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I don’t want to touch this entry “biactegory”, but in case anyone cares I’d suggest to:
fix and streamline the language in the first two paragraphs.
add some minimum of formatting.
add the missing hyperlink to bimodule category.
Is there more than one reference using the term “biactegory”? If not, it might be better to absorb this one pointer into “bimodule category”.
Capucci and Gavranović have some new material in their review https://arxiv.org/abs/2203.16351 under the name biactegories as well (see abstract).
Urs, the subject is the same so I can do the merger under the name bimodule category if you like. However the entry biactegory is from 2010 and bimodule category started in 2014 so I would rather move the newer material into older entry and then rename if you agree ? I think that it is better to hold the page line with longer Lab history. Lab is a pioneering platform and the entry history lines should reflect that whenever possible. 2008 when the Lab started is for the subject higher categories like early history and now is the minstream. For now I will just add Capucci’s reference and then merge this way if you agree.
The definition of the notion is mentioned (I do not know under which name) also by Yetter, much earlier than my 2006 preprint “Biactegories” and Justin’s thesis work. The fact that they form a tricategory is in Justin’s work and stated as a theorem without proof in my Georg. Math. J. 2009 contribution, but not in 2006 Biactegories preprint (where some steps toward are spelled out). It was too cumbersome (huge amount of checking categorical identities) and Justin’s approach is different but it seems shorter to me to achieve the goal.
I wrote a recommendation for Justin (based on our correspondence etc. after we learned of each other’s work) when he was applying to some postdoc and later heard that he left math after that, and your link at your new page contribution Justin Greenough makes me happy to see that he is still doing academic work/teaching.
Entry bimodule at present does everything in terms of rings that is bilinear version, rather than two-sided action of monoids. In that sense, biactegory is not a categorification (as no linear/additive monoidal categories and additive functors are assumed, in the game). Do we want to have the term bimodule categories linear/additive or not (as Etingof et al use the term) ? If not the entries could be merged otherwise it would be wrong. There is also a term 2-birepresentation category.
Do we want to have the term bimodule categories linear/additive or not (as Etingof et al use the term) ?
It seems sensible to me not to differentiate the two, and simply mention that the definition makes sense in an enriched context.
I agree. We should emphasize that we do not mean ring/algebra theoretic/linear version of module. I think it is not only enrichment, the additional conditions like exactness are taken by linear people. When I joined in 2004 the actegory terminology it was because there were three big groups of people at the time. One were who were using -category meaning -action category rather than more standard -enriched category what was a turn off. Second was talking actegory (Munich and Australia). The third was talking module but in all their talks and examples they were linear etc. So, only the second group had no problems with conotations.
I think it is not only enrichment, the additional conditions like exactness are taken by linear people.
Ah, I see. Plausibly it is a “pseudobimodule” in a 2-category of additive categories instead.
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