  The recent "honeycomb code" is a fault-tolerant quantum memory defined by a
sequence of checks which implements a nontrivial automorphism of the toric
code. We argue that a general framework to understand this code is to consider
continuous adiabatic paths of gapped Hamiltonians and we give a conjectured
description of the fundamental group and second and third homotopy groups of
this space in two spatial dimensions. A single cycle of such a path can
implement some automorphism of the topological order of that Hamiltonian. We
construct such paths for arbitrary automorphisms of two-dimensional doubled
topological order. Then, realizing this in the case of the toric code, we turn
this path back into a sequence of checks, constructing an automorphism code
closely related to the honeycomb code.
