Introduction to Homotopy Theory- PART 1: UNIVERSAL CONSTRUCTIONS
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- čas přidán 2. 08. 2024
- The goal of this series is to develop homotopy theory from a categorical perspective, alongside the theory of model categories. We do this with the hope of eventually developing stable homotopy theory, a personal goal a passion of mine. I'm going to follow nLab's notes, but I hope to add to it through my explanations!
Prior experience in algebraic topology and category theory may be useful, but if not, most of the concepts can be picked up quickly.
PLEASE leave any misconceptions I had or inaccuracies in my video in the comments, and I'll pin them!
References:
**nLab
ncatlab.org/nlab/show/Introdu...
**forum questions that were very insightful and helpful
math.stackexchange.com/questi...
cstheory.stackexchange.com/qu...
**animation library
github.com/3b1b/manim
Music:
► Artist Attribution
• Music By: "KaizanBlu"
• Track Name: "Remember (Extended Mix)"
• CZcams Track Link: bit.ly/31Ma5s0
• Spotify Track Link: spoti.fi/2NUH3xZ
![[Intro to HoTT - OLD] Martin-Löf Type Theory: Speaking the Language](/img/n.gif)
So excited to find this playlist! I've been crawling the nLab for many years. Its sooo difficult to learn all by myself. Lots of wasted time re-building things to convince myself they are "right". The synthetic approach to homotopy is too seductive not to study, though. So we suffer.
I really want to make video lectures like you, using Manin. I don't know much coding so it's also a project to gain familiarity with Python. This is really inspiring work you are doing!
brilliant!! can’t really find any other words. thank you for making this!
WOW THIS IS SO AMAZING
Incredible tutorial
Thank you so much!
Great content bro
6:49 I think colimits are called direct limits while limits are inverse limits. Since it's confusing, most people just don't call them that anymore
This is awesome! I love the background music and the presentation!
I have a question though. Why should an inclusion from 3 to 5 be unique? this doesn't seem right
♥️
I've taken a point-set topology course (though no algebraic topology yet), and I was noticing that there were several theorems in the function-spaces section of the book that were requiring "locally compact Hausdorff" that seemed to be of the effect of some of that exponential mappings stuff near the end of your video (though my textbook wasn't phrased in categorical language). Is the Hausdorff axiom not needed? Or are you simply assuming it (as I've heard some authors do)?
That's a great detail to point out- the Hausdorff assumption isn't strictly necessary. The property we were requiring was that "every open neighborhood of every point contains a compact neighborhood". It just happens that when a space is Hausdorff, all (4 or so) definitions of local compactness agree, and so we get the desired property.
@@rooney5395 Got it. That's not how locally compact is defined in my textbook (Munkres) but that property is shown equivalent to their definition under the assumption of Hausdorffness.
Nice! But why the background music?
I hate when people record videos forgetting about audio quality
Dude turn up the mic.
I believe there's someone playing in the same room. I can't believe those mouse clicks are from the music. Sorry, I don't know why but this stole a lot of my attention.
never ever play music while talking like this.