Grothendieck toposes and their role in Mathematics

classic Classic list List threaded Threaded
7 messages Options
Reply | Threaded
Open this post in threaded view
|

Grothendieck toposes and their role in Mathematics

jon zingale
A pretty good lecture on *classifying toposes* and their role in mathematics.
Olivia Caramello discusses how toposes came to be studied, how they came to
be understood as providing a *semantic core* for a mathematical theory, and
how they provide a suitably general context for studying the symmetries
(invariants) of a given theory.

https://www.youtube.com/watch?v=8xWZpec9pwM



--
Sent from: http://friam.471366.n2.nabble.com/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
FRIAM-COMIC http://friam-comic.blogspot.com/ 
Reply | Threaded
Open this post in threaded view
|

Re: Grothendieck toposes and their role in Mathematics

Frank Wimberly-2
Did you realize that Lawvere doesn't define the term "topos" until page 352?

Frank

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 9, 2020, 10:08 AM Jon Zingale <[hidden email]> wrote:
A pretty good lecture on *classifying toposes* and their role in mathematics.
Olivia Caramello discusses how toposes came to be studied, how they came to
be understood as providing a *semantic core* for a mathematical theory, and
how they provide a suitably general context for studying the symmetries
(invariants) of a given theory.

https://www.youtube.com/watch?v=8xWZpec9pwM



--
Sent from: http://friam.471366.n2.nabble.com/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
FRIAM-COMIC http://friam-comic.blogspot.com/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
FRIAM-COMIC http://friam-comic.blogspot.com/ 
Reply | Threaded
Open this post in threaded view
|

Re: Grothendieck toposes and their role in Mathematics

jon zingale
Ha, yeah. They spend much of the book developing categories that are
simultaneously rich enough to be topos-theoretically interesting and simple
enough to reason about their properties/consequences. Recently, another
friam member got me thinking about locales[Ɏ], the toy categories presented
by Lawvere and Schanuel have been helpful to me in reasoning about them.

[Ɏ] From https://ncatlab.org/nlab/show/locale: A locale is, intuitively,
like a topological space that may or may not have enough points (or even any
points at all).



--
Sent from: http://friam.471366.n2.nabble.com/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
FRIAM-COMIC http://friam-comic.blogspot.com/ 
Reply | Threaded
Open this post in threaded view
|

Re: Grothendieck toposes and their role in Mathematics

gepr
Mine's on pg 348. 1997 edition. Mac Lane's is on pg 106 (2nd edition). It would be interesting to know whether the choice(s) were made to introduce the concept earlier later based on the trajectory of the text or the intuitive naturalness of the concept. Oddly, I like the definition Mac Lane gives in the Appendix (pg 289) better than the one on pg 106. But I don't really understand any of it. I've placed Jon's 2 youtube recommendations in The Queue.

On 7/9/20 9:49 AM, Jon Zingale wrote:
> Ha, yeah. They spend much of the book developing categories that are
> simultaneously rich enough to be topos-theoretically interesting and simple
> enough to reason about their properties/consequences. Recently, another
> friam member got me thinking about locales[Ɏ], the toy categories presented
> by Lawvere and Schanuel have been helpful to me in reasoning about them.
>
> [Ɏ] From https://ncatlab.org/nlab/show/locale: A locale is, intuitively,
> like a topological space that may or may not have enough points (or even any
> points at all).


On 7/9/20 9:26 AM, Frank Wimberly wrote:
> Did you realize that Lawvere doesn't define the term "topos" until page 352?


--
☣ uǝlƃ

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
FRIAM-COMIC http://friam-comic.blogspot.com/ 
uǝʃƃ ⊥ glen
Reply | Threaded
Open this post in threaded view
|

Re: Grothendieck toposes and their role in Mathematics

thompnickson2
In reply to this post by jon zingale
Watched a bit of this video.

Sounds like a mathematical description of metaphor -- heh, heh.

Nick

Nicholas Thompson
Emeritus Professor of Ethology and Psychology
Clark University
[hidden email]
https://wordpress.clarku.edu/nthompson/
 


-----Original Message-----
From: Friam <[hidden email]> On Behalf Of Jon Zingale
Sent: Thursday, July 9, 2020 10:08 AM
To: [hidden email]
Subject: [FRIAM] Grothendieck toposes and their role in Mathematics

A pretty good lecture on *classifying toposes* and their role in
mathematics.
Olivia Caramello discusses how toposes came to be studied, how they came to
be understood as providing a *semantic core* for a mathematical theory, and
how they provide a suitably general context for studying the symmetries
(invariants) of a given theory.

https://www.youtube.com/watch?v=8xWZpec9pwM



--
Sent from: http://friam.471366.n2.nabble.com/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam un/subscribe
http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
FRIAM-COMIC http://friam-comic.blogspot.com/ 


- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
FRIAM-COMIC http://friam-comic.blogspot.com/ 
Reply | Threaded
Open this post in threaded view
|

Re: Grothendieck toposes and their role in Mathematics

gepr
Not just *a* math description of metaphor ... THE math description of metaphor! The Grand Unified Model for metaphor. The more category theory you learn, the closer you'll get to God.

On 7/9/20 11:16 AM, [hidden email] wrote:
> Watched a bit of this video.
>
> Sounds like a mathematical description of metaphor -- heh, heh.

--
☣ uǝlƃ

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
FRIAM-COMIC http://friam-comic.blogspot.com/ 
uǝʃƃ ⊥ glen
Reply | Threaded
Open this post in threaded view
|

Re: Grothendieck toposes and their role in Mathematics

jon zingale
haha, amazing!



--
Sent from: http://friam.471366.n2.nabble.com/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
FRIAM-COMIC http://friam-comic.blogspot.com/