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    • CommentRowNumber1.
    • CommentAuthorFinnLawler
    • CommentTimeAug 8th 2010

    Added a proof of the pasting lemma to pullback, and the corresponding lemma to comma object (also added the construction by pullbacks and cotensors there).

    • CommentRowNumber2.
    • CommentAuthorMarc
    • CommentTimeAug 10th 2010
    Actually the statement of your proposition is wrong. If the right square in

    A --> B --> C
    | | |
    v v v
    D --> E --> F

    is a pullback then the two statements

    (i) the outer rectangle is a pullback
    (ii) the left square is a pullback

    are equivalent. But it may well happen that both (i) and (ii) hold,
    without the right square being a pullback.

    For instance let i: A --> B be a split mono with retraction p: B --> A
    and consider

    A ==== A ==== A
    | | |
    | |i |
    | v |
    A ---> B ---> A
    i p

    where all unnamed maps are identity maps. Then the left square and
    the outer rectangle are pullbacks but the right square cannot be
    a pullback unless i was already an isomorphism.
    • CommentRowNumber3.
    • CommentAuthorFinnLawler
    • CommentTimeAug 10th 2010
    • (edited Aug 10th 2010)

    Well spotted. The mistake in the ’proof’ is in confusing cones over (def,cf)(d \to e \to f, c \to f) and cones over (de,ef,cf)(d \to e, e \to f, c \to f). I think they are the same iff the right square is a pullback.

    Edit: I made the same mistake at comma object. Both fixed now, I think.

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