Stratified Cohomological Quantum Codes via Colimits in Ch(R)

Abstract

We introduce \emph{stratified colimit codes}: stabiliser codes obtained by taking the degree-wise colimit $\mathcal C_\bullet(X):=\operatorname*{colim}_{\sigma\in X}F(\sigma)$ of a functor $F\colon X\to\mathbf{Ch}(R)$ from a finite poset into the category of chain complexes over a commutative ring~$R$. Axioms requiring only transitivity and boundary-compatibility of the morphisms in $F$ ensure that $\partial^2=0$, so the homology $H_\bullet$ and cohomology $H^\bullet$ furnish the usual CSS $Z$- and $X$-type logical sectors; torsion in $H_\bullet$ classifies qudit charges via the universal coefficient sequence. Varying $F$ recovers classical surface and color codes, $\mathbb{RP}^2$ torsion codes, twisted toric families with rate $k\sim d$, and X-cube style fracton models, all without referencing an ambient cell complex. Matrix Smith normal form (PID case) and sparse Gaussian elimination (field case) compute $H_\bullet$ directly, giving LDPC parameters that inherit the sparsity of $F$. Because the construction is ring agnostic and functorial, it extends naturally to code surgery (push-outs) and, at the next categorical level, to bicomplex domain walls. Stratified colimit codes therefore supply a concise algebraic chassis for designing, classifying, and decoding topological and fractal quantum codes without ever drawing a lattice.