Toots for trebor@types.pl account

Written by Trebor on 2025-02-05 at 09:36

Consider the unit 3-sphere given by coordinates (x,y,z,w). The subspace (x^2 + y^2 ≤ 1/2) is a solid torus, and we can project it to a disk, and then onto a 2-sphere. We map the rest of the 3-sphere to the basepoint of the 2-sphere.

Now this map is the Hopf fibration. And in my opinion the easiest way to see this is via the cup product structure in the James construction. But is there a way to directly compute the homotopy to the standard Hopf fibration?

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Written by Trebor on 2025-01-31 at 15:30

It's difficult to think about K(G,2) if it's defined as the truncation of suspension of K(G,1). Is there a better way?

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Written by Trebor on 2025-01-31 at 15:14

wait... can it be...

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Written by Trebor on 2025-01-31 at 15:05

K(G,1) can be defined as the type of G-torsors, and addition corresponds to taking the tensor product (in the obvious way). So what's the multiplication when G is a ring?

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Written by Trebor on 2025-01-24 at 19:45

Is it possible to relate the permutohedra to E_n-algebras? I feel like there should be a compelling narrative like A_n-algebras with Stasheff associahedra...

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Written by Trebor on 2025-01-23 at 11:41

nvm I found the arxiv link https://arxiv.org/abs/1610.01134

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Written by Trebor on 2025-01-23 at 11:40

Is there a paper accompanying this talk? https://www.mpim-bonn.mpg.de/de/node/6499 (The quaternionic Hopf fibration in HoTT via a modified Cayley-Dickson construction)

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Written by Trebor on 2025-01-22 at 15:28

A sneak peek of what I'm cooking up

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Written by Trebor on 2025-01-21 at 11:58

color spaces are so hard

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Shared by Trebor on 2025-01-20 at 20:31 (original by Talia Ringer)

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Written by Trebor on 2025-01-12 at 05:26

What's the free category equipped with an (extensional) reflexive object U^U = U?

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Written by Trebor on 2025-01-08 at 04:50

petition to normalize the notation X : C where C is a category to mean the construction that follows functorially depend on X

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Written by Trebor on 2025-01-07 at 10:58

Some PL theorist should investigate the semantics of Typst show rules... It's very challenging to design a system flexible enough to transform content the way people want, while retaining clear semantics

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Written by Trebor on 2025-01-04 at 13:58

Given a category C, the yoneda embedding has a left adjoint with a left adjoint with a left adjoint with a left adjoint iff C is Set

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Written by Trebor on 2024-12-29 at 05:56

By "is", I mean at least their types (which are 2-groupoids) are homotopy equivalent. Preferrably the obvious n-categories associated are also equivalent.

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Written by Trebor on 2024-12-29 at 05:47

I'm pretty convinced that a (pseudo)monoid of monoidal categories is a braided monoidal category but now I'll have to check by hand

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Written by Trebor on 2024-12-27 at 14:16

To be fair, it's interesting in its own right to investigate exactly how much extra structure is there. For example, a tricategory with one object, one morphism and one 2-morphism is a commutative monoid with some extra elements being selected, and all functors will preserve those elements!

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Written by Trebor on 2024-12-27 at 14:15

This periodic table thing is a scam. The correct thing to do is to take iterated (homotopy-coherent) monoids, so Mon(Mon(Cat)) yields the correct bicategory of braided categories. A 4-category with one object and one morphism happens to provide two associative products compatible with each other, but it makes no guarantee about not having any extra structure or property. And it's just lucky that for small cases no such things are introduced, just like how it's lucky that bicategories can all be made strict.

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Written by Trebor on 2024-12-26 at 11:01

Is there a low-tech proof that braided monoidal categories are the same as doubly monoidal categories? It seems to me that, in a doubly monoidal category, you get a natural isomorphism between the two monidal products, but not canonically so. So there is an extra choice of taking the isomorphism clockwise or counterclockwise, independent of the braiding. Isn't that more data?

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Shared by Trebor on 2024-12-19 at 01:26 (original by Tesla Zhang)

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