  A number of background independent quantizations procedures have recently
been employed in 4d nonperturbative quantum gravity. We investigate and
illustrate these techniques and their relation in the context of a simple 2d
topological theory. We discuss canonical quantization, loop or spin network
states, path integral quantization over a discretization of the manifold, spin
foam formulation, as well as the fully background independent definition of the
theory using an auxiliary field theory on a group manifold. While several of
these techniques have already been applied to this theory by Witten, the last
one is novel: it allows us to give a precise meaning to the sum over
topologies, and to compute background-independent and, in fact,
"manifold-independent" transition amplitudes. These transition amplitudes play
the role of Wightman functions of the theory. They are physical observable
quantities, and the canonical structure of the theory can be reconstructed from
them via a C* algebraic GNS construction. We expect an analogous structure to
be relevant in 4d quantum gravity.
