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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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    $\begingroup$ At least, for an instance of inconsistency in category theory you can look at mathoverflow.net/questions/302298/… $\endgroup$
    – Sam Hopkins
    Commented Jul 26, 2022 at 23:13
  • $\begingroup$ Would the fact that any category admitting all products is a poset count? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 27, 2022 at 5:01
  • $\begingroup$ @მამუკაჯიბლაძე It need not even admit all products; just products indexed over its hom set suffice for Freyd’s argument to go through. $\endgroup$
    – Alec Rhea
    Commented Jul 27, 2022 at 6:01
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    $\begingroup$ Having different opinions about CH or AC is a paradox? $\endgroup$
    – Asaf Karagila ♦
    Commented Jul 28, 2022 at 5:01

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