Related papers: Normalisation for the fundamental crossed complex …

We proved the so called complex bounds for multimodal, infinitely renormalizable analytic maps with bounded combinatorics: deep renormalizations have polynomial-like extensions with definite modulus. The complex bounds is the first step to…

Given two finite abstract simplicial complexes A and B, one can define a new simplicial complex on the set of simplicial maps from A to B. After adding two technicalities, we call this complex Homsc(A, B). We prove the following dichotomy:…

Complex systems consist of interacting units whose interactions may be pairwise, involving two units, or higher-order, involving more than two units simultaneously. Graphs capture pairwise interactions and represent such systems as…

We introduce the notion of Whitehead sequence which is defined for a base category together with a system of abstract actions over it. In the classical case of groups and group actions the Whitehead sequences are precisely the…

We introduce the notion of 3-crossed module, which extends the notions of 1-crossed module (Whitehead) and 2-crossed module (Conduch\'e). We show that the category of 3-crossed modules is equivalent to the category of simplicial groups…

Prolongations of a group extension can be studied in a more general situation that we call group extensions of the co-type of a crossed module. Cohomology classification of such extensions is obtained by applying the obstruction theory of…

We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…

The main properties of the crossed product in the category of Hopf algebras are investigated. Let $A$ and $H$ be two Hopf algebras connected by two morphism of coalgebras $\triangleright : H\ot A \to A$, $f:H\ot H\to A$. The crossed product…

This article investigates the homotopy theory of simplicial commutative algebras with a view to homological applications.

A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…

A Rado simplicial complex X is a generalisation of the well-known Rado graph. X is a countable simplicial complex which contains any countable simplicial complex as its induced subcomplex. The Rado simplicial complex is highly symmetric, it…

We study the existence and left properness of transferred model structures for "monoid-like" objects in monoidal model categories. These include genuine monoids, but also all kinds of operads as for instance symmetric, cyclic, modular,…

The purpose of this note is to point out that simplicial methods and the well-known Dold-Kan construction in simplicial homotopy theory can be fruitfully applied to convert link homology theories into homotopy theories. Dold and Kan prove…

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong…

The "simplicial complexes" and "join" (*) today used within combinatorics aren't the classical concepts, cf. Spanier (1966) p. 108-9, but, exept for \emptyset, complexes having {\emptyset} as a subcomplex resp. \Sigma1 * \Sigma2 := {\sigma1…

Let $M$ be a closed manifold and $\mathcal{A} \subseteq H^1_{\mathrm{dR}}(M)$ a polytope. For each $a \in \mathcal{A}$ we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope $\mathcal{A}$. The…

We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is…

We construct Morse homology groups associated with any regular function on a smooth complex algebraic variety, allowing singular and non-compact critical loci. These groups are generated by critical points of a certain large pertubation of…

For a system of non-homogeneous polynomials it was constructed explicit complex morphism of a dual complex to the Koszul complex into the Koszul complex. If the ideal of these polynomials is 0-dimensional, then this mapping is a homotopic…

In this article, we give a characterisation of crossed homomorphisms on Lie superalgebras as a Maurer-Cartan element of a graded Lie algebra. Using this characterisation we study cohomology of these crossed homomorphisms. As an application…