In this series of papers, we investigate the projective framework initiated by Jerzy Kijowski (1977) and Andrzej Okołów (2009, 2013, 2014), which describes the states of a quantum (field) theory as projective families of density matrices. A short reading guide to the series can be found in  [27] . The present first paper aims at clarifying the classical structures that underlies this formalism, namely projective limits of symplectic manifolds  [27, subsection 2.1] . In particular, this allows us to discuss accurately the issues hindering an easy implementation of the dynamics in this context, and to formulate a strategy for overcoming them  [27, subsection 4.1] .