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Stream: theory: category theory

Topic: pullback of equalizers

Callan McGill (Sep 13 2022 at 16:19):

I recently was thinking about pullbacks of equalizers. If one writes the equalizer as the pullback of $\langle f, g \rangle : X \rightarrow Y \times Y$ and $\Delta : Y \rightarrow Y \times Y$ then (I believe!) equalizers being preserved under pullbacks follows from the "pullback cube lemma" - the pullback of a pullback square on each side of a cube is itself a pullback. Do you need pullbacks in the category for this result to be true?

John Baez (Sep 13 2022 at 21:29):

By "this result" do you mean equalizers being preserved under pullbacks, or the pullback cube lemma?

I guess the former.

Callan McGill (Sep 13 2022 at 23:25):

I meant equalizers being preserved under pullback, apologies for lack of clarity.

John Baez (Sep 14 2022 at 07:52):

Okay, just checking. So your question is whether this is true even when the category doesn't necessarily have pullbacks other than those required for stating the result?