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    • CommentRowNumber1.
    • CommentAuthorPieter
    • CommentTimeFeb 2nd 2019

    I’ve noticed that the page on projection does not mention the more generic definition of projection used on wikipedia: https://en.wikipedia.org/wiki/Projection_(mathematics)

    Is there a particular reason for this?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 2nd 2019

    It’s mentioned in the “linear algebra” section, in the case of vector spaces. I’ve never heard that terminology used outside of linear algebra, and since the nLab is mainly about category theory the categorical meaning is more relevant. However, we could certainly mention the other meaning.

    • CommentRowNumber3.
    • CommentAuthorPieter
    • CommentTimeFeb 3rd 2019

    Hm… I don’t recognize what I mean in the linear algebra section.

    On wikipedia it says that a map f:XYf : X \rightarrow Y is a projection if for any ZZ and map g:ZYg : Z \rightarrow Y there exists a map h:ZXh : Z \rightarrow X such that fh=gf h = g.

    That sounds like a categorical definition to me, and seems to have some meaning on its own. Does it go by different names as well, perhaps?

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 3rd 2019

    That looks like a split epimorphism by taking Z=YZ=Y and g=id Yg=id_Y.

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 3rd 2019
    • (edited Feb 3rd 2019)

    (Duplicate)

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 3rd 2019

    I was looking at the first line of https://en.wikipedia.org/wiki/Projection_(mathematics)

    In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition (or, in other words, which is idempotent)

    I don’t see the definition you mention in #3.

    • CommentRowNumber7.
    • CommentAuthorPieter
    • CommentTimeFeb 3rd 2019

    I was looking at the commutativity diagram a bit lower… but I now realize that the definition next to it is a bit more extensive than just this universal property.

    There seems to be a relation with the notion of retract.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 3rd 2019

    Right, a splitting of an idempotent is an split epimorphism, i.e. a retraction, and any split epimorphism is a splitting of the idempotent induced on its domain.

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