[FOM] FOM Digest, Vol 37, Issue 32, question on axiom of choice in categories.

John Bell jbell at uwo.ca
Tue Jan 31 14:27:23 EST 2006

>I would like to discuss about the Axiom of Choice in
> Category Theory.
> If I am not mistaken, it is formulated the following
> way:
> "Let C and D be (small) categories such that C is not
> empty and D is discrete. Let F be a functor from C to
> D. There exists a functor G from D to C such that
> FGF=F".
> Can someone explain why  D is required to be discrete?

To see this, consider partially ordered sets as categories; functors between
these are order-preserving maps. Also a poset is discrete as a category iff
its ordering is trivial. Now let 2 be the set {0,1} with the ordering 0 <=
0, 0 <= 1, 1 <= 1 and let 2* the same set with the trivial ordering 0 <= 0, 
1 <= 1. Then the map f: 2* --> 2 which sends 0 to 0 and 1 to 1 is
order-preserving, epi and mono. From the existence of an order preserving
map g: 2 --> 2* such that fgf = f it would follow that f is an order
isomorphism (with inverse g); but 2* and 2 are obviously not isomorphic
since the former is discrete and the latter is not. 

-- John Bell

Professor John L. Bell
Department of Philosophy
University of Western Ontario
London, Ontario
Canada N6A 3K7

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