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nLab > Latest Changes: horizontal categorification

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  1.  
    • CommentRowNumber1.
    • CommentAuthorEvan Patterson
    • CommentTimeNov 29th 2018

    I’m making a correction about enriched vs internal categories.

    An algebroid is a category enriched in Vect, according to the nLab page on algebroids. Meanwhile, a category internal to Vect is what Baez and Crans call a “2-vector space”.

    diff, v18, current

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeJul 21st 2022

    this article uses query boxes, which is against the recommendation of the “Anything I shouldn’t do?” section of the writing in the nLab article. Should the query boxes be removed?

  3. based upon the advice given here I’m going ahead and removing the query boxes from the articles:

    +–{: .query} David Roberts: How do Lie algebroids fit into this framework?

    Urs Schreiber: at a rough level it is clear that the base space of a Lie algebroid has to be regardedas the “space of objects”. Certainly a Lie algebroid over a point is precisely a Lie algebra.

    But for a more precise statement one needs a more conceptual way to think of Lie algebroids. I am claiming at ∞-Lie algebroid (schreiber) that there is a way to regard Lie algebroids precisely as certain kinds of synthetically smooth groupoids, namely those all whose morphisms have “infinitesimal extension” in some sense. In such a picture Lie algebroids are on the same footing as Lie groupoids and are precisely the many-object version of Lie algebras = infinitesimal Lie groups.

    =–

    +–{: .query} Mike: Is there (and do we want there to be) a general rule about whether an X-oid means an internal category whose one-object version is an X, or an enriched category whose one-object version is an X? The examples above seem to be taking the “internal” side, but the Cafe discussion on “ringoids” was about the “enriched” version; the two are very different! And “dg-algebroid” was suggested for dg-category, which is an enriched oidification of a dg-algebra, but a Hopf algebroid is an internal oidification of a Hopf algebra.

    Urs: good point. We should say this at the beginning and split the list of examples in two sorts =–

    Anonymous

    diff, v25, current

    • CommentRowNumber4.
    • CommentAuthorGuest
    • CommentTimeJul 25th 2022

    Does horizontal categorification yield strict categories or univalent categories?

    • CommentRowNumber5.
    • CommentAuthormaxsnew
    • CommentTimeJan 11th 2023

    Mention the analogy to untyped/typed languages.

    diff, v28, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2023

    have hyperlinked the main keywords in your paragraph (here)

    diff, v29, current

    • CommentRowNumber7.
    • CommentAuthorvarkor
    • CommentTimeFeb 7th 2023

    Added the example of monads.

    diff, v30, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 7th 2023

    Applied the pedantic fix which I always apply in this situation (not picking at you, specifically, but hoping my appeal will be heard eventually):

    Namely, speaking of

    categorification of a monad

    does not type-check, does it.

    I have changed it to

    the categorification of the notion of monads

    I’d be fine with abbreviating this to

    the categorification of monads

    as ordinary people do; but among category/type-theorists we might as well say it properly. :-)

    diff, v31, current

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