I am trying to prove something about free profinite modules, but the sticking point is entirely an issue about profinite spaces. Specifically, I need to express a profinite space $X$ in a very specific way.
Let $X = lim_{\leftarrow} X_i$ where $X_i$ are profinite spaces, i.e., each $X_i$ decomposes as $lim_{\leftarrow} X_{i j}$ (where $j$ are not necessarily picked from index sets of the same cardinality) where each $X_{i j}$ is finite. Is there a way view these specific $X_{i j}$ as an inverse system whose inverse limit is then $X$?
Say the $X_i$ are indexed by $I$. Then it seems reasonable to try and take an indexing set $J$ which is the "largest" among the index sets needed to write the $X_i$'s as limits of $X_{i j}$'s, and then index all of the $X_{i j}$'s by $I \times J$, and any "extra" $X_{i j}$'s needed can just be repeats of the already-available ones. But even if the issue of how exactly to index the $X_{i j}$'s is settled, the problem I have is getting maps from, say, $X_{i_1 j_1}$ to $X_{i_2 j_2}$ where $i_1 \geq i_2$ and both sets involved are infinite.
There's an "obvious-looking" way to define something like that if I draw a diagram of the situation (basically, take anything in the preimage of the element under the map $X_{i_1} \rightarrow X_{i_1 j_1}$ and then map that to $X_{i_2}$ in the already existing way, and on to $X_{i_2 j_2}$ using $X_{i_2}$'s projection map), but it's not clear that that will be well-defined. It seems necessary to get this to work at least for "large enough" $j_1$ given a fixed $j_2$ in order to have the desired inverse system.
If the eventual goal is to show that the inverse limit of profinite spaces is profinite, then we don't need to try to do this in detail, but use that $X$ is profinite if $X$ is zero-dimensional Hausdorff and compact, and it's a well-known fact that these three properties are all preserved by inverse limits (because it's a closed (in a Hausdorff setting) subspace of the product).
There should maybe be some canonical way to see that the inverse limit of inverse limit spaces is again an inverse limit of the same spaces by combining the diagrams in some way, but others might have a good idea there (or a reference?). Maybe the index set $K = \{(i,j): i \in I, j \in J_i\}$ works (where $J_i$ is the index set used for $X_i$ with spaces $X_{i,j}$, with space $X_{(i,j)} = X_{i,j}$ as associated space and order $k_1 \le k_2$, where $(k_1 = (i_1, j_1), k_2 = (i_2, j_2)$ iff $i_1 \le i_2$ or ($i_1 = i_2$ and $j_1 \le j_2$, the latter in $J_{i_1} = J_{i_2}$), the problem (IMHO) being that I don't see an easy map from $X_{(i_2,j_2)}$ to $X_{(i_1, j_1)}$...