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charset="utf-8" Content-Transfer-Encoding: quoted-printable oh this observation about the universal property of the tensor product seem= s to me at least as important as the terminology. in 1937! on the other han= d, weyl's group theory was out for almost 10 years and it is threaded with = when representations are equivalent, and what can be concluded in terms of = the characters alone, and with looking at groups categorically, from outsid= e... the music was in the air :) On Tue, Dec 19, 2023 at 2:11=E2=80=AFPM David Roberts > wrote: If you go back to Whitney's announcement in PNAS, and the subsequent paper,= titled "On products in a complex", you can see where he started using the = phrase, even an even more non-technical manner: "In recent years the existence of certain products in a complex K has been = much studied, combining a p-chain and a q-chain to form a (p+q)-chain. For such products to have topological significance, certain conditions shou= ld be satisfied; it seems that (PI) and (P2) below (=C2=A75) are the most n= atural." "If we wish to study products of L^r and L^s into L^{r+s}, using coefficien= t groups G in L^r and H in L^s, then we most naturally use the coefficient group GH in L^{r+s}; the theory with any other coefficient group in L^{r+s}= follows from this" (from the 1937 PNAS paper https://doi.org/10.1073/pnas.23.5.285) The second quote is the first hint of the tensor product of groups, denoted= here by GH, which are defined (apparently with their universal property!) = thus: "11. Tensor products of groups. - The definition of "group pairs" may also = be split into two parts, as follows. Given the (abelian) groups G and H, th= e *tensor product* GH is the group with generators gh (defined abstractly),= and, for relations, all those obtainable with the two distributive laws. T= hen the possible definitions of group pairs G and H with respect to the gro= up Z are obtained merely by choosing the possible homomorphisms of GH into = Z." In the subsequent fuller article (submitted mid-1937) we find: "p-cell times a q-cell far away from the p-cell should certainly give nothi= ng; hence (P1) of =C2=A75 is a natural assumption." "Hence \delta(A\cup B) must be expressible in terms such as \delta A \cup B= and A \cup \delta B. (P2) is the natural form." "The maps Sd and Sd* of L^p(K) and L^p(K*) into L^p(K') are defined in the = natural manner;" (from the 1938 Annals paper https://doi.org/10.2307/1968795) As a bonus, we have the sentence "It is natural to call two parametrized cu= rves equivalent if one can be obtained from the other by a change of parame= ter (preserving orientation)." in Whitney's 1937 Compositio paper "On regul= ar closed curves in the plane" http://www.numdam.org/item?id=3DCM_1937__4__= 276_0 And, even "If we are satisfied with a function F of class C^r, we would nat= urally use (e') in place of (d), (e) and (f)." from his 1936 Annals paper "= Differentiable Manifolds" https://doi.org/10.2307/1968482. In this paper Wh= itney also uses "required nature", "general nature" when talking about cert= ain maps he wishes to construct. On this issue of "canonical" maps, earlier this year I managed to trace the= origins back to the late 1930s, and have a hypothesis as to how it was cho= sen, based on the term from matrix theory. I hope to write up a short artic= le on this before I am too much older. The chain is roughly: Ehresmann -> B= ourbaki -> Eilenberg/Cartan/Weil -> Serre -> Grothendieck, though the middl= e steps are somewhat nonlinear as all of the named mathematicians were invo= lved in Bourbaki at various stages. Certainly Eilenberg and Cartan's book h= elped disseminate the notion, as did Grothendieck's T=C3=B4hoku and then EG= A1, as well as Bourbaki's texts. I didn't track the usage *forward* through= 1960s CT literature, though, to see who was using it (eg Reports of the Mi= dwest seminar, La Jolla/Batelle proceedings etc) All the best, David Roberts Webpage: https://ncatlab.org/nlab/show/David+Roberts Blog: https://thehighergeometer.wordpress.com On Tue, 19 Dec 2023 at 22:56, Dusko Pavlovic > wrote: CAUTION: External email. Only click on links or open attachments from trust= ed senders. ________________________________ 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 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 --000000000000839672060cf74caa Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
oh this observation about the universal property of the te= nsor product seems to me at least as important as the terminology. in 1937!= on the other hand, weyl's group theory was out for almost 10 years and it = is threaded with when representations are equivalent, and what can be concluded in terms of the characters alone= , and with looking at groups categorically, from outside... the music was i= n the air :)

On Tue, Dec 19, 2023 at 2:11=E2=80=AF= PM David Roberts <droberts.6= 5537@gmail.com> wrote:
If you go back to Whitney's announcement in PNAS, and the subsequent p= aper, titled "On products in a complex", you can see where he sta= rted using the phrase, even an even more non-technical manner:

"In recent years the existence of cert= ain products in a complex K has been much studied, combining a p-chain and = a q-chain to form a (p+q)-chain.
For such products to have topological signi= ficance, certain conditions should be satisfied; it seems that (PI) and (P2= ) below (=C2=A75) are the most natural."

"If we wish to study products of L^r a= nd L^s into L^{r+s}, using coefficient groups G in L^r and H in L^s, then w= e most naturally use the coefficient
group GH in L^{r+s}; the theory with any other coefficient group in L^{r+s}= follows from this"

(from the 1937 PNAS paper https://doi.org/10.1073/pnas.23.5.285)

The second quote is the first hint of the tensor product of groups, de= noted here by GH, which are defined (apparently with their universal proper= ty!) thus:

"11. Tensor products of groups. - The = definition of "group pairs" may also be split into two parts, as = follows. Given the (abelian) groups G and H, the *tensor product* GH is the= group with generators gh (defined abstractly), and, for relations, all those obtainable with the two distributive laws. Then t= he possible definitions of group pairs G and H with respect to the group Z = are obtained merely by choosing the possible homomorphisms of GH into Z.&qu= ot;

In the subsequent fuller article (submitted mid-1937) we find:

"p-cell times a q-cell far away from t= he p-cell should certainly give nothing; hence (P1) of =C2=A75 is a natural= assumption."

"Hence \delta(A\cup B) must be express= ible in terms such as \delta A \cup B and A \cup \delta B. (P2) is the natu= ral form."

"The maps Sd and Sd* of L^p(K) and L^p= (K*) into L^p(K') are defined in the natural manner;"

(from the 1938 Annals paper https://doi.org/10.2307/1968795)

As a bonus, we have the sentence "It is natural to call two param= etrized curves equivalent if one can be obtained from the other by a change= of parameter (preserving orientation)." in Whitney's 1937 Compositio = paper "On regular closed curves in the plane" http://www.numdam.org/item?id=3DCM_1937__4__276_0

And, even "If we are satisfied with a function F of class C^r, we= would naturally use (e') in place of (d), (e) and (f)." from his 1936= Annals paper "Differentiable Manifolds" https://doi.org/10.2307/1968482. In this paper Whitney also uses "= required nature", "general nature" when talking about certai= n maps he wishes to construct.

On this issue of "canonical" maps, earlier this year I manag= ed to trace the origins back to the late 1930s, and have a hypothesis as to= how it was chosen, based on the term from matrix theory. I hope to write u= p a short article on this before I am too much older. The chain is roughly: Ehresmann -> Bourbaki -> Eilenberg= /Cartan/Weil -> Serre -> Grothendieck, though the middle steps are so= mewhat nonlinear as all of the named mathematicians were involved in Bourba= ki at various stages. Certainly Eilenberg and Cartan's book helped disseminate the notion, as did Grothendieck's T=C3=B4= hoku and then EGA1, as well as Bourbaki's texts. I didn't track the usage *= forward* through 1960s CT literature, though, to see who was using it (eg R= eports of the Midwest seminar, La Jolla/Batelle proceedings etc)

All the best,



On Tue, 19 Dec 2023 at 22:56, Dusko P= avlovic <duskgoo@= gmail.com> wrote:
CAUTION: External ema= il. Only click on links or open attachments from trusted senders.

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 <jenkakuznecov@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

 
 
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