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সায়ন্তন রায় (Jun 05 2025 at 15:53):Of course, completeness of a category implies finite-completeness of the same. However, I was wondering under which (non-trivial) conditions the converse statement is true?
Mike Shulman (Jun 05 2025 at 16:22):The only general condition I can think of offhand is if the category is a finite poset.
Morgan Rogers (he/him) (Jun 05 2025 at 16:37):If you already know that the category has some other limits and is sufficiently "cogenerated" under them from a small enough collection of objects, you can get this result. The dual situation is that of finitely accessible categories, which are generated under filtered colimits from a collection of compact objects. A finitely accessible category is cocomplete if and only if it has finite colimits (in which case it is called a locally finitely presentable (LFP) category). The dual classes of category are the ones relevant to you.
সায়ন্তন রায় (Jun 06 2025 at 03:30):Thank you very much, both of you. I think that I was looking for something like what @Morgan Rogers (he/him) mentioned. Could you please also tell me where I can find a proof of this result?
Mike Shulman (Jun 06 2025 at 05:03):Do you really need the small cogeneration? I would have guessed that if you have cofiltered limits and finite limits then you automatically have all small limits, regardless. The limit of any diagram should be the cofiltered limit of the limits of its finite subdiagrams.
Morgan Rogers (he/him) (Jun 06 2025 at 06:30):That does seem sensible @Mike Shulman. @সায়ন্তন রায় you can find at least the result I stated in Adamek and Rosicky's book.
সায়ন্তন রায় (Jun 08 2025 at 05:50):I have checked the details, and, as has been pointed out by @Mike Shulman one does not need the hypothesis of small cogeneration.
সায়ন্তন রায় (Jun 08 2025 at 05:51):Unless I am missing something, the same proof seems to work for weak limits as well.