EMCSR | 2014

Book of Abstracts | 405-408

ISSN 2227-7803


The system as emergent process

in Topos Theory

Michael Heather

Northumbria University, Newcastle NE2 1XE, UK,



Nick Rossiter

Northumbria University, Newcastle NE2 1XE, UK,



Keywords: Emergence; Design; Open System Theory; Open Holistic Systems; Life Systems; Process;

Topos Theory; Category Theory; Cartesian Closed Category; Adjointness; Terminal Object; Category of

the Ultimate; Quantum Reality; Robert Rosen; Alfred North Whitehead.

1 Overview

Emergence is a fundamental component of the current understanding of science but as a

top-down’ holistic mechanism it would be a contradiction in terms to call it foundational.

Emergence describes the global devolution of the world everywhere adjoint to local

evolution. Emergence is often as at this EMCSR meeting bracketed with the word ‘design’.

To grasp a rigorous understanding of these twin terms as part of science they need to be

expressed as formal concepts. Systems Theory hardly justifies its existence as a theory if it

cannot be expressed formally. Formal definitions are not always easy and particularly

difficult when they lie at the cutting edge of science in terra incognita. Emergence is usually

treated as a process and design as possessing an open structure. Words like ‘process’ ,

structure’, and ‘open’ more fortunately do have a track record to draw on.

Emergence as allied to design belongs to open system theory. It is the process that comes

out of ‘openness’. To be scientific systems theory has to be underpinned by rigorous logic.

Classical logic tends to be stuck in closed systems and does not avail much for open

systems. Yet open systems comprise the vast majority of problems in current areas that

system theory is today called to address. The logic of openness is by its very nature the

logic of the third way. Such logic is very difficult to represent in classical logic because

classical logic is Boolean and only operates two-way. We earlier drew attention to the

difficulties even to define ‘open systems’1 let alone to understand them formally. A cause

célèbre for emergent design is to explain the existence of life. The pioneer a way ahead of

his time in the study of life systems was Robert Rosen who recommended a shift from set

theory to category theory:

the natural habitat for discussing . . . specific modelling relations”. (Rosen 1991 p 153)

Rosen’s informal diagram is reproduced in figure 1. Emergence is in effect the ‘implication’2

on the right of his diagram. We followed Rosen’s prescription for category theory to express

the logic of social systems3 . It turns out that Rosen’s informal diagram was an early

attempt to represent the archetypal process of universal adjointness in a cartesian closed

category as shown between any two categories in our corresponding formal diagram of

figure 2. This shows the contravariant free and underlying functors between the left and

right structures. Given any one of these fixes the other three uniquely when expressed this

way. Therefore a particular right structure emerges by a particular choice of the free functor

(F). This is both awesome and trivial in that every choice we make determines the next

unique state of the world.

1 See, Heather & Rossiter (2006)

2 In Rosen’s context ‘implication’ was the emergence of life.

3 See, Heather & Rossiter (2008)

However as a mathematical model is restricted by the limitations of set theory we have

found it necessary to follow Alfred North Whitehead in his seminal work on Process &

Reality and ascend up two levels from models to metaphysics4. This brings us to the

highest possible level of category theory which is topos theory. The topos like the category

is another of Aristotle’s insights. From a conventional ‘bottom-up’ approach this is a terminal

object but in the reality of quantum theory it is the starting point. The topos is the formal

structure where every entity effects every other entity directly and indirectly through every

other object, which describes the structure of the physical Universe. Grothendieck of the

anarchic Bourbaki group of mathematicians in France was possibly the first to grasp the

concept of the topos as a ‘mathematical universe of universes’. However that is one level

too short to reach the level of Whitehead’s metaphysics. We need to go up to the category

of category of categories (the level of the double power set in naive set theory) to reach the

top level of closure, the topos as in figure 3. Here there are the three levels of category.

The double headed arrow represents the pair of contravariant functors as in figure 2. Also

such a pair exists between each level. Consequently the left to right structure of figure 2

represents the process of emergence.

4 See, Heather & Rossiter (2014).


Heather, Michael, & Rossiter, Nick, (2008). The Logic for Social Systems, EMCSR-2008: European

Meeting on Cybernetics and Systems Research, Symposium Sociocybernetic Models: Conceptual

and Formal Approaches, University Vienna, 25-28 March II 653-658

Heather Michael & Rossiter Nick, (2014) Formal Representation of Process & Reality in the

metaphysical language of Category Theory: Whitehead's relational theory of space (in the press)

Rosen, R, (1991) Life Itself, A Comprehensive Inquiry into the Nature, Origin, and Fabrication of Life

Columbia University Press, New York

Rossiter, Nick, & Heather, Michael, (2006) Free and Open Systems Theory, EMCSR-2006,

Cybernetics and Systems, 18th European Meeting on Cybernetics and Systems Research,

University of Vienna, 18-21 April 2006, Trappl, R, (ed) 1 27-32

About the Authors

Michael Heather and Nick Rossiter have collaborated on over 150 papers in the theory of information

systems and related topics, concentrating mainly in the application of category theory.