Continuous quivers of type A (I) foundations

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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 (J Algebra Appl (2015). https://doi.org/10.1142/S0219498815500668, Decomposition of persistence modules, to appear in Proceedings of the American Mathematical Society, preprint: https://arxiv.org/pdf/1811.08946.pdf ). 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.