http://popups.lib.uliege.be/1373-5411/index.php?id=2947 Index terms fr 0 http://popups.lib.uliege.be/1373-5411/index.php?id=4449 Anticipation is a property of any system and resides in its semantics as a duality of the system itself. The relationship is an adjointness between levels, requiring contravariancy. The intension/extension levels are impredicative in nature but this recursive characteristic can be represented formally in category theory. This paper focuses on the vital role of contravariancy in adjointness, permitting a structured re-ordering of the categories involved. A worked example of a three-level architecture for an information system is provided, illustrating the alternation of intension/extension pairs, the adjointness of two-way functors between each level, the (bi)functors for linking intension to extension and the locally cartesian closed structure of the underlying categories. The dynamic anticipatory aspect of contravariant mapping, relative to static covariant mapping, is highlighted, reinforcing the view that contravariancy underpins anticipation. Fri, 11 Oct 2024 10:54:21 +0200 Fri, 11 Oct 2024 10:54:29 +0200 http://popups.lib.uliege.be/1373-5411/index.php?id=4449 http://popups.lib.uliege.be/1373-5411/index.php?id=2944 Every object in existence has its type. Every subject in language has its predicate. Every intention in logic has its extension. Each therefore has two levels but with the fundamental problem of the relationship between the two. The formalism of set theory cannot guarantee the two are co-extensive. That has to be imposed by the axiom of extensibility, which is inadequate for types as shown by Bertrand Russell's ramified type theory, for language as by Henri Poincare's impredication and for intention unless satisfying Port Royal's definitive concept. An anticipatory system is usually defined to contain its own future state. What is its type? What is its predicate? What is its extension? Set theory can well represent formally the weak anticipatory system, that is in a model of itself. However we have previously shown that the metaphysics of process category theory is needed to represent strong anticipation. Time belongs to extension not intention. The apparent prediction of strong anticipation is really in the structure of its predication. The typing of anticipation arises from a combination of c5 and μ - respectively (co) multiplication of the ( co )monad induced by adjointness of the system's own process. As a property of Cartesian closed categories this predication has significance for all typing in general systems theory including even in the definition of time itself. Tue, 03 Sep 2024 15:57:17 +0200 Tue, 03 Sep 2024 15:57:28 +0200 http://popups.lib.uliege.be/1373-5411/index.php?id=2944