# 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)?