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I have removed the following discussion box from stuff, structure, property – because the entry text above it no longer contained the word that the discussion is about :-)
[begin forwarded discussion]
+–{: .query} Mike: Maybe you all had this out somewhere that I haven’t read, but in the English I am accustomed to speak, “property” is not a mass noun. So you can “forget a property” or “forget properties” but you can’t “forget property.”
Toby: Well, ’property’ can be a mass noun in English, but not in this sense. Also, if we were to invent an entirely new word for the concept, it would surely be a mass noun. Together, these may explain why it's easy to slip into talking this way, but I agree that it's probably better to use the plural count noun here. =–
[end forwarded discussion]
A few weeks back on the categories mailing list there was discussion about what the right definition is for a functor to be regarded as “forgetting structure”. It starts with a message from Jean Bénabou from Wed, Feb 8, 2017 at 9:03 AM, subject “Terminology”. (Sorry,I have trouble locating it in the online list archive. )
The issue is which extra conditions to demand beyond faithfulness. After this suggestion came up in the discussion, in the message Thu, Feb 9, 2017 at 5:38 PM Bénabou said that this is “obiously not enough”.
People suggest extra conditions: a) reflects isos and b) is an isofibration, which is what seems to find consensus.
Eventually of course the nLab entry gets a mentioning, and since it talks just about faithfulness, it gets bad press.
I have seen the word structure used with various meanings; and sometimes without any meaning at all.In particular in the nLab.
(Bénabou Sun, Feb 12, 2017 at 8:00 AM.)
I don’t have energy for this. But if anyone cares, I think it would be good to expand the Lab entries structure and stuff, structure, property to reflect this issue better.
For an old reference, Andrée Ehresmann wrote (13th Feb):
Thus your proposed definition ” p is a structure … if it is faithful and transportable ” exactly corresponds to Charles’ definition of a “foncteur d’homomorphismes”. (Cf. for instance def. 19, p. 71 of his book “Catégories et Structures”, Dunod, 1965.)
and for even older, AE wrote (10th Feb):
For Charles Ehresmann, the answer to Jean’s question was that p be a “homomorphism functor”, a notion he already defined in his 1957 paper “Gattungen in Lokalen Strukturen”,
which is “faithful+amnestic” (both of which Bénabou apparently thought too weak). For a newer reference, Streicher wrote (9th Feb):
in Remark 13.18 of their book on “Algebraic Theories” Adamek, Rosicky and Vitale suggest the following conditions
1) p faithful (what they call “concrete over X”)
2) p-vertical isos are identities (what they call “amnestic”))
3) p is an isofibration (what they call “transportable”)
(and which Bénabou accepted)
And since I’m going to get up for work in a few hours, I can’t say I can edit the entry in any coherent way myself, either.
Shouldn’t we hear first from the people involved in the original description? It was the result of many months (years?) of discussion, if I recall. Mike took notes from John Baez’s lectures and wrote an appendix for Lectures on n-Categories and Cohomology, and Toby was involved in the original discussion, so we shouldn’t have to wait long.
Maybe Benabou has a good case, but I rather hear it made first.
Shouldn’t we hear first from the people involved in the original description?
I hope these people will join in in editing the entry. But in any case the entries need to mention the possible further conditions, especially since they appear in standard textbook literature.
As a matter of policy I no longer read any thread started by Benabou. (-: As for structure, reflecting isos is obviously too strong, since topological spaces fail it. Amnestic and isofibration are both very reasonable conditions; the problem is that they are not equivalence-invariant. Perhaps there is room both for notions of “forgetting structure” and “strictly forgetting structure” (or “forgetting strict structure” or something, I’m not sure what the best grammar would be)?
There was a small amount of discussion about the difference between algebraic structure and that of topological spaces. Needless to say, I haven’t seen the promised follow-up thread by B. It may not have passed moderation, or just never written, like so many other things he has done privately.
Irrespective of Bénabou’s opinions, there is definitions of functors forgetting structure in Ehresmann’s 1957 article, and in “The Joy of Categories” and in Adamek-Rosicky-Vitale and all these demand more than just a faithful functor. Our entries should reflect that and say what’s going on with these variants.
I have tweaked and expanded the minimal additions here and here. Still not good, but better than nothing.
I really need to leave it at that for the moment. I trust that somebody here cares enough to do justice to the topic and its reflection in the literature.
By the way, this here was the argument given why faithful+amnestic is still too weak to capture the concept “forgets structure”:
I think faithful+amnestic is too weak. if is contained in the inclusion satisfies these two properties . Let be the category of finite sets and be the full subcategory of finite ordinals. Do you know any structure on finite sets such that has that structure and some other singleton doesn’t? Thus if an inclusion is a structure should be a replete subcategory of .
Since we explicitly adopt the principle of equivalence in this wiki, it’s reasonable to give the Baez et al. account as the main one. I’ve added a note that in stricter cases more may be required.
Insisting on any condition that isn't equivalence-invariant is wrong from the nPOV. Certainly the nLab should mention things that appear in the literature, and if there is an accepted definition of extra-structure-up-to-isomorphism (I would call this ‘strict extra structure’), then we should mention that (it is the corresponding notion for strict categories). But every faithful functor is, up to equivalence, the inclusion of a subcategory.
Okay, I have added one more sentence here. (Violating my pledge that I should be doing something else.)
Yeah I’d rather be hearing about what’s happening in Regensberg. What are these ubiquitous Gorenstwin ring spectra?
I’d rather be hearing
Sure, but let me just clarify. It may be fine personally to declare that one disregards certain people and their opinions, but since we are running a public site, in parts about category theory and prominently associated with the names of the regulars here, I found it disconcerting that when a fair number of category theorists discuss a basic concept from category theory, not only could none simply point to an Lab page to resolve the debate, but moreover the only mentioning the Lab gets is as a snide remark. A good site is such that even the non-sympathetic people can’t disregard it’s objective content. So apart from such politics, the event highlights that the entries on “structure” which we have failed (and still fail) to explain why the definitions they give are right, or at least good.
So what more do you want? An explanation at every relevant page as to the arguments for and against strictness and weakness? Why isn’t a link to a page which speaks to that plus mention of conditions that someone wanting strictness might use enough?
I dare say if we sampled people we’d find all kinds of objections as to partiality. Removing laden terms such as ’evil’ has no doubt been a good idea (not that that is anywhere near complete), as with talk of extra ’baggage’ at material set theory. But you’re never going to be able to defend an appreciable fraction of your decisions.
We responded to this criticism by modifying two pages in a non-confrontational tone. I don’t see what hasn’t been done.
I think the additions are good.
I tweaked the grammar (and removed an ‘at least’ that I think is unnecessary given what has now been added after it).
I also added more text to amnestic functor (and slightly to isofibration) explaining how they are trivial up to equivalence.
Thanks!
I tried my hand on giving the entry an Idea-section that conveys a bit more of the idea. Leaves room for much improvement.
Added a section
Formulation in homotopy type theory
A map between homotopy types is equivalent to a dependent sum projection . If the types are n-types, forgets -stuff.
I expanded out the two original references (which were previously only hinted at) and moved them to the top of the list:
Charles Ehresmann, Gattungen in Lokalen Strukturen, Jahresbericht der Deutschen Mathematiker-Vereinigung 60 (1958) 49-77 [dml:146434]
Charles Ehresmann, Catégories et Structures, Séminaire Ehresmann. Topologie et géométrie différentielle 6 (1964) 1-31 [numdam:SE_1964__6__A8_0, dml:112200]
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