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Can all non-concrete categories be constructed by taking a concrete category and identifying some of its morphisms?

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Or rather: Given any category, can we find a suitable concrete category such that ... Personally, I doubt that. –  Hagen von Eitzen Dec 4 '12 at 10:06
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All small categories are concrete. Categories that fail to be locally small cannot possibly be quotients of concrete categories, but it is possible to obtain non-locally-small categories by localising concrete categories. –  Zhen Lin Dec 4 '12 at 10:33

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The answer to this question is Yes you can. The relevant paper is Ludek Kucera, Every category is a factorization of a concrete one, Journal of Pure and Applied Algebra, Volume 1, Issue 4, December 1971, Pages 373-376. You may read the paper here: http://www.sciencedirect.com/science/journal/00224049/1/4 .

However, this is a formal construction that does not seem to be of much use in concrete problems (such as working in homotopy theory). Also this used the axiom of class choice in Godel-Bernays-Von von neumann set theory.

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