Re: Isbell & MacLane on the insufficiency on skeletal categories

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

Vaughan,

Your quote runs two of my paragraphs together.  .Certainly the point about
N does not need Isbell's argument.

Colin


On Thu, Jun 6, 2013 at 6:00 PM, Vaughan Pratt <pratt@cs.stanford.edu> wrote:

> 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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