G-Man (Half-Life) refers to simply as his "employers". His mysterious nature has made him an icon of the "Half-Life" series, with his identity and motives remaining almost completely unexplained. The G-Man's name comes from "G-Man", an American colloquialism meaning "Government Man". However, within the "Half-Life" games, he is identified by this name only in the program code and the multiplayer profile menu as a default name option, not within the story itself; his name is derived from his "Half-Life" model and entity name and has since been reused in subsequent games of the series. His name has been confirmed and he has
G-structure on a manifold misleading: it suggests that "H" is a subgroup of "G", which is often the case, but need not be (for example for spin structures): it's properly called a lifting. More abstractly, ""G"-bundles over "X"" is a functor in "G": given a map "H" → "G", one gets a map from "H"-bundles to "G"-bundles by inducing (as above). Reduction of the structure group of a "G"-bundle "B" is choosing an "H"-bundle whose image is "B". The inducing map from "H"-bundles to "G"-bundles is in general neither onto nor one-to-one, so the structure group cannot always be reduced, and when it can,