25 May 17:47 2013

## Re: Isbell & MacLane on the insufficiency on skeletal categories

Colin McLarty <colin.mclarty <at> case.edu>

2013-05-25 15:47:42 GMT

2013-05-25 15:47:42 GMT

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.(Continue reading)