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David Corfield (Mar 03 2025 at 09:53):That's fun. Somebody has taken up a -Category Café suggestion and worked it into a paper.
At some point during that interesting discussion on the relationship between pyknoticity/condenseness and cohesion, I tried to initiate a thread on ultracategories.
This then spun out to me trying to figure out whether there might be some 2-monad approach to them in Ultracategories and 2-Monads (in what might be the longest single-authored thread in Café history).
In the course of this speculation, I pose a question to @Mike Shulman on another thread on whether they could be seen as generalized multicategories, and he duly comes up with an answer.
So now in
the author claims to have properly established results in this area:
We show a result inspired by a conjecture by Shulman claiming that ultracategories as defined by Lurie are normal colax algebras for a certain pseudo-monad on the category of categories CAT. Such definition allows us to regard left and right ultrafunctors as defined by Lurie as instances of lax/colax algebras morphisms.
David Corfield (Mar 03 2025 at 10:11):I wonder this new work sits with this article:
The first attempt of providing a more conceptual framing to understand ultrastructures and ultracategories was given by Marmolejo in his PhD thesis [Mar95]. He was probably the first to trace some geometric aspects of the construction of ultraproducts, introducing the notion of Łoś category. We will see that the blueprint of our approach is essentially the same of Marmolejo’s, even though we have a quite different way of encoding the same idea.
In more recent years Lurie [Lur] revisited the notion of ultracategory, proposing a morally similar, but technically different notion of ultrastructure, ultracategory and ultrafunctor with respect to Makkai’s one. Both Makkai’s and Lurie’s notions are justified by the fact that they manage to deliver the most compelling theorems of this theory. Yet, none of these notions appears definitive when read or encountered for the first time for several reasons. The main one being that the definition of ultracategory is in both cases very heavy, and comes together with axioms whose choice seems quite arbitrary; and indeed the two authors make different choices.
Ivan Di Liberti (Mar 03 2025 at 10:14):Very good question. So many good young people are thinking about this topic these days. I guess I'll wait for their next paper on the arXiv to know!
Jean-Baptiste Vienney (Mar 03 2025 at 13:05):Ali is one of my fellow phd student at uOttawa. The world is small :joy:
Mike Shulman (Mar 03 2025 at 17:48):Cool!