Volume 9, Issue 4

We describe a duality theory for finite simplicial complexes that gives isomorphisms between the (reduced) homology of the complex and the (reduced) cohomology of the dual.

Two different definitions of a chain theory 𝐊∗ lead to the sameclass of derived homology theories ℎ∗( ). On the category of CW pairs, these are those homology theories 𝐄∗ admitting a classifying simplicial abelian group spectrum 𝐄. So one has a functor from the category of chain theories into the category of simplicial abelian group spectra.

The category of crossed complexes gives an algebraic model of CW -complexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating non-abelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions for amalgamated sums and HNN-extensions of groups, and so obtain computations of higher homotopical syzygies in these cases.

We develop a general approach to adjunctions satisfying the admissibility condition useful for Boolean Galois Theories, i. e. for Galois Theories whose Galois (pre)groupoids are profinite. Various examples and applications are briefly described.

Some properties on stem extensions and stem covers of Leibniz algebras are studied by means of exact sequences associated to a central extension of Leibniz algebras. We obtain particular results in case of perfect Leibniz algebras.

The notion of central series for groups with action on itself is introduced. An analogue of Witt's construction is given for such groups. A certain condition is found for the action and the corresponding category is defined. It is proved that the above construction defines a functor from this category to the category of Lie–Leibniz algebras and in particular to Leibniz algebras; also the restriction of this functor on the category of groups leads us to Lie algebras and gives the result of Witt.

This note gives a categorical development arising from a theorem of A. A. Klyachko relating the Lie operad to roots of unity. We examine the “substitude” structure on the groupoid 𝐂 whose homsets are the cyclic groups. The roots of unity representations of the cyclic groups form a Lie algebra for a certain oplax monoidal structure on the category of linear representations of 𝐂.

The K -theory of rings can be defined in terms of nonabelian derived functors as described in [Nonabelian derived functors and algebraic K-theory. Springer, 1973]; see also the books [Algebraic K-theory. Kluwer Academic Publishers Group, 1995] and [Non-abelian homological algebra and its applications. Kluwer Academic Publishers, 1997] of Inassaridze for a similar approach. In fact both Swan's theory and Quillen's theory can be described this way. The equivalence of both K -theories is proved by Gersten [Comm. Algebra 1: 39–64, 1974]. In this paper we give a proof using these descriptions that involve nonabelian derived functors.

We use the theory of nonabelian derived functors to prove that certain Baer invariants of a group 𝐺 are torsion when 𝐺 has torsion second integral homology. We use this result to show that if such a group has torsion-free abelianisation then the Lie algebra formed from the quotients of the lower central series of 𝐺 is isomorphic to the free Lie algebra on 𝐺 𝑎𝑏 . We end the paper with some related remarks about precrossed modules and partial Lie algebras.

Given a categorical crossed module ℍ → 𝐺, where 𝐺 is a group, we show that the category of derivations, Der (𝐺, ℍ), from 𝐺 into ℍ has a natural monoidal structure. We introduce the Whitehead categorical group of derivations as the Picard category of Der (𝐺, ℍ) and then we characterize the invertible derivations, with respect to the tensor product, in this monoidal category.

We propose a new point of view of the Spencer cohomology appearing in the formal theory of differential equations based on a dual approach via comodules. It allows us to relate the Spencer cohomology with standard constructions in homological algebra and, in particular, to express it as a Cotor. We discuss concrete methods for its construction based on homological perturbation theory.

The signature of the Poincaré duality of compact topological manifolds with local system of coefficients can be described as a natural invariant of nondegenerate symmetric quadratic forms defined on a category of infinite dimensional linear spaces. The objects of this category are linear spaces of the form W = V ⊕ V * where V is abstarct linear space with countable base. The space W is considered with minimal natural topology. The symmetric quadratic form on the space W is generated by the Poincaré duality homomorphism on the abstract cochain group induced by nerves of the system of atlases of charts on the topological manifold.

This paper is a detailed version of the note with the same title ([Piriou and Schwartz, Note to CRAS: 2002]). It treats a result related to what is commonly referred to as the artinian conjecture (or finiteness conjecture). This conjecture can be stated in the following way. Consider the category ℱ of functors from the category of finite dimensional vector spaces over the two element field to that of all vector spaces. Consider its full subcategory of functors whose injective envelopes are finite direct sums of indecomposable injectives. The conjecture is that this subcategory is abelian. In our circumstances the only point to prove is that it is stable under quotients (that this formulation is equivalent to the usual one is easy but not formal). The result proved in this paper shows that the subobject lattices of standard injectives of the category are “as simple as possible” in what concerns the weight filtrations of unstable modules. It is shown that the filtrations by weights and socles are compatible in an appropriate sense. In addition to the recalled notions and facts, the appendix contains a result showing that certain unstable modules are cyclic.

The notion of Quillen factorization system is obtained by strengthening the left and right lifting properties in a Quillen model category to the unique diagonalization property. An equivalent description of this notion is given in terms of a double factorization system which decomposes each arrow uniquely into three factors. The free category with Quillen factorization system over a given category is described.