Triple Brackets and Lax Morphism Categories

Various aspects of the traditional homotopy theory of topological spaces may be developed in an arbitrary 2-category C with zeros. In particular certain secondary composition operations called box brackets recently have been defined for C; these are similar to, but extend, the familiar Toda brackets in the topological case. In this paper we introduce further the notion of a suspension functor in C and explore the ramifications of relativizing the theory in terms of the associated lax morphism category of C, denoted mC. Four operations associated to a 3-box diagram are introduced and relations among them are clarified. The results and insights obtained, while by nature somewhat technical, yield effective and efficient techniques for computing many operations of Toda bracket type. We illustrate by recording some computations from the homotopy groups of spheres. Also the properties of a new operation, the 2-sided matrix Toda bracket, are explored.     

TODA BRACKETS AND THE CATEGORY OF HOMOTOPY PAIRS

A new treatment is given of the cylinder-web diagram and associated diagonal sequences in homotopy pair theory. The efficiency of the diagram as a machine for computing homotopy pair groups is enhanced by a result that traces the path of a Toda bracket element through the arrows of the diagram. The diagonal factorization problem for a homotopy pair class is studied and related to the behaviour of Toda brackets. A necessary and sufficient condition for the vanishing of a Toda bracket is obtained.     

Higher homotopy operations

We provide a general definition of higher homotopy operations, encompassing most known cases, including higher Massey and Whitehead products, and long Toda brackets. These operations are defined in terms of the W-construction of Boardman and Vogt, applied to the appropriate diagram category; we also show how some classical families of polyhedra (including simplices, cubes, associahedra, and permutahedra) arise in this way. Second author partially supported by grant GA AV R 1019203.     

A Coherent Homotopy Category of 2-track Commutative Cubes

We consider a category \({\mathcal H}^{\ominus \otimes}\) (the homotopy category of homotopy squares) whose objects are homotopy commutative squares of spaces and whose morphisms are cubical diagrams subject to a coherent homotopy relation. The main result characterises the isomorphisms of \({\mathcal H}^{\ominus \otimes}\) to be the cube morphisms whose forward arrows are homotopy equivalences. As a first application of the new category we give a direct 2-track theoretic definition of the quaternary Toda bracket operation.     

ON CUP-ONE AND TODA BRACKET

Abstract

The definition of the cup-one construction in various cases is recalled. A variation expressible in terms of the Toda bracket is given, enabling nontriviality of Toda brackets of a certain type to be established in a rather simple way from knowledge of the behaviour of cup-one.     

Derived brackets and sh Leibniz algebras

We develop a general framework for the construction of various derived brackets. We show that suitably deforming the differential of a graded Leibniz algebra extends the derived bracket construction and leads to the notion of strong homotopy (sh) Leibniz algebra. We discuss the connections among homotopy algebra theory, deformation theory and derived brackets. We prove that the derived bracket construction induces a map from suitably defined deformation theory equivalence classes to the isomorphism classes of sh Leibniz algebras.     

Toda Brackets and Cup-One Squares for Ring Spectra

In this article we prove the laws of Toda brackets on the homotopy groups of a connective ring spectrum and the laws of the cup-one square in the homotopy groups of a commutative connective ring spectrum.     

Cores of Simplicial Complexes

We introduce a method to reduce the study of the topology of a simplicial complex to that of a simpler one. Applying this method to complexes arising from graphs, we give topological meaning to classical graph invariants. As a consequence, we answer some questions raised in (Ehrenborg and Hetyei in Eur. J. Comb. 27(6):906–923, 2006) on the independence complex and the dominance complex of a forest and obtain improved algorithms to compute their homotopy types.     

Poisson pencils, algebraic integrability, and separation of variables

In this paper we review a recently introduced method for solving the Hamilton-Jacobi equations by the method of Separation of Variables. This method is based on the notion of pencil of Poisson brackets and on the bihamiltonian approach to integrable systems. We discuss how separability conditions can be intrinsically characterized within such a geometrical set-up, the definition of the separation coordinates being encompassed in the bihamiltonian structure itself. We finally discuss these constructions studying in details a particular example, based on a generalization of the classical Toda Lattice.     

N=(1|1) Supersymmetric Dispersionless Toda Lattice Hierarchy

Generalizing the graded commutator in superalgebras, we propose a new bracket operation on the space of graded operators with an involution. We study properties of this operation and show that the Lax representation of the two-dimensional N=(1|1) supersymmetric Toda lattice hierarchy can be realized via the generalized bracket operation; this is important in constructing the semiclassical (continuum) limit of this hierarchy. We construct the continuum limit of the N=(1|1) Toda lattice hierarchy, the dispersionless N=(1|1) Toda hierarchy. In this limit, we obtain the Lax representation, with the generalized graded bracket becoming the corresponding Poisson bracket on the graded phase superspace. We find bosonic symmetries of the dispersionless N=(1|1) supersymmetric Toda equation.     

On the Cohomology of Categories, Universal Toda Brackets and Homotopy Pairs

A category of homotopy pairs is characterised by a cohomology class which generalizes the notion of Toda bracket. Explicit computations of such cohomology classes are described.     

Smash Products for Secondary Homotopy Groups

We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered. The second author was partially supported by the Spanish Ministry of Education and Science under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research contract.     

The Poisson bracket compatible with the classical reflection equation algebra

We introduce a family of compatible Poisson brackets on the space of 2 × 2 polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the XXX Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top.      

Multi‐Hamiltonian structure for the finite defocusing Ablowitz‐Ladik equation

We study the Poisson structure associated to the defocusing Ablowitz‐Ladik equation from a functional‐analytical point of view by reexpressing the Poisson bracket in terms of the associated Carathéodory function. Using this expression, we are able to introduce a family of compatible Poisson brackets that form a multi‐Hamiltonian structure for the Ablowitz‐Ladik equation. Furthermore, we show using some of these new Poisson brackets that the Geronimus relations between orthogonal polynomials on the unit circle and those on the interval define an algebraic and symplectic mapping between the Ablowitz‐Ladik and Toda hierarchies. © 2008 Wiley Periodicals, Inc.     

Homotopical algebra in homotopical categories

We develop here a version of abstract homotopical algebra based onhomotopy kernels andcokernels, which are particular homotopy limits and colimits. These notions are introduced in anh-category, a sort of two-dimensional context more general than a 2-category, abstracting thenearly 2-categorical properties of topological spaces, continuous maps and homotopies. A setting which applies also, at different extents, to cubical or simplicial sets, chain complexes, chain algebras, ... and in which homotopical algebra can be established as a two-dimensional enrichment of homological algebra.Actually, a hierarchy of notions ofh-,h1-, ...h4-categories is introduced, through progressive enrichment of thevertical structure of homotopies, so that the strongest notion,h4-category, is a sort of relaxed 2-category. After investigating homotopy pullbacks and homotopical diagrammatical lemmas in these settings, we introduceright semihomotopical categories, ash-categories provided with terminal object and homotopy cokernels (mapping cones), andright homotopical categories, provided also with anh4-structure and verifying second-order regularity properties forh-cokernels.In these frames we study the Puppe sequence of a map, its comparison with the sequence of iterated homotopy cokernels and theh-cogroup structure of the suspension endofunctor. Left (semi-) homotopical categories, based on homotopy kernels, give the fibration sequence of a map and theh-group of loops. Finally, the self-dual notion of homotopical categories is considered, together with their stability properties.Lavoro esequito nell'ambito dei progetti di ricerca del MURST.     

Fibrations of topological stacks

In this note we define fibrations of topological stacks and establish their main properties. When restricted to topological spaces, our notion of fibration coincides with the classical one. We prove various standard results about fibrations (long exact sequence for homotopy groups, Leray–Serre and Eilenberg–Moore spectral sequences, etc.). We prove various criteria for a morphism of topological stacks to be a fibration, and use these to produce examples of fibrations. We prove that every morphism of topological stacks factors through a fibration and construct the homotopy fiber of a morphism of topological stacks. As an immediate consequence of the machinery we develop, we also prove van Kampen?s theorem for fundamental groups of topological stacks.     

CATEGORICAL HOMOTOPY

We present a homotopy theory of small categories. In a work of this nature there is a need to give a theory which is clear and which shows the methods of work in this field. It is also necessary to prove theorems which place the theory within the general framework of homotopy, i.e. particularly to liaise with the homotopy of topological spaces and with abstract homotopy theories. Firstly we define the important notion of finite functor on which the theory is based. Next we introduce a type of fibred category fitting to the work on homotopy. After having studied the paths and loops of a category, we consider homotopy between functors. Finally, we demonstrate the possibility of obtaining homotopy groups before taking into consideration the relations between categorical and topological homotopy.     

On the functoriality of cohomology of categories

In this paper we show that the Baues-Wirsching complex used to define cohomology of categories is a 2-functor from a certain 2-category of natural systems of abelian groups to the 2-category of chain complexes, chain homomorphisms and relative homotopy classes of chain homotopies. As a consequence we derive (co)localization theorems for this cohomology.