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Daniel Teixeira (Jun 13 2024 at 15:31):If we forget size issues then we can localize a category at any class of morphisms by zig-zags. No fractions or further conditions needed; these only address size issues.
Is there an analogous procedure for localizing a 2-category at a class of 1-morphisms? Pronk does it for some 2-fractions condition, some people do it for model categories, but we haven't found a general procedure. It wouldn't seem that crazy to write it down, except that it seems hard to formally turn "morphisms into equivalences", since that's like extra data
Morgan Rogers (he/him) (Jun 13 2024 at 15:50):You can do it abstractly, taking the 2-categorical pushout of the wide subcategory you want to invert along its groupoidification. I don't know if that helps?
Daniel Teixeira (Jun 13 2024 at 16:21):Yes existence would suffice. However, your suggestion seems to respect conservation of difficulty since "groupoidification" is localizing the wide subcategory at all morphisms...
Kevin Carlson (Jun 13 2024 at 18:36):You can do better than that: let be the inclusion of the walking arrow into the walking equivalence. Given a class of morphisms in a 2-category you want to invert, you get a map with as many factors in the coproducts as there are elements of Then just push out this map along ! There’s no need to put these arrows together into a wide sub-2-category or anything as detailed as that, as long as you don’t care about actually, like, understanding anything detailed about the result.
Daniel Teixeira (Jun 13 2024 at 18:38):this sounds reasonable!