Categorified trace for module tensor categories over braided tensor categories (Q321896)

A monoidal category \(\mathcal{M}=(\mathcal{M},\odot, J)\), with homs enriched in a closed braided monoidal category \(\mathcal{C}=(\mathcal{C},\otimes,I)\), is pivotal when every object \(A\) has a right dual \(A^{*}\) and there is further supplied a monoidal (enriched) natural isomorphism \(\varphi_A : A\cong A^{**}\). The monoidal functor \(\mathrm{Tr}_{\mathcal{C}}= \mathcal{M}(J,-) : \mathcal{M}\to \mathcal{C}\) taken together with the natural isomorphisms NEWLINE\[NEWLINE\tau_{A,B} : \mathrm{Tr}_{\mathcal{C}}(A\odot B)\cong \mathcal{M}(J,A^{**}\odot B)\cong \mathcal{M}(A^{*},B)\cong \mathrm{Tr}_{\mathcal{C}}(B\odot A)NEWLINE\]NEWLINE captures much about \(\mathcal{M}\). Each monoid (= algebra) \(A\) in \(\mathcal{M}\) is taken to a monoid \(\mathrm{Tr}_{\mathcal{C}}(A)\) in \(\mathcal{C}\) since \(\mathrm{Tr}_{\mathcal{C}}\) is monoidal. For monoids \(A\) and \(B\) in \(\mathcal{M}\), there is no natural monoid structure on \(A\odot B\) since we assume no commutativity in \(\mathcal{M}\). However, by convolution using the comonoid \(A^{*}\), the object \(\mathrm{Tr}_{\mathcal{C}}(A\odot B)\cong \mathcal{M}(A^{*},B)\) is a monoid in \(\mathcal{C}\). Under further assumptions, the authors prove that \(\mathrm{Tr}_{\mathcal{C}}(A\odot B)\in \mathcal{C}\) is separable if \(A\) and \(B\) are. A large contribution of the paper is to establish a faithful diagramatic calculus of strings on cylinders, where the traciator corresponds to wrapping a string around the back of a cylinder.