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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

  9. There are a number of useful examples of horizontal categorification, but I think the page needs some pointers to a general theory of when constructions can be horizontally categorified.

    Patrick Nicodemus

    diff, v33, current

  10. The page contains this comment:
    > It has rightly been remarked that groupoids are more fundamental than groups, algebroids are more fundamental than algebras, etc. Hence in a better world, the suffix would be characterizing the one-object special cases, not the general concepts.
    I'd like to challenge this claim. In my view, groupoids are not more fundamental than groups. I imagine that the argument is something like "the setting of groupoids is a more general setting than the theory of groups in which the same signature is valid", i.e., one can write $f \cdot (g \cdot h) = (f\cdot g)\cdot h$, $id \cdot f = f$, $f \cdot (f^{-1}) = id$, and so on.
    I have a hard time accepting the argument that a strictly complex structure can be more fundamental.
    I imagine that one could iterate this argument: if groupoids are more fundamental than groups, then the horizontal categorifications of groupoids (2-groupoids?) are more fundamental than groupoids, and so on, so in the limit, infinity-groupoids are the most fundamental thing of all. Indeed such a perspective would be consistent with the general outlook of this forum, so maybe some people here really believe that, but I don't find it plausible.

    Another counterargument: I claim that Abelian group theory is a part of group theory. A study of Abelian groups is an important part of group theory and its applications to field theory, every textbook on group theory treats the theory of Abelian groups. Similarly with commutative ring theory and commutative algebra, every modern book on algebra treats commutative algebra. But there is not a natural generalization of the commutativity condition to groupoids or algebroids. Therefore I argue that the theory of groupoids does not realistically subsume the theory of groups, nor algebroid theory the theory of algebras, because one cannot appropriately generalize essential parts of these theories, and so the theory of groupoids is not strictly more fundamental than that of groups.
    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2025
    • (edited Jan 15th 2025)

    Thanks for bringing this up.

    Note that this is an ancient entry which has not received real attention in a couple of decades. It would deserve to be edited more.

    Looking specifically at the paragraph you are referring to —- “It has rightly been remarked… In a better world…” (from revision 4 in 2008) — I agree that it’s not useful.

    A quick technical observation to make on this point is that the inclusion of groups into groupoids, namely as the one-object delooping groupoids, is not 2-faithful and in this sense is not a generalization (only the inclusion into pointed groupoids is, as discussed at looping and delooping).

    In any case, I have deleted that paragraph now.

    diff, v34, current

  12. Good idea to remove the paragraph, but just to point out that neither of the counter-arguments quite stand up. Picard groupoids are a quite exact analogue for groupoids to what abelian groups are for groups; and there is no horizontal categorification of groupoids as such (in particular, 2-groupoids are a vertical categorification of groupoids, not a horizontal one).

    anonymous poster

    diff, v35, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2025

    Just to say that Picard groupoids are pointed. In fact they carry considerable extra structure (being abelian 2-groups, really).

  14. One could point out that the previous poster’s counter-argument does not quite stand-up in a simpler way by just observing that one can take coproducts in the category of groupoids equivalent to abelian groups (viewed as one-object groups); note that these are not the same as groupoids enriched over abelian groups. This is as exact a direct analogy for abelian groups vs groups for groupoids as one could wish for!

    But if defining horizontal categorification as something whose one-object version recovers what one is categorifying, then it is also true that Picard groupoids horizontally categorify abelian groups, and Picard groupoids play the right role structurally in the vertically categorified setting that abelian groups do for sets. This kind of dichotomy between horizontal and vertical is very typical for situations where internalisation is relevant (groups internal to groups are abelian groups, and 2-groups are groups internal to groupoids, …).

    anonymous poster

    diff, v36, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2025

    As I said:

    Picard groupoids are abelian 2-groups (groupoids equipped with abelian group structure), and as such they clearly generalize abelian groups.

    In particular, Picard groupoids are pointed groupoids, and the inclusion of groups (abelian or not) into pointed groupoids is 2-faithful and hence qualifies as a “generalization”.

    What is not 2-faithful is the inclusion of groups into general groupoids, which is one way to make precise the statement that, without further qualification, groupoids are not a generalization of groups.

    I have now made this a remark in the entry (here).

    In the course of this, I also touched wording and formatting throughout the Examples-section of the entry.

    diff, v37, current

  16. I don’t pointedness really is of any significance here; of course one cannot hope for fully faithfulness in the situation described, given that one is ignoring the 2-categorical aspect (conjugations) in the domain! If one regards groups as assembling into a 2-category whose 2-arrows are conjugations (i.e. a 2-arrow from a group homomorphism ff to another one f′f' is given by a hh such that h −1fh=f′h^{-1} f h = f'), i.e. just regarding groups as a full sub-2-category of groupoids in the same way as monoids are a full sub 2-category of categories, then of course one gets the fully faithfulness; no pointedness involved anywhere. It is true as you say that one can impose pointedness and observe that one obtains fully faithfulness then too, but the very simple observations I was making in reply to the original poster are not at all dependent on this point of view (in particular, not dependent upon regarding Picard groupoids as pointed). My previous comment was meant simply to clarify why I was regarding Picard groupoids as appropriate to consider in the context of horizontal categorification, when as you pointed out there is also an aspect of vertical categorificaton.

    anonymous poster

    diff, v39, current

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