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Tim Hosgood (Dec 06 2022 at 17:03):here's a vague question: bicategories look globular, and double categories look cubical; we could think of bicategories as double categories but where vertical morphisms (I'll call these proarrows to avoid horizontal/vertical confusion) are only allowed to be identities; is there some intermediary notion that looks simplicial? i.e. a double category where the only cells are those whose "internal" source (or target, but pick one of the two and fix it, I think) is an identity proarrow?
Tim Hosgood (Dec 06 2022 at 17:04):visually: if i pinch both vertical sides of a square then i get a circle, but if i just pinch one then i get a triangle
Mike Shulman (Dec 06 2022 at 17:04):How would you compose them?
Tim Hosgood (Dec 06 2022 at 17:04):no idea, that's what i've been stuck on!
Mike Shulman (Dec 06 2022 at 17:04):That seems like an indication the answer is no...
Mike Shulman (Dec 06 2022 at 17:05):I mean, you could just say "I have a double category in which the horizontal target of every 2-cell is an identity"
Tim Hosgood (Dec 06 2022 at 17:05):hmm, that could be true, but it seems mysterious to me that there are nice globular and cubical structures, but no simplicial-looking one
Mike Shulman (Dec 06 2022 at 17:06):It makes sense to me. To have a composition structure in any direction, you need both sources and targets.
Nathaniel Virgo (Dec 07 2022 at 03:15):Random speculation, but maybe the simplicial version is about the case where source and target don't matter. You can compose simplices with each other to get simplicial complexes, but those don't involve any notion of "to" and "from".
This makes me wonder if dagger categories are relevant, since they somehow weaken the notion of "to" and "from" in a category. Is there an existing higher-dimensional analog of dagger categories, where there's a dagger structure at each level?
Matteo Capucci (he/him) (Dec 07 2022 at 08:33):@Carlos Zapata-Carratala is working on ternary compositional structures
Matteo Capucci (he/him) (Dec 07 2022 at 08:39):I thought myself of this question Tim, and got stuck here:
image.png
Matteo Capucci (he/him) (Dec 07 2022 at 08:44):though can't exclude there exists other ways to compose them that work
Matteo Capucci (he/him) (Dec 07 2022 at 08:44):Nathaniel Virgo said:
Random speculation, but maybe the simplicial version is about the case where source and target don't matter. You can compose simplices with each other to get simplicial complexes, but those don't involve any notion of "to" and "from".
This makes me wonder if dagger categories are relevant, since they somehow weaken the notion of "to" and "from" in a category. Is there an existing higher-dimensional analog of dagger categories, where there's a dagger structure at each level?
IIRC @Carlos Zapata-Carratala does indeed have a ternary compositional structure for three-way isomorphisms
John Baez (Dec 07 2022 at 09:25):In my paper with James Dolan on the cobordism hypothesis we argued that for -categories it's wise to demand that morphisms have adjoints at each level except the top (-morphism) level, where it's impossible - so we give morphisms daggers only on the top level. Being an adjoint is a property, while the dagger is an extra structure.
John Baez (Dec 07 2022 at 09:55):You could also require that morphisms have inverses at the top level - that's also dagger-like, and it's a mere property.
Carlos Zapata-Carratala (Dec 07 2022 at 11:09):Matteo Capucci (he/him) said:
Carlos Zapata-Carratala is working on ternary compositional structures
I am indeed working on higher-order compositionality. Thanks for the heads up!
Matteo Capucci (he/him) said:
I thought myself of this question Tim, and got stuck here:
image.png
This looks intriguing. As a composition of 2-morphisms that would be a quaternary operation. I haven't ventured into the quaternary world much, but the "crossed pattern" of the directions of 2-morphisms (3 down, 1 up) evokes the composition of semiheapoids. Semiheapoids can be defined similarly to (semi)categories via operations on quivers. The key difference is that composition is fundamentally ternary and is based on the following "non-circulatory" diagram:
Which results in a natural associativity-like axiom (note the twist in the middle term):
Incidentally to @Nathaniel Virgo's comment, dagger categories are, in fact, the prime example of semiheapoids where the ternary composition is defined via
This structure is quite source-and-targety for a ternary operation, however, most higher-order compositional structures that are currently poorly understood and under study demand a departure from sequentiality and notions of input/output, to/from, etc. Indeed, simplicial set unions/splicings (rewrite systems) can be regarded as well-defined higher-order compositional structures where there are no obvious notion of morphisms going "to" and "from" anywhere.
As @Matteo Capucci (he/him) points out, I found some examples of ternary compositions of "3-way isomorphisms" or "trisomorphisms" (a commuting triangle of isomorphisms) that somewhat realize higher-order non-sequential simplicial-like composition in ordinary categories. You can find the details in Section 4.4 of my recent paper:
Invitation to Higher-Artiy Science
Here are some diagrams from that section that illustrate the non-sequential nature of "ternary trisomorphism compositions":
By the way, I will be talking about this topic in a couple of week at the SYCO 10 meeting in Edinburgh. Looking forward to in-person discussions!
Matteo Capucci (he/him) (Dec 07 2022 at 12:36):Lovely!