The 2-Deligne tensor product (Q6185380)

The paper under review studies a higher version of the classical Deligne tensor product for (compact) semisimple 2-categories, called the 2-Deligne tensor product. The author proves that the 2-Deligne tensor product of two compact semisimple tensor 2-categories is again a compact semisimple tensor 2-category. Throughout, let \(\Bbbk\) be a perfect field. Section 1 recalls notions and definitions about the classical \emph{Deligne tensor product}. Given two abelian categories \(\mathcal{C}\) and \(\mathcal{D}\), their Deligne tensor product comprises -- if it exists -- an abelian \(\mathcal{C} \boxtimes \mathcal{D}\), together with a bilinear functor \[ \mathcal{C} \otimes \mathcal{D} \to \mathcal{C} \boxtimes \mathcal{D} \] that is right-exact in each variable, such that \[ \mathsf{Rex}(\mathcal{C} \boxtimes \mathcal{D}, \mathcal{E}) \simeq \mathsf{Rex}_2(\mathcal{C}, \mathcal{D}; \mathcal{E}) := \mathsf{Rex}(\mathcal{C}, \mathsf{Rex}(\mathcal{D}, \mathcal{E})) \] is an equivalence of categories, for all other abelian categories \(\mathcal{E}\). One can show that the Deligne product of finite semisimple \(\Bbbk\)-linear (tensor) categories \(\mathcal{C}\) and \(\mathcal{D}\) exists, and is equivalent to the \emph{Cauchy completion} (completion under direct sums and splitting of idempotents) \(\mathrm{Cau}(\mathcal{C} \otimes \mathcal{D})\) of their external tensor product \(\mathcal{C} \otimes \mathcal{D}\) -- that is, objects of \(\mathcal{C} \otimes \mathcal{D}\) are pairs, and a hom-space is the tensor product of the individual hom-spaces. This category is itself a finite semisimple \(\Bbbk\)-linear (tensor) category. It is this construction that, in Theorem 3.7, is lifted to the 2-Deligne tensor product. Section 2 introduces various notions needed for this. The chosen generalisations of splitting idempotents to 2-categories are \emph{condensations} and \emph{condensation monads}; see [\textit{D. Gaiotto} and \textit{T. Johnson-Freyd}, ``Condensations in higher categories'', Preprint, \url{arXiv:1905.09566}]. A notion of Cauchy completion of a locally Cauchy complete 2-category is constructed in the spirit of the Karoubi envelope of [loc cit]. The notion of a \textit{compact} semisimple 2-category from [\textit{T. D. Décoppet}, Trans. Am. Math. Soc. 376, No. 12, 8309--8336 (2023; Zbl 1525.18022)] is used as the basis for the investigation. Section 3 defines the \textit{completed tensor product} of \(\Bbbk\)-linear 2-categories, proves existence, and that it has an analogue of the universal property of the Deligne tensor product. This culminates in Theorem 3.7, which proves that the 2-Deligne tensor product \(\mathfrak{C} \boxdot \mathfrak{D}\) of two compact semisimple 2-categories \(\mathfrak{C}\) and \(\mathfrak{D}\) exists. Section 4 explores the properties of the 2-Deligne tensor product -- in particular, it is shown that \[ \mathrm{Hom}_{\mathfrak{C}, \mathfrak{D}}(C_1 \boxdot D_1, C_2 \boxdot D_2) \simeq \mathrm{Hom}_{\mathfrak{C}}(C_1, C_2) \boxtimes \mathrm{Hom}_{\mathfrak{D}} (D_1, D_2). \] Section 5 extends the previous construction to include monoidal structure. Theorem 5.6 proves that for compact semisimple tensor 2-categories \(\mathfrak{C}\) and \(\mathfrak{D}\), their 2-Deligne tensor product \(\mathfrak{C} \boxdot \mathfrak{D}\) is again a compact semisimple tensor 2-category, and the 2-functor \(\boxdot \colon \mathfrak{C} \times \mathfrak{D} \to \mathfrak{C} \boxdot \mathfrak{D}\) is monoidal.