General Landau–Ginzburg matrix-factorization data

Let $k$ be a field, let $(Y,{\mathcal{O}}_Y)$ be a ringed space with commutative structure sheaf of $k$-algebras, and let $W\in \mathrm{\Gamma }(Y,{\mathcal{O}}_Y)$. A matrix factorization of $W$ is a finite-rank locally free $\mathbb{Z}\mathrm{/}2$-graded ${\mathcal{O}}_Y$-module $E=E^{\bar{0}}\oplus E^{\bar{1}}$ together with an odd ${\mathcal{O}}_Y$-linear endomorphism ${\delta }_E$ such that ${\delta }_E^2=W{\mathrm{id}}_E$. A morphism of parity $p\in \mathbb{Z}\mathrm{/}2$ from $E$ to another factorization $F$ is a global section of the parity-$p$ part of the sheaf $\mathcal{H}o{m}_{{\mathcal{O}}_Y}(E,F)$. Composition is composition of sheaf maps.

This definition records the objects and graded morphism modules only. The differential and category axioms are separate mathematical claims; no quotient, idempotent completion, equivalence with a singularity category, or generation claim is included here.