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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2010

    added to equalizer statement and proof that a category has equalizers if it has pullbcks and products

    • CommentRowNumber2.
    • CommentAuthorNikolajK
    • CommentTimeMay 21st 2016
    • (edited May 21st 2016)

    With reference to the last diagram, what is an example for a pullback S× f,gSS\times_{f,g}S that isn’t isomorphic to the equalizer? I think another way of asking this is asking for an example where do the two projections out of this pullback objects differ by more than an isomorphism.

    And if the category has a terminal object, are this pullback and the equalizer already isomorphic?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMay 22nd 2016

    Let f,gf,g both be the unique morphism 212\to 1 in SetSet. Then S× f,gSS\times_{f,g} S is 2×2=42\times 2 = 4, while the equalizer is 22.

    • CommentRowNumber4.
    • CommentAuthorJohn Baez
    • CommentTimeApr 22nd 2019

    I changed the final theorem from

    If a category has products and equalizers, then it has limits.

    to

    If a category has equalizers and finite products, then it has finite limits.

    and I changed an earlier proposition from

    A category has equalizers if it has products and pullbacks.

    to

    A category has equalizers if it has binary products and pullbacks.

    diff, v14, current

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 22nd 2019

    Why not kill two birds with one stone by putting “finite” in parentheses (in appropriate places) in proposition 3.2? The change of the first doesn’t seem entirely warranted to me.

  1. Added that every equaliser is a monomorphism.

    diff, v15, current

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 17th 2021

    Simplified the description of equalizers in terms of products and pullbacks.

    diff, v19, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 27th 2021

    Added this:

    Equalizers were defined in the paper

    for any finite collection of parallel morphisms. The paper refers to them as left equalizers, whereas coequalizers are referred to as right equalizers.

    diff, v22, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMar 27th 2021

    Excellent – thanks!

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2021

    added pointer to:

    diff, v24, current

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