  We study the category of KM fans - a "stacky" generalization of the category
of fans considered in toric geometry - and its various realization functors to
"geometric" categories. The "purest" such realization takes the form of a
functor from KM fans to the 2-category of stacks over the category of fine
fans, in the "characteristic-zero-\'etale" topology. In the algebraic setting,
over a field of characteristic zero, we have a realization functor from KM fans
to (log) Deligne-Mumford stacks. We prove that this realization functor gives
rise to an equivalence of categories between (lattice) KM fans and an
appropriate category of toric DM stacks. Finally, we have a differential
realization functor to the category of (positive) log differentiable spaces.
Unlike the other realizations, the differential realization of a stacky fan is
an "actual" log differentiable space, not a stack. Our main results are
generalizations of "classical" toric geometry, as well as a characterization of
"when a map of KM fans is a torsor". The latter is used to explain the
relationship between our theory and the "stacky fans" of Geraschenko and
Satriano.
