Approximability for sSpaces of Lagrangian Submanifolds

I will discuss a recent notion - called categorical approximability - that extends the property of precompactness of a metric space. This is applied to certain spaces of Lagrangian submanifolds endowed with the spectral metric, such as exact Lagrangians in cotangent disk bundles or equators on the 2-sphere. While approximable, these spaces are oten not  precompact. The talk is based on recent joint work with Giovanni Amborsioni and Paul Biran.