Re: Isbell & MacLane on the insufficiency on skeletal categories

[Vaughan Pratt <pratt@cs.stanford.edu>]

On 5/25/2013 8:47 AM, Colin McLarty wrote:
> 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).

Why do you need Isbell's long argument, or even any monoidal structure
on Set, to obtain a contradiction here?  Just use that NxN is a product
and observe that the pair (3,4) in NxN (as a map from 1 to NxN) would
have to be both 3 and 4 (as maps from 1 to N) when the projections are
the identity.

The inconsistency found by Isbell works even when the projections seem
quite reasonable, namely when taken to be the three projections from the
ternary product XxYxZ whose elements are triples (x,y,z) (as the
necessary meaning of identity of associativity a: Xx(YxZ) --> (XxY)xZ).
   One can in fact consistently equip Skel(FinSet) with such structure.
Isbell shows that extending this to infinite sets breaks down, namely by
creating additional equations not encountered with finite sets due to
interference between binary and ternary product resulting from the
identification of N with NxN.

One can get close to a skeleton of Set using tau-categories as per
section 1.493 of Cats & Alligators; N becomes the ordinal omega, whose
square is a distinct object albeit still isomorphic to omega.  The full
subcategory of finite ordinals is (isomorphic to) Skel(FinSet).

Vaughan Pratt


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