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Matteo Capucci (he/him) (Mar 20 2025 at 10:10):At [[bireflective subcategory]], the nLab says:
image.png
though in the paper they say:
image.png
which seems to me to corresponds to an adjoint string
Who is right? What are subcategories with both adjoints to inclusion called?
Nathanael Arkor (Mar 20 2025 at 10:56):Aren't Kelly & Lawvere the ones who introduce that terminology? In that case, they are "right" by definition. The comment on the nLab about essential localizations should be removed.
Nathanael Arkor (Mar 20 2025 at 11:03):What are subcategories with both adjoints to inclusion called?
The mistaken reference to essential localizations aside, the nLab is correct in that the term "bireflective subcategory" is used in several different senses, among them being both reflective and coreflective (e.g. in Wilbur's Reflective and coreflective hulls in the category of locally convex spaces). I think it is reasonable to use that term as long as you explain what you mean by it.