I will introduce ideal webs = ideal bipartite graphs on decorated surfaces. They are the simplest possible higher rank analogs of ideal triangulations of decorated surfaces.  

I will explain how to use them to define cluster atlases on several flavors of character varieties for the groups of type A_m, equivariant under the action of the mapping class group, and how to define a related family of 3d Calabi-Yau categories, also equivariant under the action of the mapping class group.

It is an expository talk, mostly following arXiv:1607.05228