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David Michael Roberts (May 03 2022 at 08:52):We know every bicategory is equivalent to a 2-category, and every tricategory is equivalent to a Gray-category.
But what do we know about semistrictification results for tetracategories à la Trimble? That is, have people verified any of the various semistrictification conjectures for n=4? I'm thinking strict everything except interchange, or strict everything except units. Degenerate versions of this may have been checked, for instance braided monoidal bicategories, but I'm feeling a bit lazy to track something down. This is pretty much at the limit of what is feasible, perhaps, clearly higher versions need different definitions of n-categories. But just as Gordon, Power and Street's work was a catalyst for finding out subtleties of (3,3)-categories (i.e. not (3,1)-categories!), might there be fun things hiding in the tetracategorical case?
John Baez (May 03 2022 at 16:11):The person to ask is Nick Gurski.
Tim Campion (May 03 2022 at 16:55):Doesn't globular rely on some conjectural semistrictification theorem? It might be illuminating to take a look at their "semistrictification hypothesis"...
John Baez (May 03 2022 at 17:00):That's true.
James Dolan had a semistrictification hypothesis. He showed it was false: it breaks down precisely for tetracategories. Day and Street had a very similar semistrictification hypothesis, that every weak n-category was equivalent to something called a "file". But this also broke down for tetracategories, for very similar reasons. So someone should check to see if globular is implicitly making the same mistake.
Todd Trimble (May 03 2022 at 22:00):Here is a relevant paper by Bourke and Gurski.
Mike Shulman (May 04 2022 at 03:05):That's very interesting, John, I didn't know (or had forgotten) about those. Is anything written down about how exactly their conjectures failed?
Mike Shulman (May 04 2022 at 03:35):The failure of full strictification of tricategories to strict 3-categories is already visible in the doubly-degenerate case of braided monoidal categories. Was the failure of Dolan's and Day-Street's semistrictification hypotheses visible in degenerate cases?
Todd Trimble (May 04 2022 at 13:43):Mike Shulman said:
That's very interesting, John, I didn't know (or had forgotten) about those. Is anything written down about how exactly their conjectures failed?
See the paper I had linked.
Mike Shulman (May 04 2022 at 14:51):Ah, thanks. I'd forgotten that they had that section 4.8 at the end. It looks like that conjecture was about the possibility of making a definition of 4-file by iterated enrichment, not about whether or not tetracategories are equivalent to an actually defined notion of 4-file. So that's a somewhat different situation from Globular, which I believe does have a complete (non-enrichment-based) definition of "semistrict n-category" for all n.
Nathanael Arkor (May 04 2022 at 14:54):Globular is based on "associative n-categories", which have weak units and interchange, but strict associativity.
Tim Campion (May 04 2022 at 14:55):Nathanael Arkor said:
Globular is based on "associative n-categories", which have weak units and interchange, but strict associativity.
Ah... so in that case I think this semistrictification hypothesis was conjectured by Carlos Simpson, right?
Mike Shulman (May 04 2022 at 15:00):I thought Simpson's hypothesis was about weak units and strict interchange (and strict associativity).
Tim Campion (May 04 2022 at 15:01):Ah right -- so the appropriate hypothesis used here seems to be strictly weaker than Simpson's conjecture.
Tim Campion (May 04 2022 at 15:01):no pun intended
Tim Campion (May 04 2022 at 15:02):A strictly weaker conjecture about making weak things strict
Tim Campion (May 04 2022 at 15:02):My head is about to explode
Nathanael Arkor (May 04 2022 at 15:04):Tim Campion said:
Ah right -- so the appropriate hypothesis used here seems to be strictly weaker than Simpson's conjecture.
Actually, if I understand correctly, their conjecture is that every n-category is equivalent to a strictly unital associative n-category.
Nathanael Arkor (May 04 2022 at 15:04):So they are conjecturing weak interchange is sufficient.
Tim Campion (May 04 2022 at 15:12):Nathanael Arkor said:
Actually, if I understand correctly, their conjecture is that every n-category is equivalent to a strictly unital associative n-category.
I feel like at this point I should actually read the literature, but I think just perusing the nlab pages here and here I see 3 different conjectures. It sounds like you're talking about the third:
Simpson's conjecture : every weak n-category is equivalent to one whose associators and interchangers are strict.
The Dorn-Douglas-Vicary hypothesis: every weak n-category is equivalent to one whose associators are strict.
The strong Dorn-Douglas-Vicary or "Gray" hypothesis: every weak n-category is equivalent to one whose associators and unitors are strict.
John Baez (May 04 2022 at 15:16):Mike Shulman said:
Ah, thanks. I'd forgotten that they had that section 4.8 at the end. It looks like that conjecture was about the possibility of making a definition of 4-file by iterated enrichment, not about whether or not tetracategories are equivalent to an actually defined notion of 4-file.
Right. Maybe I should have made that clear. Jim Dolan's attempted recursive definition of semistrict n-categories via iterated enrichment broke down at n = 4 for essentially the same reasons.
So that's a somewhat different situation from Globular, which I believe does have a complete (non-enrichment-based) definition of "semistrict n-category" for all n.
I hope so. I haven't checked to see if their definition is consistent. :upside_down:
But I hope Jamie Vicary et al have figured out exactly how and why their approach gets around the problem that Day and Street and Dolan encountered.
Nathanael Arkor (May 04 2022 at 15:17):At this point, it seems reasonable to consider the "X-strictness conjecture" for X any subset of {U, A, I}, meaning "every weak n-category is equivalent to a weak n-category with strict X" (e.g. Simpson's conjecture takes X = { A, I }).
Nathanael Arkor (May 04 2022 at 15:18):And if I understand correctly the only established result is that the {U, A, I}-strictness conjecture is false.
Nathanael Arkor (May 04 2022 at 15:18):And there are a few positive results for other X at n = 3.
Amar Hadzihasanovic (May 05 2022 at 10:37):Simon Henry has proven that weak n-groupoids are modelled by n-groupoids with strict associativity and interchange + "non-algebraic" weak units (given by a fibrancy condition), but with a restriction on what the composable pasting diagrams are (compared with "globular" n-categories).
Amar Hadzihasanovic (May 05 2022 at 10:37):Then in this paper I have a precise conjecture (Conjecture 6.45) that a similar model of n-categories with strict associativity, interchange and algebraic weak units is a model of all weak n-categories.
Amar Hadzihasanovic (May 05 2022 at 10:39):I also know that Simona Paoli has been making attempts at versions of Simpson's conjecture, but I do not understand her work well enough to tell how close she is.
Amar Hadzihasanovic (May 05 2022 at 10:41):(By "precise conjecture" I mean it's not of the form "there is 'some' equivalence with a model of weak n-categories", as most of these conjectures are, but of the form "these specific strictification morphisms are weak equivalences").
Amar Hadzihasanovic (May 05 2022 at 10:45):I think Henry's result is the strongest and best strictification result at the moment, but to appreciate it one needs to give up the idea that n-categories with "globular, binary composition operations" are paradigmatic.
Amar Hadzihasanovic (May 05 2022 at 10:47):I've grown to think that this is actually wrong; I think that strict 4-categories are both too strict and not strict enough, in that there are some "directed 4-homotopies" that are not identified by the axioms of 4-categories, but should actually be identified.
Amar Hadzihasanovic (May 05 2022 at 10:50):(The example has been pointed out to me by Felix Loubaton, but its interpretation as a failure of 4-category axioms is mine and debatable.)
Nathanael Arkor (May 05 2022 at 14:09):Amar Hadzihasanovic said:
I've grown to think that this is actually wrong; I think that strict 4-categories are both too strict and not strict enough, in that there are some "directed 4-homotopies" that are not identified by the axioms of 4-categories, but should actually be identified.
Can you say precisely which extra axioms you think ought to be added?
Amar Hadzihasanovic (May 05 2022 at 14:47):Something to the effect that, if you can apply a pair of cells to disjoint parts of a pasting diagram in either order, then the order should not matter. As it stands, this fails in general for 4-cells applied to 3d pasting diagrams in 4-categories.
Amar Hadzihasanovic (May 05 2022 at 14:52):I think the "directed homotopy hypothesis", which for me is the guiding principle to higher categories, leads to the idea that diagrams in an n-category should be "directed balls" in it, which should be topological balls + a cell structure + a well-formed orientation.
In the example above, you have two 4-diagrams defining the exact same "directed 4-ball" (I can make this precise) but corresponding to different 4-morphisms.
Amar Hadzihasanovic (May 05 2022 at 14:55):Btw I have no idea if the kind of equality I suggest is even finitely axiomatisable as a "globular theory".
Nathanael Arkor (May 05 2022 at 15:07):Amar Hadzihasanovic said:
Something to the effect that, if you can apply a pair of cells to disjoint parts of a pasting diagram in either order, then the order should not matter. As it stands, this fails in general for 4-cells applied to 3d pasting diagrams in 4-categories.
Hmm, interesting. Do you know if there are similar issues for n-fold categories? That is, some identifications that seem natural to make, but are not implied by the axioms for, say, quadruple categories. In other words: is this specifically a problem with globular structures, or more generally structured defined naïvely using iteration?
Nathanael Arkor (May 05 2022 at 15:10):(I appreciate this is a very loosely-defined question.)
Amar Hadzihasanovic (May 05 2022 at 15:38):Well, considering that this is a problem with what one thinks the "right" diagrams are for a certain structure, there is some debate that already double categories fail in that respect, because they cannot compose the "pinwheel".
Amar Hadzihasanovic (May 05 2022 at 15:39):(Although things go wrong in different ways -- in a 4-category you may have "two possible composites for the same diagram" and in a double category you may have "no composites".)
Amar Hadzihasanovic (May 05 2022 at 15:40):I don't know enough about n-fold categories to say if there is a closer example.
Jamie Vicary (Dec 10 2022 at 22:13):John Baez said:
Mike Shulman said:
So that's a somewhat different situation from Globular, which I believe does have a complete (non-enrichment-based) definition of "semistrict n-category" for all n.
I hope so. I haven't checked to see if their definition is consistent. :upside_down:
But I hope Jamie Vicary et al have figured out exactly how and why their approach gets around the problem that Day and Street and Dolan encountered.
I didn't know much about the history of this interaction of Bourke, Dolan and Street, and I have just this evening done some reading to educate myself. As far as I now understand these points, it seems that they concern the appropriate notion of enrichment for an iterative internal definition of semistrict n-category. However our various definitions of semistrict n-category (arXiv:2007.08307, arXiv:2109.01513, arXiv:2205.08952, also Christoph Dorn's thesis) do not work in this way; we don't use any enrichment of semistrict (n+1)-categories in semistrict n-categories, nor do we build any tensor product of semistrict n-categories.
So at first glance it seems that we conveniently sidestep these issues. If there is nonetheless some way to express this historic problem in a way that we can then query against our definitions, we would be keen to do that.
John Baez (Dec 10 2022 at 22:22):Hi! For the problematic earlier attempts, the problem kicks in at the level of tetracategories. So, if you can check that your concept of semistrict tetracategory (or whatever you call it) gives a concept of tetagroupoid that adequately models homotopy 4-types, that'll be strong evidence that you're doing things right. I guess there are a bunch of easier checks as well.