Hi category gurus and categorists I have many questions about category theory but i start with one. 1> What are smooth functors and proper functors, originating in pursuing stacks? Both nontechnically and technicaly. I know they are dual to each other and that they are characterized by cohomological properties inspired by the proper or smooth base change theorem in algebraic geometry, but what is the relation? (I don't know the statement of the theorems) Finally, what are smooth and proper functors good for? Are smooth and proper functors fibrations and cofibrations or Grothendieck fibrations and Grothendieck op-fibrations in some model categories or derivators? The only thing i could find about smooth and proper functors on internet is the last entrance in http://golem.ph.utexas.edu/category/2008/01/geometric_representation_theor_1... Best regards Rafael Borowiecki