Inessential directed maps and directed homotopy equivalences

A directed space is a topological space $X$ together with a subspace $\vec {P}(X)\subset X^I$ of directed paths on $X$ . A symmetry of a directed space should therefore respect both the topology of the underlying space and the topology of the associated spaces $\vec {P}(X)_-^+$ of directed paths between a source ( $-$ ) and a target ( $+$ )—up to homotopy. If it is, moreover, homotopic to the identity map—in a directed sense—such a symmetry will be called an inessential d-map, and the paper explores the algebra and topology of inessential d-maps. Comparing two d-spaces $X$ and $Y$ ‘up to symmetry’ yields the notion of a directed homotopy equivalence between them. Under appropriate conditions, all directed homotopy equivalences are shown to satisfy a 2-out-of-3 property. Our notion of directed homotopy equivalence does not agree completely with the one defined in Goubault (2017, arxiv:1709:05702v2) and Goubault, Farber and Sagnier (2020, J. Appl. Comput. Topol. 4, 11–27); the deviation is motivated by examples. Nevertheless, directed topological complexity, introduced in Goubault, Farber and Sagnier (2020) is shown to be invariant under our notion of directed homotopy equivalence. Finally, we show that directed homotopy equivalences result in isomorphisms on the pair component categories of directed spaces introduced in Goubault, Farber and Sagnier (2020).

JOINT PROJECTS – EFFECTIVE TOOLS FOR FACILITATING THE DEVELOPMENT OF GLOBAL EDUCATIONAL PROCESSES

Contemporary educational processes have to be in tune with the increasing global tendencies. All the world countries strive to unite under the umbrella term “global village”, which implies interconnected and almost unified political, economic, cultural, juridical and even educational spheres of life. Special grave challenges are faced by contemporary post-Soviet countries. They have to undergo the post-communist era, to struggle for the survival via radical correctional policy and to interweave globally oriented as well as capitalistically oriented strategies. The greatest challenge is posed to the educational sphere. Many post-Soviet higher educational institutions strive to join the European Higher Education Area. The given institutions try to interflow for the genuine overcoming of the existed “insurmountable” barriers. They consolidate collaboration and cooperation on different research and educational programs. Many newly-created joint projects appear and “play a key role in creating the rightful architecture of modern society. A central point in this process is the enquiry to identify… most urgent problems in national educational systems”(Pourtskhvanidze, 2016), to determine the strategies of functioning in multilingual society and to create a universal educational model of post-communist area via finding a balance between national and worldly. The given paper presents innovative university projects, which belong to two different worlds: the highly developed capitalistic world (Italian international project “Linguaggi e attività Produttive”) and post-Soviet “capitalistically-directed” space (Georgian-Ukrainian “DIMTEGU”). “Linguaggi e attività Produttive” and “DIMTEGU” have different nature and individual ways of development. However, both of them present innovative mechanisms of the preservation of multiculturalism and plurilingualism. The promotion of multiculturalism, facilitation of the preservation of plurilingualism, introduction of multilingual teacher education, development of appropriate curricula and teaching materials, suggestion of the fruitful ways of the adaptation to the global challenges – these are the major issues of our paper and crucial problems of today’s educational world.