Gray tensor product and Kontsevich's Swiss-Cheese conjecture

We study connections between two seemingly very distant constructions: Gray-product of higher categories and famous Kontsevich Swiss-Cheese conjecture. Gray-product of 2-categories is known for almost 50 years and it is an extremely important construction in 2-category theory. It was proved by Crans and later by Bourke and Gurski that a naive analogue of Gray-product in higher dimensions does not exist. Nevertheless, there is a conjecture that there exists a weaker version of this product in all dimensions such that it descends to a closed structure on homotopy level. Swiss-Cheese conjecture was proposed by Fields medalist M. Kontsevich in 1998 to handle a problem of the existence of higher order Hochschild complexes. It is geometrical in nature and is very important in deformation quantisation theory. In the thesis we outline a surprising relationship between these two important conjectures, which was not observed before. Namely, the existence of a homotopically closed Gray-product of V-enriched categories implies the Swiss-Cheese conjecture in V. We provide a full proof of this statement for V = Set, Ab and Cat using the idea of categorification.