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James Deikun (Mar 27 2025 at 14:45):Why is Rezk completion usually specified as localizing with respect to the projection from the walking isomorphism to the point (by nature an acyclic fibration) rather than one of the injections of the point into the walking isomorphism (by nature an acyclic cofibration)? It seems like they should be generating the same strongly saturated class by 2-out-of-3, but for reflections the class of morphisms that gets squashed is naturally cofibrantly generated rather than fibrantly generated so it feels like using generating acyclic cofibrations should generalize better to similar situations.
Mike Shulman (Mar 27 2025 at 15:45):It's much easier to answer questions like this if you give specific citations to papers that use terminology in the way you're asking about rather than using vague and arguable words like "usually".
James Deikun (Mar 27 2025 at 16:32):Hm, trying to come up with the references you asked for I discovered that some sources including the nLab use the projection, but Rezk's original paper https://arxiv.org/abs/math/9811037 used an injection to define the model structure for complete Segal spaces, so it's evidently doable.
Mike Shulman (Mar 27 2025 at 18:50):That's another good reason to track down references -- sometimes you discover that what you thought was "usual" isn't at all.
Mike Shulman (Mar 27 2025 at 18:55):One reason that sometimes people choose to localize at a map to the point is that localizations at maps to the point, i.e. "nullifications" at objects, have some better properties than general localizations. For instance, in HoTT a nullification is a modality, whereas a general localization is only a reflective subuniverse. I don't know if this is anyone's motivation in the case of Rezk completion though.