Prolongations, Suspensions and Telescopes

Abstract

Autonomous differential equations induced by continuous vector fields usually appear in non-smooth mechanics and other scientific contexts. For these type of equations, given an initial condition, one has existence theorems but, in general, the uniqueness of the solution can not be ensured. For continuous vector fields, the equation solutions do not generally present a continuous flow structure; one particular but interesting case, occurs when under some initial conditions one can ensure existence of solutions and uniqueness in forward time obtaining in this case continuous semi-flows. The discretization and return Poincaré techniques induce the corresponding discrete flows and semi-flows and some inverse methods as the suspension can construct a flow from a discrete flow or semi-flow. The objective of this work is to give categorical models for the diverse phase spaces of continuous and discrete semi-flows and flows and for the relations between these different phase spaces. We also introduce some new constructions such as the prolongation of continuous and discrete semi-flows and the telescopic functors. We consider small Top-categories (weakly enriched over the category Top of topological spaces) and we take as categorical models of the solutions of these differential equations some categories of continuous functors from a small Top-category to the category of topological spaces. Moreover, the processes of discretizations, suspensions, prolongations, et cetera are described in terms of adjoint functors. The main contributions of this paper are the construction of a tensor product associated to a functor between small Top-categories and the interpretation of prolongations, suspensions and telescopes as particular cases of this general tensor product. In general, the paper is focused on the establishment of links between category theory and dynamical systems more than on the study of differential equations using some categorical terminology.