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Example 4 (skeleton of monoidal category) has format/wording issues I think.
EDIT: now fixed
Incidentally, the first example (center construction not functorial) is a special case of the second (Aut-group construction not functorial): The center of $G$ is the automorphisms of $id_{B G}$.
(And both are functorial on the core. The issue is that the would-be functor needs to act by conjugation.)
@Ulrik, thanks. But then let’s expand just a little to do it justice:
I have moved the example of the center-construction to after that of the automorphism group construction, and then I have added the explanation of how the automorphism group is functorial, so that readers are not actively misled by this entry of “counterexamples”.
Here is how the first item reads now:
Sending an object $x \in \mathcal{C}$ of a category $\mathcal{C}$ to its automorphism group $Aut_{\mathcal{C}}(x)$ (or its endomorphism monoid $End_{\mathcal{C}}(x)$ ) does not in general extend to a functor from $\mathcal{C}$ to Groups.
It does however extend to a functor on the core of $\mathcal{C}$ (the maximal groupoid inside it, keeping only the isomorphisms of $\mathcal{C}$), where it sends morphisms (now constrained to be isomorphisms) to their conjugation action:
$\array{ Core(\mathcal{C}) & \overset{ \;\;\; Aut_{\mathcal{C}} \;\;\; }{ \longrightarrow } & Groups \\ x & & Aut_{\mathcal{C}}(x) \\ \big\downarrow {}^{{}_{\mathrlap{ \gamma }}} & \mapsto & \big\downarrow {}^{{}_{\mathrlap{ ad_\gamma \colon g \mapsto \gamma \circ g \circ \gamma^{-1} }}} & \\ y & & Aut_{\mathcal{C}}(y) \,. }$For example, if $\mathcal{C} = \Pi_1(\mathcal{X})$ is the fundamental groupoid of a topological space (which thus coincides with its core already), then the automorphism groups of its objects $x \in X$ are the fundamental groups $\pi_1(X,x)$ at these basepoints, which famously are functorial under conjugation by paths in $X$.
That said, I’ll register a complaint about this entry
(which you are free to ignore, as I will ignore any replies, not meaning to debate this):
A decent counter-example is a counter-example to a decent statement.
In topology, where this habit of listing counter-examples originates, there really are plenty would-be propositions whose truths one is seriously led to wonder about but which are in fact wrong, and the way to prove them wrong is to find one counter-example.
But here in the list so far, various of the alleged counter-examples are countering silly statements. The first item, “forming automorphism groups is not functorial” is a counter-example only to the claim that “Every construction is functorial in the first sense that comes to mind.”
Therefore I think listing these as counter-examples, and explicitly so in a claimed tradition of counter-examples in topology and algebra, is putting the practice of category theory in a bad light.
This issue aside, the entry ends up being a list of random factoids, which to the extend that they are interesting are not being done justice here. Rather than dropping an off-the-cuff remark here and walk away, I’d rather that authors would take their idea to the respective entry, write \begin{example} \label{MyExample} ... \end{example}
there and give a decent coherent and self-contained discussion (providing some details/proof) that one could later actually refer to from elsewhere.
I agree with Urs in #11. The page is introduced as though cases are counterexamples to some vague claim that in category theory
things just work out without thinking too hard.
Couldn’t we have a better name for this page? Something conveying that things often need a little more thought, as Bourbaki used to mark with the dangerous bend symbol.
But then, as Urs says in his second point, this becomes an odd jumble of facts. Imagine Bourbaki made a compilation of their dangerous bends.
If there were regular patterns linking some of these together, that would be more interesting.
11 Urs
A decent counter-example is a counter-example to a decent statement.
I agree and add a slightly different statement that traditionally counterexamples are examples of objects witnessing failing statements which take some ingenuity to construct or observe. In many cases they are even mathematically generic, rather than exceptions, but without significant effort it is hard to observe them (say, non-measurable sets) or to prove their key properties.
I am not sure if counterexamples are an old tradition only in topology – analysis has quite a tradition as well as foundations.
I myself started about 10 years ago writing a note on counterexamples in localization theory, but I am not sure when I will finish it, rarely looking at the manuscript (it is about 7-8 counterexamples which I find important to observe). One of them is actually sketched in my 2005. survey in London Math. Soc. LNS on noncommutative localization. I found an example of a subset $S$ in a unital ring $R$ with a set $A$ of ring generators and a set $S_0$ multiplicatively generating $S$, such that one can check the Ore conditions for pairs of elements in $S_0$ and $A$ and still the set is not Ore. This is against the folklore that the Ore conditions should be checked only on generators, often applied in the literature. I even had this wrong statement in an appendix of my thesis with a proof having a gap. Few months later I tried to fix the gap and during a night of trying that I succeeded to construct an example where, using the diamond lemma, one can prove that the Ore condition anyway fails. Under mild additional assumptions the Ore condition is indeed sufficient to check on generators. I was quite shocked when I found the counterexample, I even thought that I used the wrong lemma essentially in my thesis (at least that was the intuition which I used in one important key result), but then I checked the place in the thesis and observed that luckily I used an explicit argument and avoided the usage of the lemma in the first place. (Alas, my LNS survey has another minor result with a missing condition, in the same section; I should once correct at least the arXiv version and note the gap elsewhere.)
So if there is agreement, maybe we could rename this entry, say to “pitfalls in basic category theory”?
There could still be another entry with actual counter-examples in category theory, too. For instance examples of limit-preserving functors between locally presentable categories not having left adjoints.
I’d be for renaming - perhaps ’basic’ is unnecessary.
What are examples of limit-preserving functors between locally presentable categories not having left adjoints counter-examples to, Urs? Was it expected for a time that such entities wouldn’t exist?
Perhaps the pitfall/counter-example distinction is somewhat vague. It seems to rely on the degree/duration of an overturned expectation.
I can see why people wouldn’t want to speak about “basic” category theory, but I feel that this is right on the point here:
Foundational subjects like category theory, type theory and logic all suffer from the fact that they have a large body of basic, straightforward, almost trivial material on which the actual theory is supported, so that newcomers tend to mistake the toying with the basic notions for engagement with the actual theory.
For instance, figuring out how some construction extends to a functor is no more a topic of category theory than organizing two apples into a set is a topic of set theory.
The negative press that category theory tends to receive (and that much type theory and logic deserves to receive, too) is largely based on the continual habit of conflating the actual theory with exploration of its basic notions.
Substantial results of category theory include the adjoint functor theorem, where counter-examples to the idea that some of its assumptions may be removed are non-trivial and interesting. We list one here. It would be great if anyone interested in counter-examples in category theory could expand the list of counter-examples there!
This page is making less and less sense to me. I can see the point of giving examples in situ to show particular pitfalls of plausible expectation, but then they can be matched to the level of expertise of a likely reader there, as with Bourbaki’s dangerous bends. A greatest hits of their dangerous bends makes little sense. Worse here with exposition aimed at different audiences.
(7). The opposite of the category of commutative von Neumann algebras has a subobject classifier and it’s finitely complete, but is not a topos since it is not cartesian closed.
surely belongs, if anywhere, to a note at topos, that some conditions do not suffice.
But enough. This page will in all likelihood slip into obscurity.
Yes, that’s what I suggested in #11 (2). If any author of an example feels it deserves a page all of it’s own, then all the better! But if that seems daunting, contributor’s are invited to add it anywhere that seems fitting and we can take care of turning the result into an !include
-file or whatever seems appropriate.
In short: Just add material to the $n$Lab as usual. :-)
For illustration, I have taken the example that Jonas Frey kindly provided and formatted a little more (following Jonas’ edit here), to show how it should be done:
The remark concerns a property of dense functors, so it goes into the entry dense functor in the section dense functor – Properties
the (counter-)example that proves the remark goes under dense functor – Examples
Check out the source code to see how the Remark and the Example point to each other.
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