Internal Reshetikhin-Turaev TQFT

A 3-dimensional topological quantum field theory (TQFT) is a symmetric monoidal functor from the category of 3-cobordisms to the category of vector spaces. Such TQFTs provide in particular numerical invariants of closed 3-manifolds such as the Reshetikhin-Turaev invariants and representations of the mapping class group of closed surfaces. In 1994, using a modular category, Turaev explains how to construct a TQFT. In this article, we describe a generalization of this construction starting from a ribbon category $\mathcal{C}$ with coend. We present a cobordism by a special kind of tangle and we associate to the latter a morphism defined between tensorial products of the coend as described by Lyubashenko in 1994. Composing with an \emph{admissible} color and using extension of Kirby calculus on 3-cobordisms, this morphism gives rise to an \emph{internal} TQFT which takes values in the symmetric monoidal subcategory of transparent objects of $\mathcal{C}$. When the category $\mathcal{C}$ is modular, this subcategory is equivalent to the category of vector spaces. When the category $\mathcal{C}$ is premodular and normalizable with invertible dimension, our TQFT is a lift of Turaev's one associated to the modularization of $\mathcal{C}$.