1. Information Systems and the Theory of Categories: Is Every Model an Anticipatory System?

2. Authors M. A. Heather, Sutherland Building, Northumbria University, NE1 8ST, m.heather@unn.ac.uk m.heather@unn.ac.uk B. Nick Rossiter, Informatics, Northumbria University, Newcastle upon Tyne, UK, NE1 8ST, nick.rossiter@unn.ac.uk

3. Exploring the Unknown Knowledge advances: – by using the known to understand the unknown. Information systems are sources of the known – but also contain the unknown by reason of yet unrealised connections between what is known Practical applications in Data Mining –looking for new rules not explicit in data schema

4. Advance of Categories The philosophy of idealism and categories, the limits and colimits of knowledge, have developed from: –classical Platonic idealism –Aristotolean categories – through modern Kantian judgments –categories of pure and applied reasoning (analytic and synthetic) – on to a postmodern formalism of topos theory

5. Requirements To deal with organisms as complex, not just simple mechanisms: –modern information systems have to cope with: dynamic, open and non-local nature of knowledge beyond set theory. Current interest with sketches.

6. Behaviour of Reactive System Reactive System Information System Slice Retract Subcategory

7. Behavioural Aspects The figure represents the possible unknown behaviour of a reactive system whether physical, biological or social. The change may not be fully understood but may be modelled in an information system by a corresponding behavioural change

8. Strength of Anticipation In the upper limiting case: –the universe is a reactive system –the information system belongs to it as a subcategory. Any other existing reactive system is a subcategory of the universe as a topos. If the information system is predictive: –termed anticipatory. Where the anticipatory is a subcategory of the reactive system it is often referred to as strong anticipation. Otherwise it is weak. However the strength of an anticipatory system is not just Boolean because the internal topos logic is Heyting. There are quantitative (  ) and qualitative (  ) degrees of sameness

9. Slice The monad gives: –idempotent isomorphism The split epimorphism provides: –extensional equivalence A freely constructed slice provides: –information through a right adjoint retract to the extent of the equivalence of the slice category, –that is any model including the predictive anticipatory system.

10. Formalisation of Anticipatory System -- Adjointness 

11. Formalisation of Anticipatory System -- Adjointness 

12. Formalisation as 2-cell -- Upper Limit 

13. Formalisation as 2-cell -- Lower Limit 

14. The question To model is to interrogate an information system. Prediction is relative to the observer and consequent to use and the type of query. Whether every model is an anticipatory system is a relativistic question of subjective locality in time and space.

15. References [1] Heather, M A, & Rossiter, B N, Locality, Weak or Strong Anticipation and Quantum Computing I. Non-locality in Quantum Theory, International Journal Computing Anticipatory Systems 13 307-326 (2002). [2] Heather, M A, & Rossiter, B N, Locality, Weak or Strong Anticipation and Quantum Computing. II. Constructivism with Category Theory, International Journal Computing Anticipatory Systems 13 327-339 (2002). [3] Johnstone, P T, Sketches of an Elephant, A Topos Theory Compendium, Oxford Logic Guides 43, Clarendon (2002). [4] Rosen, R, Life Itself, A Comprehensive Inquiry into the Nature, Origin, and Fabrication of Life, Columbia University Press, New York (1991).