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I edited subobject slightly and added the statement that in an accessible category $C$ every poset of subobjects is small.
The collection of subobjects (in the sense of equivalence classes) may be a proper class, even in abelian categories. One related entry is property sup. (edit: added this link into “related entries”; btw, I do not understand listing the entry itself there).
The collection of subobjects (in the sense of equivalence classes) may be a proper class, even in abelian categories.
Let’s see, is this meant to be in contradiction to saying that in an accessible category it is a set?
This is (if you read carefuly) an introductory statement to property sup which is about further restriction to the chains of subobjects, still intuitively surprising. I think that accessible abelian categories do not need to satisfy the property sup. I do not know what happens with the chain conditions when we talk about chains in a proper class. Dividing into accessible and nonaccessible does not solve the problem of understanding the behaviour of bounds on chains.
at subobject I have started a new Properties-subsection Limits and colimits of subobjects with some basics on joins/pushouts and meets/fiber products etc.
… and now also with the full proofs.
added list of basic examples to subobject.
I removed redirects for subtype and subalgebra (and their plurals), since neither string ‘typ’ nor ‘gebr’ appears anywhere on the page.
Actually, that edit didn't take. I keep getting Cloudfare HTTP errors. Interestingly, the last attempt did manage to break the redirects, even though it didn't save the new page source.
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