I see, thanks.

Yes, that statement seems clearly wrong, I have deleted it.

For the record, the claim was that:

A functor $F \colon C \to D$ is conservative if and only if $F(f)$ being an identity morphism implies that $f$ is an isomorphism.

A minimal counterexample (of the kind that you, Gnampfissimo, already mentioned): The functor

$\{ a \to b\} \longrightarrow \{ a \overset{\sim}{\leftrightarrow} b \}$reflects identities but is not conservative.

Now I was trying to figure out what alternative statement may have been intended, but I am not sure. If anyone knows, let’s fix the statement and add it back in.

]]>Ah yes that explains it!

Re: revision 27

I was talking of Proposition 2.3 in that version, currently Proposition 2.5. ]]>

I have trouble identifying which other sentence it is that you say has an issue (looking at the version diff is not helpful).

Could you give the item number?

(You can also add anchors into the source, such as “`{#PossibleTypoHere}`

” and then let us hunt that anchor, or even point to it via “`https://ncatlab.org/nlab/show/conservative+functor#PossibleTypoHere`

”)

Thanks for the alert.

I think that word “faithful” was a typo for an intended “fully faithful”. I have fixed that case (here).

Also added a pointer to one place where all these cases are discussed and proved:

]]>@John: in revision 27 you added that invertibility of preimages of identities implies conservativity. Aren't localizations counterexamples?

@Urs: in revision 28 you added that faithful conservative functors are pseudomonic. Aren't there non-full subgroupoids? ]]>

Added a simple criterion for conservativity and rearranged “Examples” section so simpler ones come first.

]]>Relate to pseudomonic functors.

]]>Mention that monadic functors are conservative.

]]>Gave more precise reference to a cited result from the Elephant.

]]>Yay, thanks Urs!

Jesse #20: Makes sense. So it’s enough to talk about a regular functor between regular categories?

]]>how did you manage to edit the nLab? It’s not letting me edit any pages right now, trying to save a page just takes me back to the edit page.

If that’s so since that incident from Friday, then deleting cookies should help.

That’s anyway what happened for me: After Adeel had fixed the editing behaviour the problem did remain on my end, but clearing cookies made it go away.

]]>@Mike #19: that’s strange—I edited the page the usual way. I tried editing the Sandbox just now and while it took a while to accept the edit, it ended up working.

Re: #18, I believe the proof only requires that you can talk about images of maps as subobjects and that the functor preserves images and meets in the subobject lattices.

]]>Also: how did you manage to edit the nLab? It’s not letting me edit any pages right now, trying to save a page just takes me back to the edit page.

]]>Thanks! Is it necessary to have pretoposes there? Is one direction or the other, at least, true more generally? (I should know that, but I don’t have time to think about it right now…)

]]>I also added the statement that for pretoposes, conservative pretopos morphisms are precisely those which induce injective maps on subobject lattices. (This is the notion of conservativity which is used in Makkai-Reyes.)

]]>Added statement that every fully faithful functor is conservative: here.

]]>I agree that structure vs property is relevant; but “having finite limits” is actually (like “being a group”) a *structure*, not a property, precisely because not all functors between categories with finite limits preserve them. It is a special sort of structure called a “property-like structure”, meaning that it is “unique when it exists” (more precisely, it is preserved by all *equivalences*), but it is still a structure.

I’m not just making a philosophical point either; this is an empirical observation about mathematical language. I’ve never ever heard a mathematician talk about a functor “preserving finite limits” when it wasn’t known or implicitly assumed that its domain and codomain had them.

]]>Could the perspective of “structure vs. property” help here? I totally agree that the question “is this function a group homomorphism?” is ill-typed unless source and target are groups; being a group homomorphism means being compatible with the group structure, which has to be given for the question to make sense.

However, at the moment, my personal reading of “$F:\mathcal{C}\to\mathcal{D}$ preserves $\mathcal{I}$-shaped limits” is still the following, which I believe is well-typed irregardless of whether $\mathcal{I}$-shaped limits actually exist in the source category: “For any diagram $G:\mathcal{I}\to\mathcal{C}$, the image under $F$ of any $G$-cone which happens to be a limiting cone is again a limiting cone.”

Maybe the difference is in saying “$F$ preserves limiting cones” (which should make sense even if there are no limiting cones) vs. “$F$ preserves limits” (which shouldn’t make sense if no limits are specified).

]]>To me, asking whether a functor preserves finite limits when its domain doesn’t have finite limits is like asking whether a function is a group homomorphism when its domain is not a group: it’s an ill-typed question. The fact that flat functors are “functors that would preserve finite limits if they existed” is a nice way to think about them, but being flat is still a different thing from preserving finite limits.

]]>Oh well, I guess you’re right. If $B G$ is a group, then flat functors into $Set$ correspond to torsors, and any functor $B G \to Set$ is conservative.

]]>I’m not sure that that’s true. Isn’t it possible to have a flat and conservative functor whose codomain has finite limits but whose domain does not?

]]>Maybe the reconciliation is that the functor $F: \mathcal{C} \to \mathbb{1}$ of #8, or its prolongation $y \circ F: \mathcal{C} \to Set$ along the Yoneda embedding $y: \mathbb{1} \to Set$, is not a flat functor. See the section in preserved limit which discusses the case where limits in the domain don’t exist.

So I presume that a suitable resolution for #6 is to say that a conservative functor $F: \mathcal{C} \to \mathcal{D}$ reflects limits that are (hypothetically) preserved, in the sense of that section. Is it true that if $F$ is conservative, then so is the corresponding left Kan $L F: Set^{\mathcal{C}^{op}} \to Set^{\mathcal{D}^{op}}$?

]]>I would say that in that case $F$ does not preserve terminal objects, or rather that it is ill-typed to even ask whether it does, because its domain does not have terminal objects.

]]>I thought so – that it only would be a language problem, I’m sure we agree on the formal mathematical statement – but can’t quite reconcile the wording with my understanding. Here is an explicit example:

Let $\mathcal{C}$ be any groupoid in which there is an object $X$ with at least two endomorphisms. Such a category doesn’t possess a terminal object. Therefore the unique functor $F : \mathcal{C} \to \mathbb{1}$ into the terminal category vacuously preserves terminal objects. Since any morphism in $\mathcal{C}$ is invertible, this functor is also conservative. But it doesn’t reflect terminal objects:

Consider the cone $X$ on the empty diagram in $\mathcal{C}$. Its image under $F$ is a limiting cone, since $F(X)$ is a terminal object in $\mathbb{1}$. But the considered cone is not.

]]>