A Quillen model for classical Morita theory and a tensor categorification of the Brauer group
Let \(\mathbb{K}\) be a commutative ring and \(Cat_{\mathbb{K}}\) be the category of small \(\mathbb{K}\)-categories. In this paper, the authors show that the category \(Cat_{\mathbb{K}}\) carries a combinatorial symmetric monoidal Quillen model structure. In particular, two \(\mathbb{K}\)-algebras become isomorphic in the associated homotopy category \(Ho(Cat_{\mathbb{K}})\) if and only if they are Morita equivalent, see [\textit{K. Morita}, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 6, 83--142 (1958; Zbl 0080.25702)]. As an application, the authors prove that the symmetric monoidal category \(Ho(Cat_{\mathbb{K}})^{\otimes}\) of \(\otimes\)-inversible objects in the homotopy category \(Ho(Cat_{\mathbb{K}})\) is a \(\otimes\)-categorification of the Brauer group \(Br(\mathbb{K})\). Finally, the authors prove that the contravariant corestriction map for finite Galois extensions also lifts to this categorification.
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