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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 29th 2010

I edited subobject slightly and added the statement that in an accessible category $C$ every poset of subobjects is small.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeDec 29th 2010
• (edited Dec 29th 2010)

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).

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeDec 29th 2010

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?

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeDec 30th 2010

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJun 26th 2011

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.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 26th 2011

… and now also with the full proofs.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeSep 11th 2012

added list of basic examples to subobject.