Re: The omega-functor omega-category

["Fred E.J. Linton" <fejlinton@usa.net>]

In his message of Mon, 04 Oct 2010 08:16:25 AM EDT, Vaughan Pratt 
<pratt@cs.stanford.edu> quibbled with what on 10/2/2010 3:03 PM, 
Michael Shulman had written:

>> 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.
> 
> How should this terminology be applied when the property-like structure
> is necessarily preserved by all morphisms?
> 
> A group can be defined as a monoid with the property that all of its
> elements have inverses.  The inverse is preserved by all morphisms.

A group can also be defined as a *semigroup* with that property.
"The inverse" need no longer be "preserved by all morphisms."
  
> A Boolean algebra can be defined as a bounded distributive lattice with
> the property that all of its elements have complements.  The complement
> is preserved by all morphisms.

Depends what you take to be a bounded lattice. Do you specify *finitary*
meets and joins, including the explicit empty ones that produce the bounds?
Or just *binary* ones, with the bounds *required* but not *specified*?
In the former situation, yes, "the complement is preserved by all morphisms."
In the latter situation, alas, no.

> Are these merely "property-like structures," or are they actual
> structures, despite being defined merely as properties?

When such a "property-like structure" *is* preserved, it is perhaps 
implicitly trying to behave like an "actual" structure, and could
certainly be harmlessly added to the actual structural specifications,
but, with so much riding on the *context* in which one is asking
about that property-like structure, I'm not yet ready just to declare
them, willy-nilly, to be "actual structures".

Cheers, -- Fred



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