Those of us who analyze and design systems that have worked, or are planned to work, in the context of these complex phenomena are well aware of serious problems in understanding and therefore dealing with these systems:

The distinction between a prediction of the appearance of a pattern of a certain class and a prediction of the appearance of a particular instance of this class ... assumes ... much greater importance when we turn from the relatively simple phenomena with which the natural sciences deal, to the more complex phenomena of life, of mind, and of society [where predictions of the latter] may not always be possible [1].

Patterns of reasoning in systems thinking are about the structure of various systems, and it is clear that while the components of such systems differ (often substantially), the kinds of structural patterns--the relationships--remain the same. The composition relationship, and, in particular, emergent properties of the composite, are the essence of understanding and explaining complex systems. Rigorous definitions of composition and of emergence have been provided in the literature. Many examples based on these definitions have been presented.

There appeared, however, to be some uneasiness in working with these relationship patterns. While the gut feeling of analysts, designers, and, perhaps more importantly, business stakeholders was that the approach was the right one for understanding and explanation, as well as for creating new artifacts, a respectable mathematical foundation is desired. This solid foundation for our gut feeling will permit better elucidation of the underlying system of concepts and constructs, and, therefore, its well-justified reuse in various contexts.

Category theory (CT) is such a mathematical foundation. Unfortunately, most CT texts are not accessible to uninitiated readers, and even those that are accessible use examples from areas that are far from the interests of system analysts and designers--this book is a most welcome exception. It provides a pedagogically excellent introduction to the structure of complex systems based on CT, explained in a manner accessible to a nonmathematician. A wealth of interesting examples from biological and social systems certainly helps in understanding, providing readers with a “feeling” for the relevant CT concepts. The authors’ emphasis is on biological systems (although business enterprises are also considered).

Vanbremeersch is a physician specializing in gerontology, while Ehresmann is a category theorist. Their cooperation has produced an extraordinary result: the book is readable and accessible to an audience having mathematical (and systems thinking) maturity, but not necessarily having any knowledge of CT. The CT concepts are introduced as needed to explain the structure of complex systems. Many figures help explain the narrative. Readers learn how CT “makes a general concept of structure possible,” and how

the evolution of living systems [including social systems] rests on a small number of prototypical operations, which are exactly those that CT can model: the formation, dissolution, comparison, and combination of relations between objects.

CT emphasizes relations and collective behavior, and the authors consistently demonstrate how these concepts are used to understand and explain complex systems, from composition and the formation of hierarchies of objects (explaining both algorithmic and observational, that is, nonalgorithmic, emergence), through system evolution (explaining how complex objects maintain their identity, how memory is organized and evolved, how several heterogeneous coregulators collectively control system dynamics, and how invariants and concepts form the dynamic semantic memory), and culminating in applications to cognition and the consciousness of living beings (including the notion of self).

The basic CT concepts used throughout are those of a sketch (unfortunately, somewhat underemphasized), a pattern, and a colimit. A sketch is a formal organization of a pattern, the arrows of which “determine the links between the components,” and which indicate, for example, “the various functions to be held in an organization,” while the pattern “determines which objects of the category fulfill these various functions.” This category “plays the part of the environment for the pattern.” Furthermore, collective links (“cooperation of several interrelated components forming a pattern”) realize collective behavior. A colimit of a pattern is a complex object that represents the organization of its lower-level components, retaining only the collective operations a pattern can realize. The authors note that “by binding together, individuals lose some autonomy, but their cooperation may generate a collective benefit, possibly by inducing differentiation and specialization of their role.” And a complex object acquires global properties not recognizable at the level of its lower components “when it admits different decompositions which are not materially connected.” The authors provide formal definitions of the relevant concepts. In this manner, CT illuminates and exactifies our intuitive understanding of systems and their structure.

The authors clearly bring out the general principles of their approach, as opposed to presentations where a concept “is interpreted on the basis of a particular representation” when the readers “do not integrate [it] into a wider conceptual framework and so are unable to use it successfully in contexts even slightly different from the original.” Their excellent thought-provoking book is well worth reading and re-reading.