Isbell & MacLane on the insufficiency on skeletal categories
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Sat, May 25, 2013 at 5:47 PM, Colin McLarty <colin.mclarty@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 corresponds to a truncated version of the Rezk completion for Segal spaces, and also to the stack completion of a prestack. --- Bas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
On Thu, 06 Jun 2013 07:45:03 PM EDT, by 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.
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/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Bas Spitters -
Colin McLarty -
Fred E.J. Linton -
Staffan Angere -
Vaughan Pratt