Directed Algebraic Topology is a recent field, deeply linked with ordinary and higher dimensional Category Theory. A ‘directed space’, e.g. an ordered topological space, has directed homotopies (which are generally non reversible) and a fundamental category (replacing the fundamental groupoid of the classical case). Finding a simple possibly finite model of the latter is a non-trivial problem, ...

In this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  We relate the cohomology of a normal subgroup $N$ of a topological group $G$ and the quotient $G/N$ to the cohomology of $G$. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-a...

Motivated by the Category Embedding Theorem, as applied to convergent automorphisms [BOst11], we unify and extend the multivariate regular variation literature by a reformulation in the language of topological dynamics. Here the natural setting are metric groups, seen as normed groups (mimicking normed vector spaces). We brie‡y study their properties as a preliminary to establishing that the Un...

In this paper, we show that the category of Mackey-complete, separated, topological convex bornological vector spaces and bounded linear maps is a differential category. Such spaces were introduced by Frölicher and Kriegl, where they were called convenient vector spaces. While much of the structure necessary to demonstrate this observation is already contained in Frölicher and Kriegl’s book, we...

Let $A$ be an abelian topological group and $B$ a trivial topological $A$-module. In this paper we define the second bilinear cohomology with a trivial coefficient. We show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. Also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...

Abstract We introduce the concept of TRAP (Traces and Permutations), which can roughly be viewed as a wheeled PROP (Products Permutations) without unit. TRAPs are equipped with horizontal concatenation partial trace maps. Continuous morphisms on an infinite-dimensional topological space smooth kernels (respectively, smoothing operators) closed manifold form but not PROP. build free objects in c...

Motivated by the recent study on categorical properties of latticevalued topology, the paper considers a generalization of the notion of topological system introduced by S. Vickers, providing an algebraic and a coalgebraic category of the new structures. As a result, the nature of the category   TopSys   of S. Vickers gets clari ed, and a metatheorem is stated, claiming that (latticevalu...

We introduce a new notion of covering projection E → X of a topological spaceX which reduces to the usual notion ifX is locally connected. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid π crs(X) and an induced category pro(π crs(X), Sets) such that for any topological space X the category of covering projections and transformations of X is equivalent to th...

‎The Mod $2$ Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations‎, ‎denoted by $Sq^n$‎, ‎between the cohomology groups with $mathbb{Z}_2$ coefficients of any topological space‎. ‎Regarding to its vector space structure over $mathbb{Z}_2$‎, ‎it has many base systems and some of the base systems can also be restricted to its sub algebras‎. ‎On the contrary‎, ‎in ...

In this paper, by using C-sequentially sign property for bifunctions, we provide sufficient conditions that ensure the existence of solutions of some vector equilibrium problems in Hausdorff topological vector spaces which ordered by a cone. The conditions which we consider are not imposed on the whole domain of the operators involved, but just on a locally segment-dense subset of the domain.