Meaning
A bigraded finite realization carrier places the realization data over both the gauge and spacetime indices.
It consists of objects C_D^(p,q), gauge and spacetime comparison operators, and mixed comparison cells. The two degrees say which kind of overlap is being tracked. The mixed cells explain how the two orders of comparison are related.
Example (illustrative)
At degree (1,1), a comparison cell can relate regional restriction after a gauge change to gauge change after regional restriction. Its source and target remain explicit even when they are isomorphic.
Scope and limits
Two indices and two operator symbols do not automatically make a strict double complex. Differential and higher coherence laws must be established in the relevant coefficient system.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 5.7.