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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.

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In section 3 of Laures-McClure's paper "Multiplicative properties of Quinn spectra" (arxiv.org/abs/0907.2367) there is some kind of axiomatization of bordism with corners - is it anything like what you're looking for? –  Tyler Lawson Oct 21 '12 at 16:54
    
@Tyler Lawson: Thank you very much! I hadn't been aware of this paper, and now I'll have a look at it. –  Daniel Moskovich Oct 22 '12 at 12:31
    
Kreck's stratifiolds him.uni-bonn.de/homepages/prof-dr-matthias-kreck/… may be close in spirit to what you are asking for. –  Eugene Lerman Oct 22 '12 at 14:09
    
@Eugene Lerman: In what sense? Could you give some details? –  Daniel Moskovich Oct 22 '12 at 16:00
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you might start with n-ads as in chapter 0 of Wall's book. –  Paul Oct 22 '12 at 17:09
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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).

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