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promted by demand from my Basic-Course-On-Category-Theory-Students I expanded the entry 2-category:
mentioned more relations to other concepts in the Idea-section;
added an Examples-section with a bunch of (classes of) examples;
added a list of references. Please add more if you can think of more!
What is the purpose of the bullet list containing category, 2-category, etc. at the top of the article? It looks weird to me there – shouldn’t a page called “2-category” start with the contents of itself? Those seem more like they would go in a “related pages” section at the bottom of the page.
Does it look weird?
Darn, I thought it was a good idea. Will have to remove these, then. I keep having this urge of putting pages into context. For cases where we have sequences of pages that all vary on the same theme, I started to collect their titles in short lists at the top of the page, kind of as a sort of little menu to choose from.
I thought that putting pages in context was one of the purposes of the floating contents links on the side.
@Urs. Perhaps that list could go elsewhere. I did think that it was not helpful where it is.
Those lists do look a bit strange at the very beginning of the page. They are a good idea, though, I think, and they complement the subject-contents links by grouping pages more by conceptual similarity than by usage.
Okay, will start moving these lists elsewhere!
Finn wrote:
They are a good idea, though, I think, and they complement the subject-contents links by grouping pages more by conceptual similarity than by usage.
That had been my feeling, that the floating TOC serves to set the scene, but does not convey the immediate idea “this entry is only one variant of a single concept, the other variants of which are…”. I felt we have a bunch of entries that in principle could be on a single page.
Darn, I thought it was a good idea. Will have to remove these, then. I keep having this urge of putting pages into context.
I also like to have the main content straight away without scrolling, while side links on the side (like floating contents) or at the bottom of the page along with references (related entries). Many times I was writing Literature and links or References and other entries or alike section at the bottom. So I think all that idea of adding more links ios always good but not at the beginning of the entry.
"this entry is only one variant of a single concept, the other variants of which are..."
There are so many variant of this statement of different flavour, like related concepts, synonyms, closely related concepts, other formalisms for the same idea etc. each of these is a different flavour. If one is putting just a list with some default meaning of the list one does not know which variant of list of variants one is giving. So it is difficult to colaborate on creating such lists. Sometimes a single top page title like algebraic theories above PRO may be worthy, but one can have several groups of titles with common intersection, and each group is defined by a different principle. So I do not see a way for one size to fit all. So maybe it would be better to have explicit explanation of a compact linear list at the bottom with references.
I don’t like these either and keep meaning to ask about them. Or rather, I like them very much! Just not where they are.
Since they’re so short, maybe you can put them on the side above the other lists of contents, but without the funky code to make it disappear?
added pointer to:
(which is a textbook on 2-category theory in itself)
An anonymous editor added the term “monoidal monoidoidoid” to the page without leaving a comment. I do not think this terminology is helpful to have on the page, both because the terminology does not appear anywhere, and because it is ambiguous: it relies on the term being parsed as “(monoidal monoidoid)oid” rather than “monoidal (monoidoidoid)”, which is not the natural reading. It does not help the nLab’s reputation for obscurity when people add terms like this jokingly.
Added link to opposite 2-category.
I have added (here) original references on the notion of strict 2-categories (following discussion there)
and also added pointer to:
Aren’t we jumping the gun just a little by asserting that the notion of 2-category is due to Benabou, not Ehresmann? The “seems to be” doesn’t ameliorate that.
We’re going to all this trouble to verify assertions with precise literature citations, we might as well be careful here too. Before saying everyone else seems to be wrong, let’s be sure.
Re. #18: I completely agree.
I have emailed Andrée Ehresmann regarding the origin of 2-categories and will let you know when I hear back.
No worries, I volunteer to immediately change the wording in all entries once you discover a 2-category in Ehresmann’s writing. :-)
For the moment all evidence points to Ehresmann defining double categories and all other authors implicitly granting that strict 2-categories are a special case.
Urs, this attitude seems counter to the great scruples you were applying before to the claims of others, where you were demanding proof based on direct evidence, not indirect or circumstantial evidence like you’re citing here. I suggest we wait until we can get our hands on the 1965 book and can have a look ourselves. (I would also accept it from Madame Ehresmann, if she said it was Benabou, but then I would want permission to quote her.)
No, we may state as a fact what we have seen, but must not state as a fact what merely we cannot rule out.
(By the latter logic we’d attribute everything to Ehresmann until proof to the contrary, which is not right – though this is how people did proceed in the past, and we should discontinue this malpractice as it is hurting the field.)
No harm is done. The moment we find writings that Ehresmann is to be credited with the notion of strict 2-categories, I’ll reword all entries immediately.
By the way, this entry here (2-category) did not mention Ehresmann at all in its 14+ years of existence – not until I added in the pointer (rev 56) a couple of hours ago.
It is only now, after 14+ years of no credit to Ehresmann, that I made it say that the two earliest clear references we know were “both apparently following the definition of double categories due to [Ehresmann]” (rev 56).
I received a response from Andrée Ehresmann, who writes:
He did not introduced himself 2-catégories, which have been introduced soon after both by his student Jean Bénabou as special double categories, and by different other authors as categories ’enriched’ in Cat
In the book “Catégories et Structures” Charles just mentions both double categories and 2_categories at the end (p. 324) in a historical Note.
Presumably by “others” she refers to Eilenberg and Kelly, who called 2-categories “hypercategories”. However, since they are familiar with the term “2-category”, despite it not appearing in print before 1965, it seems most likely that their citation is simply incorrect, and the intended reference was indeed the 1965 book Catégories et structures. In any case, it is confirmed that the paper of Bénabou is the first reference for 2-categories, and the concept may be attributed to him.
I must amend my previous message: in fact, Maranda independently introduced 2-categories in the same year as Bénabou, in his paper Formal categories on enriched categories, where he called them “categories of the second type”.
I believe the definition of strict 2-category should thus be jointly attributed to Bénabou and Maranda.
Thanks!! I have added pointer to Maranda: here
It’s interesting that the two articles introducing strict 2-categories immediately understand them as -enriched categories and in fact coincide with the two articles that introduce enriched categories in the first place. So the conception of higher and of enriched category theory went hand-in-hand!
With that out of the way, maybe we can use the momentum to also boost the referencing for the actual notion of bicategories:
Currently our entry offers Bénabou 1967 as the only original reference, though that already carries a title suitable for a review (“Introduction to…”).
Is this all we have before Kelly & Street 1974 write a “Review of…”?
Thank you, varkor! Nice to have that settled definitively.
Tweaked the references section. My understanding is that Eilenberg and Kelly intended to cite the definition of 2-category in the 1965 book of Ehresmann (who knew of the definition from Bénabou and/or Maranda), and mistakenly thought that Ehresmann invented the concept. However, to make matters more confusing, their citation is incorrect (presumably they assumed it was also in the earlier text, when it is not). Evidence towards them not independently formulating the definition themselves, based on the definition of double category, is the fact they know that Ehresmann calls them “2-categories”, which is Bénabou’s terminology.
Where you say:
The fundamental structure inherent to the 2-category of categories was first analysed in:
- Roger Godement, Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, Godement-TopologieAlgebrique.pdf:file]
let’s add a page number! Do you mean to refer to the discussion on p. 269 (first page of the appendix)?
All right, thanks, so you do mean p. 269.
After the words “the fundamental structure of the 2-category of categories”, I have made precise:
(namely the vertical composition, horizontal composition and the whiskering of natural transformations)
(and I should go add that reference also to these entries).
(Remarkable all the structure that Godement considered in passing (2-categories, monads, simplicial objects, monadic resolutions) towards his goal of resolving sheaves.)
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