Fajstrup:Directed Algebraic Topology an (eng)

This monograph presents an application of concepts and methods from algebraic topology to models of concurrent processes in computer science and their analysis.

Taking well-known discrete models for concurrent processes in resource management as a point of departure, the book goes on to refine combinatorial and topological models. In the process, it develops tools and invariants for the new discipline directed algebraic topology, which is driven by fundamental research interests as well as by applications, primarily in the static analysis of concurrent programs.

The state space of a concurrent program is described as a higher-dimensional space, the topology of which encodes the essential properties of the system. In order to analyse all possible executions in the state space, more than “just” the topological properties have to be considered: Execution paths need to respect a partial order given by the time flow. As a result, tools and concepts from topologyhave to be extended to take privileged directions into account.

The target audience for this book consists of graduate students, researchers and practitioners in the field, mathematicians and computer scientists alike.

“The book under review introduces a more recently developed true-concurrency model, which applies concepts and methods from algebraic topology. ... I applaud the authors for having delivered what I believe is the first book in a very promising field. I highly recommend this book to mathematicians specializing in topology and seeking a challenging new application, and to those concurrency theorists who have a firm background in category theory.” (Gerald Lüttgen, Mathematical Reviews, February, 2017)

“The book contains a large number of illustrative examples along with many diagrams that serve to illustrate the ideas and make the book quite easy to follow. Indeed the diagrams are an essential guide to the geometrical ideas on which the book is based. ... The book can be recommended to both mathematicians and computer scientists as presenting an introduction to an area where further collaboration could be fruitful.” (J. P. E. Hodgson, Computing Reviews, computingreviews.com, July, 2016)