errata

errata

[Eduardo J. Dubuc]

of course,   I meant  "not kosher' or "non kosher"


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Re: errata

[David Roberts]

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

[Michael Shulman]

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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Re: errata

[Eduardo J. Dubuc]

the problem is that "kosher" (or not kosher) is a word much more universally
known than "trefe".   e.d.

Robert Knighten wrote:
> Eduardo J. Dubuc writes:
> > of course,   I meant  "not kosher' or "non kosher"
>
> The word you want is "trefe" :-)
>
> -- Bob
>


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The omega-functor omega-category

[David Leduc]

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

[Michael Shulman]

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

[Eduardo J. Dubuc]

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

[Eduardo J. Dubuc]

>
> 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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RE: errata

[Marta Bunge]

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

[Eduardo Dubuc]

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"