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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Cited by 3 (1 self)
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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.
KOSZUL DUALITY OF EnOPERADS
"... Abstract. The goal of this paper is to prove a Koszul duality result for Enoperads in differential graded modules over a ring. The case of an E1operad, which is equivalent to the associative operad, is classical. For n> 1, the homology of an Enoperad is identified with the nGerstenhaber operad ..."
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Abstract. The goal of this paper is to prove a Koszul duality result for Enoperads in differential graded modules over a ring. The case of an E1operad, which is equivalent to the associative operad, is classical. For n> 1, the homology of an Enoperad is identified with the nGerstenhaber operad and forms another well known Koszul operad. Our main theorem asserts that an operadic cobar construction on the dual cooperad of an Enoperad En defines a cofibrant model of En. This cofibrant model gives a realization at the chain level of the minimal model of the nGerstenhaber operad arising from Koszul duality. Most models of Enoperads in differential graded modules come in nested sequences E1 ⊂ E2 ⊂ · · · ⊂ E ∞ homotopically equivalent to the sequence of the chain operads of little cubes. In our main theorem, we also define a model of the operad embeddings En−1 ↩ → En at the level of cobar constructions.