Re: The omega-functor omega-category

[Michael Shulman <shulman@math.uchicago.edu>]

I personally prefer to say that "unique choice structure" is something
"in between" property and structure.  Kelly and Lack dubbed it
"Property-like structure" in their paper with that title.  The
difference is exactly as you say: property-like structure is unique
(up to unique isomorphism) when it exists, but is not necessarily
"preserved" by all morphisms.  In terms of forgetful functors,
property-like structure corresponds to a functor which is
*pseudomonic*, i.e. faithful, and full-on-isomorphisms.  Another nice
example is that being a monoid is a "property" of a semigroup, i.e. a
semigroup can have at most one identity element, but a semigroup
homomorphism between monoids need not be a monoid homomorphism.

Mike

On Fri, Oct 1, 2010 at 7:22 AM, Steve Vickers <s.j.vickers@cs.bham.ac.uk> wrote:
> 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.
>

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