You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Graham Manuell (Apr 26 2024 at 08:03):On the nlab page https://ncatlab.org/nlab/show/coherence+theorem#relating_coherence_theorems there is the claim:
One thing to beware of is that even for structures whose free-algebra coherence theorem is of the form “all diagrams commute,” it does not necessarily follow that all such algebras can be fully strictified.
Does anyone have an example where this fails?
Matteo Capucci (he/him) (Apr 26 2024 at 09:26):Maybe this: https://arxiv.org/abs/1005.1520 ?
Amar Hadzihasanovic (Apr 26 2024 at 09:30):I thought of it but I'm not sure in what sense the semistrict 3-categories used as a counterexample here would "have a coherence theorem of the form 'all diagrams commute'".
Graham Manuell (Apr 26 2024 at 09:34):I wouldn't have thought all diagrams for tricategories commute, since it's not true for braided monoidal categories, but it's possible I'm missing some subtlety.
Nathanael Arkor (Apr 26 2024 at 09:45):It seems that comment was added by @Mike Shulman in revision 4: perhaps he had an example in mind he could share.
Rémy Tuyéras (Apr 26 2024 at 09:48):Graham Manuell said:
On the nlab page https://ncatlab.org/nlab/show/coherence+theorem#relating_coherence_theorems there is the claim:
One thing to beware of is that even for structures whose free-algebra coherence theorem is of the form “all diagrams commute,” it does not necessarily follow that all such algebras can be fully strictified.
Does anyone have an example where this fails?
Is this not referring to the fact that tricategories do not strictify to strict-3-categories (the fully strictified version) but instead Gray-categories?
EDIT: as said in the paragraph above in the nlab page
EDIT 2: a construction that may be relevant to investigate here is in a paper from Dimitri Ara (https://arxiv.org/abs/1206.2941) and more specifically the construction in Remark 4.7
Philip Saville (Apr 26 2024 at 11:19):Graham Manuell said:
I wouldn't have thought all diagrams for tricategories commute, since it's not true for braided monoidal categories, but it's possible I'm missing some subtlety.
In his Coherence in three-dimensional category theory book Gurski proves an 'all diagrams commute' statement for tricategories by showing that certain freely-generated tricategories are triequivalent to 3-categories (Sec 10.1). The subtlety is what you're allowed to be free on: Gurski also shows that if you have non-equal endo-1-cells in a monoidal bicategory then you can construct a diagram of structural 2-cells that coherence does not force to commute (Sec 10.3). So the 'all' diagrams only applies to those where you've started with some basic objects and 1-cells, then freely added the monoidal structure after
Mike Shulman (Apr 26 2024 at 15:41):I think tricategories are what I had in mind, with "free" meaning free on a higher graph, not necessarily a computad.