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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJul 10th 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAccording to [[stuff, structure, property]], we say that a functor "forgets structure" if it is faithful, "forgets properties" if it is fully faithful, "forgets property-like structure" if it is pseudomonic, and so on. But what if I have a functor that is not faithful, but is conservative? Is there anything I can say that it forgets? I am motivated by one of the comments on [this question](http://mathoverflow.net/questions/173546/why-localize-spaces-with-respect-to-homology/173627) which suggests that the localization of a space $X$ with respect to a homology theory $E$ can be thought of as "the $E$-homology of $X$ equipped with as much structure as possible". The sense in which this is true is that $E$-localization is a factorization of $E$-homology as a localization (i.e. a coinverter) followed by a conservative functor. So it's not "structure" in the same sense as stuff-structure-property, but the factorization seems analogous to the "Postnikov towers" used there.

   According to stuff, structure, property, we say that a functor “forgets structure” if it is faithful, “forgets properties” if it is fully faithful, “forgets property-like structure” if it is pseudomonic, and so on. But what if I have a functor that is not faithful, but is conservative? Is there anything I can say that it forgets?

   I am motivated by one of the comments on this question which suggests that the localization of a space $X$ with respect to a homology theory $E$ can be thought of as “the $E$-homology of $X$ equipped with as much structure as possible”. The sense in which this is true is that $E$-localization is a factorization of $E$-homology as a localization (i.e. a coinverter) followed by a conservative functor. So it’s not “structure” in the same sense as stuff-structure-property, but the factorization seems analogous to the “Postnikov towers” used there.