Re: Isbell & MacLane on the insufficiency on skeletal categories

[Colin McLarty <colin.mclarty@case.edu>]

There is no problem, either in practice or in formal ETCS or CCAF
foundations, with assuming there is only one countably infinite set, call
it N.  Then indeed every function N-->N has graph N>-->NxN for some monic
N>-->NxN.  But there are uncountably many different monics N>-->NxN so it
does not follow that all functions N-->N are equal.  Different monics to
NxN can have identical domains.

Of course is also follows that NxN=N.  But it does not follow, and in fact
it is refutable, that the projection functions are the identity function
1_N.

Isbell's argument is on p. 164 of my copy of CfWM (1998).  On the plainest
reading, it shows you could assume Xx(YxZ)=(XxY)xZ as sets, for all sets
nX,Y,Z, but even then it is contradictory to suppose the associativity
function  Xx(YxZ) --> (XxY)xZ is always the identity function.

This shows we cannot simultaneously maintain:

1)  There is a category Set^2 with the usual properties of a functor
category.
2)  Isomorphic objects are equal in all categories.

Mac~Lane concludes we cannot accept the sweeping skeletal principle 2.

I will say some higher category theorists promote another option.  They
would keep 2, by rejecting 1, by saying there are not functor categories in
the standard (1-categorical) sense, but only some infinity-categorical
analogue.  I do not know if that has ever been systematically spelled out
though of course there are projects like Makkai's advocacy of FOLDS that
are meant to go that way.

Colin


On Fri, May 24, 2013 at 6:34 PM, Staffan Angere <staffan.angere@fil.lu.se>wrote:

> Dear Category theorists,
>
> in Categories for the Working Mathematician, page 164, MacLane relates an
> argument, "due to Isbell", why one cannot identify all isomorphic objects.
> I  have not, however, been able to find any publication of Isbell that
> contains the argument. Does anyone here know if he published it?
>
> I also have a question about the argument itself: why is it made the way
> MacLane does it, rather than just though noticing that all functions from
> countable sets are countable, and thus themselves countable, and so
> isomorphic  to any other countable set? It seems like it would follow
> directly from this that any functions on the natural numbers have to be
> equal, if isomorphic (i.e. equinumerious) sets are identical. Why is
> MacLane doing all the "detours" though products, epics, etc.?
>
> Thanks in advance,
> Staffan
>

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