"First" use of 'Category theory' to describe our field

"First" use of 'Category theory' to describe our field

[David Roberts]

Hi all,

the (idle) question is: when did the phrase 'category theory' catch on
for the field? Clearly it didn't leap from either of the heads of
Eilenberg or Mac Lane full-grown, since they used the phrase 'General
theory of natural equivalences'. There are the old 'Reports of the
Midwest Category Seminar' lecture notes (the first in 1967), which
hints that 'category theory' wasn't quite the name in use.

Even more interesting: who was the first "category theorist", by that name?

Answers referring to verifiable sources would be best.

Thoughts?

David



David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com


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Re: "First" use of 'Category theory' to describe our field

[Ross Street]

Dear David

From memory, the Pure Mathematics Honours (USyd) course that Max Kelly taught in 1965
was called ``Category Theory''. It concentrated on different kinds of morphisms
and factorizations in a category, and finished with adjoint functors.
Also John Gray's (UIllinois) 1968-69 graduate course had that name.

From Eilenberg I heard that each person using categories should have their own category of expertise.
I told this to John Gray who said that was fine; the time had come for that  category to be Cat.

I would suggest that the first category theorists to think of themselves as  such were Eilenberg's students at Columbia.
However, Mac Lane was definitely a category theorist.

This is probably not the verifiable stuff you were seeking.

Ross


On 10 Jul 2019, at 10:01 PM, David Roberts <droberts.65537@gmail.com<mailto:droberts.65537@gmail.com>> wrote:

Hi all,

the (idle) question is: when did the phrase 'category theory' catch on
for the field? Clearly it didn't leap from either of the heads of
Eilenberg or Mac Lane full-grown, since they used the phrase 'General
theory of natural equivalences'. There are the old 'Reports of the
Midwest Category Seminar' lecture notes (the first in 1967), which
hints that 'category theory' wasn't quite the name in use.

Even more interesting: who was the first "category theorist", by that name?

Answers referring to verifiable sources would be best.

Thoughts?

David



David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://protect-au.mimecast.com/s/VE16C91W8rCN283MHOg1ul?domain=ncatlab.org>
Blog: https://thehighergeometer.wordpress.com<https://protect-au.mimecast.com/s/wpBIC0YKgRsgmjoMIOaf6Z?domain=thehighergeometer.wordpress.com>



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: "First" use of 'Category theory' to describe our field

[La Monte H. P. Yarroll]

There's a nice Wikipedia article on "General Abstract Nonsense", but not on
the more formal name.

On Fri, Jul 12, 2019, 3:45 PM Ross Street <ross.street@mq.edu.au> wrote:

> Dear David
>
> From memory, the Pure Mathematics Honours (USyd) course that Max Kelly
> taught in 1965
> was called ``Category Theory''. It concentrated on different kinds of
> morphisms
> and factorizations in a category, and finished with adjoint functors.
> Also John Gray's (UIllinois) 1968-69 graduate course had that name.
>
> From Eilenberg I heard that each person using categories should have their
> own category of expertise.
> I told this to John Gray who said that was fine; the time had come for
> that  category to be Cat.
>
> I would suggest that the first category theorists to think of themselves
> as  such were Eilenberg's students at Columbia.
> However, Mac Lane was definitely a category theorist.
>
> This is probably not the verifiable stuff you were seeking.
>
> Ross
>
>
> On 10 Jul 2019, at 10:01 PM, David Roberts <droberts.65537@gmail.com
> <mailto:droberts.65537@gmail.com>> wrote:
>
> Hi all,
>
> the (idle) question is: when did the phrase 'category theory' catch on
> for the field? Clearly it didn't leap from either of the heads of
> Eilenberg or Mac Lane full-grown, since they used the phrase 'General
> theory of natural equivalences'. There are the old 'Reports of the
> Midwest Category Seminar' lecture notes (the first in 1967), which
> hints that 'category theory' wasn't quite the name in use.
>
> Even more interesting: who was the first "category theorist", by that name?
>
> Answers referring to verifiable sources would be best.
>
> Thoughts?
>
> David
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: "First" use of 'Category theory' to describe our field

[Vaughan Pratt]

  > From memory, the Pure Mathematics Honours (USyd) course that Max Kelly
taught in 1965 was called ``Category Theory''.

While I don't recall the course title, my recollection is that the year Max
taught category theory to Ross and other third year students was 1964.  Max
was going to move on to algebraic topology in 1965 (our honours year, 14 in
that class including Brian Day and Henry Irgang) until he realized we'd
never seen point set topology so he settled for that.

The first homework for the latter was to enumerate the T0 spaces up to
three points (or perhaps four, which is still feasible for a homework
though tedious).  In those days I was much more at home with combinatorics
than algebra so that was a fun homework for me.  Maybe that's why I ended
up in computer science.  I didn't really warm to algebra until 1979, in
particular universal algebra courtesy of Rasiowa and Sikorski, despite
having taught a groups-rings-fields course at MIT in 1972, and it was a
couple of years more before I started to come to grips with categories.

Vaughan


On Sat, Jul 13, 2019 at 12:06 PM La Monte H. P. Yarroll <
piggy.yarroll@gmail.com> wrote:

> There's a nice Wikipedia article on "General Abstract Nonsense", but not on
> the more formal name.
>
> On Fri, Jul 12, 2019, 3:45 PM Ross Street <ross.street@mq.edu.au> wrote:
>
>> Dear David
>>
>> From memory, the Pure Mathematics Honours (USyd) course that Max Kelly
>> taught in 1965
>> was called ``Category Theory''. It concentrated on different kinds of
>> morphisms
>> and factorizations in a category, and finished with adjoint functors.
>> Also John Gray's (UIllinois) 1968-69 graduate course had that name.
>>
>> From Eilenberg I heard that each person using categories should have
> their
>> own category of expertise.
>> I told this to John Gray who said that was fine; the time had come for
>> that  category to be Cat.
>>
>> I would suggest that the first category theorists to think of themselves
>> as  such were Eilenberg's students at Columbia.
>> However, Mac Lane was definitely a category theorist.
>>
>> This is probably not the verifiable stuff you were seeking.
>>
>> Ross
>>
>>
>> On 10 Jul 2019, at 10:01 PM, David Roberts <droberts.65537@gmail.com
>> <mailto:droberts.65537@gmail.com>> wrote:
>>
>> Hi all,
>>
>> the (idle) question is: when did the phrase 'category theory' catch on
>> for the field? Clearly it didn't leap from either of the heads of
>> Eilenberg or Mac Lane full-grown, since they used the phrase 'General
>> theory of natural equivalences'. There are the old 'Reports of the
>> Midwest Category Seminar' lecture notes (the first in 1967), which
>> hints that 'category theory' wasn't quite the name in use.
>>
>> Even more interesting: who was the first "category theorist", by that
> name?
>>
>> Answers referring to verifiable sources would be best.
>>
>> Thoughts?
>>
>> David
>>

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Re: "First" use of 'Category theory' to describe our field

[David Roberts]

Dear George,

thanks for supplying that quote about Eilenberg. After emails with
Peter May I had tracked down a secondary source that did not cite that
Mac Lane article you give, so it's good to know the provenance of the
claim.

As far as people studying categories acknowledging the field by name
goes, Peter Hilton in the intro to the Battelle conference proceedings
in 1968 (titled Category theory, Homology theory and their
applications, LNM 86, 92, and 99), writes thus in vol 1:

"The object of this conference was to bring together research workers
in the fields of category theory and homology theory and those who
applied the results of these theories to their own mathematical
disciplines within algebra or topology. Thus this was not, and was not
intended to be, a tightly specialized conference in categorical
algebra (by comparison with the Midwest Seminars), the expectation of
its organizers being that the roles of category theory and homology
theory within mathematics would emerge the more clearly from the
conference and that the interplays of these theories with other parts
of mathematics would be highlighted."

Interestingly, Mac Lane contributed an article titled "Possible
programs for categorists" (note: "categorists", not category
theorists), in which he writes

"Category theory today is both a specialty and a generality.
Specialities are the many particular fields in which current
Mathematical knowledge and folklore develops; a new specialty arises
in a field when the knowledge in that field and its prospects of
further development demand full time workers. In the last six or eight
years, category theory has become a flourishing specialty."

So we might have a lower bound of 1960-62 according to Mac Lane's
written estimate as to when category theory started to 'became a
flourishing speciality' (around the time of Freyd's thesis, it seems).

Going back a few years in the publication record, the 1965 La Jolla
conference was published as "Proceedings of the Conference on
Categorical Algebra" (https://doi.org/10.1007/978-3-642-99902-4), so
perhaps it was a lot more focussed in nature. However, the
introduction states

"The editors hope to have achieved a representative, if incomplete,
cover­ age of the present activities in Categorical Algebra within the
United States by bringing together this group of mathematicians and by
solici­ting the articles contained in this volume. They also hope that
these Proceedings indicate the trend of research in Categorical
Algebra in this country."

So it looks like 'categorical algebra' was at least a working phrase
(modulo having to satisfy the United States Air Force Office of
Scientific Research, which I read elsewhere was not pleased with
having funded such abstract work, and promised to never fund such a
conference again)

In between these two there is the "Seminar on Triples and Categorical
Homology Theory" (LNM 80):

"The papers in this volume were presented to the seminar on category
theory held during the academic year 1966-67 at the Forschungsinstitut
für Mathematik of the Eidgenossische Technische Hochschule, Zürich."

Someone pointed out off-list the reference to which Colin McLarty alluded:

Rosen, Robert. 1958. “The Representation of Biological Systems from
the Standpoint of  the Theory of Categories.” Bulletin of Mathematical
Biophysics 20 (4): 317–42.

in which he talks of "the theory of categories and functors" in his
abstract, and closes with

"The application of category theory to more general kinds of systems
becomes correspondingly more complicated, but at the very least, we
hope to have indicated in the foregoing that the notion of systems
introduced here can be put on a rigorous basis and that the results
obtained by using those notions can be formally justified."

Thanks to all who replied here and elsewhere.

Best regards, and apologies for so many bit of historical trivia,

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com
On Mon, 15 Jul 2019 at 00:19, George Janelidze
<george.janelidze@uct.ac.za> wrote:
>
> Dear Colleagues,
>
> I would like to add three remarks to this discussion:
>
> 1. In his paper "Samuel Eilenberg and Categories" (Journal of Pure and
> Applied Algebra 168 (2002) 127–131), Saunders Mac Lane, talking about [S.
> Eilenberg and S. Mac Lane, General theory of natural equivalences,
> Transactions of the American Mathematical Society 58, 2 (1945) 231-294]
> says:
>
>      "...At the time, Sammy stated firmly that this would be the only paper
> needed for category theory. Probably what he had in mind was that the trio
> of notions - category, functor, and natural transformation - was enough to
> make good applications possible; in particular it was enough to formulate
> the axiomatic treatment of homology theory carried out in the famous
> Eilenberg--Steenrod text “Foundations of Algebraic Topology”.
>      This initial paper on category theory was certainly a “far out”
> endeavor; it might not have seen the light of day! Also the terminology was
> largely purloined: “category” from Kant, “natural” from vector spaces and
> “functor” from Carnap. (It was used in a different sense in Carnap’s
> influential book “Logical Syntax of Language”; I had reviewed the English
> translation of the book (in the Bulletin, AMS) and had spotted some errors;
> since Carnap never acknowledged my finding, I did not mind using his
> terminology.)
>      Sammy’s initial idea that one paper would be enough turned out to be
> wildly wrong. Other basic examples such as adjoint functors were developed;
> at Columbia University Sammy subsequently inspired and guided a remarkable
> group of young mathematicians who took up category theory: John Gray, Daniel
> Kan, Bill Lawvere, Mike Barr, Jon Beck, Alex Heller, Peter Freyd, and many
> others. Sammy and I were very fortunate in our students and associates..."
>
> 2. We celebrated 50th Anniversary of Category Theory in 1995 twice: in
> Halifax (Canada) and then in Cambridge (UK). In particular, the webpage
> https://www.mta.ca/~cat-dist/ct95.html says:
>
> "...Fifty years after the paper which founded Category Theory and
> twenty-five years after the discovery of Elementary Topos Theory, the
> Category Theory community met in Halifax..."
>
> 3. Yes, the title "General theory of natural equivalences" has no categories
> in it, and one might have different opinions on "which paper has the most
> important contribution in transforming 'language' into 'theory'" (what about
> [S. Mac Lane, Duality for groups, Bulletin of the American Mathematical
> Society 56 (1950) 485-516]?). But I think the citations above clearly
> suggest to say that Category Theory was 'officially' born in 1945, and let
> us hope to celebrate its 100th Anniversary in 2045!
>
> Of course all this means no disrespect for great contributions of
> non-North-American authors mentioned (or not mentioned) in various messages
> on this topic.
>
> Best regards,
> George
>
> Disclaimer - University of Cape Town This email is subject to UCT policies and email disclaimer published on our website at http://www.uct.ac.za/main/email-disclaimer or obtainable from +27 21 650 9111. If this email is not  related to the business of UCT, it is sent by the sender in an individual capacity. Please report security incidents or abuse via https://csirt.uct.ac.za/page/report-an-incident.php.


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Re: "First" use of 'Category theory' to describe our field

[David Roberts]

Dear Ross,

no, that's pretty good! It's mildly surprising that it took ~20 years
for the name to 'stick', but maybe less so given that the field grew
slowly to start.

Thanks,

David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com

On Thu, 11 Jul 2019 at 08:21, Ross Street <ross.street@mq.edu.au> wrote:
>
> Dear David
>
> From memory, the Pure Mathematics Honours (USyd) course that Max Kelly taught in 1965
> was called ``Category Theory''. It concentrated on different kinds of morphisms
> and factorizations in a category, and finished with adjoint functors.
> Also John Gray's (UIllinois) 1968-69 graduate course had that name.
>
> From Eilenberg I heard that each person using categories should have their own category of expertise.
> I told this to John Gray who said that was fine; the time had come for that category to be Cat.
>
> I would suggest that the first category theorists to think of themselves as such were Eilenberg's students at Columbia.
> However, Mac Lane was definitely a category theorist.
>
> This is probably not the verifiable stuff you were seeking.
>
> Ross
>
>
> On 10 Jul 2019, at 10:01 PM, David Roberts <droberts.65537@gmail.com> wrote:
>
> Hi all,
>
> the (idle) question is: when did the phrase 'category theory' catch on
> for the field? Clearly it didn't leap from either of the heads of
> Eilenberg or Mac Lane full-grown, since they used the phrase 'General
> theory of natural equivalences'. There are the old 'Reports of the
> Midwest Category Seminar' lecture notes (the first in 1967), which
> hints that 'category theory' wasn't quite the name in use.
>
> Even more interesting: who was the first "category theorist", by that name?
>
> Answers referring to verifiable sources would be best.
>
> Thoughts?
>
> David
>
>
>
> David Roberts
> Webpage: https://ncatlab.org/nlab/show/David+Roberts
> Blog: https://thehighergeometer.wordpress.com
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Fwd: categories: Re: "First" use of 'Category theory' to describe our field

[David Roberts]

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Hi all,

It's been a few years, but it came to my attention in the past 24 hours that Freyd's 1960 PhD thesis was titled "Functor theory", at Princeton, supervised by Steenrod and Buchsbaum. I suspect it was largely influenced by Buchsbaum, being a student of Eilenberg, and who essentially introduced abelian categories, with which Freyd's PhD was largely about; more precisely, embedding theorems and hence a study of suitable functor categories.

It's not a strong data point, but I think it lends weight to the conjecture that the name "category theory" really didn't emerge quite yet at that time, at least in print.

All the best,
David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/ASQIC5QP8ySmZvPZHzREss?domain=ncatlab.org>
Blog: https://thehighergeometer.wordpress.com<https://url.au.m.mimecastprotect.com/s/S2dSC6XQ68f2rD6ru6Le5O?domain=thehighergeometer.wordpress.com>


---------- Forwarded message ---------
From: David Roberts <droberts.65537@gmail.com<mailto:droberts.65537@gmail.com>>
Date: Sat, 13 Jul 2019 at 05:13
Subject: categories: Re: "First" use of 'Category theory' to describe our field
To: Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>>
Cc: categories@mta.ca<mailto:categories@mta.ca> list <categories@mta.ca<mailto:categories@mta.ca>>


Dear Ross,

no, that's pretty good! It's mildly surprising that it took ~20 years
for the name to 'stick', but maybe less so given that the field grew
slowly to start.

Thanks,

David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/ASQIC5QP8ySmZvPZHzREss?domain=ncatlab.org>
Blog: https://thehighergeometer.wordpress.com<https://url.au.m.mimecastprotect.com/s/S2dSC6XQ68f2rD6ru6Le5O?domain=thehighergeometer.wordpress.com>

On Thu, 11 Jul 2019 at 08:21, Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>> wrote:
>
> Dear David
>
> From memory, the Pure Mathematics Honours (USyd) course that Max Kelly taught in 1965
> was called ``Category Theory''. It concentrated on different kinds of morphisms
> and factorizations in a category, and finished with adjoint functors.
> Also John Gray's (UIllinois) 1968-69 graduate course had that name.
>
> From Eilenberg I heard that each person using categories should have their own category of expertise.
> I told this to John Gray who said that was fine; the time had come for that category to be Cat.
>
> I would suggest that the first category theorists to think of themselves as such were Eilenberg's students at Columbia.
> However, Mac Lane was definitely a category theorist.
>
> This is probably not the verifiable stuff you were seeking.
>
> Ross
>
>
> On 10 Jul 2019, at 10:01 PM, David Roberts <droberts.65537@gmail.com<mailto:droberts.65537@gmail.com>> wrote:
>
> Hi all,
>
> the (idle) question is: when did the phrase 'category theory' catch on
> for the field? Clearly it didn't leap from either of the heads of
> Eilenberg or Mac Lane full-grown, since they used the phrase 'General
> theory of natural equivalences'. There are the old 'Reports of the
> Midwest Category Seminar' lecture notes (the first in 1967), which
> hints that 'category theory' wasn't quite the name in use.
>
> Even more interesting: who was the first "category theorist", by that name?
>
> Answers referring to verifiable sources would be best.
>
> Thoughts?
>
> David
>
>
>
> David Roberts
> Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/ASQIC5QP8ySmZvPZHzREss?domain=ncatlab.org>
> Blog: https://thehighergeometer.wordpress.com<https://url.au.m.mimecastprotect.com/s/S2dSC6XQ68f2rD6ru6Le5O?domain=thehighergeometer.wordpress.com>
>


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Re: categories: Re: "First" use of 'Category theory' to describe our field

[Michael Barr, Prof.]

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I don't know when I first heard the term "category theory".  Interestingly, the 1965 conference in La Jolla called itself a conference on categorical algebra.  The n-lab page says it was "on categorical algebra, nominally, but really about fundamental category theory as such, to a large extent."

I know when I first heard the word category in this context.  In 1959-60 I took a course on homological algebra from Dave Harrison (my eventual thesis advisor) and he talked about categories at length.  I might mention that Dave was Peter Freyd's undergraduate advisor at Brown where Peter really discovered his abelian categories embedding theorem.

Dave also predicted that by the turn of century, categories would have replaced sets as the foundation of mathematics.  We are not there yet.  In those days, category theory and homological algebra were closely entwined.  I didn't turn from the latter to the former until I spent 6 months in Zurich in 1967 at the Mathematische Forschunginstitut (which we sometimes called the Eckmann-Hilton).

Michael
________________________________
From: David Roberts <droberts.65537@gmail.com>
Sent: Thursday, February 22, 2024 9:24 PM
To: categories@mq.edu.au <categories@mq.edu.au>; Ross Street <ross.street@mq.edu.au>
Subject: Fwd: categories: Re: "First" use of 'Category theory' to describe our field

Hi all,

It's been a few years, but it came to my attention in the past 24 hours that Freyd's 1960 PhD thesis was titled "Functor theory", at Princeton, supervised by Steenrod and Buchsbaum. I suspect it was largely influenced by Buchsbaum, being a student of Eilenberg, and who essentially introduced abelian categories, with which Freyd's PhD was largely about; more precisely, embedding theorems and hence a study of suitable functor categories.

It's not a strong data point, but I think it lends weight to the conjecture that the name "category theory" really didn't emerge quite yet at that time, at least in print.

All the best,
David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/zThhCzvkmpflRRO3u4eio6?domain=ncatlab.org>
Blog: https://thehighergeometer.wordpress.com<https://url.au.m.mimecastprotect.com/s/fyLVCANpnDCy99GgT8NBa5?domain=thehighergeometer.wordpress.com>


---------- Forwarded message ---------
From: David Roberts <droberts.65537@gmail.com<mailto:droberts.65537@gmail.com>>
Date: Sat, 13 Jul 2019 at 05:13
Subject: categories: Re: "First" use of 'Category theory' to describe our field
To: Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>>
Cc: categories@mta.ca<mailto:categories@mta.ca> list <categories@mta.ca<mailto:categories@mta.ca>>


Dear Ross,

no, that's pretty good! It's mildly surprising that it took ~20 years
for the name to 'stick', but maybe less so given that the field grew
slowly to start.

Thanks,

David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/zThhCzvkmpflRRO3u4eio6?domain=ncatlab.org>
Blog: https://thehighergeometer.wordpress.com<https://url.au.m.mimecastprotect.com/s/fyLVCANpnDCy99GgT8NBa5?domain=thehighergeometer.wordpress.com>

On Thu, 11 Jul 2019 at 08:21, Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>> wrote:
>
> Dear David
>
> From memory, the Pure Mathematics Honours (USyd) course that Max Kelly taught in 1965
> was called ``Category Theory''. It concentrated on different kinds of morphisms
> and factorizations in a category, and finished with adjoint functors.
> Also John Gray's (UIllinois) 1968-69 graduate course had that name.
>
> From Eilenberg I heard that each person using categories should have their own category of expertise.
> I told this to John Gray who said that was fine; the time had come for that category to be Cat.
>
> I would suggest that the first category theorists to think of themselves as such were Eilenberg's students at Columbia.
> However, Mac Lane was definitely a category theorist.
>
> This is probably not the verifiable stuff you were seeking.
>
> Ross
>
>
> On 10 Jul 2019, at 10:01 PM, David Roberts <droberts.65537@gmail.com<mailto:droberts.65537@gmail.com>> wrote:
>
> Hi all,
>
> the (idle) question is: when did the phrase 'category theory' catch on
> for the field? Clearly it didn't leap from either of the heads of
> Eilenberg or Mac Lane full-grown, since they used the phrase 'General
> theory of natural equivalences'. There are the old 'Reports of the
> Midwest Category Seminar' lecture notes (the first in 1967), which
> hints that 'category theory' wasn't quite the name in use.
>
> Even more interesting: who was the first "category theorist", by that name?
>
> Answers referring to verifiable sources would be best.
>
> Thoughts?
>
> David
>
>
>
> David Roberts
> Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/zThhCzvkmpflRRO3u4eio6?domain=ncatlab.org>
> Blog: https://thehighergeometer.wordpress.com<https://url.au.m.mimecastprotect.com/s/fyLVCANpnDCy99GgT8NBa5?domain=thehighergeometer.wordpress.com>
>


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Re: categories: Re: "First" use of 'Category theory' to describe our field

[Michael Barr, Prof.]

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I just heard from Peter Freyd who tells me that some of things in my earlier message were wrong.  The person at Brown who most influenced him was David Buchsbaum who later became one of his thesis acvisors at Princeton (Steenrod was the other).  And in fact, Peter told me that Dave Harrison may have first heard of categories from him.  Buchsbaum (a student of Eilenberg's) wrote a paper called something like Exact categories, giving essentially the definition of abelian categories.  The latter name is probably due to Grothendieck in his famous Tohoku paper.  Later on, not knowing that term had been used already, I used exact categories for a non-additive version.  At least exact+additive is equivalent to abelian.

Peter is not using email these days so if you want his story you have to talk (or zoom) with him.  BTW, an English translation of Grothendieck's paper prepared by my wife with some help from me is available on my web site.

Michael
________________________________
From: Michael Barr, Prof. <barr.michael@mcgill.ca>
Sent: Friday, February 23, 2024 12:07 PM
To: David Roberts <droberts.65537@gmail.com>; categories@mq.edu.au <categories@mq.edu.au>; Ross Street <ross.street@mq.edu.au>
Subject: Re: categories: Re: "First" use of 'Category theory' to describe our field

I don't know when I first heard the term "category theory".  Interestingly, the 1965 conference in La Jolla called itself a conference on categorical algebra.  The n-lab page says it was "on categorical algebra, nominally, but really about fundamental category theory as such, to a large extent."

I know when I first heard the word category in this context.  In 1959-60 I took a course on homological algebra from Dave Harrison (my eventual thesis advisor) and he talked about categories at length.  I might mention that Dave was Peter Freyd's undergraduate advisor at Brown where Peter really discovered his abelian categories embedding theorem.

Dave also predicted that by the turn of century, categories would have replaced sets as the foundation of mathematics.  We are not there yet.  In those days, category theory and homological algebra were closely entwined.  I didn't turn from the latter to the former until I spent 6 months in Zurich in 1967 at the Mathematische Forschunginstitut (which we sometimes called the Eckmann-Hilton).

Michael
________________________________
From: David Roberts <droberts.65537@gmail.com>
Sent: Thursday, February 22, 2024 9:24 PM
To: categories@mq.edu.au <categories@mq.edu.au>; Ross Street <ross.street@mq.edu.au>
Subject: Fwd: categories: Re: "First" use of 'Category theory' to describe our field

Hi all,

It's been a few years, but it came to my attention in the past 24 hours that Freyd's 1960 PhD thesis was titled "Functor theory", at Princeton, supervised by Steenrod and Buchsbaum. I suspect it was largely influenced by Buchsbaum, being a student of Eilenberg, and who essentially introduced abelian categories, with which Freyd's PhD was largely about; more precisely, embedding theorems and hence a study of suitable functor categories.

It's not a strong data point, but I think it lends weight to the conjecture that the name "category theory" really didn't emerge quite yet at that time, at least in print.

All the best,
David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/dStYCE8wlRCrWn93FNpPFb?domain=ncatlab.org>
Blog: https://thehighergeometer.wordpress.com<https://url.au.m.mimecastprotect.com/s/bGvOCGv0Z6fxJqY1S7hl_o?domain=thehighergeometer.wordpress.com>


---------- Forwarded message ---------
From: David Roberts <droberts.65537@gmail.com<mailto:droberts.65537@gmail.com>>
Date: Sat, 13 Jul 2019 at 05:13
Subject: categories: Re: "First" use of 'Category theory' to describe our field
To: Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>>
Cc: categories@mta.ca<mailto:categories@mta.ca> list <categories@mta.ca<mailto:categories@mta.ca>>


Dear Ross,

no, that's pretty good! It's mildly surprising that it took ~20 years
for the name to 'stick', but maybe less so given that the field grew
slowly to start.

Thanks,

David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/dStYCE8wlRCrWn93FNpPFb?domain=ncatlab.org>
Blog: https://thehighergeometer.wordpress.com<https://url.au.m.mimecastprotect.com/s/bGvOCGv0Z6fxJqY1S7hl_o?domain=thehighergeometer.wordpress.com>

On Thu, 11 Jul 2019 at 08:21, Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>> wrote:
>
> Dear David
>
> From memory, the Pure Mathematics Honours (USyd) course that Max Kelly taught in 1965
> was called ``Category Theory''. It concentrated on different kinds of morphisms
> and factorizations in a category, and finished with adjoint functors.
> Also John Gray's (UIllinois) 1968-69 graduate course had that name.
>
> From Eilenberg I heard that each person using categories should have their own category of expertise.
> I told this to John Gray who said that was fine; the time had come for that category to be Cat.
>
> I would suggest that the first category theorists to think of themselves as such were Eilenberg's students at Columbia.
> However, Mac Lane was definitely a category theorist.
>
> This is probably not the verifiable stuff you were seeking.
>
> Ross
>
>
> On 10 Jul 2019, at 10:01 PM, David Roberts <droberts.65537@gmail.com<mailto:droberts.65537@gmail.com>> wrote:
>
> Hi all,
>
> the (idle) question is: when did the phrase 'category theory' catch on
> for the field? Clearly it didn't leap from either of the heads of
> Eilenberg or Mac Lane full-grown, since they used the phrase 'General
> theory of natural equivalences'. There are the old 'Reports of the
> Midwest Category Seminar' lecture notes (the first in 1967), which
> hints that 'category theory' wasn't quite the name in use.
>
> Even more interesting: who was the first "category theorist", by that name?
>
> Answers referring to verifiable sources would be best.
>
> Thoughts?
>
> David
>
>
>
> David Roberts
> Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/dStYCE8wlRCrWn93FNpPFb?domain=ncatlab.org>
> Blog: https://thehighergeometer.wordpress.com<https://url.au.m.mimecastprotect.com/s/bGvOCGv0Z6fxJqY1S7hl_o?domain=thehighergeometer.wordpress.com>
>


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Re: categories: Re: "First" use of 'Category theory' to describe our field

[Colin McLarty]

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Systems biologist Robert Rosen used the term "category theory" on p. 340 of    Rosen, R. 1958. The Representation of Biological Systems from the Stand- point of the
Theory of Categories. Bulletin of Mathematical Biophysics 20, 317– 342.


On Thu, Feb 22, 2024 at 10:09 PM David Roberts <droberts.65537@gmail.com<mailto:droberts.65537@gmail.com>> wrote:
Hi all,

It's been a few years, but it came to my attention in the past 24 hours that Freyd's 1960 PhD thesis was titled "Functor theory", at Princeton, supervised by Steenrod and Buchsbaum. I suspect it was largely influenced by Buchsbaum, being a student of Eilenberg, and who essentially introduced abelian categories, with which Freyd's PhD was largely about; more precisely, embedding theorems and hence a study of suitable functor categories.

It's not a strong data point, but I think it lends weight to the conjecture that the name "category theory" really didn't emerge quite yet at that time, at least in print.

All the best,
David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/QJDdCBNqgBCQVDn4Sz0Ghn?domain=ncatlab.org>
Blog: https://thehighergeometer.wordpress.com<https://url.au.m.mimecastprotect.com/s/IuqoCD1vRkC2BoxVH5kh1P?domain=thehighergeometer.wordpress.com>


---------- Forwarded message ---------
From: David Roberts <droberts.65537@gmail.com<mailto:droberts.65537@gmail.com>>
Date: Sat, 13 Jul 2019 at 05:13
Subject: categories: Re: "First" use of 'Category theory' to describe our field
To: Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>>
Cc: categories@mta.ca<mailto:categories@mta.ca> list <categories@mta.ca<mailto:categories@mta.ca>>


Dear Ross,

no, that's pretty good! It's mildly surprising that it took ~20 years
for the name to 'stick', but maybe less so given that the field grew
slowly to start.

Thanks,

David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/QJDdCBNqgBCQVDn4Sz0Ghn?domain=ncatlab.org>
Blog: https://thehighergeometer.wordpress.com<https://url.au.m.mimecastprotect.com/s/IuqoCD1vRkC2BoxVH5kh1P?domain=thehighergeometer.wordpress.com>

On Thu, 11 Jul 2019 at 08:21, Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>> wrote:
>
> Dear David
>
> From memory, the Pure Mathematics Honours (USyd) course that Max Kelly taught in 1965
> was called ``Category Theory''. It concentrated on different kinds of morphisms
> and factorizations in a category, and finished with adjoint functors.
> Also John Gray's (UIllinois) 1968-69 graduate course had that name.
>
> From Eilenberg I heard that each person using categories should have their own category of expertise.
> I told this to John Gray who said that was fine; the time had come for that category to be Cat.
>
> I would suggest that the first category theorists to think of themselves as such were Eilenberg's students at Columbia.
> However, Mac Lane was definitely a category theorist.
>
> This is probably not the verifiable stuff you were seeking.
>
> Ross
>
>
> On 10 Jul 2019, at 10:01 PM, David Roberts <droberts.65537@gmail.com<mailto:droberts.65537@gmail.com>> wrote:
>
> Hi all,
>
> the (idle) question is: when did the phrase 'category theory' catch on
> for the field? Clearly it didn't leap from either of the heads of
> Eilenberg or Mac Lane full-grown, since they used the phrase 'General
> theory of natural equivalences'. There are the old 'Reports of the
> Midwest Category Seminar' lecture notes (the first in 1967), which
> hints that 'category theory' wasn't quite the name in use.
>
> Even more interesting: who was the first "category theorist", by that name?
>
> Answers referring to verifiable sources would be best.
>
> Thoughts?
>
> David
>
>
>
> David Roberts
> Webpage: https://ncatlab.org/nlab/show/David+Roberts<https://url.au.m.mimecastprotect.com/s/QJDdCBNqgBCQVDn4Sz0Ghn?domain=ncatlab.org>
> Blog: https://thehighergeometer.wordpress.com<https://url.au.m.mimecastprotect.com/s/IuqoCD1vRkC2BoxVH5kh1P?domain=thehighergeometer.wordpress.com>
>


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Re: "First" use of 'Category theory' to describe our field

[David Roberts]

To add another data point, there is the 1966 book

M. Hasse und L. Michler, Theorie der Kategorien, Berlin, VEB Deutscher
Verlag der Wissenschaften, 1966

which is mentioned in this report

https://www.mfo.de/document/0908a/OWR_2009_08.pdf ("Mini-Workshop:
Category Theory and Related Fields: History and Prospects")

This ties for year of publication with

Brinkmann, H.-B., Puppe, D., Kategorien und Funktoren
(Berlin/Heidelberg/New York, LNM 18, 1966

And following

Ehresmann, Charles, Catégories et structures, Paris: Dunod 1965

(perhaps ironically, Charles Ehresmann was already doing *internal*
category theory to some extent by 1959. However: in the linked 2009
OWR Ageron writes "characterizations of Ehresmann as being a category
theorist... misses the point, scientifically and sociologically: he
cannot be characterized better than being during all of its life a
structure theorist.")

Mike Barr mentioned privately the 1966 meeting at Oberwolfach

Tagung über Kategorien 17. - 24. Juli 1966

("Conference on categories")

So I guess mid-1960s is likely to be the date.

David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com

On Thu, 11 Jul 2019 at 08:21, Ross Street <ross.street@mq.edu.au> wrote:
>
> Dear David
>
> From memory, the Pure Mathematics Honours (USyd) course that Max Kelly taught in 1965
> was called ``Category Theory''. It concentrated on different kinds of morphisms
> and factorizations in a category, and finished with adjoint functors.
> Also John Gray's (UIllinois) 1968-69 graduate course had that name.
>
> From Eilenberg I heard that each person using categories should have their own category of expertise.
> I told this to John Gray who said that was fine; the time had come for that category to be Cat.
>
> I would suggest that the first category theorists to think of themselves as such were Eilenberg's students at Columbia.
> However, Mac Lane was definitely a category theorist.
>
> This is probably not the verifiable stuff you were seeking.
>
> Ross
>
>
> On 10 Jul 2019, at 10:01 PM, David Roberts <droberts.65537@gmail.com> wrote:
>
> Hi all,
>
> the (idle) question is: when did the phrase 'category theory' catch on
> for the field? Clearly it didn't leap from either of the heads of
> Eilenberg or Mac Lane full-grown, since they used the phrase 'General
> theory of natural equivalences'. There are the old 'Reports of the
> Midwest Category Seminar' lecture notes (the first in 1967), which
> hints that 'category theory' wasn't quite the name in use.
>
> Even more interesting: who was the first "category theorist", by that name?
>
> Answers referring to verifiable sources would be best.
>
> Thoughts?
>
> David
>
>
>
> David Roberts
> Webpage: https://ncatlab.org/nlab/show/David+Roberts
> Blog: https://thehighergeometer.wordpress.com
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: "First" use of 'Category theory' to describe our field

[Johannes Huebschmann]

In the preface of Eilenberg-Mac Lane: Collected works,
EML write (on p.ix/x):

Category theory was thus born very quietly in 1945.
...
It was only with Grothendieck's famous Tohoku paper in 1957
that the theory came into full bloom.

Grothendieck already exploited, among other things,
abelian categories.

The term category derives form the ancient Greek verb

kategoreo.


Johannes



----- Mail original -----
De: "David Roberts" <droberts.65537@gmail.com>
À: "categories@mta.ca list" <categories@mta.ca>
Envoyé: Mercredi 10 Juillet 2019 14:01:16
Objet: categories: "First" use of 'Category theory' to describe our field

Hi all,

the (idle) question is: when did the phrase 'category theory' catch on
for the field? Clearly it didn't leap from either of the heads of
Eilenberg or Mac Lane full-grown, since they used the phrase 'General
theory of natural equivalences'. There are the old 'Reports of the
Midwest Category Seminar' lecture notes (the first in 1967), which
hints that 'category theory' wasn't quite the name in use.

Even more interesting: who was the first "category theorist", by that name?

Answers referring to verifiable sources would be best.

Thoughts?

David



David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: "First" use of 'Category theory' to describe our field

[Peter May]

For Saunders, the terminology of category theory came from Kant.?? From
Wikipedia:

In Kant's philosophy, a category (German: Categorie in the original or
Kategorie in modern German) is a *pure concept of the understanding
(Verstand)*.

Etc.?? It may be relevant that Saunders was very influenced by his time
at Gottingen.
In any case, the term category theory was second nature to him. Although
that was well before my time, I'm quite sure he used the term pretty
much from the beginning.

On 7/10/19 7:01 AM, David Roberts wrote:
> Hi all,
>
> the (idle) question is: when did the phrase 'category theory' catch on
> for the field? Clearly it didn't leap from either of the heads of
> Eilenberg or Mac Lane full-grown, since they used the phrase 'General
> theory of natural equivalences'. There are the old 'Reports of the
> Midwest Category Seminar' lecture notes (the first in 1967), which
> hints that 'category theory' wasn't quite the name in use.
>
> Even more interesting: who was the first "category theorist", by that name?
>
> Answers referring to verifiable sources would be best.
>
> Thoughts?
>
> David
>
>
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: "First" use of 'Category theory' to describe our field

[Colin McLarty]

I don't think anyone used the term category theory (in the mathematical
sense) before  mathematical biologist Robert Rosen in 1958.  That is the
earliest reference known to Math Reviews.

Not surprising, it was slow to spread from Rosen into the field.  Saunders
mentioned "the theory of categories"  once in his 1963 book Homology (p.
34).  In that book he writes repeatedly of set theory, group theory,
homology theory, and many others, but not once "category theory".  One
other datum: Freyd's 1960 dissertation was titled "Functor theory," but
"category theory" occurs in the introduction to the 1964 book version.
Category theory was a very common term by the late 1960s.

Good question about "category theorist."  I have no idea but expect it came
several years later.  It is not used very formally, and searching it on
Math Reviews gives only 15 hits in the entire database.

Colin


On 7/10/19 7:01 AM, David Roberts wrote:
>> Hi all,
>>
>> the (idle) question is: when did the phrase 'category theory' catch on
>> for the field? Clearly it didn't leap from either of the heads of
>> Eilenberg or Mac Lane full-grown, since they used the phrase 'General
>> theory of natural equivalences'. There are the old 'Reports of the
>> Midwest Category Seminar' lecture notes (the first in 1967), which
>> hints that 'category theory' wasn't quite the name in use.
>>
>> Even more interesting: who was the first "category theorist", by that
> name?
>>
>> Answers referring to verifiable sources would be best.
>>
>> Thoughts?
>>
>> David
>>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: "First" use of 'Category theory' to describe our field

[David Roberts]

Dear Peter,

>For Saunders, the terminology of category theory came from Kant.

this is not what I mean. That 'category' came from Kant is well-known.
But when was the field itself called 'category theory'? That is, when
was it sufficiently established to warrant its own name, rather than
just be a method in algebraic topology/abstract algebra? Some early
books were titled 'Categories and functors' (albeit in German), rather
than 'Category theory', though we got there in the end!

Certainly by the publication of

Proceedings Sydney Category Theory Seminar 1972 /1973 (Springer LNM 420)

we have 'Category theory' in print (in a title in English), though
Ross pointed out Max Kelly's honours-level course "category theory" in
Sydney in 1965 (in principle one could track down the university
archives...). But 'category theory' as a phrase appears nowhere in the
1945 paper.

This was all just idle curiosity, though, so I'm happy to receive
replies off-list if the moderator deems this all too frivolous.

Best regards,
David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com

On Thu, 11 Jul 2019 at 22:28, Peter May <may@math.uchicago.edu> wrote:
>
> For Saunders, the terminology of category theory came from Kant.  From Wikipedia:
>
> In Kant's philosophy, a category (German: Categorie in the original or Kategorie in modern German) is a pure concept of the understanding (Verstand).
>
> Etc.  It may be relevant that Saunders was very influenced by his time at Gottingen.
> In any case, the term category theory was second nature to him.  Although that was well before my time, I'm quite sure he used the term pretty much from the beginning.
>
> On 7/10/19 7:01 AM, David Roberts wrote:
>
> Hi all,
>
> the (idle) question is: when did the phrase 'category theory' catch on
> for the field? Clearly it didn't leap from either of the heads of
> Eilenberg or Mac Lane full-grown, since they used the phrase 'General
> theory of natural equivalences'. There are the old 'Reports of the
> Midwest Category Seminar' lecture notes (the first in 1967), which
> hints that 'category theory' wasn't quite the name in use.
>
> Even more interesting: who was the first "category theorist", by that name?
>
> Answers referring to verifiable sources would be best.
>
> Thoughts?
>
> David
>
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: "First" use of 'Category theory' to describe our field

[Johannes Huebschmann]

Dear All

The phrase

"the terminology of category theory came from Kant"

oversimplifies the situation.

Aristotle (Peri ton kategorion)
discusses categories.
Kant uses categories to mediate his thought
(Kritik der Urteilskraft).
Saunders Mac Lane's adviser in Goettingen was
Paul Bernays. Bernays knew ancient Greek philosophy
very well.

During my student's time
at the ETH I still had occasion to talk
to Paul Bernays (he then was in his 80s).
He regularly attended the
logic seminar and even contributed to the discussion.

As for the terminology "functor" I vaguely remember
this derives from Carnap but I may be wrong
and perhaps my memory fails.
Perhaps someone knows better.

Also, in German, when you teach a course entitled "Kategorien"
or "Kategorien und Funktoren",
that synonymously means "Kategorientheorie".
For example, D. Puppe taught such a course in the 1960s,
and that was the origin of the Brinkmann-Puppe LNM.


Best regards

Johannes


----- Mail original -----
De: "David Roberts" <droberts.65537@gmail.com>
À: "Peter May" <may@math.uchicago.edu>
Cc: "categories@mta.ca list" <categories@mta.ca>
Envoyé: Jeudi 11 Juillet 2019 15:12:10
Objet: categories: Re: "First" use of 'Category theory' to describe our field

Dear Peter,

>For Saunders, the terminology of category theory came from Kant.

this is not what I mean. That 'category' came from Kant is well-known.
But when was the field itself called 'category theory'? That is, when
was it sufficiently established to warrant its own name, rather than
just be a method in algebraic topology/abstract algebra? Some early
books were titled 'Categories and functors' (albeit in German), rather
than 'Category theory', though we got there in the end!

Certainly by the publication of

Proceedings Sydney Category Theory Seminar 1972 /1973 (Springer LNM 420)

we have 'Category theory' in print (in a title in English), though
Ross pointed out Max Kelly's honours-level course "category theory" in
Sydney in 1965 (in principle one could track down the university
archives...). But 'category theory' as a phrase appears nowhere in the
1945 paper.

This was all just idle curiosity, though, so I'm happy to receive
replies off-list if the moderator deems this all too frivolous.

Best regards,
David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com

On Thu, 11 Jul 2019 at 22:28, Peter May <may@math.uchicago.edu> wrote:
>
> For Saunders, the terminology of category theory came from Kant.  From Wikipedia:
>
> In Kant's philosophy, a category (German: Categorie in the original or Kategorie in modern German) is a pure concept of the understanding (Verstand).
>
> Etc.  It may be relevant that Saunders was very influenced by his time at  Gottingen.
> In any case, the term category theory was second nature to him.  Although  that was well before my time, I'm quite sure he used the term pretty much from the beginning.
>

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Re: "First" use of 'Category theory' to describe our field

[Tadeusz Litak]

> As for the terminology "functor" I vaguely remember
> this derives from Carnap but I may be wrong
> and perhaps my memory fails.
> Perhaps someone knows better.
This is indeed commonly accepted original inspiration; I've never heard about Eilenberg or Mac???Lane protesting against
this explanation (maybe somebody knows better?). See for example:

https://plato.stanford.edu/entries/category-theory/#2

which seems a better summary of historical origins than anything you might find on wikipedia.

> The central notion at the time, as their title indicates, was that of natural transformation. In order to give a
> general definition of the latter, they defined functor, borrowing the term from Carnap, and in order to define
> functor, they borrowed the word ???category??? from the philosophy of Aristotle, Kant, and C. S. Peirce, but redefining it
> mathematically.


As regards Mathematical Reviews, while this is an interesting observation, I am not sure if it settles definitively the
question?? who and when started to see category theory as a field in its own right. There is no guarantee that the
authors of these reviews paid particular attention to terminological issues, or that they were sympathetic enough to the
goals of papers under discussion. In particular, that they shared the views of reviewed authors that mathematics needs a
new subdiscipline.

I haven't checked if the phrase "category theory" was ever used in the 1950's books of Eilenberg & Steenrod or Cartan &
Eilenberg. But Kan's 1958 paper on adjoint functors contains, by a quick count, 120 occurrences of the term "category"
on merely 36 pages and Grothendieck's 1957 T??hoku paper---over 200 (it is almost four times as large that of Kan
though). It'd seem that if you talk about a mathematical entity so much, you are developing its theory.

In fact, the 1945 paper itself does seem to suggest quite openly that a grand unifying foundational theory is the
ultimate goal. It is enough to read the final paragraphs of its intro:


> In a metamathematical sense our theory provides general concepts applicable to all branches of abstract mathematics,
> and so contributes to the current trend towards uniform treatment of different mathematical disciplines. In
> particular, it provides opportunities for the comparison of constructions and of the isomorphisms occurring in
> different branches of mathematics; in this way it may occasionally suggest new results by analogy. (...)
> This may be regarded as a continuation of the Klein Erlanger Programm, in the sense that a geometrical space with its
> group of transformations is generalized to a category with its algebra of mappings.

Best,
t.



On 13.07.19 11:45, Johannes Huebschmann wrote:
> Dear All
>
> The phrase
>
> "the terminology of category theory came from Kant"
>
> oversimplifies the situation.
>
> Aristotle (Peri ton kategorion)
> discusses categories.
> Kant uses categories to mediate his thought
> (Kritik der Urteilskraft).
> Saunders Mac Lane's adviser in Goettingen was
> Paul Bernays. Bernays knew ancient Greek philosophy
> very well.
>
> During my student's time
> at the ETH I still had occasion to talk
> to Paul Bernays (he then was in his 80s).
> He regularly attended the
> logic seminar and even contributed to the discussion.
>
> As for the terminology "functor" I vaguely remember
> this derives from Carnap but I may be wrong
> and perhaps my memory fails.
> Perhaps someone knows better.
>
> Also, in German, when you teach a course entitled "Kategorien"
> or "Kategorien und Funktoren",
> that synonymously means "Kategorientheorie".
> For example, D. Puppe taught such a course in the 1960s,
> and that was the origin of the Brinkmann-Puppe LNM.
>
>
> Best regards
>
> Johannes
>
>

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Re: "First" use of 'Category theory' to describe our field

[Tadeusz Litak]

PPS. I see now that the 1958 paper of Rosen is indeed listed as a reference in Kurosh et al. 1960 "Foundations of the
theory of categories" (and, of course, Rosen too uses "the theory of categories" in the title). Curiously, however, it
seems there is no quotation of Rosen in the body of the paper and it's not clear to me that it was Rosen who convinced
them this is a separate research area. Throughout, they refer to "the theory of categories" as an already established
field, impossible to summarize in one article.

Furthermore, the introduction ends with one an expression of?? "gratitude to the participants in the seminar on the
theory of categories, which worked under the direction of one of the authors in Moscow University". Its participants
apparently produced quite a few papers listed by MacLane. Not sure when it started, but I doubt if there is any example
of an earlier seminar devoted to (foundations of) category theory. As I said earlier and as mentioned by MacLane, in
1960 Kurosh also lectured about the subject in Prague.


> PS. Actually, I stand corrected. I found out now that MacLane's "Concepts and Categories in Perspective"
>
> http://www.ams.org/samplings/math-history/hmath1-maclane25.pdf
>
> discusses in some detail when exactly he started to see "category theory" as a viable field for further research in
> its own right, rather than just "a handy language" or "a conceptual view of parts of mathematics".
>
> He claims that for him, 1963 was the critical year. Details to be found in Section 17 of the paper linked above. He
> says, for example, that being invited as the 1963 AMS Colloquium Lecturer, he chose to give four lectures on category
> theory (more specifically, he spoke about "Categorical Algebra"), whereas a year or two later he would have spoken
> about homological algebra or algebraic topology.
>
> Notably, one can also find there a list of early papers on category theory, which illustrates how the field suddenly
> exploded in early 1960's.?? I can't find Robert Rosen being mentioned, so it might be that MacLane missed the first
> occurrence of the phrase "category theory" in print. But he mentions, for example, a 1960 paper by a mathematician
> from the Kurosh school (M. S. Calenko) whose title is "On the foundations of the theory of categories".
> Another very early paper with "theory of categories"?? or "category theory" in the title is a 1962 one by V??ra Trnkov??.
>
> Actually, MacLane seems to omit an even earlier reference (quoted, for example, by Eckmann and Hilton 1962, who also
> speak occasionally about "theory of categories"):
>
> Kurosh, A. G., A. Kh. Livshits and E. G. Shul'geifer: Foundations of the theory of categories. Uspekhi Matem. Nauk
> XV,6, 1 (1960).
>
> Is this the first paper which openly mentions category theory in its title then? It seems that researchers influenced
> by Kurosh (who also lectured in Prague in 1960)?? were the earliest ones who had the courage to write papers explicitly
> on "the theory of categories".?? I guess, e.g., Ji???? Ad??mek?? or Ale?? Pultr could have much more to say here.
>
>
>
> On 14.07.19 17:58, Tadeusz Litak wrote:
>>> As for the terminology "functor" I vaguely remember
>>> this derives from Carnap but I may be wrong
>>> and perhaps my memory fails.
>>> Perhaps someone knows better.
>> This is indeed commonly accepted original inspiration; I've never heard about Eilenberg or Mac???Lane protesting
>> against this explanation (maybe somebody knows better?). See for example:
>>
>> https://plato.stanford.edu/entries/category-theory/#2
>>
>> which seems a better summary of historical origins than anything you might find on wikipedia.
>>
>>> The central notion at the time, as their title indicates, was that of natural transformation. In order to give a
>>> general definition of the latter, they defined functor, borrowing the term from Carnap, and in order to define
>>> functor, they borrowed the word ???category??? from the philosophy of Aristotle, Kant, and C. S. Peirce, but redefining
>>> it mathematically.
>>
>>
>> As regards Mathematical Reviews, while this is an interesting observation, I am not sure if it settles definitively
>> the question?? who and when started to see category theory as a field in its own right. There is no guarantee that the
>> authors of these reviews paid particular attention to terminological issues, or that they were sympathetic enough to
>> the goals of papers under discussion. In particular, that they shared the views of reviewed authors that mathematics
>> needs a new subdiscipline.
>>
>> I haven't checked if the phrase "category theory" was ever used in the 1950's books of Eilenberg & Steenrod or Cartan
>> & Eilenberg. But Kan's 1958 paper on adjoint functors contains, by a quick count, 120 occurrences of the term
>> "category" on merely 36 pages and Grothendieck's 1957 T??hoku paper---over 200 (it is almost four times as large that
>> of Kan though). It'd seem that if you talk about a mathematical entity so much, you are developing its theory.
>>
>> In fact, the 1945 paper itself does seem to suggest quite openly that a grand unifying foundational theory is the
>> ultimate goal. It is enough to read the final paragraphs of its intro:
>>
>>
>>> In a metamathematical sense our theory provides general concepts applicable to all branches of abstract mathematics,
>>> and so contributes to the current trend towards uniform treatment of different mathematical disciplines. In
>>> particular, it provides opportunities for the comparison of constructions and of the isomorphisms occurring in
>>> different branches of mathematics; in this way it may occasionally suggest new results by analogy. (...)
>>> This may be regarded as a continuation of the Klein Erlanger Programm, in the sense that a geometrical space with
>>> its group of transformations is generalized to a category with its algebra of mappings.
>>
>> Best,
>> t.
>>
>>

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