Building on Schlessinger's work, we define a framework for studying geometric
deformation problems which allows us to systematize the relationship between
the local and global tangent and obstruction spaces of a deformation problem.
Starting from Schlessinger's functors of Artin rings, we proceed in two steps:
we replace functors to sets by categories fibered in groupoids, allowing us to
keep track of automorphisms, and we work with deformation problems naturally
associated to a scheme X, and which naturally localize on X, so that we can
formalize the local behavior. The first step is already carried out by Rim in
the context of his homogeneous groupoids, but we develop the theory
substantially further. In this setting, many statements known for a range of
specific deformation problems can be proved in full generality, under very
general stack-like hypotheses.Comment: 46 pages. Comments welcom
