* errata
@ 2010-09-14 1:02 Eduardo J. Dubuc
2010-09-14 22:10 ` errata David Roberts
[not found] ` <19600.5343.491002.650226@zeus.knighten.org>
0 siblings, 2 replies; 7+ messages in thread
From: Eduardo J. Dubuc @ 2010-09-14 1:02 UTC (permalink / raw)
To: Categories list
of course, I meant "not kosher' or "non kosher"
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* Re: errata
2010-09-14 1:02 errata Eduardo J. Dubuc
@ 2010-09-14 22:10 ` David Roberts
2010-09-15 4:48 ` errata Michael Shulman
[not found] ` <19600.5343.491002.650226@zeus.knighten.org>
1 sibling, 1 reply; 7+ messages in thread
From: David Roberts @ 2010-09-14 22:10 UTC (permalink / raw)
To: Eduardo J. Dubuc; +Cc: Categories list
Actually this is the sort of thing I was trying to think of - it
shifts the focus to the positive (we can talk about kosher, as opposed
to 'non-evil'). It also has an element of incongruity, so that like
'evil' people stop for a moment and go 'hang on...'. And like the
concept of kosher, people who are not that way inclined can live quite
happily without it.
David
On 14 September 2010 10:32, Eduardo J. Dubuc <edubuc@dm.uba.ar> wrote:
> of course, I meant "not kosher' or "non kosher"
>
>
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* Re: errata
2010-09-14 22:10 ` errata David Roberts
@ 2010-09-15 4:48 ` Michael Shulman
0 siblings, 0 replies; 7+ messages in thread
From: Michael Shulman @ 2010-09-15 4:48 UTC (permalink / raw)
To: David Roberts; +Cc: Eduardo J. Dubuc, Categories list
And of course instead of "evil" one could say "treif".
On Tue, Sep 14, 2010 at 3:10 PM, David Roberts
<droberts@maths.adelaide.edu.au> wrote:
> Actually this is the sort of thing I was trying to think of - it
> shifts the focus to the positive (we can talk about kosher, as opposed
> to 'non-evil'). It also has an element of incongruity, so that like
> 'evil' people stop for a moment and go 'hang on...'. And like the
> concept of kosher, people who are not that way inclined can live quite
> happily without it.
>
> David
>
>
> On 14 September 2010 10:32, Eduardo J. Dubuc <edubuc@dm.uba.ar> wrote:
>> of course, I meant "not kosher' or "non kosher"
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* The omega-functor omega-category
@ 2010-09-23 10:07 David Leduc
2010-10-04 18:41 ` Michael Shulman
[not found] ` <20101007010252.EA0FDCF26@mailscan2.ncs.mcgill.ca>
0 siblings, 2 replies; 7+ 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
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* Re: The omega-functor omega-category
@ 2010-10-04 18:41 ` Michael Shulman
2010-10-05 15:42 ` property_vs_structure Eduardo J. Dubuc
0 siblings, 1 reply; 7+ messages in thread
From: Michael Shulman @ 2010-10-04 18:41 UTC (permalink / raw)
To: Vaughan Pratt; +Cc: categories
By definition (at least according to the usage under discussion),
something necessarily preserved by all morphisms is a "property,"
although it can also be regarded as a particular degenerate case of a
structure and, I guess, also a degenerate case of a property-like
structure.
property = forgetful functor is full and faithful
structure = forgetful functor is faithful
property-like structure = forgetful functor is pseudomonic
http://ncatlab.org/nlab/show/stuff%2C+structure%2C+property
Mike
On Mon, Oct 4, 2010 at 12:52 AM, Vaughan Pratt <pratt@cs.stanford.edu> wrote:
>
> On 10/2/2010 3:03 PM, Michael Shulman wrote:
>>
>> 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 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.
>
> Are these merely "property-like structures," or are they actual
> structures, despite being defined merely as properties?
>
> Vaughan
>
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* property_vs_structure
2010-10-04 18:41 ` Michael Shulman
@ 2010-10-05 15:42 ` Eduardo J. Dubuc
2010-10-06 12:34 ` errata Eduardo J. Dubuc
0 siblings, 1 reply; 7+ messages in thread
From: Eduardo J. Dubuc @ 2010-10-05 15:42 UTC (permalink / raw)
To: categories
Michael Shulman wrote:
>
> property = forgetful functor is full and faithful
> structure = forgetful functor is faithful
> property-like structure = forgetful functor is pseudomonic
>
On the thread "property" "structure" "property-like structure" and may be
some other etceteras.
I put on the table the following example to be analyzed:
Let f: X --> B a continuous function of topological spaces:
[assume surjective to simplify, and if b \in B, write X_b for the fiber
X_b = f^-1(b)].
Then, we have the two familiar definitions a) and b):
f is "fefesse" if given b \in B, then
a) for each x \in X_b, there is U, b \in U, such that
b) there is U, b \in U, such that for each x \in X_b,
there is V, x \in V, and f|V : V --> U homeo.
(the non commuting quantifiers again !)
a) fefesse = local homeomorphism
b) fefesse = covering map
Well, both are "properties" of a continuous function, but they are not of the
same kind.
in b) is hidden a structure, namely a trivialization structure associated to
an open cover of B.
If B is locally connected, then "covering map" behaves like a perfectly pure
property.
The difference is only manifest when the space B is not locally connected. In
this case we may have homeomorphisms from X to X over B which do not preserve
this structure (Spanier, Algebraic Topology).
have fun ! e.d.
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* errata
2010-10-05 15:42 ` property_vs_structure Eduardo J. Dubuc
@ 2010-10-06 12:34 ` Eduardo J. Dubuc
0 siblings, 0 replies; 7+ messages in thread
From: Eduardo J. Dubuc @ 2010-10-06 12:34 UTC (permalink / raw)
To: Eduardo J. Dubuc; +Cc: categories
>
> The difference is only manifest when the space B is not locally
> connected. In
> this case we may have homeomorphisms from X to X over B which do not
> preserve
> this structure (Spanier, Algebraic Topology).
>
>
is not quite it should be,
there is a clear notion of isomorphism of trivialization structure, and a same
space X over B may have non isomorphic structures. Alternatively, a continuous
function over B does not necessarily preserve the trivialization structures.
however, if B is locally connected, trivialization structures are like a pure
property of X.
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[parent not found: <20101007010252.EA0FDCF26@mailscan2.ncs.mcgill.ca>]
* RE: errata
[not found] ` <20101007010252.EA0FDCF26@mailscan2.ncs.mcgill.ca>
@ 2010-10-07 23:46 ` Marta Bunge
0 siblings, 0 replies; 7+ messages in thread
From: Marta Bunge @ 2010-10-07 23:46 UTC (permalink / raw)
To: Eduardo Dubuc; +Cc: categories
Even in the locally connected case there are several non isomorphic trivialization structures. The difference is that, in that case, there is a canonical one.
> Date: Wed, 6 Oct 2010 09:34:51 -0300
> From: edubuc@dm.uba.ar
> To: edubuc@dm.uba.ar
> CC: categories@mta.ca
> Subject: categories: errata
>
>
>>
>> The difference is only manifest when the space B is not locally
>> connected. In
>> this case we may have homeomorphisms from X to X over B which do not
>> preserve
>> this structure (Spanier, Algebraic Topology).
>>
>>
>
> is not quite it should be,
>
> there is a clear notion of isomorphism of trivialization structure, and a same
> space X over B may have non isomorphic structures. Alternatively, a continuous
> function over B does not necessarily preserve the trivialization structures.
>
> however, if B is locally connected, trivialization structures are like a pure
> property of X.
>
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* errata
@ 2007-10-10 13:58 Eduardo Dubuc
0 siblings, 0 replies; 7+ messages in thread
From: Eduardo Dubuc @ 2007-10-10 13:58 UTC (permalink / raw)
To: categories
in my posting:
"this is essentially the difference between filterness and cofilterness,
with all what it means"
it should be:
"this is essentially the difference between filterness and
pseudofilterness, with all what it means"
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2010-09-14 1:02 errata Eduardo J. Dubuc
2010-09-14 22:10 ` errata David Roberts
2010-09-15 4:48 ` errata Michael Shulman
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2010-09-15 2:07 ` errata Eduardo J. Dubuc
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2010-09-23 10:07 The omega-functor omega-category David Leduc
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2010-10-05 15:42 ` property_vs_structure Eduardo J. Dubuc
2010-10-06 12:34 ` errata Eduardo J. Dubuc
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2010-10-07 23:46 ` errata Marta Bunge
2007-10-10 13:58 errata Eduardo Dubuc
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