Hodge duality is linear transformation, among fully antisymmetric tensor fields on a smooth manifold, derived from the action of the alternating algebra on a metric. That's an awful lot of jargon – which I'll try to explain briefly below, but you can also follow links to fuller explanations – so now here's a diagram that I'll use in describing the relationships among the parts:

… when I get round to writing more on this …[unfinished]

Written by Eddy.