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created a bare minimum at sharp map
I added in Rezk’s original definition. It agrees with the intuitive definition in the case of a right proper model category, but in general there seems to be a subtle difference.
Ah, thanks!
@ZhenLin: Where did you actually find this particular definition (i.e., base changes are also homotopy base changes) in Rezk’s paper?
I am currently looking at Rezk’s paper and he defines sharp maps at the beginning of Section 2 and his definition is different from the one that is currently used in the article sharp map, and in fact one can construct examples of model categories (necessarily not right proper) where these two definitions are actually different.
If the model category is right proper, then Rezk’s Proposition 2.7 does establish the equivalence of these two definitions.
@ZhenLin: Sorry, I guess I misunderstood which definition was actually added by you.
Anyway, the current article has two nonequivalent definitions, and only the second one describes sharp maps.
The definition in the “Idea” section is also a very important notion (in fact, I consider it to be more important than sharp maps), but I don’t know a name for it.
It only coincides with sharp maps for right proper model categories, but in general it’s very different from sharp maps.
In fact, I asked a question about this on MathOverflow: http://mathoverflow.net/questions/181807/reference-for-maps-whose-pushouts-are-also-homotopy-pushouts but so far there are no replies.
I ended up calling these maps “i-fibrations”, and in my paper with Jakob Scholbach (http://arxiv.org/abs/1410.5675) we investigate the dual notion of i-cofibration quite extensively, in particular, we prove that (acyclic) i-cofibrations are weakly saturated under some mild additional conditions on the underlying model category and the class of acyclic i-cofibrations coincides with the class of couniversal weak equivalence (i.e., maps whose cobase changes are weak equivalences).
My fault. Fixed it.
The intuitive definition is given by Rezk in the introduction:
In this paper we study a class of maps called sharp maps. In our context, a map $f : X \to Y$ will be called sharp if for each base-change of $f$ along any map into the base $Y$ the resulting pullback square is homotopy cartesian.
I am inclined to agree that this is the more important notion, but it has the disadvantage of requiring a (good) notion of homotopy pullback.
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