Relative De Rham cohomology in diffeological spaces

Diffeological spaces extend the category of smooth manifolds and provide a flexible framework for studying geometric and topological structures on spaces that may fail to be manifolds in the classical sense. In recent years, De Rham cohomology has been successfully developed in this setting, revealing new connections between differential forms, smooth maps, and categorical constructions.
In this communication, we present recent developments concerning De Rham cohomology and its relative version for diffeological spaces. We discuss two approaches to relative De Rham cohomology associated with smooth maps and examine their fundamental properties. Particular attention is paid to the difficulties that arise from the absence of partitions of unity in general diffeological spaces, a feature that distinguishes this theory from its classical counterpart.
Our approach is based on categorical methods. Instead of using open coverings and their intersections, we work with generating families of plots together with pull-back and join constructions. These tools provide a natural framework for studying relative cohomological structures and allow classical ideas to be reformulated in the diffeological context.
Within this setting, we obtain a new viewpoint on Mayer–Vietoris type sequences and analyze the conditions under which long exact sequences in cohomology can be constructed. We also discuss the existence of relative cup products and show how these constructions extend familiar cohomological operations beyond the category of smooth manifolds.
The ideas presented in this communication are based on recent joint work concerning relative De Rham cohomology and related cohomological constructions in diffeological spaces.