The Density Theorem and Its ‘Suspension’: Mathematical Models of Complex Systems Considered from the ‘From-Within’ Mode of Being

A fundamental concern of complex systems science is the relationship between the part and the whole. The density theorem in category theory can be interpreted as a statement about this part–whole relationship. In the first half, I will explain this intuitively, taking directed networks as an example. In the second half, I will explain that one formulation of the ‘from-within’ mode of being—important for understanding biological systems—is obtained by ‘suspending’ the density theorem, and, building on this, I will introduce examples of research such as the self-organization of gene regulatory networks toward critical states.