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Ralph Sarkis (Jan 26 2021 at 15:14):With @Liam Chung and another student, we are starting this week a reading group on this book. Since I have been enjoying this community very much, I thought I could share our progress here and maybe recruit other members that want to learn about topos theory. Our first meeting is Firday 2:30pm EST and we will only discuss the intro of the book, our background in CT and possibly meeting arrangements. I'll try to keep this topic updated so that if you cannot join now, you can catch up later.
We chose this book because it seemed to be a very beginner friendly introduction to toposes, but if the reading group goes well, we will probably continue our journey into topos theory with other resources so any recommendation will be appreciated. I already have in mind this paper that was advertised here as a nice application and Olivia Caramello's book because I think I'd like to know more about toposes being used for unification.
Cheers!
Fawzi Hreiki (Jan 26 2021 at 15:23):Just a warning, Olivia’s book is not exactly an introductory book on topos theory. It’s focused more so on classifying toposes.
While it gives some background, you’re probably better off reading a more general topos theory book first, eg Sheaves in Geometry and Logic or Toposes, Triples, and Theories.
Fawzi Hreiki (Jan 26 2021 at 15:25):The second of these is available for free from TAC.
Ralph Sarkis (Jan 26 2021 at 15:34):Fawzi Hreiki said:
Just a warning, Olivia’s book is not exactly an introductory book on topos theory. It’s focused more so on classifying toposes.
While it gives some background, you’re probably better off reading a more general topos theory book first, eg Sheaves in Geometry and Logic or Toposes, Triples, and Theories.
Oh I didn't know that, I though it was just an intro with a different flavour. Thanks Fawzi! I started reading SGL last year and didn't like it but I've read some parts of TTT on monads and enjoyed it.
John Baez (Jan 26 2021 at 15:35):I list a bunch of introductory texts on topos theory here:
and this page is also a very short introduction to topos theory.
Fawzi Hreiki (Jan 26 2021 at 15:39):Also, this introduction by Tom Leinster is pretty clear and easy going.
John Baez (Jan 26 2021 at 15:40):I'll add that to my list.
John Baez (Jan 26 2021 at 15:41):I'm amazed that Ralph can like TTT but not SGL. I found TTT to be extremely scary the first five-and-a-half times I tried to read it.
John Baez (Jan 26 2021 at 15:43):Now I like TTT as a kind of reference. It's a funny book in some ways because it's about three different topics in category theory, so if you want to learn topos theory then just about 1/3 is about that... though of course everything is connected.
John Baez (Jan 26 2021 at 15:44):It's sort of like "three important concepts in category theory that have this in common: they begin with the letter T".
John Baez (Jan 26 2021 at 15:45):(Well, Lawvere theories and triples have a lot in common.)
Fawzi Hreiki (Jan 26 2021 at 15:45):The content on sketches is pretty interesting but yeah it's certainly a very condensed book.
John Baez (Jan 26 2021 at 15:45):I find it all quite interesting now... now that I get the basic ideas and know why I care about them.
Ralph Sarkis (Jan 26 2021 at 15:49):John Baez said:
I'm amazed that Ralph can like TTT but not SGL. I found TTT to be extremely scary the first five-and-a-half times I tried to read it.
:shrug: I only read about monads there as a secondary resource (I used Emily Riehl's book for basic CT), but I enjoyed the style of their proofs.
Morgan Rogers (he/him) (Jan 26 2021 at 15:55):Ralph Sarkis said:
With Liam Chung and another student, we are starting this week a reading group on this book.
I hadn't heard of this one! I'm rather concerned by their introduction:
The fully fledged notion of a Grothendieck topos seems formidable to the beginner. [...] The usual alternative is to start with the axiomatic approach to topos theory, delving into elementary toposes from the beginning. [...] But elementary toposes are an axiomatization of Grothendieck toposes and we are back to the previous difficulty! A few years ago, Lawvere suggested another alternative: to introduce topos theory through presheaf toposes or, equivalently, C-sets.
First, Lawvere was one of those involved in coming up with the definition of elementary toposes, and second, presheaf toposes are such a natural special case of Grothendieck toposes (you literally have to define presheaf toposes in order to define sheaf/Grothendieck toposes) that I find it extremely hard to believe that Lawvere "suggested" this "alternative" entry point into topos theory...
John Baez (Jan 26 2021 at 15:57):Ralph Sarkis said:
John Baez said:
I'm amazed that Ralph can like TTT but not SGL. I found TTT to be extremely scary the first five-and-a-half times I tried to read it.
:shrug: I only read about monads there as a secondary resource (I used Emily Riehl's book for basic CT), but I enjoyed the style of their proofs.
Okay: if you already knew why monads were important, I can imagine you enjoying it. I love this book now because it provides a lot of technical details about the topics it covers. At first it was overwhelming.
John Baez (Jan 26 2021 at 15:58):Morgan Rogers (he/him) said:
Ralph Sarkis said:
With Liam Chung and another student, we are starting this week a reading group on this book.
I hadn't heard of this one! I'm rather concerned by their introduction:
The fully fledged notion of a Grothendieck topos seems formidable to the beginner. [...] The usual alternative is to start with the axiomatic approach to topos theory, delving into elementary toposes from the beginning. [...] But elementary toposes are an axiomatization of Grothendieck toposes and we are back to the previous difficulty! A few years ago, Lawvere suggested another alternative: to introduce topos theory through presheaf toposes or, equivalently, C-sets.
First, Lawvere was one of those involved in coming up with the definition of elementary toposes, and second, presheaf toposes are such a natural special case of Grothendieck toposes (you literally have to define presheaf toposes in order to define sheaf/Grothendieck toposes) that I find it extremely hard to believe that Lawvere "suggested" this "alternative" entry point into topos theory...
I don't know if Lawvere suggested it, but I like this approach: explain the basic concepts of topos theory using presheaf categories as your go-to examples, before getting into Grothendieck topoi.
Morgan Rogers (he/him) (Jan 26 2021 at 16:02):That's what Sheaves in Geometry and Logic does, but it is true that the category theory competence expected for that book is significantly higher!
Morgan Rogers (he/him) (Jan 26 2021 at 16:06):There's no point me being overly critical, though. As long as the theoretical content is accurate and it gets people into topos theory, they can say anything they like :wink:
John Baez (Jan 26 2021 at 16:06):Yes, SGL assumes more category theory and also assumes the reader will be interested and maybe a bit knowledgeable about logic and topology. (It's hard to imagine enjoying this book if you'd never studied sheaves before.)
Fawzi Hreiki (Jan 26 2021 at 16:09):I think if someone hasn't seen sheaves before or dealt with manifolds/schemes/etc... sheaf toposes can be quite difficult and unmotivated.
John Baez (Jan 26 2021 at 16:09):Since I already loved logic and sheaves, SGL was great for me. But we can imagine readers who don't have much mathematical experience and want a more self-contained introduction to topos theory.
John Baez (Jan 26 2021 at 16:09):So, there need to be many different books....
Morgan Rogers (he/him) (Jan 26 2021 at 16:10):Ralph Sarkis said:
We chose this book because it seemed to be a very beginner friendly introduction to toposes, but if the reading group goes well, we will probably continue our journey into topos theory with other resources so any recommendation will be appreciated. I already have in mind this paper that was advertised here as a nice application.
Oh hey, I just noticed that that's my paper with Jens! :tada: I hope you enjoy it, and both of us are here on Zulip if you end up having follow-up questions :grinning_face_with_smiling_eyes:
Ralph Sarkis (Jan 26 2021 at 16:11):Nice, thanks!
Dylan Braithwaite (Feb 01 2021 at 18:58):I’d be keen to join in if this is still going ahead @Ralph Sarkis. I’ve been working through SGL for a while, but I think it would be good to supplement it with a more introductory text, and Generic figures and their glueings looks pretty cool!
Ralph Sarkis (Feb 01 2021 at 19:01):Nice! We are reading chapter 1 and doing a couple of exercise each for Friday. I'll DM you the details.
Dylan Braithwaite (Feb 01 2021 at 19:02):Great, thanks!
Naso (Feb 03 2021 at 06:13):@Ralph Sarkis i'm also really keen, i've been reading this book and SGL on my own & would love to discuss with others. only problem is my timezone is utc+10 so it may be too early for me (5:30am i think).
Ralph Sarkis (Feb 03 2021 at 07:58):Oof... Yeah we are already accommodating UTC-5, UTC and UTC+1, so it is hard. I tried to use a website to find a suitable meeting time and coincidentally, our meeting time is already the (unique) closest that it could work. For instance, if you are observing daylight saving time right now, we could move the meeting 30 minutes ahead† to 3:00pmEST, you might be able to join us at 7:00am your time, but I don't know if you'd want to wake up this early in the weekend.
Naso (Feb 04 2021 at 00:47):@Ralph Sarkis thank you: i'm in brisbane so it would be 6AM for me, but i will try to adjust my schedule as i'm really interested in this. i guess i can go back to sleep after the meeting... lol. thanks for the amusing video