Since the early developments of Quantum theory, theoretical efforts have been made to connect classical physics and quantum mechanics. Niels Bohr laid out the most notable effort in the early days by presenting the "Correspondence" principle which requires that the predictions of quantum mechanics must reproduce the predictions of classical physics in the limit of large quantum numbers. The guidelines prescribed by the Bohr correspondence principle are nevertheless limiting, i.e. this framework does not shed light on of the mechanisms behind the classification of quantum systems. In particular, it does not explain the absence of quantum coherences (superpositions) at the macroscopic scale. It was not until the pioneering work of H. Dieter Zeh (1970)  E.Joos and W. H. Zurek that the theoretical framework now known as "Decoherence" theory was created and used to explain the dynamical disappearance of quantum coherence as a byproduct of the interaction of a quantum system with its environment (open quantum systems). Although the work of H. D. Zeh, E. Joos, W. H. Zurek, and others had great success in theoretically explaining the absence of quantum coherences at the macroscopic scale, further reconciling quantum theory with classical theory,  it nevertheless did not explain how classical objectivity emerges from the quantum; i.e. if measurement at the quantum scale affects the state of a quantum system how is it that ultimately things seem to have a measurer-independent-state at the macroscopic scale. It was not until W.H. Zurek  (2003) with the introduction of his theory of "Quantum Darwinism" that Zurek took the first steps in creating a framework for the study of the dynamical emergence of objectivity via entropic measure conditions and his operational definition of objectivity. Some of the requirements for W. H. Zurek's definition of objectivity in W. H. Zurek et al's work are qualitative; it is because of this that it is not possible to fully study the emergence of objectivity in a rigorous sense within the Quantum Darwinism framework. In 2013 the Horodecki group, Korbicz, and collaborators took some of the main ideas of Quantum Darwinism and constructed a theoretical framework for the study of objectivity emerging dynamically from the quantum; they provide a fully mathematical definition for objectivity, Gdansk-Objectivity,  and the object of focus in this theory become the so-called Spectrum Broadcast Structures (SBS), whose dynamical appearance are indicators that some system finds itself in a Gdansk-Objective state. SBS theory has had much success in modeling how one may retrieve  Gdansk-objectivity from a particular application of the theory of open quantum systems. In this talk, we will primarily focus on presenting SBS theory, highlighting some of the mathematical challenges embedded in it, and discussing some recent results.