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Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A \hookrightarrow X$ is a cofibration, or that $(X,A)$ is a neighborhood deformation retract (NDR) pair. All maps $f:(X,A) \to (Y,B)$ are continuous and satisfy $f(A) \subset B$.

Consider a sequence $\mathcal{S}$ given by: $$\ldots \to (X_{n-1},A_{n-1}) \stackrel{d_{n-1}}{\to} (X_n,A_n) \stackrel{d_n}{\to} (X_{n+1},A_{n+1}) \to \ldots$$ so that $d_n \circ d_{n-1}(X_{n-1}) \subset A_{n+1}$ for each $n$.

Has this object been defined and studied? If so, where?

One can associate to this sequence of space-pairs a "homology" which takes values in the category of space-pairs. That is, define $$\mathcal{HT}_n(\mathcal{S}) = \left(d_n^{-1}(A_{n+1}),d_{n-1}(X_{n-1})\right).$$ The construction is functorial if one considers the obvious analogue of "chain maps" in the category which contains objects like $\mathcal{S}$. I'm wondering if basic definitions and properties, etc. of this or related constructions have been set down somewhere.

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Where did you meet this structure? In a concrete problem or just out of curiosity? – Johannes Ebert Jun 22 '13 at 9:33
Johannes: somewhere in between. I'd call it concretely-motivated curiosity. My thesis work involved simplifying a filtered cell complex via discrete Morse theory in a way that preserved algebraic topological invariants (persistent homology groups). As such, I find the process of constructing massive chain groups and then taking a huge quotient somewhat inefficient, and was wondering if there is a "compact representation" (both words used non-technically) of some object in the category of topological spaces itself from which one could derive the invariants whenever necessary. – Vidit Nanda Jun 22 '13 at 10:22
@Vidit: You mention a "filtered cell complex". The book "Nonabelian algebraic topology" (2011) see my web page, , builds algebraic topology directly from filtered spaces, via homotopically defined functors. Problem 16.1.17 is about relating these methods to Morse Theory. – Ronnie Brown Jun 22 '13 at 10:48
Thank you, Ronnie. I already have your book(s), being a fan if the groupoid point of view, so I will take a look. – Vidit Nanda Jun 22 '13 at 10:53
Ronnie, I'm not sure I see an "obvious" bifiltration anywhere in this question. Could you explain why that would be the natural thing to do? I am happy to discuss this via email if things get too intricate for comment boxes. – Vidit Nanda Jun 23 '13 at 15:57

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