Algebraic string topology from the neighborhood of infinity

We construct and study an algebraic analogue of the loop coproduct in string topology, also known as the Goresky-Hingston coproduct. Our algebraic setup, which under this analogy takes the place of the complex of chains on the free loop space of a possibly non-simply connected manifold, is the Hochschild chain complex of a smooth $A_{\infty}$-category equipped with a pre-Calabi-Yau structure and a trivialization of a version of the Chern character of its diagonal bimodule. The algebraic analogue of the loop coproduct is part of a more general mapping cone construction, which we describe in terms of the categorical formal punctured neighborhood of infinity associated to the underlying smooth $A_\infty$-category. We use a graphical formalism for $A_\infty$-categories and bimodules to describe explicit models for the operations and homotopies involved. We also compute explicitly the algebraic coproduct in the context of the string topology of spheres.