I have read that in the category of groups every monomorphism is an equalizer. The proof I found uses the notion of amalgamation, which I don't know, and refers to a book on group theory by Kuros which I don't have... Can you please explain me why in the category of groups every mono is an equalizer? I will accept both an answer using amalgamation and answers using other (possibly easier) methods
2026-05-05 15:48:42.1777996122
Every monomorphism is an equalizer
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This is not true. In the category of groups every kernel is a mono, but there are monos which are not kernels.
Edit: The question is changed. Any equalizer in a category is monic, but the converse need not be true in general. It is true however for the category of groups; all monos in this category are regular: see here.
Literature: Mitchell: Theory of Categories; it contains a proof that every monomorphism is an equalizer in Grps, due to Eilenberg-Moore.