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philip hackney (May 14 2026 at 18:15):a couple blog posts (the first of which was written roughly a year ago), both related to 2-Segal spaces:
(there was only one other post before this, from more than a decade ago!)
philip hackney (Jul 01 2026 at 15:50):This week I'm making some small revisions to "On partial groups of small order" (which I posted on arxiv about a bit more than a month ago), to get it ready for journal submission.
philip hackney (Jul 01 2026 at 15:54):This paper is about a computer enumeration of Chermak's "partial groups", which I prefer to view as a certain kind of symmetric simplicial set. They are like groups except the n-to-1 multiplications are only partially defined (in a controlled way). I'm still a bit shocked that I could do this at all, though there are a LOT of partial groups, way more than I would've guessed (over 179 million of order ≤ 10).
philip hackney (Jul 01 2026 at 15:57):More concretely, a partial group is a symmetric simplicial set which 1) is reduced X₀ = *, and 2) has the usual Segal maps being injective functions. [Groupoids are essentially symmetric sets where the Segal maps are bijective functions.]
philip hackney (Jul 02 2026 at 17:17):One thing that's fun about having 140GB of partial groups sitting on your computer is you can run experiments on them. For instance, you might notice that every 4-Segal partial group of order ≤ 10 is 2-coskeletal. Then you can try to prove the that it's just always true! (OK I originally noticed this for order ≤ 9, which is only ~30MB of partial groups.)
philip hackney (Jul 02 2026 at 17:20):But what you might not notice is that your proof of 4-Segal => 2-coskeletal doesn't depend much on 2. For that you need Justin Lynd to point out that maybe you really proved 2k-Segal => k-coskeletal (for k≥2). :upside_down:
philip hackney (Jul 08 2026 at 13:20):The updated version of this enumeration paper is now available on the arxiv and submitted somewhere. I wrote a new abstract. The paper has definitely been improved based on comments from friends. (and me rereading it and groaning at a couple things)
philip hackney (Jul 08 2026 at 13:22):At the same time, I also updated a related paper with Salati & Lynd on arxiv, which is now called binary partial groups. (previously "partial groups as partial groups"). It's about how to regard unital partial magmas with inverses as partial groups in two ways -- a 2-skeletal way, and a 2-coskeletal way.
philip hackney (Jul 08 2026 at 13:29):What I learned writing the BPG paper really had a big impact on the enumeration project. Earlier algorithms were a bit "top down" and much less capable, but the best algorithm I know starts by enumerating all the BPGs carried on a fixed involutive set. And then once you have a fixed BPG, you try to make 3-dimensional extensions of the underlying 2D-partial group, then 4D-extensions of each 3D-partial group, etc. Works pretty well for small orders.
philip hackney (Jul 24 2026 at 13:19):Here's a fact about simplicial sets: X is isomorphic to the nerve of a category if and only if X is 2-coskeletal and the Segal maps in dimensions 2 and 3 are bijections. A variant of this is on nlab but instead with strict fillers for inner 2 & 3 dimensional horns.
philip hackney (Jul 24 2026 at 13:28):A version of this is also known for 2-Segal sets. Bergner–Osorno–Ozornova–Rovelli–Scheimbauer showed that every 2-Segal set is 3-coskeletal (Corollary 1.7) and somewhat later Walker Stern showed (Proposition 4.9) that if a simplicial set X is 3-coskeletal and satisfies the 2-Segal conditions corresponding to the square and the pentagon (these are conditions on sets of 3 and 4 simplices), then it is 2-Segal.
philip hackney (Jul 24 2026 at 13:32):I'm finishing up a note that does something similar for d-Segal sets: a simplicial set is d-Segal iff it's "d-Segal in dimensions d+1 and d+2" and is (d+1)-coskeletal. (The real theorem is a bit better than this, splitting into upper/lower variants, and for the reverse direction only asking for (d+2)-coskeletal.)
philip hackney (Jul 24 2026 at 13:37):This is a question Stern asked a long time ago, and I made some partial progress more than three years ago but didn't ever push through to the end -- it was a bit too annoying, too many cases to deal with.
philip hackney (Jul 24 2026 at 13:44):Also this isn't true at all for simplicial spaces/anima/∞-groupoids. If M is a space, then the associated constant simplicial space is automatically d-Segal for all d. But that constant simplicial space being n-coskeletal boils down, more or less, to the iterated loop space being contractible. So we can break it with an Eilenberg–MacLane space or a sphere.
philip hackney (Jul 30 2026 at 10:02):The preprint of this coskeletality result is now up on arxiv.
While working on the paper I did some calculations of d-Segality for a couple standard simplicial-set models of the n-sphere, namely and , which are 2n-Segal and (n+1)-Segal, respectively (and no lower... except the first case when n is odd you can do a little bit better). Here is a blog post which states this more precisely, and has a linked PDF with details.