Applying Map to two disconnected levels

Questions
1

I am want to apply the Map function to a nested list, via:
Map[f, {{a1, b1}, {a2, b2}, {a3, b3}}, {0, 2}] and get the result:
f[{f[{f[a1], f[b1]}], f[{f[a2], f[b2]}], f[{f[a3], f[b3]}]}]

However, I want the function f to only be applied at levels 0 and level 2. Map is applying f on levels 0 to 2. How is this accomplished? The result l want is:
f[{ {f[a1], f[b1]}], f[{f[a2], f[b2]}], f[{f[a3], f[b3]} }]

How can this be done? Given the generality of the software, I assumed this was possible. Thanks.

2

It seems that you cannot do it directly with Map, but there are workarounds:

mat = {{a1, b1}, {a2, b2}, {a3, b3}};
f@Map[f, mat, {2}]
Fold[Map[f, #1, {#2}] &, mat, {2, 0}]
3

Using Apply for level 0:

mat = {{a1, b1}, {a2, b2}, {a3, b3}};
f @@ Map[f, mat, {2}]

(* f[{f[a1],f[b1]},{f[a2],f[b2]},{f[a3],f[b3]}] *)
4

Your expected result is syntactically invalid, and so I’m actually not sure how you want to apply f. It seems like some combination of Map-ing and Applying. Also, I see f at three levels in your expected result, but you say you only want to apply it at two levels. Please correct your expected output or clarify your question.