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

Michael. Heather, Northumbria University, Newcastle upon Tyne, UK, NE1 8ST, m.heather@unn.ac.uk

B. Nick Rossiter, Informatics, Northumbria University, nick.rossiter@unn.ac.uk

**Keywords:** information systems, theory of categories, anticipatory strength,
formal models, analytic and synthetic reasoning, monads and sketches.

**Abstract**

Knowledge advances by using the known to understand the unknown. Information systems are important repositories and sources of the known but also contain the unknown by reason of yet unrealised connections between what is known. The philosophy of idealism and categories, the limits and colimits of knowledge, have developed from classical Platonic idealism and Aristotolean categories through modern Kantian judgments and categories of pure and applied reasoning both analytic and synthetic on to a postmodern formalism of topos theory [3].

*Subcategory*

j jj

j

To deal with organisms as complex, not just simple mechanisms [4], modern information systems have to cope with the dynamic, open and non-local nature of knowledge beyond set theory [1,2]. This is exemplified in the current interest with sketches. 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.

The systems in this figure may be formalised as 2-cell categories in a topos. In the upper limiting case the universe is a reactive system and 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 it may be 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.

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. 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.

**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).