Connected Simple Systems And The Conley Functor

. We give a short and elementary proof that the Conley index is a connected simple system. We use this fact to construct the Conley functor. 1. Introduction. The Conley index is a topological tool used in the qualitative theory of differential equations and dynamical systems (see [1, 10, 4]). In the simplest setting it takes the form of an object of a certain category (homotopy category of metric spaces, category of graded moduli etc.), which is known up to an isomorphism. This lack of precision is caused by the fact that there are many so called index pairs used in the construction of the index but up to an isomorphism one can extract some common information from them, which is by the definition the index. In a more subtle approach to the Conley index, essential in many applications, one shows that the isomorphisms are not quite arbitrary but form a connected simple system, i.e. a small category with the property that there is exactly one morphism between any two objects. The first p..