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Finally split two-sided fibration off of Grothendieck fibration. Thanks to Emily Riehl for adding the definitions here.
I have copied to two-sided fibration the statement of theorem 2.3.2 on page 5 of Emily Riehl’s text.
Then I added a discussion of how the construction of $E$ from the profunctor $P$ that she gives there is equivalently obtained by homming $[k] \star [k]$ into the collage of $P$.
But I have one problem: what is the $E$ obtained this way actually? Possibly it is too late for me and I am not concentrating enough, but I see $E$ as sitting over $B^{op} \times A$, not over $B \times A$.
In fact, it seems to me that the construction of $E$ from $P : B^{op} \times A \to Set$ that is given in the proof of theorem 2.3.2 is just the construction of the category of elements/Grothendieck construction of $P$, and that sits over $B^{op} \times A$, not over $B \times A$.
Sorry, i removed my addition again (am keeping it elsewhere, nothing is lost). Maybe there is a typo bugging me, or I am being dense. I’ll look into this tomorrow, when I am fresh again.
It seems to me that the last two lines of the proof of theorem 2.3.2 here should read differently.
I have written out how I think the proof needs to go instead at two-sided fibration.
(Sorry to belabor this simple point in such a lengthy fashion.)
I think in summary we can say: the two-sided fibration corresponding to a profunctor is the category of sections of its collage.
I wrote:
I think in summary we can say: the two-sided fibration corresponding to a profunctor is the category of sections of its collage.
Just spent a few minutes with Christian Blohmann on the blackboard. It seems we agreed that the construction of a span of simplicial sets from a simplicial set $p : K \to \Delta[1]$ over the interval is indeed precisely this: the simplicial set of sections of $p$, whose $k$-cells are morphisms $\Delta[k] \times \Delta[1] \to K$ subject to the evident constraints.
But I’ll try to check the precise details later on.
Mike just pointed me here. Urs, you’re right that the morphisms over B are backwards in the construction in my notes. Sorry about that! I’ve checked the version on the nLab and everything seems to be correct. I’ll fix the notes now.
I like the intuition that the span associated to a profunctor is the sections of its collage. Thinking about simplicial sets though: the construction you mention above (k-simplices of the span being k-“cylinders” whose outer k-dimensional faces lie over 0 and 1 respectively) is a bit smaller than the one you’ve written about elsewhere (k-simplices of the span being maps $\Delta^k \star \Delta^k \rightarrow K$ whose restrictions to the outer k-dimensional faces lie respectively over 0 and 1). Of course, if the original map $K \rightarrow \Delta^1$ is a mid-fibration, then you can fill each k-cylinder to a (2k+1)-simplex of the right type. Is this what you meant by “the construction of a span…is indeed precisely this”?
There is a nice other characterization: the 2-sided discrete fibration $E_F$ associated to a profunctor $F:B^{op}\times A\to Set$ is also the comma category of its collage regarded as a cospan $A \to K_F \leftarrow B$. And dually the collage is the cocomma category of the 2-sided discrete fibration regarded as a span $A \leftarrow E_F \to B$. This makes it look kind of like the equivalence between congruences and quotient objects in an exact category.
Mike just pointed me here. Urs, you’re right that the morphisms over B are backwards in the construction in my notes. Sorry about that! I’ve checked the version on the nLab and everything seems to be correct. I’ll fix the notes now.
Thanks for letting me know!
I like the intuition that the span associated to a profunctor is the sections of its collage. Thinking about simplicial sets though: the construction you mention above (k-simplices of the span being k-“cylinders” whose outer k-dimensional faces lie over 0 and 1 respectively) is a bit smaller than the one you’ve written about elsewhere
Yes, I keep thinking about this, but still haven’t quite made up my mind. The problem is that I also still have not seen the detailed writeup of Christian Blohmann and Chenchang-Zhu. But the idea is that given the simplicial set $K \to \Delta[1]$ they look at its $k$-simplices $\Delta[k] \to K \to \Delta[1]$ and observe that the composite map $\Delta[k] \to \Delta[1]$ equips the vertices of $\Delta[k]$ with a “bipartitioning” given by whether they map to $\{0\}$ or to $\{1\}$ and that this equips $K$ with the structure of a bi-augmented-simplicial-set $K_{\bullet, \bullet}$ where the first index counts vertices sitting over $\{0\}$ (which may be none) and the second counts vertices over $\{1\}$ (which also may be none). The diagonal $(K_{p,p})_{p \in \mathbb{N}}$ they take to be the tip of the corresponding span.
I’ve edited two-sided fibration somewhat, moving the stuff on discrete fibrations into its own section and adding the definition of a two-sided fibration in an arbitrary finitely-complete bicategory.
For the past while I’ve been staring at the definition of two-sided fibrations in Cat and trying to relate that to the notion of $M$-algebra. I mentioned this in another thread, but I still haven’t completely worked it out, so I’ve put in what I have for what it’s worth. Maybe others could look over it and see what they think.
Thanks! This is great. I did a bit of reorganizing, and added another characterization which I like: two-sided fibrations from A to B are exactly the opfibrations in the 2-category Fib(A) over the fibration $A\times B\to A$, and dually. But I didn’t really think about the M-algebras yet.
That’s very nice (I didn’t even realise that product projections were bifibrations). Do you know if it has been published anywhere, or is it folklore, or what?
I’ve never seen or heard of it anywhere before, so it might be due to me. I stumbled across it when thinking about 2-categorical logic and categories of fibrations as the 2-categorical version of “slice categories”. I wouldn’t be surprised if someone else had noticed it earlier, though.
I’ve finished off the proof of the proposition at two-sided fibration characterizing two-sided fibrations in Cat. I changed the third condition in the proposition to follow Street’s Conspectus of variable categories — the existing one was probably equivalent, but I couldn’t quite see how off the top of my head. There are quite a few isomorphisms left implicit in the proof, which I think makes it much more readable, but I’d appreciate it if others could glance through it and see if I’ve missed or glossed over anything important.
Thanks! Looking back at the previously stated condition, I see that it’s clearly wrong: it would imply that any vertical arrow is invertible! If it was me who originally wrote that condition, I’m sorry. Anyway, I’m glad you fixed it.
I found the characterization of #10 in a recently-TAC-Reprinted preprint of Bourn and Penon from 1978, so I’ve added a reference to two-sided fibration.
Thanks. That looks like a very interesting paper!
Yes, I thought so too. Their notion of catéade seems to be the same as what you call a 2-congruence. In the preface they mention John Bourke’s talk at CT09 on exact 2-categories, but I don’t think he’s published on that (I met John at the last PSSL in Oxford, but never thought to ask him about it). There are some similar ideas in a later paper by Bourn, La tour de fibrations exactes des $n$-catégories, Cahiers 25(4), 1984; I don’t know if it’s relevant, but I found it interesting.
Also, in their appendix Bourn and Penon mention the idea of turning a 2-monad into a lax-idempotent one. I know you and Geoff Cruttwell were working on that in connection with your generalized multicategories paper. Have you written anything on it?
Their notion of catéade seems to be the same as what you call a 2-congruence.
Yes. I wish I read French more fluently.
I know you and Geoff Cruttwell were working on that in connection with your generalized multicategories paper.
No, not yet. But when we do write it down, it’s essentially just going to be a generalization to our context (virtual double categories) of what Hermida did in “From coherent structures to universal properties”.
They seem equally reasonable to me. If it was me who wrote “two-sided discrete fibration” first in the entry, probably it was just because it rolled off my tongue slightly more easily. (Most or all of the literature says something like “fibration from A to B”, which is okay when it works, but not when you want to discuss the general concept and distinguish it from one-sided fibrations.)
Perhaps you have in mind that a two-sided discrete fibration is a two-sided fibration first, with the property of being discrete, whereas a “discrete fibration” sounds like a one-sided thing that can’t then have the property of “being two-sided”? I can see that argument, but I can also not see it. (-: Already one could argue that a “fibration” is already a one-sided thing that can’t then have the property of “being two-sided”; certainly if we say “fibration” unqualified then (in many communities, at least) it means the one-sided version. So if we are to accept “two-sided fibration” (as well as other red-herring words like “Street fibration”, etc.) then the word “fibration” must have multiple possible meanings (two-sided, one-sided, op-, Street, etc.) but gets specialized to a “default” meaning (one-sided, contravariant) if none of those qualifiers are given. And in that case it seems one could equally well argue that the same is true of “discrete fibration”: by default it means one-sided, but that default can be overridden by a qualifier like “two-sided”.
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