Congruence lattices of ideals in categories and (partial) semigroups This monograph 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. Keywords: Categories semigroups congruences H congruences lattices ideals chains of ideals diagram categories partition categories Brauer categories Temperley Lieb categories Jones categories transformation categories linear categories projective linear categories partial braid categories