There is a lot of non-obvious and controversial topics and questions in set theory. From its begining in the first half of 20th century it have generated many paradoxes. For example there are different opinions on continuum hypothesis among researches. Some of this topics influenced not only math, but philosophy as well. My question is: are there any theorems/hypothesis/paradoxes of this kind in category theory (higher category theory, etc) ? I don't ask if paradoxes like Russell's paradox could happen in category theory, I mean something that is essentially peculiar to category theory.
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7At least, for an instance of inconsistency in category theory you can look at https://mathoverflow.net/questions/302298/is-fefermans-unlimited-category-theory-dead – Sam Hopkins Jul 26 '22 at 23:13
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Would the fact that any category admitting all products is a poset count? – მამუკა ჯიბლაძე Jul 27 '22 at 05:01
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@მამუკაჯიბლაძე It need not even admit all products; just products indexed over its hom set suffice for Freyd’s argument to go through. – Alec Rhea Jul 27 '22 at 06:01
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3Having different opinions about CH or AC is a paradox? – Asaf Karagila Jul 28 '22 at 05:01