La$_{2-x}$Ba$_x$CuO$_4$ as a superconducting Rosetta Stone

The high-temperature superconductivity in layered cuprates discovered by Bednorz and M\"uller arrived as a shock. Gradually, the presence of competing orders, such as antiferromagnetism and charge order, were discovered; however, the relationship to the superconductivity has been confusing. It so happens that the original cuprate superconductor family La$_{2-x}$Ba$_x$CuO$_4$ contains all of the relevant phases, with extreme competition among them, and analysis of these phases provides strong clues to the nature of the superconductivity in cuprates.