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• CommentRowNumber1.
• CommentAuthorpaikeos
• CommentTimeMay 6th 2020
Before asking a question, I would like to clarify two things.
Firstly, my ability to write in English is not a high level, and therefore I use a translator.
Secondly, I’m only going to study category theory, and therefore my question may in principle be incorrect.
But let's move on to the question itself:
Is it possible to well define the concept of “weak α-category”, where α is an arbitrary ordinal?
• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 6th 2020
• (edited May 6th 2020)

You can define it by transfinite induction, together with a functor that sends weak β-categories to weak α-categories for all β<α and possibly with a truncation functor that sends weak α-categories to weak β-categories for all β<α: a weak α+1-category is a category weakly enriched in weak α-categories, and for a limit ordinal α, a weak α-category can be defined in two different, incompatible ways: as a weak β-category for any β<α, or as a compatible family of weak β-categories, one for each β<α, so that for any β’<β” the β”-th category truncates to the β’-th category.