Re: equality is beautiful

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

On 9 Jan 2010, at 03:29, Joyal, André wrote:

> Dear All
>
> Many people seem to distrust the equality
> relation between the objects of a (large) category.
> Is this a philosophical conundrum or a mathematical problem?
>
> Can we define a notion of (large) category without supposing
> that its (large) set of objects has a diagonal

Dear Andre,

Can I explore this with regard to topologies?

Suppose we compare FinSet with Set, defining FinSet very small with N  
for its object space. The object diagonal N -> NxN is an open inclusion.

Now look at Set. The natural topology on the class of sets, as the  
Ind-completion of FinSet, is the one whose sheaves are given by the  
object classifier. Thus continuous maps from it are functorial and  
preserve filtered colimits (the categorical analogue of Scott  
continuity).

(This introduces a confusing issue. Functors from the category Set  
are already determined by their object maps. But it is a special  
category in which the morphism space is the comma object got from two  
copies of the identity map on the object space - we are using the  
fact that the category of spaces is actually a 2-category, using the  
specialization morphisms. FinSet was certainly not of this kind.)

Now the object diagonal is not even an inclusion, since it is not  
full. I would speculate, by analogy with what I know for ideal  
completions, that it is essential but not locally connected.

I don't really know what to make of this, but it does seem that there  
are topological distinctions to be made between the two categories  
based on object equality.

Now let me wonder about classifying toposes.

I love using them (as I did above), but they always seem slightly  
fuzzy because they are really defined only up to equivalence. So I  
certainly would distrust the object equality. I think the discussion  
becomes slightly sharper in terms of arithmetic universes (as  
mentioned by Paul Taylor) instead of toposes.

Given a base AU A0, there are two obvious places to look for an  
object classifier (representing the class of sets) a la Grothendieck  
topos theory. First, there is A0[U], the AU freely generated over A0  
by an object U. This can be constructed by universal algebra, and  
then is a classifying A0-AU (for the theory with one sort and no  
predicates, functions or axioms) characterized up to isomorphism with  
respect to strict A0-AU homomorphisms. However, its object equality  
depends rather delicately on the precise structure used to  
characterized AUs. For example, I suspect it will differ according as  
AUs are taken to have canonical pullbacks or canonical binary  
products and equalizers.

On the other hand, sheaf theory would suggest using the category Presh 
(FinSet^op) of internal diagrams over the internal FinSet in A0. I  
conjecture that this (is an AU and) is equivalent but not isomorphic  
to A0[U]. Hence there are issues of object equality when one compares  
them. (Milly Maietti and I are looking at "subspaces" in this AU  
setting, and again are having to be rather careful about equality and  
the distinction between strict and non-strict AU homomorphisms.)

A further issue, seen in AUs but not toposes, is that A0[U] can be  
internalized in A0, as the internal AU in A0 freely generated by one  
object. The comparison with the external A0[U] will presumably raise  
truth v. proof issues similar to those that you have already  
investigated for the initial AU and Goedel's Theorem.

To summarize:

(1) Granted the existence of the object diagonals, they may still  
have different topological characteristics for different kinds of  
categories.
(2) A universal characterization (such as for classifying toposes)  
that is only up to equivalence will make it difficult to rely on  
object equality.
(3) In arithmetic universes we can perhaps (allowing for the fact  
that the theory is not fully developed yet) see situations where the  
object equality definitely exists, but is sensitive to inessential  
differences.

Best regards,

Steve Vickers.




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