#categorification

[quanticloop]

saying that “the diagramme commutes” is actually an abuse of notation.

[julianrachman-blog]

Brief on "Category-Theoretic Proof of Higman's Lemma Admits Applicability and Constructive Proof"

Brief on “Category-Theoretic Proof of Higman’s Lemma Admits Applicability and Constructive Proof”

Over the course of the next month, I will be talking about my research and break everything apart one by one. This post is a brief overview of my research. The next post on Monday will be about monads, Kleisli categories, and Ellenberg-Moore categories.

I am using category theory in my paper through categorification to formulate Higman’s Lemma. Then I am developing a new constructive proof for…

View On WordPress

[eatsleepmath]

For example, why is 6/2 = 3? We can take a 6-element set \(S\) with a free action of the group \(G = \mathbb{Z}/(2)\) and identify all the elements in each orbit to obtain a three-element set \(S/G\).

John Baez, From Finite Sets to Feynman Diagrams

[eatsleepmath]

Now, given a category C, we may ‘decategorify’ it by forgetting about the morphisms and pretending that isomorphic objects are equal. We are left with a set (or class) whose elements are isomorphism classes of objects of C. This process is dangerous, because it destroys useful information. It amounts to forgetting which road we took from x to y, and just remembering that we got there. Sometimes this is actually useful, but most of the time people do it unconsciously, out of mathematical naivete. We write equations, when we really should specify isomorphisms. ‘Categorification’ is the attempt to undo this mistake. Like any attempt to restore lost information, it not a completely systematic process. Its importance is that it brings to light previously hidden mathematical structures, and clarifies things that would otherwise remain mysterious. It seems strange and complicated at first, but ultimately the goal is to make things simpler.

John Baez, From Finite Sets to Feynman Diagrams

[eka-mark]

a nice side effect of studying homotopy theory, especially in the context of higher category theory, is that now I can't help but see equals signs as little tiny stretches of road

so for example "2 + 3 = 5" isn't saying that "2 + 3" and "5" are somehow the same thing, it's saying that there's a two-way path between them