Associative algebras and broken lines

Inspired by Morse theory, we introduce a topological stack Broken, which we refer to as the moduli stack of broken lines. We show that Broken can be presented as a Lie groupoid with corners and provide a combinatorial description of sheaves on Broken with values in any compactly generated infinity-category C. Moreover, we show that factorizable C-valued sheaves (with respect to a natural semigroup structure on the stack Broken) can be identified with nonunital A-infinity-algebras in C. This is a first step in a program whose goal is to present an `equation-free' construction of the Morse complex associated to a compact Riemannian manifold.