Directed homotopy modules
Eric Goubault
TL;DR
This work proposes a foundation for directed homotopy by modeling traces and directed paths via absorption monoids and trace monoids, interpreting directed homotopy groups as modules over these coefficient monoids. It develops a robust categorical framework of modules over absorption monoids, including products, coproducts, quotients, and coefficient-change functors (restriction, extension, and co-extension) with adjunctions. The authors construct i-traces and Kan simplicial structures to define directed homotopy modules, establishing functoriality from directed spaces to module categories and highlighting a path toward a unified, higher-categorical treatment of directed (co)homology and homotopy. The concluding notes sketch future work on dihomotopy sequences and the relation to existing directed-homology formalisms, aiming to situate directed topology within framed bicategories and higher-category theory.
Abstract
In this short note, we argue that directed homotopy can be given the structure of generalized modules, over particular monoids. This is part of a general attempt for refoundation of directed topology.