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

- the right adjoint functor $\iota$ is fully faithful,

which means equivalently that

- the adjunction counit is a natural isomorphism: $\epsilon \;\colon\; L \circ \iota \overset{\sim}{\rightarrow} id_{\mathcal{A}}$

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.

I think you’d want the “inclusion” A → C to be fully faithful anyway, rather than the “reflector” C → A.

]]>Besides not being invariant under equivalence of categories, this misses the structure in the adjunction between the functors $A \to C$ and $C \to A$.

]]>That is, in the simplest form the claim (conjecture) is:

A full subcategory A of C is reflexive

iff

in the 1-category of categories

there is a retraction A->C->A such that C->A 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?

]]>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?

]]>Removed the *other* “apocalypticists” later down the page.

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.

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.

]]>