Smooth toric Deligne-Mumford stacks

We give a geometric definition of smooth toric Deligne-Mumford stacks using the action of a “torus”. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks