  The Bohrification program is an attempt to interpret Bohr's mature doctrine
of classical concepts as well as his earlier correspondence principle in the
operator-algebraic formulation of quantum theory pioneered by von Neumann. In
particular, this involves the study of commutative C*-algebras in relationship
to noncommutative ones. This relationship may take the form of either exact
Bohrification, in which one studies commutative unital C*-subalgebras of a
given noncommutative C*-algebra, or asymptotic Bohrification, which involves
deformations of commutative C*-algebras into noncommutative ones. Implementing
the doctrine of classical concepts, exact Bohrification is an appropriate
framework for the Kochen-Specker Theorem and for (intuitionistic) quantum
logic, culminating in the topos-theoretic approach to quantum mechanics that
has been developed since 1998. Asymptotic Bohrification was inspired by the
correspondence principle and forms the right conceptual and mathematical
framework for the explanation of the emergence of the classical world from
quantum theory (incorporating the measurement problem and the closely related
issue of spontaneous symmetry breaking). The Born rule may be derived from both
exact and asymptotic Bohrification, which reflects the Janus faces of
probability as applying to individual random events and to relative
frequencies, respectively. We review the history, the goals, and the
achievements of this program so far.
