Find a functorial zig-zag of spaces

Question

This is a rather broad question. Suppose you have an ordinary category $C$ (for example, $\Delta$), and two diagrams $X_{\bullet}, Y_{\bullet} : C \to \textrm{Top}$. Suppose also that $X_c$ is homotopy equivalent to $Y_c$ for all $c \in C$. Are there techniques to find functorial zig-zags: $$ X_{\bullet} \leftarrow Z_{\bullet} \to Y_{\bullet} $$ that are objectwise equivalences?

I am aware of one more-or-less general technique known in folklore as the "Berger argument", though I don't have a reference. I vaguely know how it works in case $X_c$ is the realization of a finite poset $P_c$ (for example, I think that for a regular CW complex one can consider $P_c$ to be its face poset). It goes like this:

1. Find a functor $K_c : P_c \to \textrm{Top}$ such that $K_c(p)$ is contractible for all $p \in P_c$ and $\varinjlim K_c \simeq Y_c$
2. Given $p \in P_c$ let $P_c^{< p} = \{q \in P_c: q < p\}$. Then show that $\varinjlim K_c |_{P_c^{< p}} \to K_c(p)$ is a cofibration
3. For all $f: c \to d$ in $C$, the map $Y_f : Y_c \to Y_d$ can be written as the colimit of a diagram map $K_f : K_c \to K_d$ (that is, the map is stratified with respect to the strata).

Then there exists such object $Z_{\bullet}$, constructed as follows: $$ X_c \simeq |N(P_c)|=\textrm{hocolim}_{P_c}(*) \leftarrow \textrm{hocolim}_{P_c}K_c \rightarrow \textrm{colim}_{P_c}K_c \simeq Y_c $$ And the maps are natural with respect to $c \in C$.

It is welcome... Any other known technique to find (functorial) zig-zags of spaces (or chains), and any precise reference of the above argument. Thanks!!

Answer 1

If the diagrams $X_\bullet$ and $Y_\bullet$ are constructed canonically, but not necessarily naturally, from spaces $X,Y$ where $X \simeq Y$, one technique to construct such a zigzag is pick $f: X \simeq Y$ and construct a diagram $\mathrm{cyl}(f)_\bullet$ based off the construction applied to the mapping cylinder $\mathrm{cyl}(f)$. If the construction of the diagram is natural with respect to inclusions, then we will have a zigzag $$X_\bullet \rightarrow \mathrm{cyl}(f)_\bullet \leftarrow Y_\bullet$$

which will often be a zigzag of equivalences. Similarly, if one was doing work in surgery theory or Waldhausen A-theory or the ilk, you might replace the role of the mapping cylinder with an H-cobordism.