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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 $S\times_{f,g}S$ 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 $f,g$ both be the unique morphism $2\to 1$ in $Set$. Then $S\times_{f,g} S$ is $2\times 2 = 4$, while the equalizer is $2$.

• 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.

• 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.