* Skeleton of a category
@ 1999-09-20 17:06 Hongseok Yang
0 siblings, 0 replies; 2+ messages in thread
From: Hongseok Yang @ 1999-09-20 17:06 UTC (permalink / raw)
To: categories
Would you let me know when the category has an equivalent skeleton? (The
definition of the skeleton subcategory that I have in mind is from
MacLane p91: a full subcategory such that for any object in the original
category, there exists a unique isomorphic object in the
skeleton subcategory.) My question is mainly about when I can use the
choice axiom without causing contradiction. For instance, I heard that the
category of abelian groups doesn't have an equivalent skeleton
subcategory.
Thank you very much,
Hongseok
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* Re: Skeleton of a category
@ 1999-09-20 17:44 Paul Taylor
0 siblings, 0 replies; 2+ messages in thread
From: Paul Taylor @ 1999-09-20 17:44 UTC (permalink / raw)
To: categories
> Would you let me know when the category has an equivalent skeleton?
"every small category has a skeleton" iff the axiom of choice holds.
See Exercise 3.26 in my book, or
http://www.dcs.qmw.ac.uk/~pt/book/html/s3e.html#e3.26
for a preorder example.
Exercise 4.37 defines "skeletal"
http://www.dcs.qmw.ac.uk/~pt/book/html/s4e.html#e4.37
Paul
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