TonyVladusich

johncarlosbaez, 10 days ago (edited 6 days ago) to random

Q: Why is an Ab-enriched star-category like the drummer in a Beatles tribute band?

A: Because it's a star-ringoid.

To make up for that pun:

'Ringoids' are actually quite important - but like 'groupoid', this term means two things. A groupoid is a category where all morphisms are isomorphisms - or morally speaking, a 'many-object group'. These are important all over mathematics. But in older literature, 'groupoid' sometimes means a set with a binary operation. Bourbaki renamed these 'magmas', and I prefer that term.

Similarly, a 'ringoid' is a 'many-object ring': that is, a category where you can add morphisms f,g: x → y, making each homset into an abelian group, and composition of morphisms gets along with addition, as follows:

f ∘ (g + h) = f ∘ g + f ∘ h
(f + g) ∘ h = f ∘ g + f ∘ h
f ∘ 0 = 0
0 ∘ f = 0

For example, the category of abelian groups is a ringoid, and so is the category of representations of any group. The slick way to summarize all the axioms here is that a ringoid is a category enriched over abelian groups, or 'Ab-category' for short.

But 'ringoid' has also been used to mean a set with binary operations + and × obeying only

f × (g + h) = f × g + f × h
(f + g) × h = f × h + g × h

These are not so important in my experience.

A 'star-category' is a category where for each morphism f: x → y we have a morphism going back, f*: y → x, obeying

f** = f
(f ∘ g)* = g* ∘ f*

So it's a category where you can turn around morphisms.

If you know all this stuff, the concept 'star-ringoid' is self-explanatory, and you might never even notice that it sounds quite silly.

dave, 22 days ago to random

Watching the Kotlin Multiplatform ‘24 keynote.

It’s a breath of fresh air after years of Apple’s super-highly-polished WWDC ones.

Specifically: people falter naturally like real humans giving talks, multiple companies involved showing some industry collaboration. Something that’s just not present at WWDC re: Swift.

matthewconroy, 22 days ago (edited 22 days ago) to mathematics

I'm surprised it took me this long to add the Reuleuax triangle to my table of isoperimetric ratios. It's curiously close to an integer. https://sites.math.washington.edu//~conroy/isoperimetrics/isoperimetrics.pdf #isoperimetric #table #shapes #mathematics

johncarlosbaez, 24 days ago to random

I want to read this book: A Darwinian Survival Guide. Sounds like a realistic view of what we need to do now. You can read an interview with one author, the biologist Daniel Brooks. A quote:

...

Daniel Brooks: What can we begin doing now that will increase the chances that those elements of technologically-dependent humanity will survive a general collapse, if that happens as a result of our unwillingness to begin to do anything effective with respect to climate change and human existence?

Peter Watts: So to be clear, you’re not talking about forestalling the collapse —

Daniel Brooks: No.

Peter Watts: — you’re talking about passing through that bottleneck and coming out the other side with some semblance of what we value intact.

Daniel Brooks: Yeah, that’s right. It is conceivable that if all of humanity suddenly decided to change its behavior, right now, we would emerge after 2050 with most everything intact, and we would be “OK.” We don’t think that’s realistic. It is a possibility, but we don’t think that’s a realistic possibility. We think that, in fact, most of humanity is committed to business as usual, and that’s what we’re really talking about: What can we begin doing now to try to shorten the period of time after the collapse, before we “recover”? In other words — and this is in analogy with Asimov’s Foundation trilogy — if we do nothing, there’s going to be a collapse and it’ll take 30,000 years for the galaxy to recover. But if we start doing things now, then it maybe only takes 1,000 years to recover. So using that analogy, what can some human beings start to do now that would shorten the period of time necessary to recover?

https://thereader.mitpress.mit.edu/the-collapse-is-coming-will-humanity-adapt/

dave, 24 days ago to random

Some days I want to defederate from the world 😅

demofox, 25 days ago to random

My video's power level is over 9000

matthewconroy, 28 days ago (edited 28 days ago) to random

Grading exams: it pains me to see some of my Calc II students using the quadratic formula to solve (b^2-3b=0). #exams #grading #calculus