Roughly speaking, algebraic topology works by reducing questions about topological objects such as manifolds and cell to questions about chain complexes.

On the topological side, although in the PL category a manifold can have a boundary and not much more, in the smooth category there is a notion of a manifold with corners, that is that every point has a neighbourhood diffeomorphic to $\mathbb{R}_{\geq 0}^n$. To go further, there is a notion of a *manifold with faces*, which adds an additional piece of stratified structure whose existence guarantees that each piece of the boundary has a smooth collar in the manifold (See Appendix A of Farber's Topology of Closed One-Forms).

On the algebraic side, there is a notion of the boundary of symmetric chain complex (I think due to Ranicki), which measures the chain-level failure of Poincare-Lefshetz duality.

Question: Is there a notion of a chain complex with corners or with faces that has been studied in the literature?

It's not hard to imagine how this would work, by mapping a symmetric chain complex with boundary to a boundary of a symmetric chain complex, for example; but I'm asking whether there is any literature on such structures.