  Motivated by applications in moduli theory, we introduce a flexible and
powerful language for expressing lower bounds on relative dimension of
morphisms of schemes, and more generally of algebraic stacks. We show that the
theory is robust and applies to a wide range of situations. Consequently, we
obtain simple tools for making dimension-based deformation arguments on moduli
spaces. Additionally, in a complementary direction we develop the basic
properties of codimension for algebraic stacks. One of our goals is to provide
a comprehensive toolkit for working transparently with dimension statements in
the context of algebraic stacks.
