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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 5th 2010

    The minute before I had entered offline territory a few days ago, I had expanded the list of examples of (commuting) diagrams at diagram.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeApr 6th 2010

    I added a few more, and also improved (I hope) the introduction.

    • CommentRowNumber3.
    • CommentAuthorEric
    • CommentTimeApr 6th 2010
    • (edited Apr 6th 2010)

    Edit: Changed this comment to a question.

    Is Quiv on diagram the category of free categories on graphs or is it the same thing as DiGraph as defined on directed graph?

    • CommentRowNumber4.
    • CommentAuthorHarry Gindi
    • CommentTimeApr 6th 2010
    I just checked the page, and it means the non-John Baez version of quiver.
    • CommentRowNumber5.
    • CommentAuthorEric
    • CommentTimeApr 6th 2010

    Thanks Harry. That's what I thought. So Quiv and DiGraph mean the same thing.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 31st 2017

    We had the definitions of limit and colimit as terminal and initial (co)cones round the wrong way, so have corrected at diagram.

    • CommentRowNumber7.
    • CommentAuthorwillymox
    • CommentTimeJun 12th 2020

    Hi I’m new here. First of all thanks for this great resource you all created. I think there might be a mistake in the component definition of a diagram here. Particularly in the definitions of limiting cone (limit) and limiting co-cone (co-limit). I think it should be stated that the limiting cone is initial among all possible cones and the limiting co-cone is terminal among all possible co-cone. Not the opposite. I might be wrong here, I’m just a hobbyist category theorist ;) Did I misunderstand something ?

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 13th 2020

    No, it’s right as it is, but I think I can understand why some might find it confusing.

    Take a simple example, where diagrams are over a discrete category with just two objects. If the diagram consists of objects X,YX, Y, then a general cone looks like

    A f g X Y\array{ A \\ \mathllap{f} \downarrow & \searrow \mathrlap{g} & \\ X & & Y }

    The limiting cone is the product X×YX \times Y together with its product projections:

    X×Y π 1 π 2 X Y\array{ & & X \times Y \\ & \mathllap{\pi_1} \swarrow & \downarrow \mathrlap{\pi_2} \\ X & & Y }

    and for any cone as in the first diagram, there is a unique map of cones to the product cone, given by a map

    A(f,g)X×Y.A \stackrel{(f, g)}{\to} X \times Y.

    Thus the product cone is terminal among all cones: terminal means that for any object there exists a unique map to the terminal.

    • CommentRowNumber9.
    • CommentAuthorwillymox
    • CommentTimeJun 13th 2020
    • (edited Jun 13th 2020)

    So we agree, the definition should say it’s terminal, not initial like it is now : over this diagram (def. 2.3) which is universal or initial among all possible cones, in that it ... <– This is from the definition of a limiting cone in definition 2.4, it should say terminal instead of initial. Same for limiting co-cone.

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 13th 2020

    I see; I was looking at a different part of the page where it was correct. You’re right.

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 13th 2020

    Fixed definition 2.4.

    diff, v34, current