Continuous quivers of type A (I) Foundations

We generalize quivers of type A to continuous quivers of type A and prove initial results about pointwise finite-dimensional representations. Among these results is the classification of those representations and a decomposition theorem, recovering results of Botnan and Crawley-Boevey. We also classify the indecomposable pointwise finite-dimensional projective representations. Finally, we prove that many of the properties of finite-dimensional representations of quivers of type A_n also hold for finitely generated representations of continuous quivers of type A. This is the self-contained foundational part of a series of works to study a generalization of continuous clusters categories and their relationship to other cluster structures of type A