Directed Algebraic Topology: Models of Non-Reversible Worlds

The field of directed algebraic topology is still quite new. It emerged in the 1990s, as homotopy theory proved not quite flexible enough to meet the demands of applications. Some real-world situations are more akin to one-way roads: there is no turning back. And thus, directed algebraic topology takes aim at understanding these non-reversible situations by developing non-reversible analogues of the ideas of homotopy theory.

This may sound simple, and in a way it is. On the other hand, the details are, well, even more detailed than the first time one studies homotopy. There is a very categorical bent to the exposition here (be ready to get your hands dirty constructing all sorts of limits, colimits and homotopies in this new setting), and unfortunately, not just one or two categories of interest are developed. The category dTop of directed topological spaces is the main working category. It’s a so-called dIP4-homotopical category, but there are also dI1, dP1 and dIP1 categories, and dI2… categories along the way. The category of cubical sets also seems to create strong interest, having some advantages over dTop, while being only a dIP1-homotopical category.

This book is divided into two parts, followed by a brief appendix on category theory. Part I, “First-order directed homotopy and homology,” introduces the setting. As the reader is told (section 1.1.5), there are several types of intrinsic symmetry in our usual category of topological spaces, some of which this subject is focused on breaking and some of which are only sometimes broken here. Part II, “Higher directed homotopy theory,” leads into the world of chain complexes and differential graded algebras, etc., ending with a discussion of weighted algebraic topology which is connected to the idea that different paths have different associated costs.

Grandis has been a pioneer in this field, and here he has collected the foundations of the subject. He does provide some words of motivation for its study (by which I mean a few paragraphs). This reader, though, would have liked some applications actually included here, as well as more references to current applied work. And I do consider myself a typical reader for this text: an algebraic topologist with an interest in parlaying my knowledge into newer currency.

Michele Intermont is an associate professor of mathematics at Kalamazoo College in Kalamazoo, MI.