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* Re: The omega-functor omega-category
@ 2010-10-04 21:00 Fred E.J. Linton
  2010-10-05 14:13 ` David Leduc
  0 siblings, 1 reply; 17+ messages in thread
From: Fred E.J. Linton @ 2010-10-04 21:00 UTC (permalink / raw)
  To: categories; +Cc: Vaughan Pratt

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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^ permalink raw reply	[flat|nested] 17+ messages in thread
* The omega-functor omega-category
@ 2010-09-23 10:07 David Leduc
  2010-09-24 15:13 ` Urs Schreiber
                   ` (5 more replies)
  0 siblings, 6 replies; 17+ messages in thread
From: David Leduc @ 2010-09-23 10:07 UTC (permalink / raw)
  To: categories

Hi,

Given two strict omega-categories C and D, how do you define the
strict omega-category of omega-functors between C and D?

Thanks,

David


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


^ permalink raw reply	[flat|nested] 17+ messages in thread

end of thread, other threads:[~2010-10-05 14:13 UTC | newest]

Thread overview: 17+ messages (download: mbox.gz / follow: Atom feed)
-- links below jump to the message on this page --
2010-10-04 21:00 The omega-functor omega-category Fred E.J. Linton
2010-10-05 14:13 ` David Leduc
  -- strict thread matches above, loose matches on Subject: below --
2010-09-23 10:07 David Leduc
2010-09-24 15:13 ` Urs Schreiber
2010-09-25  1:40   ` Ross Street
     [not found] ` <BF755983-D6D4-469B-9206-C6B275699C3F@mq.edu.au>
2010-09-25 11:22   ` Urs Schreiber
2010-09-26  2:00     ` David Leduc
     [not found] ` <1216D94C-81A2-49A7-95B3-7543B315A54B@mq.edu.au>
2010-09-26  5:00   ` David Leduc
     [not found] ` <E1P0Oe6-0005AL-SX@mlist.mta.ca>
2010-09-28  1:11   ` David Leduc
2010-09-29  1:09     ` John Baez
2010-09-30  0:29       ` David Leduc
     [not found] ` <AANLkTi=LXzu13GU8ZmB=+-qGTQmV3S-bDp6h+dJJ1xNJ@mail.gmail.com>
2010-09-30  3:10   ` John Baez
2010-10-01 14:22     ` Steve Vickers
2010-10-02 22:03       ` Michael Shulman
2010-10-04  7:52         ` Vaughan Pratt
2010-10-04 18:41           ` Michael Shulman
     [not found] ` <AANLkTin3LqPLuMFD-xSEuRK9TqBUbHrRRxrdcOEykJJo@mail.gmail.com>
2010-10-03 22:11   ` Michael Shulman

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