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1. • CommentRowNumber1.
   • CommentAuthorpaikeos
   • CommentTimeMay 6th 2020
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   Author: paikeos
   Format: TextBefore 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?

   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?
2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 6th 2020
   • (edited May 6th 2020)
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   Author: Dmitri Pavlov
   Format: MarkdownItexYou 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.

   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.