Colin McLarty | 25 May 17:47 2013

Re: Isbell & MacLane on the insufficiency on skeletal categories

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.
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Bas Spitters | 26 May 22:26 2013
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Re: Isbell & MacLane on the insufficiency on skeletal categories

On Sat, May 25, 2013 at 5:47 PM, Colin McLarty <colin.mclarty <at> case.edu> wrote:
> 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.
>
> 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.

Such an approach has recently been developed here:

http://arxiv.org/abs/1303.0584
Univalent categories and the Rezk completion
Benedikt Ahrens, Chris Kapulkin, Michael Shulman
---
     We develop category theory within Univalent Foundations, which is
a foundational system for mathematics based on a homotopical
interpretation of dependent type theory. In this system, we propose a
definition of "category" for which equality and equivalence of
categories agree. Such categories satisfy a version of the Univalence
Axiom, saying that the type of isomorphisms between any two objects is
equivalent to the identity type between these objects; we call them
"saturated" or "univalent" categories. Moreover, we show that any
category is weakly equivalent to a univalent one in a universal way.
In homotopical and higher-categorical semantics, this construction
(Continue reading)

Vaughan Pratt | 7 Jun 00:00 2013
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Re: Isbell & MacLane on the insufficiency on skeletal categories

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

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Fred E.J. Linton | 7 Jun 05:15 2013
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Re: Isbell & MacLane on the insufficiency on skeletal categories

On Thu, 06 Jun 2013 07:45:03 PM EDT, by Vaughan Pratt
<pratt <at> 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.

Even more convincing: The equalizer of those two projections from N x N
must be the diagonal in N x N. But if those projections are equal, their
equalizer is all of N x N. Thus every map to N x N factors through the
diagonal there, i.e., no matter what the object A, for every pair of maps
f, g: A --> N, we must have f = g. It will follow that N is terminal.

[Or was that your argument, Vaughan, that I somehow did not recognize?] 

Cheers, -- Fred

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]


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