  To date weak gravitational lensing surveys have typically been restricted to
small fields of view, such that the $\textit{flat-sky approximation}$ has been
sufficiently satisfied. However, with Stage IV surveys ($\textit{e.g. LSST}$
and $\textit{Euclid}$) imminent, extending mass-mapping techniques to the
sphere is a fundamental necessity. As such, we extend the sparse hierarchical
Bayesian mass-mapping formalism presented in previous work to the spherical
sky. For the first time, this allows us to construct $\textit{maximum a
posteriori}$ spherical weak lensing dark-matter mass-maps, with principled
Bayesian uncertainties, without imposing or assuming Gaussianty. We solve the
spherical mass-mapping inverse problem in the analysis setting adopting a
sparsity promoting Laplace-type wavelet prior, though this theoretical
framework supports all log-concave posteriors. Our spherical mass-mapping
formalism facilitates principled statistical interpretation of reconstructions.
We apply our framework to convergence reconstruction on high resolution N-body
simulations with pseudo-Euclid masking, polluted with a variety of realistic
noise levels, and show a significant increase in reconstruction fidelity
compared to standard approaches. Furthermore we perform the largest joint
reconstruction to date of the majority of publicly available shear
observational datasets (combining DESY1, KiDS450 and CFHTLens) and find that
our formalism recovers a convergence map with significantly enhanced
small-scale detail. Within our Bayesian framework we validate, in a
statistically rigorous manner, the community's intuition regarding the need to
smooth spherical Kaiser-Squires estimates to provide physically meaningful
convergence maps. Such approaches cannot reveal the small-scale physical
structures that we recover within our framework.
