RE: "op"_Fred_and_Thurston

RE: "op"_Fred_and_Thurston

[RONALD BROWN]

Dear Category Theorists, 

In view of the discussion of the Thurston article, you may be interested  in a much lower key article about mathematics and its methodology, addressed  also to beginners, and to educators. I have met one professor of maths who  claimed "mathematics has no need of methodology", which is odd since most human activities seem to benefit from such a discussion. which can be at various levels depending on the audience. 

http://www.groupoids.org.uk/methmat.html

I met Fred a few times at some category theory meetings, and always found him easy to get on with and with a firm idea on what the subject was about! And a nice sense of humour!


Best wishes

Ronnie




----Original message----
From : joyal.andre@uqam.ca
Date : 09/09/2017 - 05:33 (GMTDT)
To : edubuc@dm.uba.ar, eriehl@math.jhu.edu, categories@mta.ca
Subject : categories: RE: "op"_Fred_and_Thurston

Dear Eduardo,

Thank you for recalling this remarkable article by Thurston.
It contains profound observations on the role of *communities* in the creation of mathematics.
Mathematical research is about developing *human understanding* of mathematics.

Thurston does not mention category theory.
I remember trying to learn algebraic topology by reading the 
"Foundations of Algebraic Topology" by Eilenberg and Steenrod. 
It is a great book, but not the right place to learn the subject.
I also tried to learn algebraic geometry by reading the 
"Elements de Geometrie Algebrique" by Grothendieck and Dieudonné.
I never became an algebraic-geometer.
It is very difficult to learn anything without direct access to the people who knows.

Best,
André

________________________________________
From: Eduardo J. Dubuc [edubuc@dm.uba.ar]
Sent: Friday, September 08, 2017 12:03 PM
To: Emily Riehl; categories@mta.ca
Subject: categories: "op"_Fred_and_Thurston

1) Two days ago by chance I come across an article of Bill Thurston:

https://arxiv.org/pdf/math/9404236.pdf

and seeing his name mentioned in this thread it occurs to me that
everybody in this list should read it. In my opinion it is an
extraordinary document about mathematics, mathematical activity and
mathematicians.

2) Respect to to subject of this thread, the formal opposite of a
category, denoted "op", is simply a notation very useful to work with
functors which are contravariant in some variables, either with the "op"
in the domain or the codomain of the functor arrow.

Notations are important, and the "op" notation is essential in the
language of categories and functors.

3) Finally, concerning Fred Linton, his death sadness me, he did
important work in the early days of category theory, but more important,
he was one of us, it was always a pleasure to encounter him, an he was a
good guy.

all the best   e.d.


On 07/09/17 14:03, Emily Riehl wrote:
>> There is one other anecdote about UACT, nothing to do with Fred, that I
>> have always loved. In the course of MSRI director Bill Thurston's
>> opening remarks, he said words to the effect that the notion of the
>> opposite of a category made him nauseous. This was the only meeting I
>> have ever attended where fully half the attendees drew in enough breath
>> to drop the air pressure by an audible amount.
>
> I?ll confess that the idea of an opposite category appearing as the codomain of a functor also makes me somewhat nauseated (the domain of course is no problem).
>
> But this said, in the interest of full disclosure, I should admit that in   a joint paper with Cheng and Gurski someone ? Eugenia, I believe? ? convinced us that the easiest way to think of a functor
>
> C x D ?> E
>
> admitting right adjoints in both variables is as a functor
>
> C x D ?> (E^op)^op
>
> because in this way (writing E? for E^op) the other two adjoints also have the form
>
> D x E? ?> C^op
>
> and
>
> E? x C ?> D^op.
>
> Such two-variable adjunctions form the vertical binary morphisms in a ?cyclic double multi category? of multivariable adjunctions and parametrized mates:
>
> https://arxiv.org/abs/1208.4520
>
> Regards,
> Emily
>
> ?
> Assistant Professor, Dept. of Mathematics
> Johns Hopkins University
> www.math.jhu.edu/~eriehl
>


[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: "op"_Fred_and_Thurston

[Joyal, André]

Dear Eduardo,

Thank you for recalling this remarkable article by Thurston.
It contains profound observations on the role of *communities* in the creation of mathematics.
Mathematical research is about developing *human understanding* of mathematics.

Thurston does not mention category theory.
I remember trying to learn algebraic topology by reading the 
"Foundations of Algebraic Topology" by Eilenberg and Steenrod. 
It is a great book, but not the right place to learn the subject.
I also tried to learn algebraic geometry by reading the 
"Elements de Geometrie Algebrique" by Grothendieck and Dieudonné.
I never became an algebraic-geometer.
It is very difficult to learn anything without direct access to the people who knows.

Best,
André

________________________________________
From: Eduardo J. Dubuc [edubuc@dm.uba.ar]
Sent: Friday, September 08, 2017 12:03 PM
To: Emily Riehl; categories@mta.ca
Subject: categories: "op"_Fred_and_Thurston

1) Two days ago by chance I come across an article of Bill Thurston:

https://arxiv.org/pdf/math/9404236.pdf

and seeing his name mentioned in this thread it occurs to me that
everybody in this list should read it. In my opinion it is an
extraordinary document about mathematics, mathematical activity and
mathematicians.

2) Respect to to subject of this thread, the formal opposite of a
category, denoted "op", is simply a notation very useful to work with
functors which are contravariant in some variables, either with the "op"
in the domain or the codomain of the functor arrow.

Notations are important, and the "op" notation is essential in the
language of categories and functors.

3) Finally, concerning Fred Linton, his death sadness me, he did
important work in the early days of category theory, but more important,
he was one of us, it was always a pleasure to encounter him, an he was a
good guy.

all the best   e.d.


On 07/09/17 14:03, Emily Riehl wrote:
>> There is one other anecdote about UACT, nothing to do with Fred, that I
>> have always loved. In the course of MSRI director Bill Thurston's
>> opening remarks, he said words to the effect that the notion of the
>> opposite of a category made him nauseous. This was the only meeting I
>> have ever attended where fully half the attendees drew in enough breath
>> to drop the air pressure by an audible amount.
>
> I?ll confess that the idea of an opposite category appearing as the codomain of a functor also makes me somewhat nauseated (the domain of course is no problem).
>
> But this said, in the interest of full disclosure, I should admit that in  a joint paper with Cheng and Gurski someone ? Eugenia, I believe? ? convinced us that the easiest way to think of a functor
>
> C x D ?> E
>
> admitting right adjoints in both variables is as a functor
>
> C x D ?> (E^op)^op
>
> because in this way (writing E? for E^op) the other two adjoints also have the form
>
> D x E? ?> C^op
>
> and
>
> E? x C ?> D^op.
>
> Such two-variable adjunctions form the vertical binary morphisms in a ?cyclic double multi category? of multivariable adjunctions and parametrized mates:
>
> https://arxiv.org/abs/1208.4520
>
> Regards,
> Emily
>
> ?
> Assistant Professor, Dept. of Mathematics
> Johns Hopkins University
> www.math.jhu.edu/~eriehl
>


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

"op"_Fred_and_Thurston

[Eduardo J. Dubuc]

1) Two days ago by chance I come across an article of Bill Thurston:

https://arxiv.org/pdf/math/9404236.pdf

and seeing his name mentioned in this thread it occurs to me that
everybody in this list should read it. In my opinion it is an
extraordinary document about mathematics, mathematical activity and
mathematicians.

2) Respect to to subject of this thread, the formal opposite of a
category, denoted "op", is simply a notation very useful to work with
functors which are contravariant in some variables, either with the "op"
in the domain or the codomain of the functor arrow.

Notations are important, and the "op" notation is essential in the
language of categories and functors.

3) Finally, concerning Fred Linton, his death sadness me, he did
important work in the early days of category theory, but more important,
he was one of us, it was always a pleasure to encounter him, an he was a
good guy.

all the best   e.d.


On 07/09/17 14:03, Emily Riehl wrote:
>> There is one other anecdote about UACT, nothing to do with Fred, that I
>> have always loved. In the course of MSRI director Bill Thurston's
>> opening remarks, he said words to the effect that the notion of the
>> opposite of a category made him nauseous. This was the only meeting I
>> have ever attended where fully half the attendees drew in enough breath
>> to drop the air pressure by an audible amount.
>
> I?ll confess that the idea of an opposite category appearing as the codomain of a functor also makes me somewhat nauseated (the domain of course is no problem).
>
> But this said, in the interest of full disclosure, I should admit that in a joint paper with Cheng and Gurski someone ? Eugenia, I believe? ? convinced us that the easiest way to think of a functor
>
> C x D ?> E
>
> admitting right adjoints in both variables is as a functor
>
> C x D ?> (E^op)^op
>
> because in this way (writing E? for E^op) the other two adjoints also have the form
>
> D x E? ?> C^op
>
> and
>
> E? x C ?> D^op.
>
> Such two-variable adjunctions form the vertical binary morphisms in a ?cyclic double multi category? of multivariable adjunctions and parametrized mates:
>
> https://arxiv.org/abs/1208.4520
>
> Regards,
> Emily
>
> ?
> Assistant Professor, Dept. of Mathematics
> Johns Hopkins University
> www.math.jhu.edu/~eriehl
>

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