Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories newpage nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes science set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthoramathew
    • CommentTimeOct 21st 2009

    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.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 21st 2009

    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.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2009

    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.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeOct 21st 2009
    Tensor category means different thing to different people. I would define it as a symmetric monoidal category with standard properties of the setup in the subfield using the term. For majority of things it includes being additive in the compatible way. In addition it may be k-linear in compatible way or strengthen additive to abelian. Very rarely tensor means just monoidal (not even symmetric). Some people use quasitensor category for any braided monoidal category.
    • CommentRowNumber5.
    • CommentAuthoramathew
    • CommentTimeOct 21st 2009

    “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?

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 21st 2009
    • (edited Oct 21st 2009)

    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.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeOct 21st 2009
    • (edited Oct 21st 2009)
    I agree with Todd. Knop has a very particular point of view and it is better if the wider discussion on it be on separate entry/link. One should add the main references (historically regular categories appeared in two articles simultaneously, one by Barr and other by Grillet in the same volume of LNM (for the history see the intro to 1977 Johnstone's book); later the concept was a bit cleaned. There are several monographs which have nice exposition/chapter on the subject.)
    • CommentRowNumber8.
    • CommentAuthoramathew
    • CommentTimeOct 22nd 2009

    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!

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeOct 22nd 2009

    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.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)