2023/08/07 by Lallouche, Mickael
#Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2308.03942
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 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 admissible color and using extension of Kirby calculus on 3-cobordisms, this morphism gives rise to an internal TQFT which takes values in the symmetric monoidal subcategory of transparent objects of C. When the category C is modular, this subcategory is equivalent to the category of vector spaces. When the category C is premodular and normalizable with invertible dimension, our TQFT is a lift of Turaev's one associated to the modularization of C.