Re: The omega-functor omega-category

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

Dear John,

There are respects in which properties are not exactly equivalent to
degenerate, "unique choice" cases of structure. It can make a difference
whether you consider something as property or structure, and one
situation where the difference enters is when you consider
homomorphisms, i.e. structure-preserving functions.

For example, finiteness of sets looks like a property, but it can also
be expressed as structure. The finiteness of a set X is, as structure,
an element T of the finite powerset of X (i.e its free semilattice) such
that x in T for all x in X. The structure, if it exists at all, is
unique: T is the whole of X.

If f: X -> Y is a function between finite sets X and Y then for f to be
a homomorphism of finite sets, i.e. for it to preserve finiteness as a
structure, means that the direct image of T_X is T_Y, i.e. f is onto.

This may look artificial, but in fact it is exactly what you are forced
to do if you wish to express finiteness in a geometric theory, as when
presenting classifying toposes. The problem is that geometric theories
are rather restricted in what properties they can express, so a frequent
solution is to convert properties into structure.

Another example is for decidable sets, i.e. those for which equality has
a Boolean complement - an inequality relation. (We are talking about
non-classical logics here.) A homomorphism then has to preserve
inequality as well as equality, and so be 1-1.

This is comparable with what you say in your paper with Shulman, if you
replace categories with classifying toposes. (After all, you use
topological ideas in your paper, and geometric logic is well adapted to
topology.) For the classifying toposes, the difference between
properties and structure is that properties correspond to subtoposes. A
subtopos inclusion is a geometric morphism that, at a first level of
approximation that ignores deeper topology, is full and faithful on
points. This matches your classification for forgetting at most
properties. But the thing about the geometric theories is that they
oblige you to work with the category of finite sets _and surjections_,
and this is what stops the functor FinSets -> Sets from being full. It
is only faithful and so forgets at most structure.

Regards,

Steve Vickers.

John Baez wrote:
> David Leduc wrote:
>
>>> I'm not sure what [_._] is supposed to mean - an internal
>>> hom functor?
>
>> This was supposed to be the "cartesian closed structure" of
>> StrictOmegaCat, but since some say it is not a structure I'm not sure
>> how to call it...
>
> Just call it the internal hom.
>
> The point is, you can just look at a category and say, yes or no,
> whether it's cartesian closed.  So cartesian closedness is a "property"
> of a category - not a "structure" that you might equip a category with
> in more than one way.
>
> Nonetheless, you can consider properties as a special case of
> structures - namely, those structures for which you have at most
> one one choice.  And if you do this you're free to speak of a cartesian
> closed "structure".
>
> Similarly, you can consider structures as a special case of "stuff".
>
> If you don't know the yoga of "properties, structure and stuff", you
> might enjoy this paper where Mike Shulman and I explain it:
>
> http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=15
>
> Best,
> jb


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