  We perform general relativistic, magnetohydrodynamic (GRMHD) simulations of
merging binary neutron stars incorporating neutrino transport and magnetic
fields. Our new radiative transport module for neutrinos adopts a general
relativistic, truncated-moment (M1) formalism. The binaries consist of two
identical, irrotational stars modeled by the SLy nuclear equation of state
(EOS). They are initially in quasicircular orbit and threaded with a poloidal
magnetic field that extends from the stellar interior into the exterior, as in
typical pulsars. We insert neutrino processes shortly after the merger and
focus on the role of neutrinos in launching a jet following the collapse of the
hypermassive neutron star (HMNS) remnant to a spinning black hole (BH). We
treat two microphysical versions: one (a "warm-up") evolving a single neutrino
species and considering only charged-current processes, and the other evolving
three species $(\nu_e, \bar{\nu}_e, \nu_{\rm x})$ and related processes. We
trace the evolution until the system reaches a quasiequilibrium state after BH
formation. We find that the BH + disk remnant eventually launches an incipient
jet. The electromagnetic Poynting luminosity is $\sim 10^{53} \rm \, erg\,
s^{-1}$, consistent with that of typical short gamma-ray bursts (sGRBs). The
effect of neutrino cooling shortens the lifetime of the HMNS, and lowers the
amplitude of the major peak of the gravitational wave (GW) power spectrum
somewhat. After BH formation, neutrinos help clear out the matter near the BH
poles, resulting in lower baryon-loaded surrounding debris. The neutrino
luminosity resides in the range $\sim 10^{52-53} \rm \,erg\,s^{-1}$ once
quasiequilibrium is achieved. Comparing with the neutrino-free models, we
observe that the inclusion of neutrinos yields similar ejecta masses and is
inefficient in carrying off additional angular momentum.
