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John Baez (Jan 17 2024 at 19:07):(The terms loose/tight avoid confusions about the convention concerning which direction is vertical and which is horizontal.)
Nathanael Arkor (Jan 17 2024 at 19:09):John Baez said:
(The terms loose/tight avoid confusions about the convention concerning which direction is vertical and which is horizontal.)
I agree this terminology is clearer, but in this setting there are three directions, and there has been no proposed analogue for "transveral" morphisms in the tight/loose nomenclature as far as I'm aware.
John Baez (Jan 17 2024 at 19:15):Flabby. :upside_down:
Morgan Rogers (he/him) (Jan 17 2024 at 19:15):Slack?
Nathanael Arkor (Jan 17 2024 at 19:19):Morgan Rogers (he/him) said:
Slack?
This was exactly the term I had been using in my notes :grinning_face_with_smiling_eyes:
Christian Williams (Jan 17 2024 at 19:40):In an equipment, a "tight" morphism can "loosen up" to form a companion or conjoint. But in three dimensions, the story is not as simple. A double functor induces a v-profunctor and an h-profunctor; but a v-profunctor does not induce an h-profunctor. So it is no longer a case of "increasing looseness".
[In my thesis I give an interpretation of an equipment as a logic, so a tight (v-)morphism is a process (or term), and a loose (h-)morphism is a relation (or judgement). Then for the "triple category" of double categories, a V-morphism is a meta-process, because it contains processes, and an H-morphism is a meta-relation, because it contains relations. For the transversal dimension, we just need a word that reminds you of "transformation"; I chose flow.]
Nathanael Arkor (Jan 17 2024 at 20:03):Christian Williams said:
In an equipment, a "tight" morphism can "loosen up" to form a companion or conjoint. But in three dimensions, the story is not as simple. A double functor induces a v-profunctor and an h-profunctor; but a v-profunctor does not induce an h-profunctor. So it is no longer a case of "increasing looseness".
Yes, perhaps "slack" is a little misleading, then.
Mike Shulman (Jan 17 2024 at 20:13):In most of the triple categories that I'm familiar with, one direction is clearly tighter than the other two, so it would make sense to call that one "tight". This tight direction is the one that Grandis and Pare draw transversally.
Mike Shulman (Jan 17 2024 at 20:14):I don't personally see a connotation in the words "slack" and "loose" that either of them is "tighter" than the other, so from that perspective at least I wouldn't object to using those two for the other two directions.
Mike Shulman (Jan 17 2024 at 20:15):Are there naturally-occurring triple categories in which the directions are linearly ordered by tightness?
Nathanael Arkor (Jan 17 2024 at 20:27):Perhaps one could form a triple category whose objects are toposes, whose tight morphisms are essential geometric morphism, whose loose morphisms are geometric morphisms, and whose slack morphisms are finitely continuous functors (e.g. taking the direction of each class of morphisms in the logical direction)?
Notification Bot (Jan 19 2024 at 08:50):11 messages were moved here from #theory: category theory > yoneda for virtual double categories by Matteo Capucci (he/him).
Matteo Capucci (he/him) (Jan 19 2024 at 08:53):Mike Shulman said:
Are there naturally-occurring triple categories in which the directions are linearly ordered by tightness?
When you run the Para construction on a double category, what you get is a triple category where the tight direction is that of the original double category and then the two loose directions are original, non-parametric loose arrows and parametric loose arrows, and the former have companions into the latter, so are 'tighter'
I don't think this fact is typical enough to be reflected in general terminology though
Matteo Capucci (he/him) (Jan 19 2024 at 08:56):It's hard to come up with a terminological choice that is also good for higher n-fold categories and doesn't commit to specific spatial directions. Grandis uses multiindices, but that terminology/notation is quite ambiguous (one might confuse 2-arrows with 2-cells aka 'double cells') imo:
image.png
Matteo Capucci (he/him) (Jan 19 2024 at 08:57):
Matteo Capucci (he/him) (Jan 19 2024 at 08:57):One could imagine using letters instead of numbers (similarly to what Christian does with h and v), and then keep talking about single, double, triple, ..., n-fold cells
Nathanael Arkor (Jan 19 2024 at 09:53):The letters still need to stand for something, though.
James Deikun (Jan 19 2024 at 10:11):They could just be A,B,C,...
Matteo Capucci (he/him) (Jan 19 2024 at 10:34):Well usually one refers to the x, y and z direction in maths. X-arrows y-arrows, z-arrows, xy-cells, and so on don't look so bad to me
Matteo Capucci (he/him) (Jan 19 2024 at 10:35):Also Grandis too puts that little legend with numbers and axes which should be enough to disambiguate directions whatever the convention is
Nathanael Arkor (Jan 19 2024 at 10:35):This has exactly the same problem as using h, v, t.
Nathanael Arkor (Jan 19 2024 at 10:36):I.e. people will not be consistent with the terminology, because it is entirely arbitrary.
Matteo Capucci (he/him) (Jan 19 2024 at 10:36):Yeah...
Matteo Capucci (he/him) (Jan 19 2024 at 10:36):I'm brainstorming
Matteo Capucci (he/him) (Jan 19 2024 at 10:37):There is a symmetry in a triple catehory though, exchanging the loose directions, which might justify/compensate for such ambiguity
Nathanael Arkor (Jan 19 2024 at 10:38):But the two non-tight 1-cells behave in different ways, e.g. their composition has different strictness.
Nathanael Arkor (Jan 19 2024 at 10:39):They are only the same in a special kind of triple category, like an intercategory.
Matteo Capucci (he/him) (Jan 19 2024 at 10:46):Uhm right...
Matteo Capucci (he/him) (Jan 19 2024 at 10:47):Tight, tight-loose/tightly loose, loose-loose/loosely loose? :grinning_face_with_smiling_eyes:
Nathanael Arkor (Jan 19 2024 at 11:03):In the context of Mike's comment, I'm still happy with tight/loose/slack, and I don't think anyone has suggested another reason not to be?
Matteo Capucci (he/him) (Jan 19 2024 at 11:15):Nathanael Arkor said:
I.e. people will not be consistent with the terminology, because it is entirely arbitrary.
How does slack/loose not suffer from this problem too?
Dylan Braithwaite (Jan 19 2024 at 11:29):Nathanael Arkor said:
But the two non-tight 1-cells behave in different ways, e.g. their composition has different strictness.
How are triple categories like this defined? I think intercategories are the weakest class of triple category I've seen formally defined
James Deikun (Jan 19 2024 at 12:41):I mean, for virtual triple categories it's worse than different strictness, the 2-cells have different shapes depending what's involved:
Nathanael Arkor (Jan 19 2024 at 14:04):Matteo Capucci (he/him) said:
Nathanael Arkor said:
I.e. people will not be consistent with the terminology, because it is entirely arbitrary.
How does slack/loose not suffer from this problem too?
Sorry, I realised afterwards this wasn't particularly clear. What I meant is that "vertical", "horizontal", "transveral", and also "x", "y", "z" are arbitrary from the perspective of the mathematics, but are completely determined from the perspective of the diagram. So if someone makes a different diagrammatic choice, the naming is different. "Tight", "loose", "slack" do not have this disadvantage. This means that once a choice is made by whoever is first to use the terminology, there is no reason for it to change.
Nathanael Arkor (Jan 19 2024 at 14:04):James Deikun said:
I mean, for virtual triple categories it's worse than different strictness, the 2-cells have different shapes depending what's involved:
- AB-cells and BC-cells have multiple inputs, 1 output, and 1 frame each side
- AC-cells have a single input, single output, and 1 frame each side.
Yes, but on the other hand, there is a unique kind of 2-cell between any two kinds of 1-cell. So it is not necessary to come up with different names for each of them.
James Deikun (Jan 19 2024 at 14:30):I still feel like my ABC terminology has some decisive advantages:
Mike Shulman (Jan 19 2024 at 17:40):Yes, the advantage of x-y-z over horizontal/vertical is that they have a clear linear ordering.
Nathanael Arkor (Jan 19 2024 at 17:55):James Deikun said:
I still feel like my ABC terminology has some decisive advantages:
I agree that this addresses the raised problems. But, subjectively, using letters doesn't appeal to me because letters are already so overloaded (e.g. I could see someone writing something like "an A-cell ", which would not be unnatural given mathematical conventions, but is a little awkward to say out loud). But I could see myself being happy with this kind of convention using a different sequence of symbols.
Mike Shulman (Jan 19 2024 at 17:58):One systematic approach would be to label cells of all dimensions by vertices of the unit cube. So the three kinds of 1-dimensional cell in a triple category would be (1,0,0)-cells, (0,1,0)-cells, and (0,0,1)-cells, while the three kinds of 2-dimensional cell would be (1,1,0)-cells, (1,0,1)-cells, and (0,1,1)-cells. Then the cell dimension is evident from the name (it's the sum of all the coordinates) as is the ordering (the ordering on the coordinates) and also the relation between dimensions (a (1,1,0)-cell has a boundary consisting of (1,0,0)-cells and (0,1,0)-cells). But this is probably too cumbersome for everyday use.
Dylan Braithwaite (Jan 19 2024 at 18:00):James Deikun said:
I mean, for virtual triple categories it's worse than different strictness, the 2-cells have different shapes depending what's involved:
- AB-cells and BC-cells have multiple inputs, 1 output, and 1 frame each side
- AC-cells have a single input, single output, and 1 frame each side.
Is that to say that one of the directions of 1-cell in a virtual triple category has only virtual composites and the other has standard weak composition? It seems like simply saying 'tight', 'loose' and 'virtual' would be fitting then in that case?
Or is there another structure where the two loose directions have different strictnesses?
Nathanael Arkor (Jan 19 2024 at 18:04):Dylan Braithwaite said:
James Deikun said:
I mean, for virtual triple categories it's worse than different strictness, the 2-cells have different shapes depending what's involved:
- AB-cells and BC-cells have multiple inputs, 1 output, and 1 frame each side
- AC-cells have a single input, single output, and 1 frame each side.
Is that to say that one of the directions of 1-cell in a virtual triple category has only virtual composites and the other has standard weak composition?
Perhaps this diagram will help (the shape of a 3-cell in a virtual triple category):
image.png
In general, only the tight morphisms compose at all in a virtual triple category.
Nathanael Arkor (Jan 19 2024 at 18:06):(The different styles of 2-cells should be read only as distinguishing between the different kinds, not as any kind of suggestion over how one should actually style them in practice.)
Mike Shulman (Jan 19 2024 at 18:18):I think there are many different possible kinds of "virtual triple category".
James Deikun (Jan 19 2024 at 18:24):Well, this is the one you could call a "free fc-multicategory multicategory". In other words you get it by iterating the Leinster prime/plus construction 3 times starting with the identity monad on Set.
Dylan Braithwaite (Jan 19 2024 at 18:25):Nathanael Arkor said:
Perhaps this diagram will help (the shape of a 3-cell in a virtual triple category):
[...]
Ahh I see. From James' description I figured only the B cells would fail to compose, if AC squares have only one input/output. But I'm guessing the idea is cubes have lists of AC cells as domains, then this fits with your diagram
James Deikun (Jan 19 2024 at 18:36):(Although in a different sense/dimension, the cubes have pasting diagrams of BC-cells as domains.)
Kevin Arlin (Jan 19 2024 at 19:41):It does seem like naming a cell by the vector of dimensions in which the cell is extended, as Mike suggests, is the most plausible approach especially to work for higher-dimensional generalizations. I suppose it's essentially the same problem in cubical homotopy theory; I wonder if the cubical type theorists or anybody else has developed conventions that could be borrowed.
Mike Shulman (Jan 19 2024 at 19:43):Cubical type theorists generally use a cube category with symmetries (transpositions), so there is no difference between the "directions".
Mike Shulman (Jan 19 2024 at 19:44):Even in cubical homotopy theory without symmetries, I think there is usually still only one kind of 1-cell.
Nathanael Arkor (Jan 19 2024 at 19:45):Kevin Arlin said:
It does seem like naming a cell by the vector of dimensions in which the cell is extended, as Mike suggests, is the most plausible approach especially to work for higher-dimensional generalizations. I suppose it's essentially the same problem in cubical homotopy theory; I wonder if the cubical type theorists or anybody else has developed conventions that could be borrowed.
For what it's worth, there is an entirely systematic naming convention by vectors (using the presentation as a generalised multicategory as James mentioned). But I agree with Mike that this notation does not seem convenient for lower dimensions, where abbreviated terminology would be helpful.
Kevin Arlin (Jan 19 2024 at 19:48):I guess I just meant insofar as naming the types of cell is essentially the same as naming the faces and iterated faces of the cube, not that cubical homotopy theorists have distinct sets of edges.
Cole Comfort (Jan 19 2024 at 22:02):John Baez said:
Flabby. :upside_down:
Wiggly, squiggly? Maybe if the existing notation doesn't generalize to higher dimension it is better to just use an index as one does in the globular setting.
Kevin Arlin (Feb 01 2024 at 21:44):Thought of this conversation again today in the context of considering the "thing" of double categories and "all" kinds of maps between them. That's something like
-Dimension 0: double categories
-Dimension 1: lax and oplax functors, double profunctors (in both - and -directions?)
-Dimension 2: lax and oplax transformations in the - and - directions, modules
-Dimension 3: Modifications and modulations
Kevin Arlin (Feb 01 2024 at 21:44):All of this structure appears locally in various papers in the literature, though I don't know if there's been any effort to put it all into one big doodad. I wanted to bring it up to observe that a really general multiple category seems to have a reasonably arbitrary poset of cells in each dimension, with the order relation giving "loosenings", and would also need some independent specification of the boundary type and shape of higher-dimensional cells (modifications can't involve modules, for instance.) This is perhaps more an issue of mathematics than of pragmatics, but it feels like it might be a more natural way to situation this kind of question than in terms of the special cases above, even including Grandis and Pare's preference of one tight dimension.