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edited reflective subcategory and expanded a bit the beginning
I added a query and done a bit more succinct sectioning: distinguishing characterizations from just properties.
replied to Zoran’s old query box query
added the theorem about reflective subcategories of cartesian closed categories from cartesian closed category
added at reflective subcategory a new subsection Accessible reflective subcategory with a quick remark on how accessible localizations of an accessible category are equivalently accessibe reflective subcategories.
This is prop. 5.5.1.2 in HTT. Can anyone give me the corresponding theorem number in Adamek-Rosicky’s book?
I can’t find it stated explicitly. But “only if” follows immediately from 2.53 (an accessibly embedded subcategory of an accessible category is accessible iff it is cone-reflective), while “if” follows immediately from 2.23 (any left or right adjoint between accessible categories is accessible).
Thanks! I have put that into the entry here.
I wonder whether it’s explicit in Makkai-Pare; I don’t have my copy of that with me in Princeton.
I replaced statement (3) in Proposition 1 to make the statements equivalent. (Statement (3) previously only had the part about idempotent monads.)
Added an example: The category of affine schemes is a reflective subcategory of the category of schemes, with the reflector given by $X \mapsto Spec \Gamma(X,\mathcal{O}_X)$.
Thanks! There is much discussion of the infinity-version of this example at function algebras on infinity-stacks (following what Toën called “affine stacks” and what Lurie now calls “coaffine stacks”).
Proposition 4.4.3 in
is a noncommutative analogue.
I have now added both of these comments to the example.
Added to reflective subcategory the observation that (assuming classical logic) $Set$ has exactly three reflective subcategories. I learned this from Kelly’s and Lawvere’s article On the complete lattice of essential localizations.
Thanks for all your additions recently!
Here is a direct pointer to the edit that, I suppose, you are announcing. (Providing direct links like this within an entry of non-trivial length makes it easier for us all to spot what you are pointing us to.)
By the way, we have an entry subterminal object. I have added cross-links with subsingleton.
Added to reflective subcategory a reference by Adámek and Rosický about non-full reflective subcategories.
Perhaps the link to the paper should not be to the pdf? And better would be a proper human-readable reference.
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