Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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.
1 to 7 of 7