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 == exterior algebra applied to vector fields == 

 The phrase ""The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions"" is misleading.  I already mentioned this above, but the phrase has not been modified yet.  The phrase makes it appear as if the exterior algebra construction is being applied to the vector space formed of such vector fields, etc.  (Particularly in the context of the ""universal construction"", it sounds as if we are now applying the construction to this infinite dimensional vector space.)  This is of course not the case, as has been discussed both here and WPM.  Rather, the exterior algebra construction is being applied pointwise to each (co)tangent space.  In other words, we have a bundle of exterior algebras, rather than the exterior algebra of the space of sections of the (co)tangent bundle.    "