  We study charge transport for zero-temperature infinite-volume gapped lattice
systems in two dimensions with short-range interactions. We show that the Hall
conductance is locally computable and is the same for all systems which are in
the same gapped phase. We provide a rigorous versions of Laughlin's
flux-insertion argument which shows that for short-range entangled systems the
Hall conductance is an integer multiple of e^2/h. We show that the Hall
conductance determines the statistics of flux insertions. For bosonic
short-range entangled systems, this implies that the Hall conductance is an
even multiple of e^2/h. Finally, we adapt a proof of quantization of the
Thouless charge pump to the case of infinite-volume gapped lattice systems in
one dimension.
