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Created a stubby poset-valued set.
Looks like a comma double category from $F$ to $P$. Objects are functions $F A \to P$ and horizontal morphisms are squares from some $F R : F A \nrightarrow F B$ to $\leq_P$ along those functions. Then you should have vertical arrows which are of the form $F f$ and make a commutative triangle and some kind of squares. I’m not very familiar with models of linear logic though so I don’t know if those notion already has a name for coherence spaces.
That was exactly my thought on reading it!! There ought to be a general theory of when comma double categories inherit structure from their inputs. Comma double categories have also been used recently to induce structure on decorated cospans, and I believe that some forms of the Dialectica construction can also be represented as comma double categories. However, I don’t have time to think about this any more right now.
Just found mention of “$H$-valued sets” on the tripos page. Are they the same? They look very similar.
They do look somewhat similar, but I don’t think they’re the same. There are no symmetry or transitivity requirements, and the morphisms between them are different.
I’ve been looking at the relationship between polynomial monads and multicategories, and the construction of the double category of T-multicategories is the same as the construction here. You have a double category of polynomials, then pick a specific polynomial monad $T$, then you take the “slice” over $T$ to get a double category where the horizontal morphisms are $T$-graphs, and then a $T$-multicategory is a monad/monoid in that (virtual?) double category. This one looks very similar: a poset is just a monoid/monad in the double category of functions and relations, the only difference is it’s a comma over that monoid because of $F$ (might be interesting to look at the monoids in this setting).
But neither of them actually looks like it fits the definition of comma double category (described for instance here) because a monoid doesn’t consitute a subcategory of the double category it’s in, so I’m not sure how to connect it to that general notion.
Oh! A monad is just a lax morphism from the unit double category. Duh! So it is the same
Ah, neat. Yes, the general sort of comma double category is a comma of an oplax functor over a lax functor, so you can put a monad on the bottom (and, if you want, a comonad on the top). I think that’s a new universal property of the double category of T-spans that I’ve never seen before; a pity it only works when T is polynomial.
Someone ought to write a paper about comma double categories, or at least an nlab page.
I’ll write up a page this week. Would have helped me avoid my mistake above.
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