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.

directlyfrom filtered spaces, via homotopically defined functors. Problem 16.1.17 is about relating these methods to Morse Theory. $\endgroup$ – Ronnie Brown Jun 22 '13 at 10:48