With the current project proposal, we will further the development of noncommutative algebraic geometry by establishing highly structured deformation complexes for a variety of higher categorical structures. We will realise this goal through the following three main objectives. In a first objective, we will develop operadic approaches to encode higher linear categories and (higher) prestacks, leading to the definition of L infinity structured Gerstenhaber-Schack type complexes based upon endomorphism operads. For this, inspired by Leinster's free completion multicategories, we will develop an underlying operadic framework of ``cubical box operads'' in a linearly enriched setup. In the second objective, we will develop an alternative approach based upon the classical operadic cohomology due to Markl. In doing so we will answer an open question posed by Markl regarding the cohomology complex for presheaves. These two approaches will be compared and combined, and we will show that they calculate a natural notion of Hochschild cohomology as desired. This will facilitate our third objective, in which we will develop Keller's arrow category argument in the newly defined setup, leading to new cohomology computation and comparison tools.