R. Brown, P.J. Higgins, R. Sivera, Nonabelian algebraic
topology: filtered spaces, crossed complexes, cubical homotopy
groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages. (August
2011). (referred to below as NAT).
(more details on NAT including a downloadable pdf) on NAT.
Hope that helps.
October 27, 2020 I hope the following comments will help to explain my references.
My general attitude is that we are interested in particular applications of homology to spaces. That means we need to calculate something, and so we need information on the space. That information will have some structure, and so we need to be able to exploit that. A methodology for this is laid out in the paper 2018 Modelling and computing homotopy types: I.
An example is cellular homology, which is conveniently studied in terms of filtered spaces $X_*$; these give rise to a homotopically defined fundamental crossed complex $\Pi( X_*)$, which goes back (with different terminology) to A.L. Blakers (!948) and to JHC Whitehead in his paper "Combinatorial homotopy II" (1949), where the filtration is the skeletal filtration of a CW-complex.
There is a higher Van Kampen Theorem for $\Pi X_*$ which allows for some specific calculations and proofs of for example relative Hurewicz Theorems. See the book NAT.
