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To the entry on [[regular category]] I added a brief note describing an application of this idea and the calculus of relations to a paper of Knop. For the future I will try to flesh this note out as well as add a page on tensor categories.
By the way, does the definition of a tensor category have to include linearity? It seems that the definitions vary depending on where one looks (e.g. whether the monoidal structure is an additive functor). Thanks.
I guess a question I would have is: why focus on this particular article for this page? There are many, many articles which deal with regular categories and their applications.
Yes, I think different authors have meant different things by "tensor category", and I personally feel it's bad and confusing terminology. For what it's worth, I've most often seen linearity included by people who choose to use the term. If I were writing the page, I'd say something like "tensor category can mean any of the following things" followed by disambiguation which links to other pages with better (or less confusing) terms.
I used to be surrounded by people from a community who said "tensor category". They all clearly started out being motivated from the Vect-enriched examples, but later some turned to more general monoidal categories and some bad terminology tended to appear then.
So I would second Todd here: let's use "monoidal category" as the default and have a remark that "tensor category" may mean different things to different people.
“I guess a question I would have is: why focus on this particular article for this page? There are many, many articles which deal with regular categories and their applications.”
Well, it was the only one I was familiar with, as of yesterday ;). More seriously, the article seemed to be well-written and surprisingly accessible (for a math paper) while also proving some interesting results. But there is room on this page (or others) for links to and descriptions of many more articles, no?
I agree that it's a nice paper, one of interest to algebraic geometers particularly. And I think I agree with you that there is room for mention of more articles. My only real concern is that we don't miss the forest for the trees (the article is one "tree"), so stylistically it might be a good idea to put it under a section "Related articles" or something and follow up with succinct descriptions of the articles mentioned, with links of course. Among such articles we might also include where the notion was first defined (which I don't happen to know, funnily enough -- was it Max Kelly who first gave it? Freyd, Barr?).
I don't want to seem like I'm imposing rules, by the way -- it's just my opinion. Of course, that's what this forum is for: to give our opinions and hopefully reach a consensus.
I’ve edited the article again to include such a section and including other references too. Unfortunately I haven’t read them and can’t write any commentary (beyond what Zoran indicated) on them (yet, anyway). I’ll also do the disambiguation article on tensor categories shortly-thanks for the clarification there!
I wrote tensor category as a disambiguation page. Please check that I've properly described the variability; in particular, where I've written ‘(symmetric)’, it's possible that both possibilities occur but also possible that either the parentheses or the entire modifier should be removed.
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