Relative dimension of morphisms and dimension for algebraic stacks

Motivated by applications in moduli theory, we introduce a theory of lower bounds on relative dimension of morphisms of schemes and algebraic stacks. The theory is robust, applies to a wide range of situations, and has strong consequences. We thus obtain simple tools for making dimension-based deformation arguments on moduli spaces. In a complementary direction, we develop the basic properties of codimension for algebraic stacks.