  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 $\mathcal{A}$ is satisfied; we call such $\mathcal{A}$
connected. The equivalence in one direction is afforded by genus zero
restriction. Our construction of the inverse equivalence is entirely
topological and can be thought of as a far reaching generalization of the
construction of modular functors from skein theory. In order to verify the
connectedness condition in practice, we prove that it can be reduced to a
single condition in genus one. Moreover, we show that cofactorizability of
$\mathcal{A}$, a condition known to be satisfied for modular categories, is
sufficient. Therefore, we recover in particular Lyubashenko's construction of a
modular functor from a (not necessarily semisimple) modular category and show
that it is determined by its genus zero part. Additionally, we exhibit modular
functors that do not come from modular categories and outline applications to
the theory of vertex operator algebras.
