Who Gives a Sheaf? Part 1: A First Example

[@rajanalexander4949]

Lucid exposition of a deviously profound concept; looking forward to the subsequent videos; thank you!

[@k-theory8604]

My video output has slowed down a bit, but I'm hoping to return to this series in the Spring!

[@kapoioBCS]

Ah yes sheaves, this specific nightmare were first appear in my life in my first algebraic geometry course 😂😂

[@k-theory8604]

Hopefully this mini series will make them less terrifying, lol

[@valentinussofa4135]

Funny comment I've ever seen. 😅

[@fanalysis6734]

Pretty good video. I especially liked that comment about there being less globally continuous functions and that telling us about the space. Then the picture with differentiation relating to smaller open subintervals is also really insightful

[@k-theory8604]

Thanks! Glad you enjoyed it.

[@RBanerj]

Very clearly explained, great video

[@Sqaarg]

Daniel Rosiak has written a very nice book on sheaves, `Sheaf Theory Through Examples'. It's filled with lots of down to earth examples, and was a joy to read through.

[@k-theory8604]

I haven't heard of it, I'll be sure to check it out, thanks!

[@aaAa-vq1bd]

yeah it’s wonderful. and it has a bit of interesting philosophical stuff too, although that part is very unfocused.

[@rajanalexander4949]

Thanks for the recommendation

[@sebastianelytron8450]

I gave enough of a sheaf to watch this video in its entirety, and I'm glad I did.

[@themasculinesociety]

I really like your teaching, you made this look easy

[@k-theory8604]

Thanks! Glad you enjoy it!

[@mathmage420]

I started reading a book on algebraic analysis and the first chapter was a review of sheaf theory and it used notation that I didn't understand. Thank you for this video, it helped me get through chapter 1

[@k-theory8604]

Glad it helped! I'll be making new videos in this series some time this week.

[@AR5ONL]

THANK YOU FOR THIS KNOWLEDGE!!! You started at the perfect spot! Right around Calc 2-3, diff eq area. THANK YOU!!! I understand!!! 😆😆😆😆😆

[@k-theory8604]

@@AR5ONL Glad this helped! Some of the later videos get pretty abstract, but eventually I'd like to bring it back down to earth with more examples from calc/diff eq

[@AR5ONL]

@ I’m committed! Thank you for the warning! I’ve got my tinfoil hat Ready!! 😆🧮 GO MATH!!

[@robertstuckey6407]

Where has this been all my life!

[@k-theory8604]

Simmering in the aether.

[@umbraemilitos]

We need more hands on specific examples in higher math.

[@curtiswfranks]

What are some examples of things which lack the compatibility of restrictions property?

[@k-theory8604]

In order for compatibility to fail, you just need to pick one triple of open sets U, V, W, with W a subset of V and V a subset of U for which Res_W^U is not the composition Res_V^U with Res_W^V. So, to this end, for each open set O, define F(O) to be whatever you want, so long as F(W) has a t least 2 elements, and define all restriction maps to be whatever maps you want, except for Res_W^U. Now, since you already picked Res_W^V and Res_V^U, simply define Res_W^U to be any map F(U) --> F(W) other than the composition of Res_W^V and Res_V^U. The idea is that in general, if you just pick random functions to be your "restriction" maps, they most likely won't follow the composition/compatibility condition.

[@willyh.r.1216]

It's like, all of these fancy math theories steal from Calculus. Thank you again, the Lone Greatest Scientific Mind, Isaac Newton.

[@knight3481]

Now do one for etale sheaf.

[@k-theory8604]

Perhaps eventually, but I think it'd be useful to have more videos on basic sheaf theory and schemes first.

[@donnypassary5798]

Haven't got into sheaf part, but I guess it has to do with "binding" presheaves together right?

[@k-theory8604]

Essentially, although it's a little more "fine-grained" that that. As we'll see in the net video, a sheaf is pre-sheaf that satisfies some extra conditions. Namely, that if you have sections defined on an open cover (so you can think about this as functions defined on open intervals of the real numbers), and they agree on overlaps, then there's a unique section (think function) on the whole space. (In our case the real numbers) So it's not really gluing together pre-sheaves, it's gluing together the sections of the presheaf.

[@christopherwithers1131]

nice

[@tomholroyd7519]

Please never mention epsilon delta again. Calculus can be done without limits. Stop using them. People hate that

[@k-theory8604]

While I'm most truly an algebraist at heart, I can't having you besmirch analysis on this channel.