  In this work, we consider the question of local Hilbert space factorization
in 2D conformal field theory. Generalizing previous work on entanglement and
open-closed TQFT, we interpret the factorization of CFT states in terms of path
integral processes that split and join the Hilbert spaces of circles and
intervals. More abstractly, these processes are cobordisms of an extended CFT
which are defined purely in terms of the OPE data. In addition to the usual
sewing axioms, we impose an entanglement boundary condition that is satisfied
by the vacuum Ishibashi state. This choice of entanglement boundary state leads
to reduced density matrices that sum over super-selection sectors, which we
identify as the CFT edge modes. Finally, we relate our factorization map to the
co-product formula for the CFT symmetry algebra, which we show is equivalent to
a Boguliubov transformation in the case of a free boson.
