Directed homotopy theory, I. The fundamental category

TL;DR

This paper introduces the fundamental category in directed homotopy theory, establishing a Seifert-van Kampen theorem and exploring algebraic models for directed spaces, with potential applications in modeling space-time and concurrent processes.

Contribution

It defines the fundamental category for directed spaces and proves a Seifert-van Kampen theorem, advancing the algebraic and topological framework of directed homotopy theory.

Findings

Fundamental category provides a new algebraic invariant for directed spaces.

Seifert-van Kampen theorem established for the fundamental category.

New algebraic shapes for small categories and d-spaces introduced.

Abstract

Directed Algebraic Topology is beginning to emerge from various applications. The basic structure we shall use for such a theory, a 'd-space', is a topological space equipped with a family of 'directed paths', closed under some operations. This allows for 'directed homotopies', generally non reversible, represented by a cylinder and cocylinder functors. The existence of 'pastings' (colimits) yields a geometric realisation of cubical sets as d-spaces, together with homotopy constructs which will be developed in a sequel. Here, the 'fundamental category' of a d-space is introduced and a 'Seifert - van Kampen' theorem proved; its homotopy invariance rests on 'directed homotopy' of categories. In the process, new shapes appear, for d-spaces but also for small categories, their elementary algebraic model. Applications of such tools are briefly considered or suggested, for objects which model a directed image, or a portion of space-time, or a concurrent process.