Results 1  10
of
25
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 ..."
Abstract

Cited by 40 (7 self)
 Add to MetaCart
(Show Context)
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
MODULI SPACE ACTIONS ON THE HOCHSCHILD Cochains Of A Frobenius Algebra I: Cell Operads
, 2006
"... This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co–chains of a Frobenius algebra. We also prove that a there is dg–PROP action of a version of Sullivan Chord diagrams ..."
Abstract

Cited by 21 (7 self)
 Add to MetaCart
(Show Context)
This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co–chains of a Frobenius algebra. We also prove that a there is dg–PROP action of a version of Sullivan Chord diagrams which acts on the normalized Hochschild cochains of a Frobenius algebra. These actions lift to operadic correlation functions on the co–cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply–connected manifold. In this first part, we set up the topological operads/PROPs and their cell models. The main theorems of this part are that there is a cell model operad for the moduli space of genus g curves with n punctures and a tangent vector at each of these punctures and that there exists a CW complex whose chains are isomorphic to a certain type of Sullivan Chord diagrams and that they form a PROP. Furthermore there exist weak versions of these structures on the topological level which all lie inside an all encompassing cyclic (rational) operad.
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 ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
(Show Context)
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.
The topological cyclic Deligne conjecture
 Algebr. Geom. Topol
"... Abstract. Let O be a cyclic topological operad with multiplication. In the framework of the cosimplicial machinery by McClure and Smith, we prove that the totalization of the cosimplicial space associated to O has an action of an operad equivalent to the framed little 2discs operad. 1. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Let O be a cyclic topological operad with multiplication. In the framework of the cosimplicial machinery by McClure and Smith, we prove that the totalization of the cosimplicial space associated to O has an action of an operad equivalent to the framed little 2discs operad. 1.
Multitensor lifting and strictly unital higher category theory
"... Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories and the Crans tensor product of Gray categories as part of this framework. We define weak ncategories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak ncategories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)categories with strict units. 1.
Homotopy coherent centers versus centers of homotopy categories. http://arxiv.org/abs/1305.3029 William G. Dwyer Department of Mathematics University of Notre Dame Notre Dame
 IN 46556 USA dwyer.1@nd.edu Markus Szymik Department of Mathematical Sciences NTNU Norwegian University of Science and Technology 7491 Trondheim NORWAY markus.szymik@math.ntnu.no
"... Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as Hochschild cohomology type invariants in contexts that are no ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as Hochschild cohomology type invariants in contexts that are not necessarily linear or stable, and we argue that they are more appropriate to higher categorical contexts than the centers of their homotopy or derived categories. Among many other things, we present an obstruction theory for realizing elements in the centers of homotopy categories, and a BousfieldKan type spectral sequence that computes the homotopy groups. Nontrivial classes of examples are given as illustration throughout. MSC: primary 18G50, 55U40, secondary 16E40, 18G40, 55S35
Dimension vs. genus: a surface realization of the little kcubes and an E∞ operad
 in Algebraic Topology – Old and New (M.M. Postnikov Memorial Conference), Banach Center Publ., Vol. 85, Polish Acad. Sci
"... Abstract. We define a new E ∞ operad StLGT ree based on surfaces with foliations which contains Ek sub–operads. We construct CW models for these operads and provide applications of these models by giving actions on Hochschild complexes –thus making contact with string topology–, by giving explicit c ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We define a new E ∞ operad StLGT ree based on surfaces with foliations which contains Ek sub–operads. We construct CW models for these operads and provide applications of these models by giving actions on Hochschild complexes –thus making contact with string topology–, by giving explicit cell representatives for the DyerLashofCohen operations for the 2cubes and by constructing new Ω spectra. The underlying novel principle is that we can trade genus in the surface representation vs. the dimension k of the little k–cubes.
Operads and Sheaf Cohomology
, 2004
"... I explain how to construct E∞ cochain algebras for computing classical sheaf cohomology and, in principle, hypercohomology, and I explain how not to construct E∞ cochain algebras for computing motivic cohomology. ..."
Abstract
 Add to MetaCart
I explain how to construct E∞ cochain algebras for computing classical sheaf cohomology and, in principle, hypercohomology, and I explain how not to construct E∞ cochain algebras for computing motivic cohomology.