Hitler: "No organised Russian State must be allowed to exist. They are brutes, and neither Bolshevism nor Tsarism makes any difference—they are brutes in a state of nature."

First time when the principle "if it is not proven there, it must be wrong" failed when applied to EGA.

There seems to be no advanced textbook on algebraic geometry which contains something along these lines.

It seems that Shinichi Mochizuki was the first to bother to explicitly formulate the valuative criteria for morphisms of schemes in terms of those schemes' functors of points...? (This is from "Foundations of p-adic Teichmüller Theory.")

In other words, there exists a biproduct diagram for any 2 objects of any category in which matrices are usually encountered, and, despite the diagram’s simplicity (just 3 objects and 4 arrows), it’s not hard to deduce from it all of matrix algebra.

Thankfully, there is no need to believe this since all you need is the fact that the category of free modules (and not just vector spaces) is additive: matrices arise from linear maps between biproducts which exist for any finite collection of objects in such a category.

Somebody (forgot who) once said that he “cannot believe that anything so ugly as multiplication of matrices is an essential part of the scheme of nature.”

Turns out it started because the classical dichotomy “matter is particle-like, light is wave-like” was shattered by quantum mechanical considerations which are best expressed using Fourier analysis on dual infinite-dimensional vector spaces and dual groups.

I used to think that it all started because autist decided “hey, since we have a theory of linear spaces, let’s have a theory of spaces of linear functionals which are defined on them as well!”

Fuck everyone for not telling me what makes duality of vector spaces interesting and important in the first place.

The fully faithfulness of the co- and contravariant Yoneda embeddings is the underlying explanation for why we can perform these two seemingly unrelated types of translations (of linear systems into matrix algebra and of abstract groups into concrete ones)

Both statements are expressions of the fact that natural transformations between represented functors correspond to morphisms between the representing objects.

"Simplifying a matrix by row operations amounts to multiplying it on the left by elementary matrices" for essentially the same reasons which make "every group isomorphic to some subgroup of a permutation group."

No, that's it, I'm taking a nap.

I was also informed that a "group" is a (set-valued) sheaf on the category of simplicial complexes such that the morphisms associated to collapses of d-simplices are bijective for d > 1 (and merely surjective for d < 1 or d = 1).

I was informed that the "derivative" of a real-valued function f:D → ℝ (D for "domain") is the Lagrangian section of the cotangent bundle of D that gives the connection form for the unique flat connection on the trivial ℝ-bundle D x ℝ for which the graph of f is parallel.

https://t.co/hFVmTOiG2H

Every "i think its funny how" on this site believes the United States declared war on and nuked the Japanese as revenge for The Rape of Nanking, and not the Japanese violation of "Vichy" French Indochina's territoriality.

You need Reddit fedora dictatorship to save you from these freaks.

Capitalism is redistribution of wealth from 140 IQ to 115 IQ, Communism is redistribution of wealth from 115 IQ to 100 and 90 IQ

The people annoyed at Lola Zamboni believe we should artificially construct society so Hawk Tuah makes more money than a Mathematics doctoral student or classical pianist. Two sides of the same woke.