To be clear I'm not stating anything that I know to be true, just a guess. Let X be a stratified space, with locally closed strata X[j] indexed on the naturals, and such that the closure of X[j] contains X[i] if and only if i >= j. I guess there is some subspace Lk(X[i,j]) < X[i] which is the boundary of a neighborhood of the singular subspace X[i+1,j] inside of X[i,j], and a map from the link to X[i,j]. Then I hope that one can describe X[i,j] as the (homotopy) union X[i] cup_{Lk} X[i+1,j].