Rewriting Homology and Filtrations in Iterated Free 2-Category Constructions

We introduce a homological framework associated with iterated free strict 2-category constructions on 2-polygraphs. Starting from the classical free/forgetful adjunction between strict 2-categories and polygraphs, we consider the repeated application of the free construction and obtain a natural filtration indexed by rewriting depth. From this filtration, we construct an associated graded object and define a boundary operator induced by truncation between successive levels. This yields a chain complex whose homology groups measure the failure of stabilization of the rewriting process under iteration. The resulting “rewriting homology” provides a way to quantify non-idempotent behavior of free higher categorical constructions in terms of homological invariants. The construction is functorial with respect to morphisms of polygraphs and fits into the general framework of categorical homological algebra. This version is a preliminary preprint intended to fix the construction and definitions for further development.