# Is there a notion of a chain complex with corners?

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
you might start with n-ads as in chapter 0 of Wall's book. – Paul Oct 22 '12 at 17:09