We describe a coordinate-free notion of conformal nets as a mathematical
model of conformal field theory. We define defects between conformal nets and
introduce composition of defects, thereby providing a notion of morphism
between conformal field theories. Altogether we characterize the algebraic
structure of the collection of conformal nets as a symmetric monoidal
tricategory. Dualizable objects of this tricategory correspond to
conformal-net-valued 3-dimensional local topological quantum field theories. We
prove that the dualizable conformal nets are the finite sums of irreducible
nets with finite \mu-index. This classification provides a variety of
3-dimensional local field theories, including local field theories associated
to central extensions of loop groups.