Information Systems and the Theory of Categories:

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 • 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. Slice Information System Retract Reactive System Behaviour of Reactive System 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).