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Christian Williams (May 22 2023 at 15:58):When you read the definition of bicategory, it's not too hard to learn how to generalize to a double category --- different data, a span of categories, but all the same coherences: a pseudomonad.
In the same way, one can read the definition of tricategory and see how to generalize to a triple category --- but I'm having trouble finding a reference. All the Grandis/Pare work is 2-weak, rather than 3-weak. Does anyone know if general triple categories have been defined?
Christian Williams (May 22 2023 at 16:27):It seems like there isn't a reference, which is surprising. Whenever I heard of triple categories in the past, somebody would say "oh yeah, Grandis/Pare did that." But their definition doesn't include tricategories; it's not the actual general concept.
Bryce Clarke (May 22 2023 at 16:59):The third sentence of this paper attributes the notion of [[triple category]] to Ehresmann.
Christian Williams (May 22 2023 at 17:14):yes, but that's the strict version, right? I'm looking for the general notion, which includes tricategories.
John Baez (May 22 2023 at 19:36):I'm pretty sure Ehresmann recursively defined strict n-tuple categories for all n, but did not study weak double categories, weak triple categories, etc.
John Baez (May 22 2023 at 19:39):If anyone has defined fully general weak triple categories, I don't know about it.
John Baez (May 22 2023 at 19:40):Here is an approach to use strict n-tuple categories to model weak n-categories: weakly globular n-fold category. But I don't think this is what Christian wants!
Kevin Arlin (May 22 2023 at 20:15):There's also Christoph Dorn's thesis on associative n-categories. I don't know it well but if I understand correctly it contains a definition of weak triple category.
Matteo Capucci (he/him) (May 22 2023 at 20:51):Christian Williams said:
yes, but that's the strict version, right? I'm looking for the general notion, which includes tricategories.
In Grandis book about Higher dimensional categories he gives the definition of weak and lax multiple categories which afaiu includes tricategories (which are transversally discrete weak triple categories)
Matteo Capucci (he/him) (May 22 2023 at 20:51):Section 6.5
Bryce Clarke (May 22 2023 at 20:58):Looking further at the paper An introduction to multiple categories , Grandis and Paré attribute the weak and lax notions of triple category to their two papers on intercategories.
Christian Williams (May 22 2023 at 21:12):Dorn's thesis looks interesting; I'll look into it, thanks.
Christian Williams (May 22 2023 at 21:19):yes, so look at Marco-Grandis-Higher-Dimensional-Categories.pdf, section 6.4.3:
pent-tri.png
Christian Williams (May 22 2023 at 21:20):when Grandis/Pare say "weak", they mean 2-weak: they have associator and unitor 2-isomorphisms, rather than equivalences, which satisfy a strict pentagon identity and triangle identity.
Christian Williams (May 22 2023 at 21:24):In particular, the unitors being isomorphisms rather than equivalences is where it no longer provides an adequate framework for double category theory: double profunctors have a composition which is only unital up to equivalence, and the component isomorphisms of that equivalence are precisely the "coYoneda" isomorphisms, e.g. for any profunctor we have
.
Christian Williams (May 22 2023 at 21:27):So the Yoneda principle, in the foundation of CT, is in the equivalences of a 3-dimensional kind of composition.
Christian Williams (May 22 2023 at 21:35):That's why it's vital to have a clear and powerful visual language, so we can actually understand this stuff.
The definition of tricategory has been "known" for decades --- meaning there are a few dozen people alive who understand the coherences of a few 4-morphisms:
triangulator.png
Christian Williams (May 22 2023 at 21:38):yet in string diagrams, the above "triangulator" modification is this invertible "rewrite rule":
tri0.png
tri1.png
Christian Williams (May 22 2023 at 21:40):it satisfies a simple equation, where sticking it on either side of an associator has a well-defined reduction.
(will draw it nicely sometime soon)
Nathanael Arkor (May 22 2023 at 21:40):It could be worth emailing Grandis and Paré to ask whether the fully weak version appears in the literature; if it does, I imagine they'd know.
Christian Williams (May 22 2023 at 21:41):that's all it is; and drawing it in string diagrams gives the general notion of a 3-monad, of which the canonical case is a triple category.
Christian Williams (May 22 2023 at 21:54):I really believe anyone can grasp these ideas, intuitively and practically, with the right language.
Matteo Capucci (he/him) (May 24 2023 at 07:09):Christian Williams said:
when Grandis/Pare say "weak", they mean 2-weak: they have associator and unitor 2-isomorphisms, rather than equivalences, which satisfy a strict pentagon identity and triangle identity.
I see!