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charset="utf-8" Content-Transfer-Encoding: quoted-printable The nice thing about this reference to Whitney is that it explains why Eile= nberg and MacLane's Kantian naming taste was applied to *categories* and *f= unctors* but abandoned when it came to 'natural transformations'. I was alw= ays wondering why we have been deprived of the pleasure of talking about, s= ay, *transcendental* transformations all these years... Whitney's intent with "natural transformations" seems to have been similar = to Godement's intent with "standard constructions". In a similar vein, peop= le used to talk about "canonical isomorphisms"... It seems that Whitney's natural homomorphisms and natural topologies were n= atural in the sense that their definitions were the only thing that you cou= ld write down in the given context. Programmers call such definitions polym= orphic. Now we know that you can do that precisely when what you are writin= g down is preserved under the homomorphisms induced by the type constructor= s. Eilenberg and MacLane noticed this phenomenon in some form, perhaps by a= ligning their homological and homotopical constructions mentioned by Mike. = To capture the homomorphisms induced by the constructions, they had to defi= ne the homomorphism part of the constructions: the functors. And to capture= the homomorphisms... etcetc. They said themselves that they introduced cat= egories to define functors and functors to define natural transformations. Whitney did use natural transformations, and he even defined the tensors be= tween which his natural transformations occur, but he doesn't seem to have = considered what his tensor constructions do with the homomorphisms. So he d= idn't have the tensor *functors*. So the naturality *idea* was in the air b= ut it hadn't quite landed yet. He talks about a group R "operating" on a gr= oup G (as per Murray-von Neumann). Could his idea of naturality account for= the difference between the families of isomorphisms V~V* and V~V**, viewed= as a field R "operating" on a suitable group G (as von Neumann did)? All students of physics learn how Galileo introduced the idea of relativity= with respect to frames of reference. But he didn't have Lorenz transformat= ions and Riemanian geometry, so the mathematical realizations of the idea o= f relativity had to wait... Whitney didn't have functors. I think the message is that concepts are in the air people breathe before t= hey figure out a way to write them down for other people. Other people figu= re out some other ways. We shouldn't take concepts personally. 2c, -- dusko On Mon, Dec 18, 2023 at 9:49=E2=80=AFAM Evgeny Kuznetsov > wrote: Here is a copy of the paper by Hassler Whitney of 1938 titled "Tensor produ= cts of abelian groups" On Mon, Dec 18, 2023, 23:32 Jean-Pierre Marquis > wrote: Most likely. Whitney uses the terms =E2=80=98natural isomorphism=E2=80=99 and =E2=80=98n= atural homomorphism=E2=80=99 as well as the terms =E2=80=98natural topology= =E2=80=99 and =E2=80=98natural neighborhood=E2=80=99 at many different plac= es in the paper. But these terms are never explicitly defined. Cheers, Jean-Pierre De : Wesley Phoa > Date : lundi, 18 d=C3=A9cembre 2023 =C3=A0 14:18 =C3=80 : Michael Barr, Prof. > Cc : categories@mq.edu.au > Objet : Re: Modification of what I said Was he referring to the paper =E2=80=9CTensor products of abelian groups=E2= =80=9D, cited in this discussion? https://mathoverflow.net/questions/287869= /history-of-natural-transformations I don=E2=80=99t have access to it either, but it=E2=80=99s on Scribd: https= ://www.scribd.com/document/172981416/Hassler-Whitney-Tensor-Products-of-Abe= lian-Groups The terms =E2=80=9Cnatural isomorphism=E2=80=9D and =E2=80=9Cnatural homomo= rphism=E2=80=9D are used on pages 500-501, and these do turn out to be natu= ral transformations, but it=E2=80=99s not obvious that he intended to expli= citly define a new formal concept. Wesley Sent from my iPad On Dec 18, 2023, at 10:00=E2=80=AFAM, Michael Barr, Prof. > wrote: =EF=BB=BF Peter Freyd claims that Hassler Whitney defined natural transformation in a= 1938 paper. I no longer have access to Math. Reviews (except by going to = McGill, which I have done only once in the last four years) so I cannot sup= ply a reference. Michael You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | Learn = more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | Learn = more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --000000000000d8fa3d060cd65dbb Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
The nice thing about this reference to Whitney is that it explains why= Eilenberg and MacLane's Kantian naming taste was applied to *categories* a= nd *functors* but abandoned when it came to 'natural transformations'. I wa= s always wondering why we have been deprived of the pleasure of talking about, say, *transcendental* transform= ations all these years...

Whitney's intent with "natural transformations" seems to hav= e been similar to Godement's intent with "standard constructions"= . In a similar vein, people used to talk about "canonical isomorphisms= "...

It seems that Whitney's natural homomorphisms and natural topologies w= ere natural in the sense that their definitions were the only thing that yo= u could write down in the given context. Programmers call such definitions = polymorphic. Now we know that you can do that precisely when what you are writing down is preserved under th= e homomorphisms induced by the type constructors. Eilenberg and MacLane not= iced this phenomenon in some form, perhaps by aligning their homological an= d homotopical constructions mentioned by Mike. To capture the homomorphisms induced by the constructions, they h= ad to define the homomorphism part of the constructions: the functors. And = to capture the homomorphisms... etcetc. They said themselves that they intr= oduced categories to define functors and functors to define natural transformations.

Whitney did use natural transformations, and he even defined the tenso= rs between which his natural transformations occur, but he doesn't seem to = have considered what his tensor constructions do with the homomorphisms. So= he didn't have the tensor *functors*. So the naturality *idea* was in the air but it hadn't quite landed yet. He= talks about a group R "operating" on a group G (as per Murray-vo= n Neumann). Could his idea of naturality account for the difference between= the families of isomorphisms V~V* and V~V**, viewed as a field R "operating" on a suitable group G (as von Ne= umann did)?

All students of physics learn how Galileo introduced the idea of relat= ivity with respect to frames of reference. But he didn't have Lorenz transf= ormations and Riemanian geometry, so the mathematical realizations of the i= dea of relativity had to wait... Whitney didn't have functors.

I think the message is that concepts are in the air people breathe bef= ore they figure out a way to write them down for other people. Other people= figure out some other ways. We shouldn't take concepts personally.

2c,
-- dusko

On Mon, Dec 18, 2023 at 9:49=E2=80=AF= AM Evgeny Kuznetsov <jenkakuz= necov@gmail.com> wrote:
Here is a copy of the paper by Hassler Whitney of 19= 38 titled "Tensor products of abelian groups"



On Mon, Dec 18, 2023, 23:32 Jean-Pier= re Marquis <jean-pierre.marquis@umontreal.ca> wrote:

Most likely.<= /p>

 

Whitney uses the terms =E2=80= =98natural isomorphism=E2=80=99 and =E2=80=98natural homomorphism=E2=80=99 = as well as the terms =E2=80=98natural topology=E2=80=99 and =E2=80=98natura= l neighborhood=E2=80=99 at many different places in the paper. But these te= rms are never explicitly defined.

 

Cheers,

 

Jean-Pierre

 

 

De : Wesley Phoa <doctorwes@gmail.= com>
Date : lundi, 18 d=C3=A9cembre 2023 =C3=A0 14:18
=C3=80 : Michael Barr, Prof. <barr= .michael@mcgill.ca>
Cc : categories@mq.edu.au <categories@mq.edu.au>
Objet : Re: Modification of what I said

Was he referring to the paper =E2=80=9CTensor produc= ts of abelian groups=E2=80=9D, cited in this discussion? https://m= athoverflow.net/questions/287869/history-of-natural-transformations<= /u>

 

I don=E2=80=99t have access to it either, but it=E2= =80=99s on Scribd: https://www.scribd.com/document/172981416/Hassler-Whi= tney-Tensor-Products-of-Abelian-Groups

 

The terms =E2=80=9Cnatural isomorphism=E2=80=9D and = =E2=80=9Cnatural homomorphism=E2=80=9D are used on pages 500-501, and these= do turn out to be natural transformations, but it=E2=80=99s not obvious th= at he intended to explicitly define a new formal concept.

 

Wesley

 

Sent from my iPad



On Dec 18, 2023, at 10:= 00=E2=80=AFAM, Michael Barr, Prof. <barr.micha= el@mcgill.ca> wrote:

=EF=BB=BF

Peter Freyd claims that Hassler Whitney def= ined natural transformation in a 1938 paper.  I no longer have access = to Math. Reviews (except by going to McGill, which I have done only once in the last four years) so I cannot supply a referen= ce.

 

Michael

 
 
You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
View group files   |   = Leave group   |   Learn more about Microsoft 365 Groups
 
 
 
You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
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