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Just wondering about the dual of localization, I see from this Greenlees and Shipley paper that there is
…the process of cellularization (sometimes known as colocalization or right localization)
nLab appears to have none of these. Is it simply a less important construction?
The sense of “localisation” used here is that of “reflective localisation”. One could equally well say “coreflective localisation” instead of “colocalisation”.
Is there a use for the dual concept in non-(co)reflective situations?
Dual of what? The dual of reflective localisation is coreflective localisation.
If colocalization in a category is equivalent to localization in the opposite category, and if not all localization is reflective, then presumably there are occasions when people are using the term ’colocalization’ where it doesn’t mean coreflective localization.
Localisation in the broad sense of inverting morphisms is a self-dual concept.
Are you talking about localisation as a weighted colimit in Cat, David?
I wasn’t thinking about it in any more precise a sense than at localization. I was just thinking about modalities, and how just as HoTT blurs a distinction that philosophers like to make between propositions and types, that modal operators applied to general types appears odd to those used to them acting only on propositions, as mentioned here. So I was just fishing about for operations which behave in a modal kind of way, and their duals.
David, the category of algebras over an idempotent monad forms a reflective localization of the ambient category, while that of an idempotent co-monad forms a co-reflective localization. And conversely. So (co-)localizations in the reflective sense are equivalently (co-)modal types for idempotent (co-)modalities.
This is discussed at idempotent monad – Properties – Algebras for an idempotent monad and localization and briefly pointed to also from reflective subcategory – Properties – As Eilenberg-Moore category of the idempotent monad.
I have added some of that discussion about modal operators expressing a “way of being” of types at the beginning of modal type.
For what it’s worth, this is certainly a notion that’s used in the stable homotopy category of spectra. To me, colocalization means “acyclization.” In other words, one can localize at a localizing subcategory (which could be thought of a category of acyclics of some homology theory, possibly), and colocalize at a colocalizing subcategory (which could be thought of as a category of locals for some homology theory). These notions are dual in a pretty meaningful way, in that there is a fiber sequence $CX\to X\to LX$, where $C$ is the colocalization functor, and $L$ is the localization functor.
To rephrase what Zhen and Jon are saying, the coinverter of a class of morphisms in $C$ is the opposite of the coinverter of the same class of morphisms in $C^{op}$. (The inverter is a totally different kind of dualization.) But often when people (especially homotopy theorists) say “localization” they mean specifically a coinverter with a fully faithful right adjoint, and this can be dualized to a coinverter with a fully faithful left adjoint, which is thereby called “colocalization”.
Thanks for all the comments. I think there’s more that has been said here than contained in the $n$Lab, especially in terms of connections between terms - coinverters, modalities, etc.
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