Talk:Timeline of category theory and related mathematics

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Recommended additions[edit]

  •  ?60s - Homotopic category theory (abstract homotopy theory)
  • William Lawvere - sets as generalized truth values
  • Nobuo Yoneda - Yoneda lemma
  • 1958 Alexander Grothendieck? - Topological categories and topological functors
  • L-categories
  • Abelian categories,preabelian categories
  • Additive categories,preadditive categories

I would appreciate help with these and to add more entries, as well as with writing wikipedia pages about the missing topics in red.

It is also ok to wish here for a few entries to include. Fotino (talk) 02:21, 5 August 2009 (UTC)Fotino


It's a very nice timeline, but these days, Wikipedia wants its articles to have references. In particular, I came here looking for information on where Ehresmann talked about strict n-categories, and I'm leaving without knowing. If you know, I'd like to know. John Baez (talk) 17:58, 26 June 2009 (UTC)

Dear John and others looking for precise references.

To put up references would almost require as much work as to write the information already included. To sort the information and to write this page easely become a difficult mess, references would add to the complexity of writing while sorting and learning some of this stuff. Currently i am not planing to give references. I would have to give references to almost all entries. But if people insist that this is more important than the content i will do it. However there is plenty of expansions and entries i have sketched to include first.

A second thing is that the reference can often be found by googling on the person, the year and the subject. This is much easier than finding the facts, like checking the solution of an equation instead of solving it.

This is what i know by a former e-mail from Ross Street regarding your question. Ehresmann (probably in his 1963 book) defined strict n-categories inductively using internal categories. I am very sure Ross means the 1965 book Categories et structures. A google search as described above quickly reveals two confirmations leading to this source.

1 An Australian conspectus of higher categories - Ross Street (his reference here is NOT 1963 but 1965)

2 A survey of definitions of n-category - Tom Leinster

I am sure Andree Ehresmann knows all the further details and i am glad you like the timeline.

-Fotino 05 July 2008

Okay, go ahead and do what you like. Wikipedia policy really demands references, but if you don't put in references, someone else will someday. That's the great thing about Wikipedia.
Thanks for the help on finding when Ehresmann invented strict n-categories. I used "Categories et structures" as a reference in the preface to my book with Peter May. John Baez (talk) 08:45, 5 July 2009 (UTC)


Leray invented the sheaf c. 1945; the formulation that a sheaf is a functor is approximately a decade after that, and so should be shifted down the line in its own place. It is quite important to get the history right in a timeline. Charles Matthews (talk) 11:27, 8 July 2009 (UTC)

By the way, Fotino, please sign like ~~~~. Charles Matthews (talk) 11:28, 8 July 2009 (UTC)

This is something i didn't knew. I am on it. Any further information would be useful since it will take time to google it.

Apropo sheaves. In Sheaf (mathematics) the timeline states:

1955 Alexander Grothendieck - ... by using injective resolutions allows direct use of sheaf cohomology on all topological spaces, as derived functors.

This is inconsistent with this timeline, in my oppinion 1955 is too early. I don't remember exactly what i had in mind since i discovered it a long time ago, but for instance both injective resolutions and derived functors were defined 1956.

Nevertheless the article about sheaves has the reference The origin of sheaf theory - Haynes Miller on its side.

A possibility is that Grothendieck learned about injective resolutions and derived functors from Cartan or Eilenberg (or even inspired them) before they published "Homological algebra". I don't know how to resolve this but luckily one year is within the error margin. Thank you for the corrections Charles. (talk) 05:54, 9 July 2009 (UTC)Fotino

This is why we need references! Christian Houzel wrote a short but good history of sheaf theory in Kashiwara and Schapira, Sheaves on Manifolds. On p. 17 he says that Grothendieck in Spring 1955 at the University of Kansas was lecturing on abelian categories. Buchsbaum published on "exact categories" in 1955 too, so Grothendieck didn't need to wait for Cartan-Eilenberg in 1956 to start the "Tohoku" point of view. There may be more history in the Grothendieck-Serre letters, but I haven't read those.
So I approve of timelines as a way of explaining mathematics, but they require serious work. Charles Matthews (talk) 18:59, 9 July 2009 (UTC)

0) I don't know french but your approval makes me smile :)

1) If i am getting you right you are suggesting that the Grothendiecks revolution (now in 1955) came before the 1956 Cartan-Eilenberg revolution, but it was published in Tohoku in 1957. Not impossible considering it may take years to get something printed. Could you reformulate this 1955 Grothendieck entry so it don't bring confusion and move it to this timeline?

2) I am editing about Leray and sheaves, please check if this is what you mean. Fotino (talk) 03:02, 10 July 2009 (UTC)Fotino

Some good luck means that Grothendieck-Serre Correspondence by Alexandre Grothendieck, Pierre Colmez, Jean Pierre Serre is available on Google Books, in French with English translations. Early in 1955 Grothendieck was still talking to Serre about FAC, but he had decided to give a course on homological algebra based only on the outline of Cartan-Eilenberg, with his own ideas. There is much about this in a letter of February 1955. Charles Matthews (talk) 14:06, 17 July 2009 (UTC)

It was a good reference. What i could see from it Grothendieck did't had the Cartan-Eilenberg book (which he wrote existed one year before 1956) but did knew approximately what it was about. How much he knew i don't know. The key word you wrote is outline! I will just have to add that Grothendieck had an outline of the book and it fits :) But i don't understand how he could use abelian categories as he invented these in 1957. Should i change it to 1955? Fotino (talk) 02:21, 5 August 2009 (UTC)Fotino

That is a very "French" question! If he had the mathematics of homological algebrs based on injective resolutions, but not yet the abelian category formulation in which the existence of enough injectives is a result to prove, what did he have? He had something quantified as "whenever" ... There are going to be some limitations anyway in such a timeline, because the "priority date" is usually the publication date. I think the serious history of what Grothendieck was doing at this period is interesting, but really it would be better to write 1955-7 for this refoundation, and place a better history of homological algebra in another article. Charles Matthews (talk) 10:56, 6 August 2009 (UTC)

The answer is often closer than you think. The first sentence of nLab/Tohoku explains it well. Mystery solved! I will include it soon. Fotino (talk) 13:13, 7 August 2009 (UTC)Fotino

I just read here that Koszul algebraized Leray's spectral sequences in 1945. But they were inwented in 1946! Does anyone know if these are the right years (seem to be) and how Koszul got hold of the new spectral sequences so fast? Fotino (talk) 22:15, 10 August 2009 (UTC)Fotino

Ehresmann and internal categories[edit]

According to his wife [1] he did not quite discover those in the late 1950s but in 1963 a weaker (as in only small?) version thereof, which his wife called in English P-structured categories (catégories structurées in the French original); these were already good enough to particularize to double categories, ordered categories etc. From the same source, it appears that the modern formulation of internal categories did not appear until 1972, and they were initially called by Ehresmann generalized structured categories. Pcap ping 20:19, 3 August 2009 (UTC)

What i think you mean is "Categories of internal categories are studied in different papers, specially in our long 1972 paper (CE III)". It does not say anything about modern internal categories. It only say about categories of internal categories and the year is not final (it could be before 1972).
Then i don't understand the difference between P-structured categories and generalized structured categories,
but since i am writing a lot, to incluede in the timeline, about Charles and Andree Ehresmann and i am e-mailing with Andree i will ask her about it. Fotino (talk) 02:21, 5 August 2009 (UTC)Fotino

I am curious too where the modern formulation of internal category first appeared. Neither MacLane nor Borceaux give a clear attribution for the origins of the concept in their books, but then both books lack historical notes (end of chapter) sections. Pcap ping 12:32, 5 August 2009 (UTC)

Some definition of internal category came up with Grothendieck; see Johnstone, Topos Theory, Ch. 2. That was already in the TDTE seminars, so around 1960. I'm not clear what anyone means here by 'modern' definition. Charles Matthews (talk) 10:49, 6 August 2009 (UTC)
If I understand this stuff correctly, Ehresmann's generalized structured categories coincide with (likely just the small) internal categories. Pcap ping 18:58, 11 August 2009 (UTC)

P-structured categories are equivalent to internal categories in H and defined as a faithful functor P:H→Set respecting pullbacks. Generalized structured categories (internal to K) are defined as the model of a sketch S in K. When S=SCat, the sketch of categories the generalized structured categories are equivalent to internal categories. Neither of these is the modern formulation as in Lie 2-algebras p15. So you asked a good question, who wrote first the axioms in the link? I can tell by the layout of the axioms that it is John Baez's formulation of them and they are taken from Borceux handbook of categorical algebra, but which source Borceux used i have no clue. Fotino (talk) 02:43, 23 August 2009 (UTC)Fotino

Stub requests[edit]

To keep this page under some sort of control, it would be better to create separate articles, if short ones, on topics that have a long paragraph in the timeline. My requests for this, firstly, would be for Abstract Stone Duality, algebraic set theory, Fukuya category and context, topos hypothesis and context. It should be clear that the weight of exposition required to place these in the timeline is rather heavy, and so we should use indirection. Charles Matthews (talk) 11:05, 6 August 2009 (UTC)

As for the longest entries this is obvious but i am in no hurry of doing it. But what did you have in mind for the topos hypothesis? I would not write an exposition of higher topos theory on this page. If it gets a separate page (which it deserves although i don't know anything about the progress of proving it) there will be nothing to write about it in the timeline. There is only one sentence, and a link to topos hypothesis should be motivated which is about one sentence also. Fotino (talk) 13:13, 7 August 2009 (UTC)Fotino
For topos hypothesis, it should be treated as we treat conjectures in general: statement placed in context (mathematical background, historical background, motivation). I chose that article title as being most interesting for "general" (highly mathematical) readers: what is this abstract question, and why do people want to solve it?
Anyway, an article on Fukuya category is probably the one most needed in connection with other topics in the encyclopedia. Charles Matthews (talk) 20:50, 7 August 2009 (UTC)

Would it be wrong to just create a page Fukaya category, copy context from the timeline there and keep what is about Fukaya categories as it is in the timeline? Then, can i call a page A-category? I was not writing so much about the special case of Fukaya categories but about A-categories. Fotino (talk) 18:49, 8 August 2009 (UTC)Fotino

It would be OK to have Fukuya category a redirect to [[A-category]]. Some technology would be needed to get the subscript into the page title. Of course you can copy timeline content into a stub article, but it will need some more work. Charles Matthews (talk) 16:14, 11 August 2009 (UTC)

I am now taking the size issue seriously since the timeline will grow more and more. I plan to move what Charles Matthews have suggested and most of Mitchell-Bénabou internal language and higher topos. All i need to know is what code creates a redirect and what technology to use to get a proper underscore in the title. Fotino (talk) 02:43, 23 August 2009 (UTC)Fotino

I replied on your talk page regarding the technical matters. Pcap ping 04:32, 23 August 2009 (UTC)
If you start making a lot of redirect to this page, please tag them with {{cattheory-stub}} as well, so interested editors can find them later in Category:Category theory stubs and expand them. Pcap ping 04:37, 23 August 2009 (UTC)

Many entries could be cut![edit]

This timeline seems to be written from the point of view of someone mainly interested in the foundations of higher category theory. I agree that this is important work, but the title of the timeline is very broad. I don't think that it is very helpful to the student trying to learn about categorical mathematics to read that researcher X proved technical result Y in 1982. I think it would be much more helpful to focus on what are widely regarded as the important results in this area.

However, there are some surprising omissions. For instance, there is no mention of algebraic K theory (!!!); Thomason does not appear at all; Segal's construction of cohomology theories from symmetric monoidal categories is not mentioned; etc.

It seems, also, that a timeline on "Category theory and related mathematics" could, in principle, include all mathematics. The authors of this timeline have a rather scattershot approach to choosing what to cover. For instance, much as I admire Richard Thomas's thesis, I don't think this work has any more to do with category theory than Donaldson's work on gauge theory, Kontsevich's work on Gromov-Witten invariants, Borcherds work on vertex algebras, or frankly any mathematics of the period.

(Also, Chern and Simons did not construct a topological field theory in 1975. Chern-Simons theory was proposed by Witten in 1989). —Preceding unsigned comment added by (talk) 03:16, 5 October 2009 (UTC)

What is a "Foundational Debate"?[edit]

Could someone write a definition for "Foundational Debate"? The term "Foundational Debate" is used many times within WikiPedia. I cannot find a definition anywhere. There is an article on "Foundationalism", but it is not clear how that is related to a "Foundational Debate". (talk) 17:04, 15 April 2016 (UTC) (talk) 17:04, 15 April 2016 (UTC)