Timeline for Adjunctions in a weak $2$-category
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Oct 14, 2019 at 16:56 | comment | added | Mike Shulman | @AlecRhea In general lax functors, even between bicategories, are quite poorly behaved: golem.ph.utexas.edu/category/2009/12/… . So I'm skeptical that you can get anything sort of "strictification-style" coherence theorem. There is a "classifying object" coherence theorem for lax morphisms in 2-monad theory (see e.g. Steve Lack's paper "Codescent objects and coherence"), but it doesn't yield a strictification theorem. | |
| Oct 14, 2019 at 16:53 | comment | added | Mike Shulman | @TimCampion The usual term for "balanced" is "unbiased" (ncatlab.org/nlab/show/biased+definition). | |
| Oct 8, 2019 at 12:18 | comment | added | Todd Trimble | Roughly speaking, my understanding is that if you have two $n$-dimensional structures (meaning that the top dimension of non-identity cells in each is $n$), then under an equivalence, a pair of mutually inverse $n$-cells in a local hom-category in one of the structures will get mapped to a pair of mutually inverse $n$-cells in a local hom-category of the other. The effects of this propagate down to lower-dimensional cells that participate in higher equivalences. | |
| Oct 8, 2019 at 4:09 | comment | added | Alec Rhea | @TimCampion The references are appreciated; I originally asked this question because Gurski uses the theory of mates in bicategories for his coherence theorem, and the starting point is adjunctions in a bicategory. Since he's working with invertible unitors/associators I'm good to go, but any additional reading is great. | |
| Oct 8, 2019 at 4:01 | comment | added | Alec Rhea | @ToddTrimble You're right about contacting Nick, thanks for the suggestion. What precisely would be an objection to an equivalence under Yoneda? Assuming we define an equivalence as a pair of lax functors with 'lax natural isomorphisms', it would essentially be saying that 'lax constraints' are only restraints superficially unless I'm mistaken, telling us that 'apparently lax' structures 'may as well be strict' at the $2$-dimensional level. | |
| Oct 8, 2019 at 3:50 | comment | added | Alec Rhea | @KevinCarlson That's interesting, I'm trying to write down the details of the fully weak version now to see how things shake out; the insight is much appreciated. | |
| Oct 7, 2019 at 20:35 | comment | added | Kevin Carlson | @TimCampion Yeah, Leinster's book I mentioned above also focuses on balanced notions. | |
| Oct 7, 2019 at 11:56 | comment | added | Tim Campion | This "balanced" notion comes up in Higher Algebra: Definition 6.2.5.1 of a "corepresentable $\infty$-operad" is what I would call a "balanced-lax symmetric monoidal $\infty$-category". So e.g. there is a map $\otimes(x_1,x_2,x_3) \to \otimes(\otimes(x_1,x_2),x_3)$ and another $\otimes(x_1,x_2,x_3) \to \otimes(x_1,\otimes(x_2,x_3))$, as opposed to a skew monoidale where there's a map $ \otimes(\otimes(x_1,x_2),x_3) \to \otimes(x_1,\otimes(x_2,x_3))$ (or maybe the other way around). | |
| Oct 7, 2019 at 11:55 | comment | added | Tim Campion | If you're really interested in some lax version of adjunctions, then adjunctions up to adjunctions, thought not exactly what you're talking about, seem worth mentioning. I haven't thought about these lax bicategories in full generality, but I've found skew monoidales a bit puzzling -- surely the "natural" notion of a lax monoidal category would be "balanced", with an $n$-ary operation $\otimes(x_1, \dots, x_n)$ for each $n$, and associator maps from various associated things out of this? | |
| Oct 5, 2019 at 17:12 | comment | added | Todd Trimble | I don't see a problem considering a construction like $yB = \text{LaxBiCat}(B, Cat)$, but the exact features would be something to work out. I'll bet Nick Gurski could tell you more. But I do have a hard time believing that a notion of equivalence by which lax constraints are rendered 'pseudo' would be deserving of the name 'equivalence'. | |
| Oct 5, 2019 at 15:31 | comment | added | Kevin Carlson | @AlecRhea I am guessing some result like this works, based on general 2-monad theory. But the notion of "equivalence" would itself have to be made more lax than the usual one. I know these lax bicategories are introduced in Leinster's book, so there does exist at least one reference... | |
| Oct 5, 2019 at 14:45 | history | edited | Alec Rhea | CC BY-SA 4.0 |
Fix error pointed out by Todd and Kevin
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| Oct 5, 2019 at 14:42 | comment | added | Alec Rhea | @ToddTrimble I stand corrected, he seems to be working with bicategories in the sense you're referring to and I have no reference besides this. I'll edit the question accordingly, thanks for the heads up. Incidentally, do you see any reason why Yoneda shouldn't hold for fully weak $2$-categories in the above sense? | |
| Oct 5, 2019 at 13:47 | comment | added | Alec Rhea | @ToddTrimble I'm working my way through "Coherence in Three Dimensional Category Theory" by Nick Gurski, and in the "Bicategorical Conventions" section in chapter one uses fully weak $2$-categories in the above sense. | |
| Oct 5, 2019 at 12:31 | comment | added | Todd Trimble | In the argot I know, a weak $n$-category refers to the sequence set, category, bicategory, tricategory, etc., and so a weak 2-category would be a bicategory (although it may be rare these days to hear it referred to as such). There may be lax bicategories, where the one-object case is sometimes called a skew monoidal category. But I thought the Yoneda lemma applied to bicategories (?). Do you have a literature reference for your last paragraph? | |
| Oct 5, 2019 at 7:28 | comment | added | Alec Rhea | @KevinCarlson I thought it was standard knowledge, my apologies. For any interested readers, the equivalence is still given by the $2$-Yoneda embedding of a weak $2$-category into its $2$-presheaf $2$-category, which is strict since the codomain of $2$-presheaves is the $2$-category of categories which is strict and composition etc. is defined pointwise in the codomain for $2$-functor $2$-categories. | |
| Oct 5, 2019 at 6:46 | comment | added | Kevin Carlson | Are you well aware that this "fully weak" or lax notion is quite rarely studied compared to the invertible unitor/associator case? You mention the equivalence with strict 2-categories as if it were very well known, but I think it's rather rarely considered in this generality. | |
| Oct 5, 2019 at 4:48 | history | asked | Alec Rhea | CC BY-SA 4.0 |