A Classification of Modular Functors via Factorization Homology

Modular functors are traditionally defined as systems of projective representations of mapping class groups of surfaces that are compatible with gluing. They can formally be described as modular algebras over central extensions of the modular surface operad, with the values of the algebra lying in a suitable symmetric monoidal $(2,1)$-category $\mathcal{S}$ of linear categories. In this paper, we prove that modular functors in $\mathcal{S}$ are equivalent to self-dual balanced braided algebras $\mathcal{A}$ in $\mathcal{S}$ (a categorification of the notion of a commutative Frobenius algebra) for which a condition formulated in terms of factorization homology with coefficients in $\mathcal{…