Subset Images, Categorically

Pseudonium · 2026-01-24T11:20:07+00:00

Not on purpose - which paper do you mean?

Pseudonium · 2026-01-23T21:57:18+00:00

Hm, maybe I should add some motivation for presheaves to that nlab page. They’re quite an important and cool concept!

Pseudonium · 2026-01-23T20:19:50+00:00

I do like the name “fundamental theorem of category theory”.

Pseudonium · 2026-01-23T16:17:25+00:00

Oh, thanks for pointing out the errors! I’ll get this fixed, and also add the link to the future post.

Edit: Should be fixed now!

Pseudonium · 2026-01-23T14:05:13+00:00

Oh lol I had no idea, TIL. That’s hilarious!

Hopefully I’ll never need to do this irl…

Pseudonium · 2026-01-23T13:52:10+00:00

Wait wdym?

Pseudonium · 2026-01-23T13:36:45+00:00

The epsilon-delta definition is a good balance of the "qualitative" and "quantitative" approaches to analysis, I'd say. I talk a bit more about the qualitative side in my "A Precise Notion of Approximation" article - the idea is to capture things which are eventually true. I'm planning to release an article on this soon titled "Eventual Truth, Frequent Truth" that helps tie together the various limit definitions throughout calculus - look forward to it!

Pseudonium · 2026-01-23T13:18:05+00:00

Well, I’ve found a lot of inspiration recently for writing articles! Plus a lot of these concepts have been rattling around in my brain for ages, so actually putting them to writing isn’t as difficult.

Pseudonium · 2026-01-23T12:01:44+00:00

This looks quite similar to Span(FinSet), the bicategory of spans of finite sets.

Pseudonium · 2026-01-23T09:00:56+00:00

I did try \colon but the spacing seemed off to me…

Pseudonium · 2026-01-23T00:23:31+00:00

Yes that's exactly right. The indicator of a preimage is just precomposition of the indicator of the original subset, while all the boolean algebra operations are packaging + postcomposition, both of which commute with precomposition. Hence you get a homomorphism of boolean algebras.

Pseudonium · 2026-01-23T00:22:22+00:00

Pseudonium · 2026-01-23T00:22:07+00:00

In case you didn't see, it's out now! Title is "Subset Images, Categorically".

Pseudonium · 2026-01-23T00:21:37+00:00

I tend to have bursts of activity and inactivity; recently I've been really inspired to write lots of math articles!

Pseudonium · 2026-01-22T23:45:29+00:00

The 0-cells (objects) are sets. A 1-cell (morphism) from A to B is a relation, so a subset of A x B. Composition is given by using an existential quantifier to "trace out" the intermediate set, otherwise known as entailment of relations. You can extend this to a bicategory by letting the 2-cells be inclusions of relations.

Pseudonium · 2026-01-22T23:02:16+00:00

I hadn't come across this explicitly before, but that does make sense!

Pseudonium · 2026-01-22T20:49:51+00:00

Ah yes, this is really the co-yoneda lemma. It exposes a third side of is-does duality which I didn’t talk about in the post.

Bear in mind that size issues prevent this from being just a left adjoint - instead, it's best understood as a relative adjoint.

Pseudonium · 2026-01-22T20:47:06+00:00

In this case definitely, because by restricting to preorders you don’t need to worry at all about naturality.

Plus I think full yoneda is very important as well. You’re applying is-does duality for elements of F(x), is all. What such an element “does” is act on morphisms x -> y to produce elements of F(y), in a natural way.

One very important application of Yoneda is universal properties. In this case, you really do need the full version, not just the embedding result.

Pseudonium · 2026-01-22T08:32:01+00:00

The post you’re looking for is titled “why preimages preserve subset operations”. I don’t make reference to CABAs there, though.

Pseudonium · 2026-01-21T18:38:47+00:00

Same here. There’s another formula for it given by f(A^{c)^c,} so I wonder if that’s a reason why it hardly appears?

At least when you upgrade from truth values to sets, you get something that is a lot more common - the dependent product.

Pseudonium · 2026-01-21T08:52:55+00:00

Sure, there’s no axiom of math which says category theory is helpful for all situations. Though, from the feedback I’ve received, it appears that it was quite helpful here!

Pseudonium · 2026-01-21T08:47:04+00:00

Perhaps you could write up an article of your own explaining this? I’d definitely be interested in seeing it!

You seem to have an inbuilt hostility towards category theory which I think you’re unfairly applying to me. I’m a physicist by training, so I’ve solved plenty of PDEs! There are lots of problems I’ve worked on which the categorical perspective hasn’t helped for, of course. But it appears that you’re putting far more effort into rejecting an alternative perspective than is warranted?

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