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nLab > Latest Changes: differential category

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  1.  
    • CommentRowNumber1.
    • CommentAuthorTim_Porter
    • CommentTimeJul 12th 2021
    • (edited Jul 12th 2021)

    Just a stub for the moment to try to introduce the notion of differential category due to Blute, Cockett and Seely.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2021

    have polished it up a little.

    (Why the extra effort to hide the full names of the authors? Author identification is and will increasingly be an issue on the web. We do a favor to authors and their legacy if we name them as uniquely as possible.)

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorTim_Porter
    • CommentTimeJul 12th 2021

    Full names are often not (in fact ore often than not) given in the publication data so a copy and paste of a reference ends up without the full names.(Of course, ideally one can add in fuller form later on.) I do not understand why in papers the full name is usually not used.

    I wholeheartedly agree about ‘legacy’.

    In fact it is not extra work to hide those names, it is extra work for find full names. :-)

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2021

    Yes, we need not follow the weird traditions in print publication of tending to make following citations a detective story – such as not identifying authors completely, not giving titles (standard in parts of physics!) and (nowadays) not giving url-s. Let’s provide all this data!

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeJul 12th 2021
    • (edited Jul 12th 2021)

    I agree … where we can find it!!!!! It took mequite some time to find Laurent Regier’s homepage. A google search gave lots of derivative pages, (Researchgate, etc.) but the Google weighting of his personal web page must have been low, at least that is what I presume.

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeJul 12th 2021

    Added a reference to a 2020 paper.

  7. I’ve added the complete definition of a (co)differential category.

    Jean-Baptiste Vienney

    diff, v5, current

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 25th 2022
    • (edited Jul 25th 2022)

    Thanks.

    I added a couple of links: chain rule and Leibniz rule. The latter redirects to Leibniz algebra, not so clear the connection to the rule. [EDIT: But there is equation (1) there]

    It would be good to have a few words on why those diagrams do what they say,e.g., chain rule.

    And we ought to have first a definition of ’differential category’.

    diff, v6, current

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 25th 2022

    Also, is there a reference for codifferential categories?

  10. Explanation of the differences between differential and codifferential categories. Example of Vect 𝕂Vect_{\mathbb{K}} and formal polynomials. I’ve explained what the diagrams mean in the definition of the derving transformation.

    For the forum: In the references, the notion of differential category is used and they say for the specific examples that this category 𝒞\mathcal{C} is such that 𝒞 op\mathcal{C}^{op} is a differential category, ie. it is a codifferential category rather than giving directly the definition of a codifferential category.

    However in a recent paper, Jean-Simon have given directly the definition of a codifferential category (“Why FHilb is Not an Interesting (Co)Differential Category” and “”). At FMCS 2022, Sacha Ikonicoff used the codifferential setting with his presentation titled “Cartesian Differential Monads” (https://pages.cpsc.ucalgary.ca/~robin/FMCS/FMCS2022/slides/Sacha.pdf), however Jean-Simon wrote the definition of a cartesian codifferential monad in the paper (https://arxiv.org/pdf/2108.04304.pdf), this is another structure than differential categories but this is the same problem.

    Codifferential definitions are more intuitive for mathematicians because monads and monoids are more natural for us than comonads and comonoids but linear logic and differential linear logic are based on the differential definition. This is why Sacha like me, we prefer the codifferential definition but Jean-Simon which has been the main contributor during the preceding years want to stay close to linear logic.

    A lot of categories are at the same time differential categories and codifferential categories, for example the **-autonomous differential categories, and this fact seems to be linked to distributions. I’m starting to disuss of this with Jean-Simon and Marie Kerjean who is more into differential linear logic and concrete models. I hope to come back with theorems to write on the nlab. Definitely differential categories are still under development.

    Jean-Baptiste Vienney

    diff, v7, current

  11. Example of differentiation in FVec 𝕂FVec_{\mathbb{K}} as a codifferential category.

    Jean-Baptiste Vienney

    diff, v7, current

    • CommentRowNumber12.
    • CommentAuthorJ-B Vienney
    • CommentTimeJul 25th 2022

    Two references using the codifferential definition added.

    diff, v9, current

    • CommentRowNumber13.
    • CommentAuthorJ-B Vienney
    • CommentTimeJul 26th 2022

    I’ve replaced the notation !! by SS for the monad. Using !! is confusing as it is a comonad in linear logic. Thus, SS corresponds more to the ?? of linear logic.

    diff, v10, current

    • CommentRowNumber14.
    • CommentAuthorJ-B Vienney
    • CommentTimeJul 28th 2022

    Just added the hyperlink to symmetric algebra.

    diff, v11, current

    • CommentRowNumber15.
    • CommentAuthorJ-B Vienney
    • CommentTimeJul 28th 2022

    Commentary to explain why the deriving transformation is at it is in relation with the unpublished notion of graded codifferential category.

    diff, v11, current

    • CommentRowNumber16.
    • CommentAuthorJ-B Vienney
    • CommentTimeJul 28th 2022

    Added hyperlink to **-autonmous category.

    diff, v12, current

    • CommentRowNumber17.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 20th 2022

    Added related page link to tangent bundle category, which I think is more appropriate than tangent category.

    diff, v15, current

    • CommentRowNumber18.
    • CommentAuthorJ-B Vienney
    • CommentTimeNov 19th 2022

    Made two separate sections for differential categories and codifferential categories. The section for differential categories is to be completed.

    diff, v16, current

    • CommentRowNumber19.
    • CommentAuthorJ-B Vienney
    • CommentTimeNov 19th 2022

    Made a section “Differential category” and moved the facts about codifferential categories in a section “Codifferential categoy”. The section “Differential category” is to be completed.

    diff, v16, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeNov 20th 2022

    I have touched the wording in the first four sentences of “Codifferential categories – Definition” (here). Please check if you agree.

    diff, v17, current

    • CommentRowNumber21.
    • CommentAuthorJ-B Vienney
    • CommentTimeNov 20th 2022

    I agree. I find it better after your edit.

    • CommentRowNumber22.
    • CommentAuthorJ-B Vienney
    • CommentTimeNov 20th 2022

    Added that the free C ∞C^\infty-ring on ℝ\mathbb{R}-vector space provides a codifferential category. Reference + link to the entry on C ∞C^\infty-ring + more to add.

    diff, v18, current

  23. Fixed typo: “\S(A)” -> “S(A)”

    pnips

    diff, v20, current

    • CommentRowNumber24.
    • CommentAuthorJ-B Vienney
    • CommentTimeMar 11th 2023

    Added some intuitions for differential categories.

    diff, v21, current

  25. The natural transformations giving comonoid structures in the first definition were backwards, so I made them go the correct direction.

    Jordan Sawdy

    diff, v24, current

1 to 25 of 25

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