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. Our algebraic analogue of the loop coproduct is part of a more general mapping cone construction, which we describe in terms of Efimov's categorical formal punctured neighborhood of infinity. We use the graphical formalism developed by Kontsevich, Takeda, and Vlassopoulos to describe explicit models for the operations and homotopies involved. We compute explicitly this algebraic coproduct in the context of string topology of spheres.