Citation: Greenough, Justin. "Monoidal 2-structure of bimodule categories." Journal of Algebra 324.8 (2010): 1818-1859.
Web: https://arxiv.org/abs/0911.4979
Tags: Mathematical, Monoidal-categories, Monoidal-categories
In this paper, Justin Greenough introduces the notion of the relative Deligne tensor product. He proves all of the relevant facts about it. Even though many of the properties are intuitive, it still takes a lot of space and real category theory to give the proofs. There are a great deal of 2-categorical commutative diagrams.
Even though this paper is a good primary source for the notion of relative Deligne tensor product, there is still a bit of history of the notion which should be appreciated. The paper
> Tambara, Daisuke. "A duality for modules over monoidal categories of representations of semisimple Hopf algebras." Journal of Algebra 241.2 (2001): 515-547.
seems to contain the basic definition in disguise. The paper
> Etingof, Pavel, Dmitri Nikshych, and Victor Ostrik. "Fusion categories and homotopy theory." Quantum topology 1.3 (2010): 209-273.
appeared before Greenough's work, but still uses the notion of relative Deligne tensor product. This is because Dmitri Nikshych was Greenough's advisor at the time, and hence was very well aware of Greenough's work. The use of the relative Deligne tensor product in the study of equivariantization was thus the direct motivation for the definition.
The notion of relative Deligne tensor product has since been given very strong physical intuition. It corresponds to the fusion of domain walls:
> Kitaev, Alexei, and Liang Kong. "Models for gapped boundaries and domain walls." Communications in Mathematical Physics 313.2 (2012): 351-373.
The main tool used in this work is the idea of the internal hom. This internal hom gives the structure of the Deligne tensor product a 2-rigid structure, which allows for the proofs of many nice results such as a 2-categorical Frobenius reciprocity.