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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.
Added an idea section. Expanded the section regarding limits and colimits in a reflective subcategory (one sentence for the case when the ambient category is complete and complete) and added a reference.
I also added two examples. The first example is that Cat is a reflective subcategory of sSet. The second is example is that for a Lawvere theory T, its category of models is a reflective subcategory of the category of all functors from T to Set. I added references for these examples.
This is my first contribution to nLab so my code hygiene may not be the best. If you have any suggestions on anything please let me know and I’ll be happy to make fixes.
Jade Master
Jade, thanks for contributing!
I made just some minor cosmetic edits to the Idea-section here that you added, such as hyperlinking more of the keywords.
Here some hints on editing:
Hyperlinks within the nLab are and should be obtained simply by including the word/phrase to be hyperlinked in double square brackets.
For instance [[Grp]]
produces Grp. Most entries have all possible variants of their titles declared as redirects. For instance [[category of groups]]
produces category of groups which gives the same link as Grp. Should the redirect not exist, you can either a) add a redicrect (look at the very bottom of the source code of any page to see how) or b) force it on the spot with a vertical slash: [[Grp|Groups]]
produces Groups, which once again links to the same page as the previous two cases.
Instiki maths is almost exactly like LaTeX maths, except for some small differences. One of them is the “feature” that consecutive letters in maths mode are rendered in roman font. This is useful when declaring category names such as in $AbGroups \to Groups$
, because this gets rendered as intended $AbGroups \to Groups$. But it means that for operators applied to variables, such as in Tc \to Td
get rendered as $Tc \to Td$ instead of as the intended $T c \to T d$. For the latter you need to introduce a whitespeace and type ‘T c \to T d“.
A typo for when we get editing back: In Theorem 4.1 “If X is a reflective subcategory…” should be “If C is a reflective subcategory…”
I added another equivalent characterization of adjunctions for which the counit $\varepsilon : L R \to 1_B$ of the adjunction is a natural isomorphism. Namely, it’s enough that there exists some natural isomorphism $L R \to 1_B$! I added a reference to Johnstone and a sketch of the proof.
I find this a delightful simplification.
added pointer to:
Following discussion in another thread (here)
I have deleted the string
apocalypticists
(apparently introduced in revision 111, but you need to look into the source code of the revision, via “rollback”, to see this)
which, in immediately following the closing =--
of an Example-environment, caused the whole remainder of the entry not to be rendered anymore.
What is the relation between the notions of a reflexive subcategory and a retract of a category?
The definitions of the two notions look similar, and the page should clarify the relationship, I think. Is it the same as a retract of a category such that the retraction is fully faithful?
Are you thinking of ordinary retracts in the 1-category of categories, or did you see another definition of “retract of categories” anywhere?
Besides not being invariant under equivalence of categories, this misses the structure in the adjunction between the functors $A \to C$ and $C \to A$.
I think you’d want the “inclusion” A → C to be fully faithful anyway, rather than the “reflector” C → A.
Let’s sum this up:
A reflective subcategory is first and foremost a pair of adjoint functors
$\mathcal{A} \underoverset {\underset{\iota}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{C}$such that
which means equivalently that
This is probably where you make the connection to retracts:
If we take $\mathcal{A}$ and $\mathcal{C}$ to be represented strict categories
and if it so happens that with that presentation the components of the adjunction counit are not just isomorphisms but in fact equalities,
(which are some strong “if”s)
then the above reflection is in particular a retract in the 1-category of strict categories. It’s still a bit more, even in that case.
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