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nLab > Latest Changes: pointed endofunctor

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  1. A pointed endofunctor is not pointed object in the endofunctor category, because the terminal object of the category of endofunctors on C is the functor which sends all objects to 1 and all morphisms to the unique morphism 1→1, where 1 is terminal object of category C.

    xiaoniu

    diff, v2, current

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJun 21st 2018

    Hmm… that’s true, given that our page pointed object is all about objects pointed by a map from the terminal object. But there’s certainly a sense in which an object equipped with a map from the unit object is also “pointed”. Maybe “monoidally pointed”?

  3. added definition of pointed endomorphism in a 2-category

    diff, v3, current

    • CommentRowNumber4.
    • CommentAuthorvarkor
    • CommentTimeJun 5th 2023

    Add a reference.

    diff, v9, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2023

    have touched wording and formatting of the first few paragraphs.

    Added the warning that the notion of pointed endofunctors is not in general a special case of the notion of pointed objects.

    diff, v10, current

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 5th 2023
    • (edited Jun 5th 2023)

    I made a related comment in the discussion page for pointed object, that there are more general notions of pointing out there, involving maps out of the monoidal unit, which specialize to maps out of the terminal object when interpreted in the doctrine of cartesian monoidal categories. The article pointed object is essentially about that specialization (even though you don’t need all finite products to state the definition).

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2023

    added pointer to:

    • Harvey Wolff, p. 234 of: Free monads and the orthogonal subcategory problem, Journal of Pure and Applied Algebra 13 3 (1978) 233-242 [doi:10.1016/0022-4049(78)90010-5]

    for an original use of the terminology

    (incidentally, another good example for why it’s a bad idea to denote terminal objects by “11”)

    diff, v11, current

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 5th 2023

    Changed the wording a bit, so as to link more smoothly with the new article pointed object in a monoidal category.

    diff, v12, current

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