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The category of sets satisfy the Schroder-Bernstein property: If there exists an injection from set $A$ to set $B$, and also from $B$ to $A$, then there exists a bijection from $A$ to $B$. But some categories do not satisfy that analogous property. I want a list of important and notable categories used in mathematics that do not satisfy that property.

user107952
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    Just to clarify, "injection" and "bijection" are not category-theoretic notions. Presumably you mean "monomorphism" and "isomorphism" here. – Pilcrow Jul 22 '24 at 15:40
  • CSB fails for the category of sets if your metatheory is intuitionistic enough: see nLab for more counterexamples. – Naïm Camille Favier Jul 22 '24 at 16:03
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    Previous discussions on MO and here: https://mathoverflow.net/questions/452811/does-mathbfcat-have-the-cantor-schr%c3%b6der-bernstein-property, https://mathoverflow.net/questions/89718/classes-of-fields-and-cantor-schr%c3%b6der-bernstein, https://mathoverflow.net/questions/343418/does-the-cantor-schr%c3%b6der-bernstein-theorem-hold-in-the-category-opposite-to-the, https://math.stackexchange.com/questions/3372047/categorical-schr%c3%b6der-bernstein-theorem, https://math.stackexchange.com/questions/2461137/a-counterexample-to-cantor-schr%c3%b6der-bernstein-in-groups, and... – Qiaochu Yuan Jul 22 '24 at 16:10
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    https://math.stackexchange.com/questions/257650/analogue-of-the-cantor-bernstein-schroeder-theorem-for-general-algebraic-structu/257726#257726, that I could find immediately. Very, very few categories satisfy CSB, it's a very strong statement. – Qiaochu Yuan Jul 22 '24 at 16:10

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