  This paper presents a unified framework for determining the congruences on a
number of monoids and categories of transformations, diagrams, matrices and
braids, and on all their ideals. The key theoretical advances present an
iterative process of stacking certain normal subgroup lattices on top of each
other to successively build congruence lattices of a chain of ideals. This is
applied to several specific categories of: transformations; order/orientation
preserving/reversing transformations; partitions; planar/annular partitions;
Brauer, Temperley--Lieb and Jones partitions; linear and projective linear
transformations; and partial braids. Special considerations are needed for
certain small ideals, and technically more intricate theoretical underpinnings
for the linear and partial braid categories.
