Element-wise test on List elements

To my knowledge, there aren't built-in versions for comparison operators that would be automatically threaded over lists. One reason for that is that Mathematica is a symbolic system, and every auto-simplification has a cost, because there may be cases when this isn't desirable.

It is relatively easy however to construct the behavior you want:

ClearAll[l];
l[f_] := Function[Null, f[##], Listable]

Now, you can call:

{{0.6, 1.2}, {5, 0.1}} ~ l[Greater] ~ 1

(* {{False, True}, {True, False}} *)

and similarly with other comparison operations.

Note that, since you didn't mention efficiency, I intentionally left this aspect out. If you have large numerical lists, there are vastly more efficient ways to perform the comparisons, making use of vectorization and packed arrays.

The BoolEval` package does exactly this. For example:

BoolEval[{0.6, 1.2} > 1]
(* Out: {0, 1} *)

and

BoolEval[{{0.6, 1.2}, {5, 0.1}} > 1]
(* Out: {{0, 1}, {1, 0}} *)

In order to return True and False instead of 0 and 1, you can append /. {0 -> False, 1 -> True}.

Depth 1

MapAt[Greater[#, 1] &, {0.6, 1.2}, {All}]

{False, True}

OR

Thread[Greater[#, 1]] & @ RandomReal[2, 10]

{True, False, False, True, True, True, False, False, False, True}

Depth 2

MapAt[Greater[#, 1] &, {{0.6, 1.2}, {5, 0.1}}, {All, All}]

{{False, True}, {True, False}}

OR

Thread[Greater[#, 1]] & /@ RandomReal[2, {3, 5}]

{{True, True, False, True, True}, 
 {True, False, False, False, False},
 {False, False, False, False, True}}

Depth n

f=MapAt[Greater[#, 1] &, #, Table[All, (Depth[#] - 1)]] &

f[{{{1.6, 0.2}, {3, 0.1}}, {{0.6, 1.2}, {5, 0.1}}}]

{{{True, False}, {True, False}}, {{False, True}, {True, False}}}

f@RandomReal[2, Range[6]]

OR

RandomReal[2, Range[4]] /. (w_Real -> w > 1)