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Seeing a thousand "fork found in kitchen" and "believe scientists" and "we all knew about AI" tags on a post with an AI incorrectly summarizing a preprint study advising thoughtful use of AI as "AI is making your brain weak and ineffective" was making me crazy i'm sorry i tried not reblogging it like four times and every time i didn't reblog it it had like an order of magnitude more notes

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why have you not changed the words on your ask button

Oh, i didn't know you could do that, thx!

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how to write math real good (algebraic-dumbass' guide)

the more commutative diagrams there are, the more correcter the math

be sure to include a rant about capitalism in the middle of the document

drop quotes that will make people laugh on the internet (e.g. "given a homomorphism, one must salivate, like Pavlov's dog, to know its kernel and image" from Rotman I think?)

use more creative variable names with the fontAwesome package

Instead of leaving proofs as an exercise, gaslight the reader into thinking it's obvious (it's not)

include an epigraph that is a lemon demon lyric

indicate crucial steps of proofs by using the word "Behold"

refer to your technical lemmas by clear and concise names that end in -inator

if writing lecture notes, include stage directions

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Taxpolicy that might actually work. For x amount of taxes paid you get some cool medal or something.

why wouldnt you increase municipal taxes on rich people. itll make it more prestigious if its more expensive. rich people love wasting money in exchange for intangible prestige

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idk why hyperrogue is actually one of the best roguelikes, the gimmick game with several mechanics designed specifically to showcase odd properties of hyperbolic geometry is actually just really good

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Have you homogenized your homogeneous objects today?

have you sheft your sheaves today?

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Hm, i'm not sure that the implication is wrong though. It is after all inherently unfair that children of rich families get showered with presents and inherentance and poor children not.

It would be interesting to see how a (large scale) society in which children are raised communally would look like.

Does anyone know of an example? Of course there are small tribes, but did there ever exist a city/nation which operated like this?

But i guess that if you want to be able to categorically forbid nepotism and still allow families, alternatively one could argue against all (major) personal property.

Monarchism is a really funny political ideology. "I wish I was governed by a nepo baby. Things would be so much better if I was governed by a nepo baby"

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Kernels and Injectivity

So one very nice thing about categories like the categories of abelian groups, vector spaces, Lie algebras, rings, etc., is that a homomorphism of such objects is injective (i.e. it is, essentially, the inclusion of some subobject) if and only if its kernel, meaning the set of elements that get mapped to the zero element, is trivial. Notable abstract nonsense enjoyer @algebraic-dualist recently linked me to a webpage featuring a method to extend this to arbitrary categories, which I thoroughly enjoy.

Let 𝒞 be a category. For an object A of 𝒞, define an equation in A to be a pair of parallel arrows f, g: X -> A into A from some object X. We refer to X as the domain of the equation. We write e: f = g to say that e is the equation in A given by the pair (f,g). Let 1(A) denote the set of all equations in A (we'll see later why we use 1 for this set). Note that if h: A -> B is an arrow in 𝒞, then it induces a function 1(A) -> 1(B) by mapping e: f = g to the equation h(e): h∘f = h∘g.

(For those worried about set theoretic size issues, later on we'll see that if you take a set of generating objects in 𝒞, it suffices to restrict to equations whose domains are in that set. If this set is small, so is the set of equations. This is possible for very many categories of interest, in particular any category monadic over a Grothendieck topos.)

Within 1(A) we can point out a special subset, namely those equations of the form e: f = f. Call such an equation a tautology, and let 0(A) ⊆ 1(A) denote the subset of tautologies. Now for an arrow h: A -> B, let its kernel be defined as the set

ker(h) = { e ∈ 1(A) : h(e) ∈ 0(B) }.

Written differently, ker(h) = h¯¹(0(B)). Note that we always have 0(A) ⊆ ker(h) for any h. Here the notation for 1 and 0 starts to make sense; they are thought of as corresponding to the unit ideal and zero ideal of a ring. In fact, the webpage continues to develop an entire theory of ideals, being defined as intersections of kernels, but let's restrict ourselves to just the kernel for this post.

Recall that the arrow h is a monomorphism if for all i, j: X -> B we have that if h∘i = h∘j, then i = j. Monomorphisms are the most common generalization of 'injectivity' and 'subobjects' to arbitrary categories, but one thing I'll sweep under the rug here is that there are also others. Note that the definition of being a monomorphism is equivalent almost by definition to the statement that ker(h) = 0(A). We've succeeded in the construction we wanted: a morphism is the inclusion of a subobject if and only if its kernel is trivial.

How do we connect this to the classical notion of the set of elements that get mapped to 0? Call an object G of 𝒞 a generator (of 𝒞) if for all objects A, B and arrows f, g: A -> B we have that if for all x: G -> A we have f∘x = g∘x, then f = g. In words, arrows in 𝒞 that are not equal can always be distinguished by precomposing them with an arrow out of G.

In the category of sets, any singleton set {∙} is a generator, because a function x: {∙} -> A is the same as just picking out an element x ∈ A, and two functions out of A are equal if and only if their images on every element of A are equal. In the category of groups, the infinite cyclic group ℤ serves the same purpose. In the category of rings, the integer polynomial ring ℤ[X] does.

Fix a generator G of 𝒞. For an object A, instead of the set of all equation 1(A), consider just the subset

1'(A) = { e ∈ 1(A) : the domain of e is G }.

Now we can consider the set 0'(A) of tautologies with domain G, and for an arrow h: A -> B we have the restricted kernel ker'(h) = ker(h) ∩ 1'(A). Here's the nice thing about all of this: we have that ker'(h) = 0'(A) if and only if ker(h) = 0(A). The latter trivially implies the former, so to prove this assume that ker'(h) = 0'(A). We only have to show that ker(h) ⊆0(A) (or equivalently that h is a monomorphism), so let e: f = g be an equation in A with domain X such that h∘f = h∘g. We want to show that f = g. Well, because G is a generator it suffices to show that f∘x = g∘x for all x: G -> A. But we have h∘f∘x = h∘g∘x by assumption, so (f∘x, g∘x) ∈ ker'(h) = 0'(A), so by definition we have f∘x = g∘x, so f = g and we're done. We have shown that h is a monomorphism if and only if ker'(h) is trivial.

The final step to connect this back to the classical kernel is to talk a little bit about adjoint functors. For all our familiar categories 𝒞 of algebraic structures, we have a forgetful functor U that maps an object A to its underlying set U(A), and a homomorphism f to its underlying set function U(f). Going the other way, we also usually have a free object functor F, which sends a set X to the 𝒞-object F(X) which is in some precise sense 'freely generated' by the elements of X. This sense is captured by the fact that we have a natural bijection

{ 𝒞-morphism F(X) -> A } ≅ { set functions X -> U(A) },

which makes F into the left adjoint of U. Recall that the singleton set {∙} is a generator in the category of sets, and that set functions {∙} -> X correspond naturally to elements of X. Using this we can specialize the above bijection to

{ 𝒞-morphism F({∙}) -> A } ≅ { elements of U(A) } = U(A).

In fact, if the forgetful functor U is faithful (as it usually is), then F({∙}) is necessarily a generator of 𝒞. This shows that elements of 1'(A) can be identified with pairs of elements of the underlying set of A; i.e. an equation of the form e: x = y with x, y ∈ A. And as a final final step, note that in our example categories, any equation x = y is equivalent to the equation x - y = 0. It follows that a homomorphism h is a monomorphism if and only if the only elements that get mapped to 0 by h are those of the form x - x (which we already have a different symbol for!), so it's exactly when the classical kernel is trivial.

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If you think you didn't do "that much work", just write the stuff down in LaTeX. I needed to come up with just "one small example". I'm not even finished with the proof of the first claim and it's already a page long. The fucking example will have like 5 pages I can see it.

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it's funny watching Firefly now, like Whedonesque dialogue was supposedly ironic and deadpan at the time but I think it reads much more sincerely today, to the point that it can comes across like that Sanctuary Moon satire show that Murderbot watches.

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Recent set theory breakthrough: in order to make Ultimate L tall and slender set theorists are now forcing beauty standards

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Remember: At advanced enough levels of set theory telling yourself "this is a set, this is a set, this is a set" actually works

I really feel like size issues should not matter in category theory. I know that is not true, particularly in thinking about completeness and cocompleteness. Somehow, though, it feels like Yoneda's Lemma and the characterisation of adjunctions in terms of a single natural isomorphism is so much more general than only applying to locally small categories.

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What about self-inverses like the identity? Would that not mean that no more objects can exist and therefore you are actually eliminating all morphisms?

Had a dream where they made division in mathematics illegal

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decentralize and clean up your life!!!

use overdrive, libby, hoopla, cloudlibrary, and kanopy instead of amazon and audible.

use firefox instead of chrome or opera (both are made with chromium, which blocks functionality for ad-blockers. firefox isn't based on chromium).

use mega or proton drive instead of google drive.

get rid of bloatware

use libreoffice instead of microsoft office suite

use vetted sites on r/FREEMEDIAHECKYEAH for free movies, books, games, etc.

use trakt or letterboxd instead of imdb.

use storygraph instead of goodreads.

use darkpatterns to find mobile game with no ads or microtransactions

use ground news to read unbiased news and find blind spots in news stories.

use mediahuman or cobalt to download music, or support your favorite artists directly through bandcamp

make youtube bearable by using mtube, newpipe, or the unhook extension on chrome, firefox, or microsoft edge

use search for a cause or ecosia to support the environment instead of google

use thriftbooks to buy new or used books (they also have manga, textbooks, home goods, CDs, DVDs, and blurays)

use flashpoint to play archived online flash games

find books, movies, games, etc. on the internet archive! for starters, here's a bunch of David Attenborough documentaries and all of the Animorphs books

burn your music onto cds

use pdf24 (available online or as a desktop app) instead of adobe

use unroll.me to clean your email inboxes

use thunderbird, mailfence, countermail, edison mail, tuta, or proton mail instead of gmail

remove bloatware on windows PC, macOS, and iOS X

remove bloatware on samsung X

use pixelfed instead of instagram or meta

use NCH suite for free software like a file converter, image editor, video editors, pdf editor, etc.

feel free to add more alternatives, resources or advice in the reblogs or replies, and i'll add them to the main post <3

last updated: march 18th 2025

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Ok, my experience may be a bit skewed, since i was focusing on set theory and a bit of category theory in my (officially 2y, de facto 3y) masters, but about ⅔ of what i was doing was research seminars and topics lecures for grad students. Of course there were also some coures you could take during the end of your undergrad which we were required to take a bit from filling up the last third.

In undergrad there definitely were set theory corses and model theory courses available, but in my opinion they do a horrible job in teaching you what set theory is *about*. In the sense that we try to balance large cardinals against fragments of choice, that we do this in order to try to construct universes of ZF with nice enough properties to serve as V (ultimate L for example), that cardinals serve as axioms limiting to constructablity, and choice as axioms expanding from constructability, about BG theory which extends ZF conservatively, how those principles can be used to go bejond class-sized categories, etc...

The undergrad courses for foundational fields just taught the formalisms, and maybe some of the simple cornerstone theorems, but even the more complicated cornerstones, like forcing, were barely touched upon, since there is simply not enough time to build them up in the available time, and setting what we learned into perspective was rarely done.

It is crazy how little math you learn in undergrad. Like, foundations wise almost nothing. In retrospect i feel that post undergrad i was closer to gymnasium level math (read: high school) than post masters. And lets not talk about phd

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Evil advisor? No, do not trust the SLANDER sowed by those who only seek disarray and confusion. Simply someone concerned with what is BEST for YOU and the REALM.

Excellent decision on the spices your Highness. The spice must flow! Excellent, excellent...

A bunch of my roommates were gathered in a room today, and when I interacted with them I had a certain "I sense danger" instinctual response to something that just felt uncanny valley about their interactions.

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Yeah, coherrence is simply what coherrence should be :P

I get that all concepts are Kan extensions, but I struggle to see the payoff of doing that.