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nLab > Latest Changes: well-pointed endofunctor

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  1.  
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2021

    Created page. Probably some of the material at transfinite construction of free algebras ought to be moved here, but I’m too lazy to do it right now.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorvarkor
    • CommentTimeJun 5th 2023

    Link to pointed endofunctor and mention the dual.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorvarkor
    • CommentTimeJun 5th 2023

    Add a reference.

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2023

    added a warning that “pointed endofunctors” are not in general pointed objects in the category of endofunctors

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorvarkor
    • CommentTimeJun 5th 2023

    While they’re not pointed objects in the sense of the current content of pointed object, I’d argue that’s because the definition of pointed object is overly restrictive as it stands. I think the definition on that page ought to be changed (so that a pointing is with respect to a category with a distinguished object). However, it’s a large page, and this seems a slightly daunting task.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2023

    Please see how you feel about the remark that I added a pointed endo-functor: here, which means to deal with this.

    I think if we rename anything, then it’s “pointed endofunctor” that would deserve to be called something like “unital endofunctor”, instead.

    (You might think that there is no “unitality” absent a multiplication, but since one also speaks of algebras over an endofunctor absent of multiplication, it should be just fine.)

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 5th 2023

    We also have the term “pointed abelian group”, referring to abelian groups equipped with an extra element, ℤ→A\mathbb{Z} \to A.

    The term “pointed endofunctor” is awfully well-established. I think varkor has a point (and I’ve been making related comments in the past few minutes, before seeing his). It would be good to think about this.

    • CommentRowNumber8.
    • CommentAuthorvarkor
    • CommentTimeJun 5th 2023

    I think if we rename anything, then it’s “pointed endofunctor” that would deserve to be called something like “unital endofunctor”, instead.

    The term “pointed object” has been used in the literature in the greater generality of objects with a morphism from the unit in a monoidal category (see here, for instance). The terminology I’m suggesting is only a minor generalisation of this usage, and fits with the general principle of the microcosm principle. For what it’s worth, I think “unital objects” rather than “pointed objects” would also be reasonable terminology (and perhaps more consistent terminology) , but “pointed” appears to be much more common in the literature. (I definitely think we ought to avoid having “unital” and “pointed” mean different things.)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2023

    Not to get distracted too much by trivialities, of course no harm is done by adding commentary about generalization to “pointed object” and then maybe corresponding commentary to coslice category. Please go ahead.

    But looking at

    (see here, for instance)

    I find that they don’t say “pointed object” but “MM-pointed object”.

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