  The category of (abstract) fans is to the category of monoids what the
category of schemes is to the category of rings: a fan is obtained by gluing
spectra of monoids along open embeddings. Here we study the basic algebraic
geometry of fans: coherent sheaves, group fans, affine maps, flat maps, proper
maps, and so forth. This study is motivated by the fact that the category of
fans lies over all other reasonably geometric categories (schemes,
differentiable spaces, analytic spaces, etc) as well as the "log" versions of
all such categories.
