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.

**Update** (6/23/2013):

Here's a question that I thought had a positive answer but now I am not so sure. One can construct an honest chain complex $\text{Ch}(\mathcal{S})$ from $\mathcal{S}$ by considering singular chains; applying the homology functor yields homology groups $H_*(\mathcal{S})$ of this chain complex.

Is there an operation on the space-pair $\mathcal{HT}_n(\mathcal{S})$ which yields the $n$-th homology group $H_n(\mathcal{S})$?

directlyfrom 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