“The search for integrative and integrating approaches in the biosciences is very important at the present time because of the rapid increases in amounts of data, knowledge and information” [1]. While too many biological books and papers add to these amounts, understanding complex systems of “life, mind, and society” [2] requires clearly defined foundational approaches. Narratives of such approaches were provided, for example, by Hayek [2] and Bunge [3] decades ago, and a solid mathematical foundation based on category theory was formulated only recently, for example, in Ehresmann and Vanbremeersch [4].

The purpose of this very enthusiastically written book is “to advance in the understanding of brain function by defining a general framework for representation based on category theory.” The author’s way of modeling cognitive systems properly follows the aforementioned texts [1,2,3,4] (although Hayek is never mentioned), and means discovering structural and functional relationships in the system in order to capture the organizing principles underlying the behavior of the system. The importance of emergent properties is stressed throughout, and the author makes an interesting observation that the emphasis of the early theoretical biologists on emergent properties of entire organisms “was far removed from the reductionist methodological approach championed by 20th century biology.” Nevertheless, at least some of the 21st century biology (for example, biosemiotics) is different, and the author, in particular, mentions in this context meaning-making as opposed to information processing.

Most of the body of the book is an appropriately structured, terse, and at times critical survey referring to 448 sources, including a short overview of Bunge’s work and substantial references to Ehresmann and Vanbremeersch [4]. The book’s approach follows from the author’s observation that, although biology textbooks “abound with accurate descriptions of thousands of molecules, organelles, and cells,” the same textbooks “give little account for the laws or general principles that govern the interactions between those molecules, organelles, or cells.” The issue that the book specifically addresses is how “the neural system encode mental objects, in particular spatial representation and memory,” and the author stresses that the dynamic nature of biological organisms has to be understood in terms of patterns at the different levels of description rather than by observing the parts in isolation. In particular, the author properly observes that the problem of how cognition arises can be studied as the formation and evolution of a special kind of pattern. The need to consider patterns rather than details in understanding and specifying complex systems is well known, for example, from [2] where Hayek also quotes G. H. Hardy’s observation that “a mathematician, like a painter or poet, is a maker of patterns.”

This is not the first book (or source) to “[conjecture] that category theory could provide the necessary concepts to bridge the gap between the different levels of brain activity” (see, for example, Ehresmann and Vanbremeersch [4,5]). Category theory probably takes care of “the common mistake of thinking that ‘throwing’ more syntax at an issue will improve the semantics” [1]. Regretfully, for an uninitiated reader, the explanation of category theory in general and of colimits in particular leaves much to be desired, and I would recommend a better choice, one with excellent examples [4] (for a flavor of the relevant category theory concepts, see Ehresmann et al. [1]). Also regretfully, some phrases in the book, such as “it is described the effect in injecting the concepts of co-product and colimit from category theory into the problem of place cell formation in the hippocampus” (p. 47) or “we reckon that biological functionality is multilevel and that there is not any priviledge (sic!) level of explanation, e.g., the genome” (p. 61), as well as numerous typographical and grammatical errors, do not help readers get a better understanding of the author’s message. Another fragment, from presenting the important concept of structure, probably goes without comment:

For any physical system, the relations between its components can be encoded into a mathematical structure. For example, the structure of a pack of six can of beers is 6 and the structure of the 12,349 telephone poles that connects Berkeley with San Bernardino (CA) is 12,349. (p. 88)

When discussing representations, the author properly notes that “representing an object is to define a collection of morphisms by which the relationships between components of the object can be precisely specified.” Further, he observes that “the neural representation of holistic perceptions are (sic!) embedded in recurrent patterns of connectivity” (this is in excellent agreement with the ideas presented by Hayek in 1952 [6]), and proposes some other interesting approaches, for example, the study of graph homomorphisms in brain connectivity patterns. He also proposes to model such network properties as clustering and modularity using the number of colimits and limits, observing that, in his opinion, the potential of category theory tools for quantitative (as opposed to qualitative) analysis has not been tackled so far. And he presents an excellent example of using colimits to understand and specify emergence and collective behavior in the hippocampus: “the cooperation of several grid cells identified with grid fields gives rise to the colimit which is a place field.” I think it would be very desirable to expand chapters 8 and 9 presenting the author’s own contributions, for instance, by comparing his approach “from cells to memories” (chapter 9) to the one described in Ehresmann and Vanbremeersch [4].

Summing up, one can only agree with the author’s emphasis on the suitability of category theory as a foundation for complex system modeling in general, leading to new and deeper insights into the structure and representational power of the brain in particular.

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