Lax additivity

Abstract. In this paper, we show that, under natural homological conditions, Hopf cyclichomology theory has excision. Mathematics Subject Classifications (2000): 16W30, 19D55, 57T05, 58B34. Key words: cyclic homology, Hopf algebras. 1. Introduction Cyclic cohomology theory for Hopf algebras was introduced by Connesand Moscovici in their ground breaking paper on transverse index theory[4] (cf. also [3, 5, 6] for a recent survey and new results). This theory can beregarded as the right noncommutative analogue of Lie algebra homologybecause of existence of a characteristic map for actions of Hopf algebrasand since it reduces to Lie algebra homology for Hopf algebras associ-ated to Lie algebras. This should be compared with the role played byordinary cyclic cohomology of noncommutative algebras and its reductionto de Rham homology for smooth commutative algebras [2]. The alge-braic underpinnings of Hopf cyclic theory however turned out to be morecomplicated. It is now understood as a special case of a general theorydeveloped in the series of papers [11, 12, 15, 16] (with the help of [1])for (co)algebras endowed with a (co)action of a Hopf algebra and where,among other things, the right notion of coefficients was introduced (cf. also[14] for a recent survey). This theory was later extended by the first author[13] to bialgebras and to a much larger class of coefficients. This extensionis rather surprising since existence of a bijective antipode was essential inconstructing the cyclic and cocyclic complexes developed for Hopf cyclichomology of both variants. By bialgebra or Hopf cyclic homology we nowmean the theory expounded in these papers mentioned above.The main results of the present paper are two excision theorems(Theorems 4.13 and 7.7) for Hopf cyclic homology of module coal-gebras and comodule algebras, respectively. A basic property of cyclic(co)homology for algebras, and coalgebras is excision. If by cyclic theory

The notions of cofibration and fibration are central to homotopy theory. We show that the defining property of a cofiber inclusion map i : A → X is equivalent to the homotopy extension property of the pair (X,A). Thus the inclusion map of a subcomplex into a CW complex is a cofiber map, and so this concept is widespread in topology. We study cofiber maps in Section 3.2 and introduce pushout squares and mapping cones. In Section 3.3 we treat fiber maps as well as pullback squares and homotopy fibers and obtain some of their basic properties. We also consider fiber bundles which are defined by maps p : E → B for which there is a locally trivializing cover of B. We prove that fiber bundles are fibrations and thus obtain examples of fiber maps by exhibiting fiber bundles. In this way we obtain many diverse fibrations in which spheres, topological groups, Stiefel manifolds and Grassmannians appear. This is done in Section 3.4, and these results will be used in Chapter 5 to calculate homotopy groups. In the last section we introduce the mapping cylinder and its dual and use them to show that any map can be factored into the composition of a cofiber map followed by a homotopy equivalence or into the composition of a homotopy equivalence followed by a fiber map. These are important techniques and highlight the major role of cofiber and fiber maps in homotopy theory.

In the present paper the notions of a -differential and a -differential module are introduced, which are, respectively, homotopically invariant analogues of the differential and the chain complex. Basic homotopic properties of -differentials and -differential modules are established. The connection between the Gugenheim-Lambe-Stasheff theory of differential perturbations in homological algebra and the construction of a -differential module is considered.

The purpose of these notes is to provide as rapid an introduction to category theory and homological algebra as possible without overwhelming the reader entirely unfamiliar with these subjects. We begin with the definition of a category, and end with the basic properties of derived functors, in particular, Tor and Ext. This was the spirit of the four lectures on which thenotes are based, although there is, needless to say, much more material contained herein an was touched on in the lectures. For example, we have included a fairly complete treatment of the basic facts pertaining to adjoint functors, including Freyd's adjoint functor theorems. Application of category theory in the direction of topos theory and logic were treated in the accompanying lectures of Tierney, and Buchsbaum in his lectures indicated some outlets for homological algebra in commutative algebra and local ring theory. We have therefore not felt compelled to emphasize any specific topic. We have, nevertheless, presented module theory as something associated with ringoids (small, additive categories) rather than with the more conventional and restrictive notion of a ring. This point of view has enabled us recently to incorporate several new examples into the traditional setting of homological algebra as found in the book of Cartan-Eilenberg [2]. One can consult [15] in this regard.

Let [Formula: see text] be a complete intersection local ring, [Formula: see text] be the Koszul complex on a minimal set of generators of [Formula: see text], and [Formula: see text] be its homology algebra. We establish exact sequences involving direct sums of the components of [Formula: see text] and express the images of the maps of these sequences as homologies of iterated mapping cones built on [Formula: see text]. As an application of this iterated mapping cone construction, we recover a minimal free resolution of the residue field [Formula: see text] over [Formula: see text], independent from the well-known resolution constructed by Tate by adjoining variables and killing cycles. Through our construction, the differential maps can be expressed explicitly as blocks of matrices, arranged in some combinatorial patterns.

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.

Chapter 13 - Population Games and Deterministic Evolutionary Dynamics

* Introduction* Knots and links in $S^3$* Grid diagrams* Grid homology* The invariance of grid homology* The unknotting number and $\tau$* Basic properties of grid homology* The slice genus and $\tau$* The oriented skein exact sequence* Grid homologies of alternating knots* Grid homology for links* Invariants of Legendrian and transverse knots* The filtered grid complex* More on the filtered chain complex* Grid homology over the integers* The holomorphic theory* Open problems* Homological algebra* Basic theorems in knot theory* Bibliography* Index

Resolutions which split off of the bar construction

Categorical versions of the mapping cone and mapping cylinder constructions from topology are given and their basic properties generalized.

Abstract“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual Rat(C*) in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra A that is isomorphic to its complete topological dual Aν. We show that A is a (quasi)Frobenius algebra if and only if A is the dual C* of a (quasi)co-Frobenius coalgebra C. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.

Chapter 29 Vectors, matrices and operations on matrices

These notes are an introduction to basic properties of Andre- Quillen homology for commutative algebras. They are an expanded version of my lectures at the summer school: Interactions between homotopy theory and algebra, University of Chicago, 26th July - 6th August, 2004. The aim is to give fairly complete proofs of characterizations of smooth homomorphisms and of locally complete intersection homomorphisms in terms of vanishing of Andre-Quillen homology. The choice of the material, and the point of view, are guided by these goals.

We study an integrable case ofn-particle Toda lattice: open chain with boundary terms containing 4 parameters. For this model we construct a Backlund transformation and prove its basic properties: canonicity, commutativity and spectrality. The Backlund transformation can be also viewed as a discretized time dynamics. Two Lax matrices are used: of order 2 and of order 2n + 2, which are mutually dual, sharing the same spectral curve.

A ring extension is a ring homomorphism that preserves identities. Motivated by the theory of Gorensteion homological algebra, we introduce the definitions of relative Gorensteion projective and relative Gorensteion injective modules over arbitrary ring extensions and study their basic properties. Let f : S → R be a ring extension. We prove that the category of all relative Gorenstein projective modules is a relative resolving subcategory of R ‐Mod , which is closed under direct sums and direct summands. Additionally, every R-module with finite relative Gorenstein projective dimension admits a relative Gorenstein projective precover. Furthermore, we provide a series of homological descriptions of relative Gorenstein projective dimensions. Dually, all the results concerning relative Gorenstein projective modules have a corresponding relative Gorenstein injective version. Additionally, we introduce the concept of quasi-Frobenius extensions, and demonstrate that quasi-Frobenius extensions encompass both semisimple extensions and Frobenius extensions.