Early intimations of categories

Early intimations of categories

[Michael Barr, Prof.]

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I have nothing to add to the Ulam question, but I did want to give some speculations on early ideas.  The first is Emmy Noether and use of groups instead of Betti numbers and torsion numbers.  A group is, after all, a category and the use of groups is the sort of abstraction that categories are.  Although Vietoris was quoted as saying that of course they knew there were groups, but it wasn’t the style to say so.  But the homology groups of a space (as well as the homotopy groups) are certainly early examples of functors.  Although I am not sure when the homomorphism induced by a continuous map was known.

But the most interesting example is Garrett Birkhoff’s lattice theory.  First let me mention that in those early days of the XXth century, homomorphisms (between groups, rings,…) were always understood to be surjective.  It was only in the 50s that people started talking about homomorphisms into, which were explicitly allowed not to be onto.  This being the case, when Birkhoff looked at groups he saw the lattice of subgroups and the lattice of quotient groups (essentially the lattice of normal subgroups) and at rings, he saw the lattice of subrings and the lattice of ideals.  I have often wondered whether, had arbitrary homomorphisms been in common use, he would have discovered category theory rather than lattice theory.

Michael


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Re: Early intimations of categories

[Johannes Huebschmann]

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Dear All

In [Baer 1934], "homomorph" (homomorphic)
does not necessarily mean homomorphic onto.
For that situation  Baer uses the terminology
"homomorph auf".

Johannes


________________________________
De: "Michael Barr, Prof." <barr.michael@mcgill.ca>
À: "categories" <categories@mq.edu.au>
Envoyé: Mercredi 12 Juin 2024 00:21:23
Objet: Early intimations of categories

I have nothing to add to the Ulam question, but I did want to give some speculations on early ideas.  The first is Emmy Noether and use of groups instead of Betti numbers and torsion numbers.  A group is, after all, a category and the use of groups is the sort of abstraction that categories are.  Although Vietoris was quoted as saying that of course they knew there were groups, but it wasn’t the style to say so.  But the homology groups of a space (as well as the homotopy groups) are certainly early examples of functors.  Although I am not sure when the homomorphism induced by a continuous map was known.

But the most interesting example is Garrett Birkhoff’s lattice theory.  First let me mention that in those early days of the XXth century, homomorphisms (between groups, rings,…) were always understood to be surjective.  It was only in the 50s that people started talking about homomorphisms into, which were explicitly allowed not to be onto.  This being the case, when Birkhoff looked at groups he saw the lattice of subgroups and the lattice of quotient groups (essentially the lattice of normal subgroups) and at rings, he saw the lattice of subrings and the lattice of ideals.  I have often wondered whether, had arbitrary homomorphisms been in common use, he would have discovered category theory rather than lattice theory.

Michael


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Re: Early intimations of categories

[Posina Venkata Rayudu]

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Dear Professor Barr,

I just thought of sharing with you the discussions we had regarding
the delayed discovery of category theory:

https://conceptualmathematics.wordpress.com/2012/06/13/from-function-to-category-prof-lawvere/<https://url.au.m.mimecastprotect.com/s/zdoYCZY146skNwYzhxuOdv?domain=conceptualmathematics.wordpress.com>

https://conceptualmathematics.wordpress.com/2012/06/13/charles-ehresmann-introduction-of-categories/<https://url.au.m.mimecastprotect.com/s/hV9uC1WLjwsPY0X8f1gb8f?domain=conceptualmathematics.wordpress.com>

https://conceptualmathematics.wordpress.com/2012/06/13/from-function-to-category-prof-barr/<https://url.au.m.mimecastprotect.com/s/9kRfC2xMRkUBw4Q7UX-jp6?domain=conceptualmathematics.wordpress.com>

https://conceptualmathematics.wordpress.com/2012/06/13/from-function-to-category-prof-brown/<https://url.au.m.mimecastprotect.com/s/JDtlC3QNl1SyQPvZsE_ySx?domain=conceptualmathematics.wordpress.com>

Thanking you,
Yours truly,
posina

On Wed, Jun 12, 2024 at 10:35 AM Michael Barr, Prof.
<barr.michael@mcgill.ca> wrote:
>
> I have nothing to add to the Ulam question, but I did want to give some speculations on early ideas. The first is Emmy Noether and use of groups instead of Betti numbers and torsion numbers. A group is, after all, a category and the use of groups is the sort of abstraction that categories are. Although Vietoris was quoted as saying that of course they knew there were groups, but it wasn’t the style to say so. But the homology groups of a space (as well as the homotopy groups) are certainly early examples of functors. Although I am not sure when the homomorphism induced by a continuous map was known.
>
>
>
> But the most interesting example is Garrett Birkhoff’s lattice theory. First let me mention that in those early days of the XXth century, homomorphisms (between groups, rings,…) were always understood to be surjective. It was only in the 50s that people started talking about homomorphisms into, which were explicitly allowed not to be onto. This being the case, when Birkhoff looked at groups he saw the lattice of subgroups and the lattice of quotient groups (essentially the lattice of normal subgroups) and at rings, he saw the lattice of subrings and the lattice of ideals. I have often wondered whether, had arbitrary homomorphisms been in common use, he would have discovered category theory rather than lattice theory.
>
>
>
> Michael
>
>
>
> You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message.
>
> View group files | Leave group | Learn more about Microsoft 365 Groups
>


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Re: Early intimations of categories

[Nath Rao]

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On 6/11/24 18:21, Michael Barr, Prof. wrote:
[...] Although I am not sure when the homomorphism induced by a continuous map was known.
[...]

I was struck by the above sentence. I thought that surely Lefschetz fixed point index was based on the induced homomorphism. But a cursory look at the two papers of Lefschetz listed in the Wikipedia page<https://url.au.m.mimecastprotect.com/s/HfgRCQnM1Wf2Gl61TxCUzI?domain=en.wikipedia.org> on this was illuminating: The 1926 paper seems to get by without even using groups. The 1937 paper goes directly to the matrix of the induced homomorphism in rational homology, by using bases.

On the positive side, Lefschetz's 1941/42 version of the AMS Colloquium notes (vol 27, not the original 1930 version, published as vol 12; I thought that AMS made the former freely available, but I could not find it in the AMS bookstore), in VII.5.11, very briefly mentions the homomorphism in homology induced by a mapping, and in his appendix in this book (on his fixed point theorem), P. A. Smith makes use of it. It seems that people primarily used Cech and Vietoris type theories based on coverings, where defining the induced homomorphism is harder than for singular (co)homology. [Lefschetz VIII.5.11 is defined for the groups based on specific coverings.]

-Nath Rao



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