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nLab > Latest Changes: finite limit

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2010

    added to finite limit the statement that a category has finite limits if it has binary products and equalizers

  2. Can someone add a reference for "Proposition 2.2. A category that has all pullbacks and a terminal object also has all finite limits." that contains a proof of the statement ?
    • CommentRowNumber3.
    • CommentAuthormaxsnew
    • CommentTimeAug 5th 2021
    1. A pullback of a diagram where all morphisms are into the terminal object is a product.
    2. You can construct the equalizer of f,g:X→Yf, g : X \to Y by taking the pullback of (id,f)(id, f) and (id,g):X→X×Y(id, g) : X \to X \times Y

    So it reduces to the products and equalizers case.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 23rd 2021

    I have added a textbook reference for the claim in the entry (this Prop.) that existence of finite limits is implied by existence of finite products+equalizers and from pullbacks+terminal object, namely Prop. 2.8.2 in Borceux Vol. 1.

    But now there seems to be an issue with the entry, which goes on to say (in this remark) something that sounds like it contradicts Borceux’s Prop. 2.8.2: Borceux says that finite limits exist if and only if those other conditions hold, while the remark in the entry makes it sound (with its “More precisely…”) that this is the case not for finite limits but for L-finite limits. (?)

    diff, v16, current

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeAug 24th 2021

    Couldn’t it be the case both for finite limits and for L-finite limits?

    It is true that “more precisely” is not really the appropriate phrase; the proposition is completely precise already, the remark is just adding to it.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2021

    Let’s see. Taken at face value, the remark says that those two conditions (existence of pullbacks etc.) imply not just the existence of finite limits, but of the larger class of L-finite limits. If that’s not in contradiction to Borceux’s claim, then it follows that existence of finite limits already implies existence of L-finite limits. If that ’s the case, then this remark should simply be moved to another spot in the entry as a stand-alone remark on finite limits.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2021
    • (edited Aug 24th 2021)

    I have re-worded that remark on L-finite limits (here) towards what I am guessing it’s saying (but I have no actual idea of what L-finite limits are, so please check and fix as necessary).

    While I was at it, I also expanded that Prop. here a fair bit, adding that functors that preserve any one of the three classes of limits also preserve the other two.

    diff, v17, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2021

    added some basic examples to the entry (here)

    diff, v17, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 25th 2021

    added to that remark a substantiating pointer to Prop. 7 in Paré 1990

    diff, v19, current

1 to 9 of 9

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