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@associativeglassdesert
Wow, thank you nLab, this is exactly what I was looking for
(Slightly exaggerated) Nlab is great and I will hear no criticism of it.
Like, I mean, it is actually great, just the not hearing criticism part is sarcastic, it's obviously not perfect. I like nlab. Nice place to click links on and get nerd sniped.
This is the proper characterization of nLab I feel.
I had a professor who once referred to it as "like Wikipedia for people who hate themselves"
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I’ll admit I’ve sent a 👍 in my time
Ragebait ideas for your math friends: tell them that their research area is just a special case of your much cooler research
guy who thinks everyone has the same deficits in education as him: haha yeah all these grad students have to know category theory even though they never teach it in school. it's like the LaTeX of grad school.
has anyone else tried this problem-solving strategy when wherever they come across an exercise and are stuck, they lay on the floor and visualise their advisor/professor asking it to them. success rate 100% (I tried it once)
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also i read the american state "missouri" which gets abbreviated to "MO" as the thom spectrum today so i may really need to graduate and take a short break from math. just a little bit
Danny what are you doing
Sometimes a vision appears to you in class and you have to draw it
Updates
Something is emerging
average first sentence of a math wikipedia page:
A snorkle basis is a particular sort of set that has some properties and is generally "nice" (in a rigorous sense) and can do many things and is very practical.
hey now, you're a hodge star,
get your frame on
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my tired brain got very fixated and latched onto the idea of dualizable objects and finiteness but since i am actually pretty terrible at anything categorical this just stunlocked me for like an hour while I sat there
Let's start a thread where we collect papers with funny titles. I'll start.
We prove that the groups associated with the Revenge Cube and the Professor's Cube can be realized as Galois groups over the rationals.
Following a remark of Lawvere, we explicitly exhibit a particularly elementary bijection between the set T of finite binary trees and the se
Batman... My homology is my cohomology backwards, Batman.........
The well known Joker $\mathcal{A}(1)$-module of Adams and Priddy is known to be realisable as the cohomology of a $1$-connected space. By at
Hot dust-obscured galaxies (hot DOGs) are a rare class of hyperluminous infrared galaxies identified with the Wide-field Infrared Survey Exp
We study admissible subcategories of the bounded derived category of a smooth projective surface that are supported on the exceptional locus
A wiggle is an embedded curve in the plane that is the attractor of an iterated function system associated to a complex parameter z. We show
It should be possible to teach homotopy theory entirely synthetically without point-set topology but I'm not sure how
I suppose you could just start with simplicial sets and never geometrically realize? That gets you to infinity categories and your spaces can be kan complexes.
Motivating things will be rough. You'll need model categories still, to be thorough. And then you prevent your students from talking to manifolds people, which is unfortunate. Also I kinda love point set topology it's weird and gets a bad rap.
There are purely axiomatic and synthetic approaches to the theory of infinity-categories, aren't there? I guess once you accept the axioms you can replace the word "space" by the word "anima" and be good, though I am certainly not a homotopy-stuff-knower for now so take this with a pinch of salt.
@hildegunst-taillemythes probably has more intelligent things to say here than me
what
why are you throwing me under the bus like that
Also: @galois-groupie yes you maybe possibly could (?) but don't. Building the idea of homotopy starting at "oohhh, continuous deformation indexed by [0, 1] with the most evident topology there is" is, imho, very important. As @locally-normal said, you need motivation. Drawings tell the whole story, and starting with manifolds is very good.
Quite long but: If I were to take sSet, i.e. the classifying topos of the theory of intervals, the archetypical combinatorial model for spaces, yada yada yada... and wanted the classical (Quillen) model structure, it is defined using geometric realization and weak homotopy equivalcen in Top. You could use Joyal's one (works really well: cofibrantly generated and fibrant objects are quasicats instead of Kan complexes), but it's not Quillen equivalent to the other, i.e. it describes a different theory.
(And that's quite an issue because you're loosing Grothendieck homotopy theory (what I mean is, test categories and shapes), and that makes younger me sad because he loved that stuff)
Also, I find it hard to motivate several notions of equivalences without the idea of homotopy equivalence, you're loosing a useful example of a recurrent pattern.
tldr; is it possible ? Honestly, theoretically probably but I don't know, you would lack a lot of stuff; pedagogically nope, impossible. Should you ? No. If you try, I'll punch you in the face. I'm serious.
@algebraic-dumbass I don't know how fully developed the synthetic approach is, I hear like Riehl Verity are working on it and there's lots of stuff out about it, but I don't know if it's a viable drop in replacement for Lurie yet. Or if that's even theoretically the goal in some senses, a model can just give you more power to work with sometimes.
Doing homotopy theory without homotopy would be kind of crazy. I suppose you have natural equivalence as an example but those are hardly easy to motivate without singular homology. I think we ignore history too much as it is.
And besides, there's something so beautiful about spaces and categories being essentially the same sort of thing. Simplicial sets have never stopped feeling a little too artificial for that beauty. Topology is so fluid. You have to work to make simplicial sets fluid, it's not so obvious. And by the time you do so much work you start thinking of chain complexes of modules as pretty fluid, but I wouldn't start there either, I'm not an algebraic geometer.
I've heard some motivic ppl are trying to do homotopy theory without homotopy, but I don't know if that counts because it's motivic
who up uhhh internalising they hom
someone needs to remind me to stop working on my master’s report late at night (or at least drink a coffee when i do) because why is this just in here
same vibes
i am being inspired to work on my senior thesis
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It's absolutely crazy that intellectual labor can wipe you out. It seems like it shouldn't be a thing, like your stores of brain juice shouldn't be able to be depleted in that way.
I feel like a wizard that's out of spell slots, and to me that's a hackish mechanical limitation put in place to try to balance the classes.
the power of a bowl of rice mixed with some fucking bullshit cannot be overstated
