complexity theory – What is the comparator circuit?

The standard circuits $AC^i$, $NC^i$ are constructed using $AND$, $OR$ and $NOT$ of various fan-ins, fan-outs and depths.

What is the comparator gate constituted from?

Structurally why is it believed $NC^i$ is not in any $CC^j$ if $igeq 1$ and why is it believed $CC^i$ is not in any $NC^j$ if $igeq 1$ (refer https://arxiv.org/abs/1208.2721 for the conjecture $NC$ and $CC$ are incomparable)?