TITLE: Is there a notion of a chain complex with corners?
QUESTION [16 upvotes]: 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.

REPLY [9 votes]: The chain complex n-ads in my 1992 CUP book Algebraic L-theory and topological manifolds are chain complexes with corners. They are the chain complex analogues of Wall's n-ads (which hark back to J.H.C. Whitehead).