added pointer to:

- Saunders MacLane, §IV.3 of:
*Categories for the Working Mathematician*, Graduate Texts in Mathematics**5**Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

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.

]]>Fix aforementioned error

]]>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…”

]]>Use $L \dashv R$ rather than $Q^* \dashv R_*$. (The latter can be confusing, because $({-})_*$ is often used to denote left adjoints, whereas $L$ and $R$ are unambiguous).

]]>Mention that full faithfulness of a right adjoint is equivalent to density of the left adjoint.

]]>These are helpful tips. I have put some names of categories in \mathsf font. I will change it to roman to match the rest of nLab. ]]>

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“.

Fixed typo in the Cat \to sSet example.

Christopher James Stough-Brown

]]>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

]]>I gave the Prop. with “alternative characterizations” a proof, by pointing to the relevant sub-propositions proved in other entries.

]]>Perhaps the link to the paper should not be to the pdf? And better would be a proper human-readable reference.

]]>Added to *reflective subcategory* a reference by Adámek and Rosický about non-full reflective subcategories.

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* 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*.

I have now added both of these comments to the example.

]]>Proposition 4.4.3 in

- A.L. Rosenberg,
*Noncommutative schemes*, Comp. Math. 112, 93–125 (1998)

is a noncommutative analogue.

]]>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”).

Added an example: The category of affine schemes is a reflective subcategory of the category of schemes, with the reflector given by $X \mapsto Spec \Gamma(X,\mathcal{O}_X)$.

]]>I replaced statement (3) in Proposition 1 to make the statements equivalent. (Statement (3) previously only had the part about idempotent monads.)

]]>I wonder whether it’s explicit in Makkai-Pare; I don’t have my copy of that with me in Princeton.

]]>Thanks! I have put that into the entry here.

]]>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).

]]>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?

]]>