  We present the first map of carbonicity, [C/Fe], for the halo system of the
Milky Way, based on a sample of over 100,000 main-sequence turnoff stars with
available spectroscopy from the Sloan Digital Sky Survey. This map, which
explores distances up to 15 kpc from the Sun, reveals clear evidence for the
dual nature of the Galactic halo, based on the spatial distribution of stellar
carbonicity. The metallicity distribution functions of stars in the inner- and
outer-halo regions of the carbonicity map reproduce those previously argued to
arise from contributions of the inner- and outer-halo populations, with peaks
at [Fe/H] = -1.5 and -2.2, respectively. From consideration of the absolute
carbon abundances for our sample, A(C), we also confirm that the
carbon-enhanced metal-poor (CEMP) stars in the outer-halo region exhibit a
higher frequency of CEMP-no stars (those with no overabundances of heavy
neutron-capture elements) than of CEMP-s stars (those with strong
overabundances of elements associated with the s-process), whereas the stars in
the inner-halo region exhibit a higher frequency of CEMP-s stars. We argue that
the contrast in the behavior of the CEMP-no and CEMP-s fractions in these
regions arises from differences in the mass distributions of the mini-halos
from which the stars of the inner- and outer-halo populations formed, which
gives rise in turn to the observed dichotomy of the Galactic halo.
