Chain Functors Research Articles

Categorical, Homological, and Homotopical Properties of Algebraic Objects

This monograph is based on the doctoral dissertation of the author defended in the Iv. Javakhishvili Tbilisi State University in 2006. It begins by developing internal category and internal category cohomology theories (equivalently, for crossed modules) in categories of groups with operations. Further, the author presents properties of actions in categories of interest, in particular, the existence of an actor in specific algebraic categories. Moreover, the reader will be introduced to a new type of algebras called noncommutative Leibniz–Poisson algebras, with their properties and cohomology theory and the relationship of new cohomologies with well-known cohomologies of underlying associative and Leibniz algebras. The author defines and studies the category of groups with an action on itself and solves two problems of J.-L. Loday. Homotopical and categorical properties of chain functors category are also examined.

The differentiable chain functor is not homotopy equivalent to the continuous chain functor

Let S ∗ and S ∗ ∞ be the functors of continuous and differentiable singular chains on the category of differentiable manifolds. We prove that the natural transformation i : S ∗ ∞ → S ∗ , which induces homology equivalences over each manifold, is not a natural homotopy equivalence.

Simplicial model category structures on the category of chain functors

The model structure on the category of chain functors Ch, developed in [4], has the main features of a simplicial model category structure, taking into account the lack of arbitrary (co-)limits in Ch. After an appropriate tensor and cotensor structure in Ch is established (§1, §3), Quillen’s axiom SM7 is verified in §5 and §6. Moreover, it turns out that in the definition of a simplicial model structure, the category of simplicial sets can be replaced by the category of simplicial spectra endowing Ch with the structure of an approximate simplicial stable model structure (= approximate ss-model structure) (§7). In §8 the model structure on Ch is shown to be proper.

THE EXISTENCE OF CERTAIN (CO-)LIMITS IN THE CATEGORY OF CHAIN FUNCTORS

Chain functors A*, B* have been introduced for calculating generalized homology theories by using chains and cycles, as one is doing for ordinary homology theories by means of chain complexes. Like chain complexes these chain functors form a category ℭh displaying interesting properties by themselves. The present paper emphasizes more the algebraic aspescts of this category ℭh. Although not every morphism f ∈ ℭh(A*, B*) in the category of chain functors admits a kernel or a cokernel, it turns out that: (1) all cofibrations have a cokernel, (2) all regular fibrations have a kernel, (3) the pushout of a cofibration along a cofibration exists in ℭh, respectively the dual statement for fibrations, (4) there are interesting results about exact sequences involving (co-)fibrations.

Strong Homology Theories as Localizations

Abstract Let 𝔎 be a category of pairs of spaces, 𝔏 ⊂ 𝔎 the category of pairs of ANRs or CW-spaces, 𝐴∗ a chain functor (e.g., one associated with a spectrum). Then the derived homology 𝑠ℎ∗ of the 𝔏-localization of 𝐴∗ is the strong homology theory on 𝔎 which is up to an isomorphism uniquely determined by the fact that 𝑠ℎ∗ | 𝔏 agrees with the derived homology of 𝐴∗ on 𝔏. This establishes a relationship between localization theory and strong homology theory (e.g., Steenrod–Sitnikov homology theory, whenever all pairs are compact metric).

Closed model category structures on the category of chain functors

In the category Ch of chain functors one can introduce fibrations (Section 3), cofibrations and weak equivalences (Section 4), satisfying all the properties of a closed model category as defined by D. Quillen except for the existence of finite limits and colimits. Nevertheless we show that there exists a canonically defined suspension—as well as a loop functor, which are invertible, turning the homotopy category Ch h into a stable category (Section 8).

Chain functors with isomorphic homology

Every chain functor Kdetermines a homology theory on a given category of topological spaces resp. of spectra H�(K�)(·) cf. §4. If K�, Lare chain functors such that H�(K�)(·)� H�(L�)(·) then there exists a third chain functor Cand transformations of chain functors K : K� ! C�, L : L� ! Cinducing isomorphisms of the associated homology theories (theorem 1.1.). Moreover the distinction between regular and irregular chain functors is introduced.

Tensor products of spectra and localizations

1) The tensor product of two spectra, different from the $\wedge$-product, is introduced in such a way that a Kunneth theorem holds. 2) Localizations of spectra are treated by using the more algebraic category of chain functors (instead of the category of CW-spectra). 3) The localization of a given chain functor ${\bf K}_*$ can up to chain homotopy be expressed by tensoring ${\bf K}_*$ with the localization of a fixed chain functor $\boldsymbol{\mathbb Z}_*$.

The Boardman category of spectra, chain complexes and (co-)localizations

This is a survey of the author‘s results about chain functors, tensor products of chain functors, (co-)localizations of chain functors and their relationship with the associated concepts for spectra in the Boardman category. Some introductory remarks about quotient categories, stable homotopy theory and application of the theory to strong homology theories are included.

Bordism theories and chain complexes

Although every generalized homology theory h ∗ is related to a chain functor C ∗ , some homology theories (like e.g. bordism theories) admit a much simpler chain theoretical description by means of chain theories with models. Algebraically these models are playing the role of relative cycles of a specific chain functor, while geometrically they behave like singular manifolds in bordism theory. A necessary and sufficient condition is formulated (using chain functors) ensuring that a homology theory is associated with a chain theory with models.

Extensions of chain functors

Let K be a category of topological spaces and P ⊂ K a subcategory of good spaces; let moreover C K (resp. C p ) be the associated category of chain functors (cf. 1.3.) on K (resp. on P ) and the restriction functor. The main theorem (theorem 1.4.2.) asserts the existence of a kind of right adjoint with the following properties: 1) s C * is c-continuous (cf. 1.2.); 2) if determine isomorphic homology theories (on P ) , then so do (now on the larger category K ); 3) determine isomorphic homology theories on P. The main application of this result consists in an existence proof of a strong homology theory sh* on K agreeing with a prescribed sh* on P Suppose that C ϵ C p determines h * (on P ) then s h * will be the derived homology of s C * Due to properties 1) - 3) of s( ), this s h * is well-defined (in fact uniquely determined by h * and the axioms of a strong homology theory)c-continuous (constituting one of these axioms, which replaces Milnor’s clusteraxiom) and agrees, up to an isomorphism, with h * on ...

Classifying spectra for generalized homology theories

The well-known fact that each generalized homology theory h* on the category of CW spaces has a classifying spectrumE which is unique up to an isomorphism in the Boardman (homotopy) category is proved by using the fact that each such h* comes from a chain functor (cf. [1] or § 9). The proof does not use S-duality nor E. H. Brown's representation theorem.

REMARKS ON UNIVERSAL COEFFICIENT THEOREMS FOR GENERALIZED HOMOLOGY THEORIES

Abstract New proofs of universal coefficient theorems for generalized homology theories (cf. ∮ 2, ∮ 3) including L. G. Brown's result, relating Brown-Douglas-Fillmore's Ext (X) with complex K-theory are presented. They are all based on a theorem asserting the existence of a chain functor for a generalized homology theory (cf. ∮ 1), which was originally designed for the construction of strong homology theories on strong shape categories.

A Chain Functor For Bordism

A Chain Functor For Bordism

A chain functor for bordism

Chains of differential graded abelian monoids are defined for bordism and cobordism theories. These chains are used to define matric Massey products and can be filtered so as to define the Adams spectral sequence. From this point of view, we prove three basic theorems which show how Massey products behave in the Adams spectral sequence.