The paper is devoted to the investigation of effective descent morphisms in categories of (co)algebras.

It is the aim of this work to study product structures on four dimensional solvable Lie algebras. We determine all possible paracomplex structures and consider the case when one of the subalgebras is an ideal. These results are applied to the case of Manin triples and complex product structures. We also analyze the three dimensional subalgebras.

Let $R$ be any associative ring with unit and let $HR$ denote thecorresponding Eilenberg--Mac Lane spectrum. We show that thecategory of algebras over the monad $X\mapsto HR\wedge X$ on thehomotopy category of spectra is equivalent to the homotopycategory associated to a model category of $HR$-module spectra, ifthe ring $R$ is a field or a subring of the rationals, but not forall rings.

The branching (resp. merging) space functor of a flow is a left Quillen functor. The associated derived functor allows to define the branching (resp. merging) homology of a flow. It is then proved that this homology theory is a dihomotopy invariant and that higher dimensional branchings (resp. mergings) satisfy a long exact sequence.

We give a new simple proof of the exactness of the complex of injective words and use it to prove Nakaoka's homology stability for symmetric groups. The methods are generalized to show acyclicity in low degrees for the complex of words in "general position".

It is proved that the homology and cohomology theories of groups and associative algebras are non-abelian derived functors of the cokernel and kernel groups of higher dimensions of their defining standard chain and cochain complexes respectively. The same results are also obtained for the relative (co)homology of groups, the mod q cohomology of groups and the cohomology of groups with operators. This allowed us to give an alternative approach to higher Hopf formulas for integral homology of groups. An axiomatic characterization of the relative cohomology of groups is given and higher relative $(n+1)$-th cohomology of groups is described in terms of n-fold extensions.

For a given category $\mathsf{C}$ and a topological space $X$, the constant stack on $X$ with stalk $\mathsf{C}$ is the stack of locally constant sheaves with values in $\mathsf{C}$. Its global objects are classified by their monodromy, a functor from the fundamental groupoid $\Pi_1(X)$ to $\mathsf{C}$. In this paper we recall these notions from the point of view of higher category theory and then define the 2-monodromy of a locally constant stack with values in a 2-category $\mathsf{C}$ as a 2-functor from the homotopy 2-groupoid $\Pi_2(X)$ to $\mathsf{C}$. We show that 2-monodromy classifies locally constant stacks on a reasonably well-behaved space $X$. As an application, we show how to recover from this classification the cohomological version of a classical theorem of Hopf, and we extend it to the non abelian case.

A rack is a set equipped with a bijective, self-right-distributive binary operation, and a quandle is a rack which satisfies anidempotency condition.

In this paper, we introduce a new definition of modules over a rack or quandle, and show that this definition includes the one studied by Etingof and Graña [9] and the more general one given by Andruskiewitsch and Graña [1]. We further show that this definition coincides with the appropriate specialisation of the definition developed by Beck [3], and hence that these objects form a suitable category of coefficient objects in which to develop homology and cohomology theories for racks and quandles.

We then develop an Abelian extension theory for racks and quandles which contains the variants developed by Carter, Elhamdadi, Kamada and Saito [6, 7] as special cases.

An action of $A$ on $X$ is a map $F\colon A\times X \to X$ such that $F\vert_X = \id \colon X\to X$. The restriction $F\vert_A$: $A \to X$ of an action is called a cyclic map. Special cases of these notions include group actions and the Gottlieb groups of a space, each of which has been studied extensively. We prove some general results about actions and their Eckmann-Hilton duals. For instance, we classify the actions on an $H$-space that are compatible with the $H$-structure. As a corollary, we prove that if any two actions $F$ and $F'$ of $A$ on $X$ have cyclic maps $f$ and $f'$ with $\Omega f = \Omega f'$, then $\Omega F$ and $\Omega F'$ give the same action of $\Omega A$ on $\Omega X$. We introduce a new notion of the category of a map $g$ and prove that $g$ is cocyclic if and only if the category is less than or equal to $1$. From this we conclude that if $g$ is cocyclic, then the Berstein-Ganea category of $g$ is $\le 1$. We also briefly discuss the relationship between a map being cyclic and its cocategory being $\le 1$.

The classifying space of a small category classifies sheaves whose values are contravariant functors from that category to sets and whose stalks are representable.

Finite quandles with n elements can be represented as n x n matrices. We show how to use these matrices to distinguish all isomorphism classes of finite quandles for a given cardinality n, as well as how to compute the automorphism group of each finite quandle. As an application, we classify finite quandles with up to 5 elements and compute the automorphism group for each quandle.