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Asad Saeeduddin (May 19 2020 at 18:55):Is there a name for lax monoidal functors whose laxity is not a full isomorphism, but is a section/retraction?
John Baez (May 19 2020 at 19:01):No. :upside_down:
John Baez (May 19 2020 at 19:01):Probably yes: probably someone has given those a name. I'm just saying I don't know it.
John Baez (May 19 2020 at 19:06):It sounds like a cool concept. In your examples, if the laxator is a section, can you choose the corresponding retraction to make the functor oplax monoidal?
John Baez (May 19 2020 at 19:06):(I say "laxator" instead of "laxity", since it does something, like an associator or unitor.)
Jules Hedges (May 19 2020 at 19:08):At some point I found a rabbit hole about monoidal functors that are lax and colax where the laxator and colaxator interact in some ways other than being inverses. There was a whole book about it.... don't know if I could find it again though
Jules Hedges (May 19 2020 at 19:11):The thing I'm thinking of is "bilax monoidal functors" and "frobenius monoidal functors" and the book is "monoidal functors, species and hopf algebras" by Aguinar and Mahajan. No idea whether that helps or not
Asad Saeeduddin (May 19 2020 at 19:12):awesome, thank you. i'll check it out
John Baez (May 19 2020 at 19:13):Wow, cool! I first ran into monoidal functors that are lax and oplax but not strong when I was thinking about the functor that turns a simplicial abelian group into a chain complex (its normalized chain complex). Now I see that that oplaxator has a left inverse, so this is an example of the situation Asad is talking about:
https://ncatlab.org/nlab/show/Eilenberg-Zilber+map
John Baez (May 19 2020 at 19:14):Oh yeah, that massive free book Monoidal Functors, Species and Hopf Algebras has a lot about "bilax" monoidal functors. It's a great book if you like combinatorics (and species). It's huge because it actually explains things.
Asad Saeeduddin (May 19 2020 at 19:15):@John Baez I'm working in the category of sets for modeling some programming concepts. It turns out that every endofunctor oplax monoidally coheres the product structure, but only some are lax monoidal. Fewer still are strong monoidal, but the margin by which there are fewer strong monoidal functors than lax monoidal functors is pretty huge. There seems to be a useful middle ground where many monoidal functors have the property of the laxator being a retraction of the oplaxator, but not vice versa.
Asad Saeeduddin (May 19 2020 at 19:25):I suspect that there is a category of strong monoidal endofunctors equipped with a left distributive law over an arbitrary monoidal functor , and that the functor composition structure on this category (unlike on the regular endofunctor category) is symmetric or at least braided
Asad Saeeduddin (May 19 2020 at 19:26):what I'm trying to figure out is whether this works with merely "retractable" monoidal functors instead of needing strong monoidal functors
John Baez (May 19 2020 at 19:27):Sounds fun!
Asad Saeeduddin (May 19 2020 at 19:29):it is! :)