  In holographic duality, a higher dimensional quantum gravity system emerges
from a lower dimensional conformal field theory (CFT) with a large number of
degrees of freedom. We propose a formulation of duality for a general causally
complete bulk spacetime region, called subalgebra-subregion duality, which
provides a framework to describe how geometric notions in the gravity system,
such as spacetime subregions, different notions of times, and causal structure,
emerge from the dual CFT. Subalgebra-subregion duality generalizes and brings
new insights into subregion-subregion duality (or equivalently entanglement
wedge reconstruction). It provides a mathematically precise definition of
subregion-subregion duality and gives an independent definition of entanglement
wedges without using entropy. Geometric properties of entanglement wedges,
including those that play a crucial role in interpreting the bulk as a quantum
error correcting code, can be understood from the duality as the geometrization
of the additivity anomaly of certain algebras. Using general boundary
subalgebras rather than those associated with geometric subregions makes it
possible to find duals for general bulk spacetime regions, including those not
touching the boundary. Applying subalgebra-subregion duality to a boundary
state describing a single-sided black hole also provides a precise way to
define mirror operators.
