The omega-functor omega-category

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

[Urs Schreiber]

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

There is the Crans-Gray tensor product on StrOmegaCat that makes it
biclosed monoidal.

So for G^n the standard n-globe regarded as a strict omega-category,
the (right/left) internal hom between strict omega-categories X and Y
is

  [X,Y ]  = Hom( X otimes G^bullet , Y ) .

See

   http://ncatlab.org/nlab/show/Crans-Gray+tensor+product

for references.

Best,
Urs


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Re: The omega-functor omega-category

[Ross Street]

Dear Urs

On 25/09/2010, at 1:13 AM, Urs Schreiber wrote:

>> Given two strict omega-categories C and D, how do you define the
>> strict omega-category of omega-functors between C and D?
>
> There is the Crans-Gray tensor product on StrOmegaCat that makes it
> biclosed monoidal.

I think David was asking about the simpler cartesian closed structure
on omega-Cat. This is constructed in

 	The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987)
283-335

for example.

Ross


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Re: The omega-functor omega-category

[Urs Schreiber]

Dear Ross,

concerning the internal hom of strict omega-categories:

>>> Given two strict omega-categories C and D, how do you define the
>>> strict omega-category of omega-functors between C and D?
>>
>> There is the Crans-Gray tensor product on StrOmegaCat that makes it
>> biclosed monoidal.
>
> I think David was asking about the simpler cartesian closed structure on
> omega-Cat. This is constructed in
>
>        The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987)
> 283-335

You know all this, but for the record I say the following:

The cartesian closed structure has an internal hom that is a
restriction of the internal hom wrt the Gray structure. Usually the
one of the Gray structure is the one of interest. It is the one closer
to the full oo-category theoretic notion (the one with no strictness
constraints whatsoever).

The Crans-Gray tensor product with its property that

  G^k otimes G^l

is k+l-dimensional is the fix in the globular model for what in the
simplicial model is automatic, namely that

   Delta^k x Delta^l

is k+l-dimensional. That this is automatic for the cartesian product
in simplicial sets but requires more work for globular sets is one of
the reasons why simplicial models for oo-categrories are more
highly-developed than globular ones: they are easier.

A good brief introduction to this is on the first few pages of Sjoerd Crans'

A tensor product for Gray-categories
  http://www.emis.de/journals/TAC/volumes/1999/n2/5-02abs.html

(After that introduction the article goes on to refine the globular
Gray tensor product to the case of _weak_  (or rather: semi-strict)
3-categories.)

But of course for the purposes of David's application (which I don't
know about) the strict version of the internal hom might be
sufficient.

Best,
Urs


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Re: The omega-functor omega-category

[David Leduc]

Dear all,

Thank you for your replies.

All the constructions I can find rely on strict omega-categories
defined as graphs with structure. If instead we define recursively a
strict omega-category as a category enriched over a strict
omega-category, is there a recursive way to define the omega-category
of omega-functors (between two fixed omega-categories)?

David


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Re: The omega-functor omega-category

[David Leduc]

Dear Ross,

> If V = 2Cat then V-Cat = 3Cat . . . and on it goes.

OK. So if V = omegaCat, then V-Cat = omegaCat.

Now how do you define (recursively) the internal-hom of omegaCat?

David


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Re: The omega-functor omega-category

[David Leduc]

> But being cartesian closed is a
> *property*, not extra *structure*,

"being cartesian closed is a property" but [_._] is structure, isn't it?


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Re: The omega-functor omega-category

[John Baez]

On Tue, Sep 28, 2010 at 9:11 AM, David Leduc <david.leduc6@googlemail.com>wrote:

>> But being cartesian closed is a
>> *property*, not extra *structure*,
>
> "being cartesian closed is a property" but [_._] is structure, isn't it?
>

I'm not sure what [_._] is supposed to mean - an internal hom functor?


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Re: The omega-functor omega-category

[David Leduc]

> I'm not sure what [_._] is supposed to mean - an internal hom functor?

This was supposed to be the "cartesian closed structure" of
StrictOmegaCat, but since some say it is not a structure I'm not sure
how to call it...


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Re: The omega-functor omega-category

[John Baez]

David Leduc wrote:

>> I'm not sure what [_._] is supposed to mean - an internal
>> hom functor?

> This was supposed to be the "cartesian closed structure" of
> StrictOmegaCat, but since some say it is not a structure I'm not sure
> how to call it...

Just call it the internal hom.

The point is, you can just look at a category and say, yes or no,
whether it's cartesian closed.  So cartesian closedness is a "property"
of a category - not a "structure" that you might equip a category with
in more than one way.

Nonetheless, you can consider properties as a special case of
structures - namely, those structures for which you have at most
one one choice.  And if you do this you're free to speak of a cartesian
closed "structure".

Similarly, you can consider structures as a special case of "stuff".

If you don't know the yoga of "properties, structure and stuff", you
might enjoy this paper where Mike Shulman and I explain it:

http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=15

Best,
jb


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Re: The omega-functor omega-category

[Steve Vickers]

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.

John Baez wrote:
> David Leduc wrote:
>
>>> I'm not sure what [_._] is supposed to mean - an internal
>>> hom functor?
>
>> This was supposed to be the "cartesian closed structure" of
>> StrictOmegaCat, but since some say it is not a structure I'm not sure
>> how to call it...
>
> Just call it the internal hom.
>
> The point is, you can just look at a category and say, yes or no,
> whether it's cartesian closed.  So cartesian closedness is a "property"
> of a category - not a "structure" that you might equip a category with
> in more than one way.
>
> Nonetheless, you can consider properties as a special case of
> structures - namely, those structures for which you have at most
> one one choice.  And if you do this you're free to speak of a cartesian
> closed "structure".
>
> Similarly, you can consider structures as a special case of "stuff".
>
> If you don't know the yoga of "properties, structure and stuff", you
> might enjoy this paper where Mike Shulman and I explain it:
>
> http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=15
>
> Best,
> jb


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Re: The omega-functor omega-category

[Michael Shulman]

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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Re: Re: The omega-functor omega-category

[Colin McLarty]

I like this discussion by Mike Shulman.  And a propos of the related
discussion of terminology I note the terms here describe mathematical
features (very well, I think) rather than focusing on whether one
*likes* the features.

2010/10/2 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.

best, Colin


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Re: The omega-functor omega-category

[Vaughan Pratt]

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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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: The omega-functor omega-category

[Michael Shulman]

Rereading my message, I realized I should perhaps clarify that not
just the name, but (as far as I know) the concept itself was
originated by Kelly and Lack.  The example of semigroups is also in
their paper, which is concerned mainly with the case when the
forgetful functor is also 2-monadic.  The resulting "property-like
2-monads" generalize both "lax-idempotent" (= "Kock-Zoberlein")
2-monads, such as those which assign colimits, and the dual
"colax-idempotent" 2-monads, such as those which assign limits.  But
they are strictly more general than either: for instance, a 2-monad
which assigns both limits and colimits is property-like, but not lax-
or colax-idempotent.

Mike

On Sun, Oct 3, 2010 at 6:32 AM, Colin McLarty <colin.mclarty@case.edu> wrote:
> I like this discussion by Mike Shulman.  And a propos of the related
> discussion of terminology I note the terms here describe mathematical
> features (very well, I think) rather than focusing on whether one
> *likes* the features.
>
> 2010/10/2 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.
>
> best, Colin
>


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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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Re: property_vs_structure

[Eduardo J. Dubuc]

I am talking about topological spaces

Marta Bunge wrote:
>
>
> 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.
>

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RE: property_vs_structure

[Marta Bunge]

Dear Eduardo,
Topological spaces or toposes, it is the same question. A space is locally connected iff its topos of sheaaves is locally connected. 
In my view, the question of whether the notion of a covering space is a structure or a property depends on the definition of covering space that one  adopts. 
If the definition is made for arbitrary spaces (as in Spanier, whom you quote), where a continuous map p from X to B is said to be a covering projection if each point of X has an open neighborhood U evenly covered by p,  then covering space is a structure, no matter what the nature of the base space is.  It so happens that, in the case of a locally connected space B, an alternative definition of a covering space can be given (as in R. Brown, Topology and groupoids) that refers directly to canonical neighborhoods of points of X (U open, connected, and each connected component  of the inverse image of U under p in X is mapped homeomorphically onto U) and, with this definition, covering space is indeed a property.  
So, in the locally connected case, the structure of covering space can be equivalently replaced by a property - but I believe that it is still a structure before those canonical choices are made. Can a structure be equivalent to a property,  yet not be a property?  
This is all I meant. I was not disputing a well known fact about covering spaces of locally connected spaces (or of toposes, for that matter).
Best regards,Marta



> Date: Thu, 7 Oct 2010 21:40:54 -0300
> From: edubuc@dm.uba.ar
> To: marta.bunge@mcgill.ca
> CC: categories@mta.ca
> Subject: Re: property_vs_structure
> 
> I am talking about topological spaces
> 
> Marta Bunge wrote:
>> 
>> 
>> 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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Re: property_vs_structure

[Eduardo J. Dubuc]

Marta Bunge wrote:
> Dear Eduardo, Topological spaces or toposes, it is the same question. A
> space is locally connected iff its topos of sheaaves is locally connected.

Of course, it is only that I wanted to focus in topological spaces to fix the
ideas and so that the  following two definitions can be compared.

ALSO, I had FORGOTTEN to say that in definition b) the V's are DISJOINT.

********
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
**********

>  In my view, the question of whether the notion of a covering space is a
> structure or a property depends on the definition of covering space that
> one adopts. If the definition is made for arbitrary spaces (as in Spanier,
> whom you quote), where a continuous map p from X to B is said to be a
> covering projection if each point of X has an open neighborhood U evenly
> covered by p, then covering space is a structure, no matter what the nature
> of the base space is.

Well, for locally connected space B (or any locally connected topos as you
pointed out), the forgetful functor into the topos of etale spaces over B is
full and faithful, and for X over B, there is only one structure (up to
isomorphism of structures).

I wanted this to be considered under the analysis:

***************
Michael Shulman wrote:
  >
  > property = forgetful functor is full and faithful
  > structure = forgetful functor is faithful
  > property-like structure = forgetful functor is pseudomonic
***************

You see, with this criteria (property = forgetful functor is full and
faithful) covering space is a property, something you do not think it is. I am
not saying who is right, just putting in evidence that it is a matter not
settled yet. May be full and faithfulness of the forgetful functor is not
enough to call a covering space to be a property of a continuous map ?

> It so happens that, in the case of a locally
> connected space B, an alternative definition of a covering space can be
> given (as in R. Brown, Topology and groupoids) that refers directly to
> canonical neighborhoods of points of X (U open, connected, and each
> connected component of the inverse image of U under p in X is mapped
> homeomorphically onto U) and, with this definition, covering space is
> indeed a property.  So, in the locally connected case, the structure of
> covering space can be equivalently replaced by a property - but I believe
> that it is still a structure before those canonical choices are made. Can a
> structure be equivalent to a property, yet not be a property?.

Well, interesting question, but first we have to settle:

   What do we mean by structure ?, and,  what do we mean by property ?.

Finally, I still do not understand what do you mean (in your first mail) by:

>>> 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.

Since in this case all trivialization structures ARE isomorphic!.

(if U and V are neighborhoods of b evenly covered, then the structures are
isomorphic in a connected W contained in the intersection)

best  e.d.


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FW: property_vs_structure

[Marta Bunge]

Dear Eduardo,
>
You ask
>
>>What do we mean by structure ?, and, what do we mean by property ?
>
On a given data, a structure is additional data on it that, if it exists, it is not necessarily unique up to isomorphism. A  property is additional data such that, if it exists, it is unique up to isomorphism (in the model theoretic sense). 
>

>You also write
>>Finally, I still do not understand what do you mean (in your first mail) by:>>
>>>>> 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.
>>
>>> Since in this case all trivialization structures ARE isomorphic!.
>>
>>> (if U and V are neighborhoods of b evenly covered, then the structures are
>>> isomorphic in a connected W contained in the intersection)
>>>
>
Precisely, it comes down to what definition of covering space one adopts.  If the general definition is adopted then, even in the locally connected  case, it is a structure, as any two trivialization structures given by  U, V, need not be isomorphic except on a connected W contained in the intersection. If, on the other hand, the specifi definition is adopted, where canonical neighborhoods (U open, connected, and each connected component of the inverse image of U under p in X is mapped homeomorphically onto U) are given as part of the structure, then the structure is a property. 
> 
I think that I have nothing else to say on his matter.
>
Best,Marta

> ----------------------------------------
>> Date: Fri, 8 Oct 2010 18:53:31 -0300
>> From: edubuc@dm.uba.ar
>> To: marta.bunge@mcgill.ca
>> CC: categories@mta.ca
>> Subject: Re: property_vs_structure
>>
>>
>>
>> Marta Bunge wrote:
>>> Dear Eduardo, Topological spaces or toposes, it is the same question. A
>>> space is locally connected iff its topos of sheaaves is locally connected.
>>
>> Of course, it is only that I wanted to focus in topological spaces to fix the
>> ideas and so that the following two definitions can be compared.
>>
>> ALSO, I had FORGOTTEN to say that in definition b) the V's are DISJOINT.
>>
>> ********
>> 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
>> **********
>>
>>> In my view, the question of whether the notion of a covering space is  a
>>> structure or a property depends on the definition of covering space that
>>> one adopts. If the definition is made for arbitrary spaces (as in Spanier,
>>> whom you quote), where a continuous map p from X to B is said to be a
>>> covering projection if each point of X has an open neighborhood U evenly
>>> covered by p, then covering space is a structure, no matter what the nature
>>> of the base space is.
>>
>> Well, for locally connected space B (or any locally connected topos as  you
>> pointed out), the forgetful functor into the topos of etale spaces over B is
>> full and faithful, and for X over B, there is only one structure (up  to
>> isomorphism of structures).
>>
>> I wanted this to be considered under the analysis:
>>
>> ***************
>> Michael Shulman wrote:
>>>
>>> property = forgetful functor is full and faithful
>>> structure = forgetful functor is faithful
>>> property-like structure = forgetful functor is pseudomonic
>> ***************
>>
>> You see, with this criteria (property = forgetful functor is full and
>> faithful) covering space is a property, something you do not think it is. I am
>> not saying who is right, just putting in evidence that it is a matter not
>> settled yet. May be full and faithfulness of the forgetful functor is not
>> enough to call a covering space to be a property of a continuous map ?
>>
>>> It so happens that, in the case of a locally
>>> connected space B, an alternative definition of a covering space can be
>>> given (as in R. Brown, Topology and groupoids) that refers directly to
>>> canonical neighborhoods of points of X (U open, connected, and each
>>> connected component of the inverse image of U under p in X is mapped
>>> homeomorphically onto U) and, with this definition, covering space is
>>> indeed a property. So, in the locally connected case, the structure  of
>>> covering space can be equivalently replaced by a property - but I believe
>>> that it is still a structure before those canonical choices are made. Can a
>>> structure be equivalent to a property, yet not be a property?.
>>
>> Well, interesting question, but first we have to settle:
>>
>> What do we mean by structure ?, and, what do we mean by property ?.
>>
>> Finally, I still do not understand what do you mean (in your first mail) by:
>>
>>>>> 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.
>>
>> Since in this case all trivialization structures ARE isomorphic!.
>>
>> (if U and V are neighborhoods of b evenly covered, then the structures  are
>> isomorphic in a connected W contained in the intersection)
>>
>> best e.d.
>
  		 	   		  

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Re: property_vs_structure

[Eduardo J. Dubuc]

Marta Bunge wrote:
> Dear Eduardo,>>You ask>>>> What do we mean by structure ?, and, what do we
> mean by property ?.>>On a given data, a structure is additional data on it
> that, if it exists, it is not necessarily unique up to isomorphism. A
> property is additional data such that, if it exists, it is unique up to
> isomorphism (in the model theoretic sense).

Well, here it is necessary first to establish what do we mean by
"isomorphism". To do this we need a way to compare the structures, that is, we
have to define morphism of structures (see below). Without this, the above is
meaningless.

People discussing structure vs property were giving examples where all this
was clear and straightforward (invertibility in a monoid, neutral element in a
semigroup, etc). My original purpose when I wrote my first mail was to
consider a less trivial example testing the following  "definitions", that I
see you subscribe above at least in what it concerns "structure" and "property":

*********
Michael Shulman wrote:
  > (**)
  > property = forgetful functor is full and faithful
  > structure = forgetful functor is faithful
  > property-like structure = forgetful functor is pseudomonic
**********

>> You also write>>>> Finally, I
> still do not understand what do you mean (in your first mail) by:
>>>>> 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.
>>> Since in this case all trivialization structures ARE isomorphic!. (if U
>>> and V are neighborhoods of b evenly covered, then the structures are
>>> isomorphic in a connected W contained in the interesection)
>>>
> Precisely, it comes down to what definition of covering space one adopts.
> If the general definition is adopted then, even in the locally connected
> case, it is a structure, as any two trivialization structures given by U,
> V, need not be isomorphic except on a connected W contained in the
> intersection. If, on the other hand, the specific definition is adopted,
> where canonical neighborhoods (U open, connected, and each connected
> component of the inverse image of U under p in X is mapped homeomorphically
> onto U) are given as part of the structure, then the structure is a
> property. >> I think that I have nothing else to say on his
> matter.>>Best,Marta

Let us analyze carefully: take definition a): "open neighborhood U of b evenly
covered", say in X and in Y (over B). Given two structures (in the same U),
the definition of morphism is clear. It is a continuous function from X to Y
over B together with some extra data. When U is not connected, a function over
B does not necessarily carries the extra data, and the forgetful functor is
not full. There are non isomorphic structures on the same X. It is not a
property. When U is connected (no loss of generality in the locally connected
case), any continuous function over B carries automatically this extra data.
The forgetful functor is full, and for a given X, there is only one structure
up to isomorphisms. It is a property according to  definition (**) above.

All this if we have a fixed U (or a fixed cover of B). But we need all
covering projections, trivialized over all possible covers.

Now, what happens if we have an structure in a different open neighborhood V
of b. How we define morphism ?. Well, we take the restriction of the
structures to an open W contained in U and in V, and we are in the previous
case. In this way, when U, V and W are all connected, any continuous function
over B carries automatically this extra data. The forgetful functor is full,
and for a given X, there is only one structure up to isomorphisms.

It seems to me that, once you define the meaning of isomorphism between
structures, in the locally connected case for a given X there is only one
structure up to isomorphism.
Consider what  you call "the specific definition" b) (U open, connected, and
each connected component of the inverse image of U under p in X is mapped
homeomorphically onto U) which makes sense only for the locally connected case
(it use connected components). Well, both definitions are  equivalent, and
they are both "properties" according to the definition of property (**) quoted
above.

Classically, covering projections were treated as spaces with a certain
property, and so defined. And everything was fine since immediately after the
definition, it is assumed local connectivity once they start to develop the
subject. In shape theory they had to deal with non locally connected spaces,
and they run into trouble. There was a "hidden structure" which can not be
ignored in the non locally connected case.

It was this hidden structure that I put into consideration in order to analyze
if the concepts of property and  structure given in (**) are or are not fine
enough.

  From my previous mails:
*****
                  local homeomorphism   covering map

Both are "properties" of a continuous function, but they are not of the same kind.
in "covering map" there is hidden a structure (not considered classically),
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.
*****
May be full and faithfulness of the forgetful functor is not enough to call a
covering space to be a property of a continuous map ?
******

best to all   e.d.

> ----------------------------------------
>> Date: Fri, 8 Oct 2010 18:53:31 -0300 From: edubuc@dm.uba.ar To:
>> marta.bunge@mcgill.ca CC: categories@mta.ca Subject: Re:
>> property_vs_structure
>>
>>
>>
>> Marta Bunge wrote:
>>> Dear Eduardo, Topological spaces or toposes, it is the same question. A
>>>  space is locally connected iff its topos of sheaaves is locally
>>> connected.
>> Of course, it is only that I wanted to focus in topological spaces to fix
>> the ideas and so that the following two definitions can be compared.
>>
>> ALSO, I had FORGOTTEN to say that in definition b) the V's are DISJOINT.
>>
>> ******** 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 **********
>>
>>> In my view, the question of whether the notion of a covering space is a
>>>  structure or a property depends on the definition of covering space
>>> that one adopts. If the definition is made for arbitrary spaces (as in
>>> Spanier, whom you quote), where a continuous map p from X to B is said
>>> to be a covering projection if each point of X has an open neighborhood
>>> U evenly covered by p, then covering space is a structure, no matter
>>> what the nature of the base space is.
>> Well, for locally connected space B (or any locally connected topos as
>> you pointed out), the forgetful functor into the topos of etale spaces
>> over B is full and faithful, and for X over B, there is only one
>> structure (up to isomorphism of structures).
>>
>> I wanted this to be considered under the analysis:
>>
>> *************** Michael Shulman wrote:
>>> property = forgetful functor is full and faithful structure = forgetful
>>> functor is faithful property-like structure = forgetful functor is
>>> pseudomonic
>> ***************
>>
>> You see, with this criteria (property = forgetful functor is full and
>> faithful) covering space is a property, something you do not think it is.
>> I am not saying who is right, just putting in evidence that it is a
>> matter not settled yet. May be full and faithfulness of the forgetful
>> functor is not enough to call a covering space to be a property of a
>> continuous map ?
>>
>>> It so happens that, in the case of a locally connected space B, an
>>> alternative definition of a covering space can be given (as in R.
>>> Brown, Topology and groupoids) that refers directly to canonical
>>> neighborhoods of points of X (U open, connected, and each connected
>>> component of the inverse image of U under p in X is mapped
>>> homeomorphically onto U) and, with this definition, covering space is
>>> indeed a property. So, in the locally connected case, the structure of
>>> covering space can be equivalently replaced by a property - but I
>>> believe that it is still a structure before those canonical choices are
>>> made. Can a structure be equivalent to a property, yet not be a
>>> property?.
>> Well, interesting question, but first we have to settle:
>>
>> What do we mean by structure ?, and, what do we mean by property ?.
>>
>> Finally, I still do not understand what do you mean (in your first mail)
>> by:
>>
>>>>> 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.
>> Since in this case all trivialization structures ARE isomorphic!.
>>
>> (if U and V are neighborhoods of b evenly covered, then the structures
>> are isomorphic in a connected W contained in the intersection)
>>
>> best e.d.
>


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Re: property_vs_structure

[George Janelidze]

Dear All,

Apologizing to those who heard this many times, I would like to say - since
we are talking about "property_vs_structure" for covering maps:

What I called "Galois structures" more than 20 years ago were exactly the
structures needed to define covering morphisms in general categories. It is
well known (although it is not clear what it really means!) that Poincare's
first ideas about covering maps were inspired by Galois theory, and so
algebra was "there" even before topology. Having also in mind Grothendieck's
work, topos-theoretic developments, and Magid's work, I am certainly not
original in saying that covering maps should belong to category theory
rather than to topology.

A Galois structure (say, on a category C) essentially consists of three
ingredients (although various modifications are possible):

(i) A functor I : C ---> X. It should better have a right adjoint, and it  is
wonderful if it is semi-left-exact or, which is almost the same in a sense,
if it is a fibration. But relative versions of these conditions involving  F
below are also good.

(ii) A class F of morphisms in C. All covering morphisms we are going to
define will be inside F.

(iii) Another class E of morphisms in C. Although usually I do not mention
it separately because I prefer to define it as the class of effective
F-descent morphisms. According to the terminology, Marta and Eduardo used  in
their messages, E should now be called a "trivialization structure".

For a given Galois structure, the covering morphisms are defined as in my
papers, and I would repeat the question about "property vs structure" for
covering morphisms as follows:

Given a category C, where the concept of a covering seems to be important,
do we want to fix "the best" Galois structure, or we should consider
several/many such structures?

It is certainly a matter of taste, but to feel the taste one surely needs
examples. In my opinion the ones listed below are especially important; they
were investigated together with several people you know, whom I was very
honoured to work with.

Example 1. C = the opposite category of commutative rings. Here we have  a
very good candidate, which is: X = the category of Stone spaces; I : C --->
X = the Boolean spectrum functor; F = the class of all morphisms in C; E =
the class of all effective descent morphisms in C. In this case covering
morphisms are the same what A. R. Magid called componentially locally
strongly separable algebras (considering an R-algebra A as a morphism A --->
R in C), and are THE most general algebras for which he developed his
"separable Galois theory" presented in [A. R. Magid, The separable Galois
theory of commutative rings, Marcel Dekker, 1974].

Example 2. C = the category of locally connected topological spaces. Here
again, we seem to have "the best candidate". It is: X = the category of
sets; I : C ---> X = the functor sending spaces to the sets of their
connected components; F = the class of local homeomorphisms of locally
connected spaces; F = the class of surjective maps from E. The covering
morphisms are then the same the covering morphisms of locally connected
spaces in the usual sense.

Example 3. C is a locally connected topos. This essentially generalizes the
previous example and everything happens as there (although here F is the
class of all morphisms and E the class of all epimorphisms in C of course).
Marta knows much more than I do about this example and its connections with
other topos-theoretic constructions; my only contribution is the short paper
[G. Janelidze, A note on Barr-Diaconescu covering theory, Contemporary
Mathematics 131, 3, 1992, 121-124].

Example 4. C = Fam(A) (or FiniteFam(A)), where A is an arbitrary category
with terminal object and "multi-pullbacks" (which simply means that C has
pullbacks). This is a further generalization of the same thing, and
everything can be repeated, but instead of "epimorphism" we should say
"effective descent morphisms" (which is the same thing in the case of a
topos). There are many non-topos-theoretic important special cases. For
instance if C is the category of all (small) categories, then the covering
morphisms are as they should be, that is functors that are discrete
fibrations and discrete opfibrations at the same time (this observation is
due to Steve Lack, although Steve never published it). If C is the category
of all (small) groupoids, then this becomes even nicer since the discrete
fibrations of groupoids are the same as discrete opfibrations, are Ronnie
Brown often tells us how nicely can they be used in homotopy theory...

Example 5. C = the category of compact Hausdorff spaces. Here "the best
candidate" seems to be: X = the category of Stone spaces; I : C ---> X
sending compact Hausdorff spaces to the Stone spaces of their connected
components; F = the class of all morphisms in C; E = the class of all
morphisms in C that are surjections. As shown in [A. Carboni, G. Janelidze,
G. M. Kelly, and R. Paré, On localization and stabilization of factorization
systems, Applied Categorical Structures 5, 1997, 1-58], the covering
morphisms here are the same as light maps in the sense of Eilenberg and
Whyburn.

Example 6. C = the category of simplicial sets. Since it is a category of
the form Fam(A), Example 4 can be used. However, [R. Brown and G. Janelidze,
Galois theory of second order covering maps of simplicial sets, Journal of
Pure and Applied Algebra 135, 1999, 23-31] gives a Galois structure that
produces a larger (new) class of covering morphisms. In that structure X is
the category of groupoids; I : C ---> X the fundamental groupoid functor,  F
the class of Kan fibrations, and E the class of surjective Kan fibrations.
Another "simplicial Galois theory" is presented in [M. Grandis and G.
Janelidze, Galois theory of simplicial complexes, Topology and its
Applications 132, 3, 2003, 281-289].

Example 7. C = the category of groups. The "most classical" candidate would
be: X = the category of abelian groups; I : C ---> X = the abelianization
functor; E = F = the class of group epimorphisms. In this case the covering
morphisms are the same as central extensions. This was my first example of a
"very-non-Grothendieck Galois theory". A bit later I realized that C can be
replaced with any variety of groups with multiple operators in the sense of
[P. J. Higgins, Groups with multiple operators, Proc. London Math. Soc.
(3)6, 1956, 366-416] and X with any subvariety in C - and then we get
central extensions relative to a subvariety in the sense of A. Fröhlich's
school (see e. g. [A. Fröhlich, Baer-invariants of algebras, Trans. AMS  109,
1963, 221-244], [A. S.-T. Lue, Baer-invariants and extensions relative to  a
variety, Proc. Cambridge Philos. Soc. 63, 1967, 569-578], [J.
Furtado-Coelho, Varieties of W-groups and associated functors, Ph.D. Thesis,
University of London, 1972]). The next step was to get rid of groups
completely and Max Kelly and I found out that the crucial property that
helps to work with generalized central extensions is congruence modularity,
and we wrote [G. Janelidze and G. M. Kelly, Galois theory and a general
notion of a central extension, Journal of Pure and Applied Algebra 97, 1994,
135-161]. Many further results were obtained by Marino Gran, partly in
collaboration with Dominique Bourn (see [M. Gran, Applications of
categorical Galois theory in universal algebra, Fields Institute
Communications 43, 2004, 243-280] and references there for what was done
until 2002/3), Tomas Everaert and Tim Van der Linden, and by Tim and Tomas
separately and together. Writing this I feel now bad not to say more about
their brilliant results, also involving higher central extensions (see in
particular [T. Everaert, M. Gran, and T. Van der Linden, Higher Hopf
formulae for homology via Galois theory, Advances in Mathematics 217, 2008,
2231-2267]). I shall gladly say more at another occasion.

Examples 8-11?. Let C be one of the following three categories: (a)
topological spaces; (b) locales; (c) toposes; (d) schemes of algebraic
geometry (although (c) is 2-dimensional). I still do not know anything like
"the best candidates"...

And finally there are trivial examples: (a) C = X, with I the identity
functor; and (b) X = 1. Taking (in both of them) E to be the class of all
morphisms in C (and suitable F) we will have: all morphisms in C are
covering morphisms in the situation (a), and only isomorphisms are covering
morphisms in the situation (b).

In particular, since "the largest" Galois structure is trivial, I would
conclude: there is no "the best" Galois structure, and one should rather
consider several/many such structures.

George



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RE: property_vs_structure

[Marta Bunge]

Dear Eduardo,>In a  previous message of mine and  in response to your  questions >
>> What do we mean by structure ?, and, what do we mean by property  ?
>
 I wrote>
>> On a given data, a structure is additional data on it such that, if it exists, it is not necessarily unique up to isomorphism. >>A property is additional data such that, if it exists, it is unique up to isomorphism (in the model theoretic sense).>
to which you replied>  
>> Well, here it is necessary first to establish what do we mean by 
>> "isomorphism". To do this we need a way to compare the structures, that is, we 
>> have to define morphism of structures (see below). Without this, the above is 
>> meaningless.>
Notice that I specified "in the model theoretic sense". In that context, the notions of a structure and of a morphism of structures are defined and it is that sense that I meant them. They are perfectly meaningful. 
>
You added> 
>> People discussing structure vs property were giving examples where all this 
>> was clear and straightforward (invertibility in a monoid, neutral element in a 
>> semigroup, etc). My original purpose when I wrote my first mail was to 
>> consider a less trivial example testing the following "definitions", that I 
>> see you subscribe above at least in what it concerns "structure" and "property":
> 
> *********
> Michael Shulman wrote:
>> (**)
>> property = forgetful functor is full and faithful
>> structure = forgetful functor is faithful
>> property-like structure = forgetful functor is pseudomonic
> **********
>
I do not "subscribe" to these notions. I simply use the well-known notions from first-order logic and model theory as I said earlier. I am aware of the difference between the locally connected  and the general case concerning covering projections and it is not the mathematics that I was disputing.  But I still have trouble following your analysis of this situation as was your purpose. The difference is that, whereas you consider the notion of a covering projection to be a property of a continuous map p from X to B which, in the general (non locally connected case) is a "hidden structure", I view the notion of a covering projection to be a structure in the first place and, in the specific locally connected case, one that may be equivalently reduced to a property. Every property is a structure, but not  every structure is  reducible to a property. 

>
All the best,  Marta



  		 	   		  

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Re: property_vs_structure

[Marta Bunge]

Dear George,



Thank you for reminding us of your old notion of Galois structure and covering morphism in general categories. Although tangentially relevant to the discussion initiated by Eduardo Dubuc, it relates to examples of properties of continuous maps of spaces (or or morphisms of  toposes) studied in my book with Jonathon Funk, which may be relevant. I sent you this privately already, but on second thoughts I think it might be useful to make it public. I begin by quoting a paragraph from your posting. 

>
>

>> Example 4. C = Fam(A) (or FiniteFam(A)), where A is an arbitrary category
>> with terminal object and "multi-pullbacks" (which simply means that C has
>> pullbacks). This is a further generalization of the same thing, and
>> everything can be repeated, but instead of "epimorphism" we should say
>> "effective descent morphisms" (which is the same thing in the case of a
>> topos). There are many non-topos-theoretic important special cases. For
>> instance if C is the category of all (small) categories, then the covering
>> morphisms are as they should be, that is functors that are discrete
>> fibrations and discrete opfibrations at the same time (this observation is
>> due to Steve Lack, although Steve never published it). If C is the category
>> of all (small) groupoids, then this becomes even nicer since the discrete
>> fibrations of groupoids are the same as discrete opfibrations, are Ronnie
>> Brown often tells us how nicely can they be used in homotopy theory...



> 
>

The notions of discrete fibration and discrete opfibration are lifted from categories to geometric morphisms of toposes (in M. Bunge and J. Funk, Singular Coverings of Toposes, LNM 1890, Springer, 2006, Chapter 9) relative to the symmetric KZ-monad called M therein for "measures" (M.Bunge and A.Carboni, JPAA 105 (1995) 233-249).  They are, respectively, the local homeomorphisms and the complete spreads (singular coverings). A local homeomorphism over a locally connected space E with defining object X  is said to be an unramified covering if it is also a complete spread. Unramified coverings generalize covering morphisms  over a locally connected space-- if X is a locally constant object of a locally connected space E,  then the corresponding local homeomorphism is a complete spread, hence an unramified covering. The class of unramified coverings is strictly larger  than the class of locally constant coverings, even over a locally connected space (J. Funk and E.D. Tymchatyn, Unramified maps, J. Geometric Topology 1(3) (2001) 249-280). Under hypotheses of the locally simply connected kind, unramified coverings are locally constant. The larger class of  unramified coverings  has some nice properties which the class of locally constant coverings fails to have -- for instance, they compose. Moreover, a van Kampen theorem holds not just for the class of locally constant coverings but also for the larger class of unramified coveirngs (M.Bunge and S. Lack, Van Kampen theorems for toposes, Advances in Mathematics 179/2 (2003) 291-317). It is clear from your theory that both classes of morphisms are instances of what you call a Galois structure on the category of  (locally connected) topological spaces. 
>


>

Best wishes,
Marta







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Re: property_vs_structure

[George Janelidze]

Dear Marta,

Many thanks, and apologizing for the delay, I am now answering:

As you know better than I do, Topos Theory has many aspects and great impact
(using these days' expression) on many areas of mathematics. But in this
message let me consider only one of its aspects, namely that the
(2-)category TOP of toposes can be considered as one of the candidates for
'the right geometric/topological category'. The long list of other possible
candidates includes

Fam(A) = the category of families of objects of a category A, such that
Fam(A) has pullbacks (e.g. every (cocomplete) locally connected topos is
such);

Loc = the category of locales,

Top = the category of topological spaces,

CHTop = the category of compact Hausdorff spaces,

LaxAlg(T,V) = the categories of lax (T,V)-algebras in the sense of M. M.
Clementino, D. Hofmann, and W. Tholen (if T is the ultrafilter monad of
Sets, and V = {0,1}, then LaxAlg(T,V) = Top by a theorem of M. Barr),

Schemes = the category of schemes in algebraic geometry,

CR^o = the opposite category of commutative rings,

and many others (I listed only those that will be mentioned below). Each of
them certainly has various (subcategories with various) Galois structures
with interesting covering morphisms. But essentially only in the cases of
CR^o, CHTop, and Fam(A) I have a feeling that the Galois structure I am
using (which is just the adjunction with Stone spaces for CR^o and for
CHTop, and with sets for Fam(A)) I am using is THE right one.

In particular the case of TOP seems to be very interesting, and probably
"the answer" would give an answer for Loc, while a "good answer" for Loc
might suggest something for TOP.

I am not sure I fully understood what you say about unramified coverings
versus locally constant coverings. Are you even saying that you found a
Galois structure on TOP, or on any subcategory of TOP, whose coverings are
exactly the unramified coverings (and the situation is non-trivial in the
sense that unramified coverings are not the same as locally constant
coverings? That would be wonderful!

Independently of that your work on study and comparing what you call local
homeomorphisms, complete spreads, and unramified coverings in TOP is
absolutely very interesting! And there should be a connection to be
understood between it and the work of Maria Manuel Clementino and Dirk
Hofmann on similar concepts in LaxAlg(T,V). You say

"The notions of discrete fibration and discrete opfibration are lifted from
categories to geometric morphisms of toposes (in M. Bunge and J. Funk,
Singular Coverings of Toposes, LNM 1890, Springer, 2006, Chapter 9) relative
to the symmetric KZ-monad called M therein for "measures" (M.Bunge and
A.Carboni, JPAA 105 (1995) 233-249). They are, respectively, the local
homeomorphisms and the complete spreads (singular coverings)..."

And this is to be compared with the following:

Just as for categories, there are discrete fibrations and discrete
opfibrations of preorders, and coverings are exactly those that are discrete
fibrations and discrete opfibrations at the same time. On the other hand
finite preorders are the same as finite topological spaces, and discrete
fibrations of finite preorders are the same as the local homeomorphisms of
finite topological spaces. This generalizes to the infinite case as follows:
Using the ultrafilter convergence, one can define discrete fibrations of
T-preorders (=lax T-algebras = T-categories), where T is the ultrafilter
monad on the category of sets; and it turns out that:

(i) the class of discrete fibrations of T-preorders is not pullback stable;

(ii) the pullback stable discrete fibrations of T-preorders are the same as
local homeomorphisms of general topological spaces (see [M. M. Clementino,
D. Hofmann, and G. Janelidze, Local Homeomorphisms via Ultrafilter
Convergence, Proc. AMS 133, 3, 2004, 917-922]).

Similarly to the case of T-preorders one can define discrete fibrations and
discrete opfibrations of lax (T,V)-algebras (for arbitrary T and V), and
this is what I would like to compare with your local homeomorphisms and
complete spreads. Of course the categories TOP and LaxAlg(T,V) are so
different that the only way to make such a comparison, would be to find
appropriate categorical definitions. I don't know how to define a local
homeomorphism categorically, but maybe something similar to the story of
separability (see [A. Carboni and G. Janelidze, Decidable (=separable)
objects and morphisms in lextensive categories, Journal of Pure and Applied
Algebra 110, 1996, 219-240] and [G. Janelidze and W. Tholen, Strongly
separable morphisms in general categories, Theory and Applications of
Categories 23, 5, 2010, 136-149]) can be done.

It is also interesting that you say:

"...Under hypotheses of the locally simply connected kind, unramified
coverings are locally constant..."

while what is done in my paper with Aurelio almost suggests to use the path
lifting property to make a separable morphism a covering morphism (the
occurrence of the path-lifting property is familiar of course, but the fact
that it is "almost suggested" categorically a kind of new).

All these problems, as well as the absence (so far!) of good Galois
structures in TOP, Loc, Top, and Schemes, are related to each other, and
more purely-categorical concepts are needed to understand these
relationships better.

Best wishes,
George



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Re: property_vs_structure

[Marta Bunge]

Dear George,
Thank you for your comments and questions.  I will have to look closely into the conditions of coverings in your sense to make sure that the off-my-hat assertion that unramified maps are an example is indeed correct. Also, your remark about the path-lifting property being "categorically forced" upon you in your work with Aurelio is particularly interesting to me as you suggested. I will most certainly look into those two points. 
I am also familiar with the T-categories and lax algebras, so that the work of Clementino, Hoffman and yourself should be easy for me to understand.  I point out (in case you do not know this) that my paper "Coherent extensions and relational algebras", Trans. AMS 197 (1974) 355-390 introduces lax adjointness,  examines examples including the T-categories of Burroni, and gives a new analysis of the example of topological spaces in this new light.  I mentioned this paper to Maria Manuel Clementino the first time I heard her talk about this subject, as it was obviously relevant. 
However, in view of my imminent trip to Buenos Aires, where I will spend most of November, I doubt that I will find the time this week to look into all of that before I return. 
With best regards,Marta

************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
McGill UniversityBurnside Hall, Office 1005
805 Sherbrooke St. West
Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810/3800  
Home: (514) 935-3618
marta.bunge@mcgill.ca 
http://www.math.mcgill.ca/~bunge/
************************************************



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Re: property_vs_structure

[Marta Bunge]

Dear George,>

I haste to correct a possible misconception arising from my previous posting, and to propose an idea in connection with it. You wrote: 

>
> I am not sure I fully understood what you say about unramified coverings
> versus locally constant coverings. Are you even saying that you found a
> Galois structure on TOP, or on any subcategory of TOP, whose coverings are
> exactly the unramified coverings (and the situation is non-trivial in the
> sense that unramified coverings are not the same as locally constant
> coverings? That would be wonderful!

>
Let C = LoCo/E, defined as in your book Galois Theories (with F. Borceux). An object p  of C with domain F is said to be a covering morphism if there exists a morphism e of effective descent in Top with codomain E such that (F,p) is split by e.  A complete spread p of C with domain F  - that is, an unramified morphism, need not be a covering morphism in C  in your sense, as we know. Whether there is a Galois structure on C whose coverings are precisely the unramified coverings without it forcing them to be identified with the locally constant coverings does not seem likely. At least we know that the class of unramified coverings in C is stable under pullbacks and has other nice properties, so C is a natural choice of universe. The fact that we have called "coverings" the unramified morphisms may then be misleading if coverings are to be tied up with Galois theory. 

>
Nevertheless, it is the case in topology that the notion of a covering in  the traditional sense has been enlarged to include branchings but not folds. A branched covering of a locally connected space E (R.H. Fox 1957)  is  the spread completion of a locally constant covering on a pure open subspace U of E, thought off as "the complement of a knot". At least in this case it is meaningful to consider the "branched fundamental groupoid" (or "knot groupoid") of E with non-singular part U. It might be of interest to consider a notion of  "generalized Galois theory" to encompass this notion of "generalized covering morphism". Let me know what you think. Relevant discussions in topos theory can be found in (Bunge-Niefield 2000), (Funk 2000),  (Bunge-Lack 2003), as well as in (Bunge-Funk 2006, 2007). 

>

With best regards,

>

Marta




************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics
McGill University
Burnside Hall, Office 1005
805 Sherbrooke St. West
Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810/3800
Home: (514) 935-3618
marta.bunge@mcgill.ca
http://www.math.mcgill.ca/~bunge/
************************************************





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Re: property_vs_structure

[George Janelidze]

Dear Marta,

I fully agree with every word you say, and I can only add:

1. After these many years I trust myself that covering morphisms should only
be defined via Galois theory. But exactly for this reason any adjective
should indicate that Galois theory is not applicable (or we do not know yet,
how to apply it). So for me "unramified coverings" is one of possible good
names for something that is not presented as coverings with respect to some
Galois theory.

2. More importantly than terminology, I think to find what you call
"generalized Galois theory" would be very interesting, and, as I already
said many times, I should study your work seriously. And again, in my
opinion the aim would be to find a general-categorical definition that gives
good examples in all (or in the most of) those categories I mentioned before
(that is, not just in Top and TOP, but also, say, in CR^o = the opposite
category of commutative rings). Such investigations will - I believe - soon
or late lead to a beautiful unification of certain big parts of algebraic
topology and algebraic geometry.

With best regards, George

----- Original Message -----
From: "Marta Bunge" <marta.bunge@mcgill.ca>
To: <categories@mta.ca>
Sent: Sunday, October 24, 2010 11:15 PM
Subject: categories: Re: property_vs_structure


Dear George,

I haste to correct a possible misconception arising from my previous
posting, and to propose an idea in connection with it. You wrote:

>
> I am not sure I fully understood what you say about unramified coverings
> versus locally constant coverings. Are you even saying that you found a
> Galois structure on TOP, or on any subcategory of TOP, whose coverings are
> exactly the unramified coverings (and the situation is non-trivial in the
> sense that unramified coverings are not the same as locally constant
> coverings? That would be wonderful!

>
Let C = LoCo/E, defined as in your book Galois Theories (with F. Borceux).
An object p of C with domain F is said to be a covering morphism if there
exists a morphism e of effective descent in Top with codomain E such that
(F,p) is split by e. A complete spread p of C with domain F - that is, an
unramified morphism, need not be a covering morphism in C  in your sense, as
we know. Whether there is a Galois structure on C whose coverings are
precisely the unramified coverings without it forcing them to be identified
with the locally constant coverings does not seem likely. At least we know
that the class of unramified coverings in C is stable under pullbacks and
has other nice properties, so C is a natural choice of universe. The fact
that we have called "coverings" the unramified morphisms may then be
misleading if coverings are to be tied up with Galois theory.


...


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Re: property_vs_structure

[Marta Bunge]

Dear George,
>
Thanks for your interesting response. Let me just comment here on your "addition 1" below. It is my contention that "unramified coverings" is not an  appropriate expression to describe those "coverings" not associated with a  Galois theory in your sense, as first, there is a specific meaning attached to it, and secondly, a "branched Galois theory" already exists informally in the subject of knot groupoids. One may possibly generalize your  Galois theories to include these phenomena. I devote one paragraph to each  contention. 

>
1. When R.H.Fox ("Covering spaces with singularities", R.H. Fox et al, editors, Algebraic Geometry and Topology: A Symposium in honor of S. Lefschetz, Princeton University Press, 1957, 243-257) introduced spreads and their completions, he had in mind what the title of his paper says,  that is, "coverings with singularities" so, not the traditional locally constant coverings. There could be "ramifications", or branchings over points of the base. But no folds. Specifically, he was thinking of branched coverings  (branching over a knot in the base) as the spread completions of locally constant coverings, in which branching points were added to the domain space. This is what led him to define a notion of spread, and then perform a completion process leading to another spread singled out among all such corresponding to a given cosheaf on the base space. The branched coverings, and more generally the complete spreads of which they are  the motivating example, are "ramified". Now, add the condition that the complete spread (e.g. a branched covering) be a local homeomorphism. This  does not force it to be locally constant, as we know, but it cannot then have ramifications. Hence the expression "unramified coverings". 
>
2. As I said in my last posting, the  "branched coverings", which are  very important in topology, yet do not correspond to any Galois theory in your sense, should correspond to a "generalized Galois theory" or to a "branched Galois theory". To support my contention, note that, in (M. Bunge and S. Lack, van Kampen theorems for toposes, Advances in Mathematics 179/2 (2003) 291-317), we obtain, as an application of the van Kampen theorems we prove therein,  a connection with the use of the automorphism group of a (universal) branched covering in the calculation of knot groups, as advocated by Fox. In particular, and the point I am making here, the expression "unramified coverings" does not describe them accurately, as there may be ramifications. For a topos E, there is a biequivalence of the 2-categories of branched coverings of E branching over an object  Y (the latter thought of as the complement of a knot K)  on the one hand, and that of all locally constant coverings of the slice topos E/Y on the other. The latter may in turn be viewed as the fundamental groupoid of E/Y, or as the knot groupoid G(K) of K. There is a "Galois theory" there not associated with coverings in your sense, that is, with locally constant coverings. 

>
All of this requires further investigation, for which I will have no time  possibly until December, due to my trip to Buenos Aires. 
>

With best regards,
>
Marta

************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
McGill UniversityBurnside Hall, Office 1005
805 Sherbrooke St. West
Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810/3800  
Home: (514) 935-3618
marta.bunge@mcgill.ca 
http://www.math.mcgill.ca/~bunge/
************************************************


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Re: property_vs_structure

[Marta Bunge]

Dear George,
>
Thanks for your interesting response. I contend that "unramified coverings" is not an appropriate expression to describe those "coverings" not associated with a Galois theory in your sense since there is a specific meaning attached to it. I would simply call them "generalized coverings".  Such would be the unramified coverings or just the complete spreads. In fact also  the branched coverings but there is a difference here. For branched coverings, the motivation of it all,  a "branched Galois theory" already exists informally in the subject of knot groupoids. Moreover,  this example gives you the reason concerning coverings and Galois theories.  I devote one paragraph to each contention. 
>
1. Terminology "unramified". When R.H.Fox introduced spreads and their completions, he had in mind what the title of his paper says, that is, "coverings with singularities" so, not the traditional locally constant coverings. There could be "ramifications", or branchings over points of the base. But no folds. Specifically, he was thinking of branched coverings   (branching over a knot in the base) as the spread completions of locally constant coverings, in which branching points were added to the domain space. This is what led him to define a notion of spread, and then perform a completion process. The branched coverings, and more generally the complete spreads of which they are the motivating example, are "ramified". Now, add the condition that the complete spread (e.g. a branched covering) be a local homeomorphism. This does not force it to be locally constant, as we know, but it cannot then have ramifications. Hence the expression "unramified coverings". That is why I suggested "generalized coverings" - for both the ramified and the unramified which are not locally constant. >
2. Do we need a generalized Galois theory to deal with the branched coverings? Perhaps but, if so, it would not be a big generalization. Indeed,   the  "branched coverings", which are very important in topology, although they do not correspond directly to any Galois theory in your sense, they do through a biequivalence. Indeed for a topos E (say), there is  a biequivalence of the 2-categories of branched coverings of E branching over an object Y (the latter thought of as the complement of a knot K)  on  the one hand, and that of all locally constant coverings of the slice topos E/Y on the other (Bunge-Niefield 2000, Funk 2000, Bunge-Lack 2003).  Now, C(E/Y) is in turn viewed as the knot groupoid G(K) of K. So, branched coverings branching over a knot is not an instance of coverings in your sense as they are not locally constant,  yet there is a Galois theory  associated with them, however, only through a non-trivial equivalence.  A strange but not unmanageable situation. 
> 
With best regards,
>
Marta

************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
McGill UniversityBurnside Hall, Office 1005
805 Sherbrooke St. West
Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810/3800  
Home: (514) 935-3618
marta.bunge@mcgill.ca 
http://www.math.mcgill.ca/~bunge/
************************************************


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Re: The omega-functor omega-category

[Fred E.J. Linton]

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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Re: The omega-functor omega-category

[David Leduc]

It's funny that each time I ask a dummy question on this mailing list,
after some time my thread ends up in a "philosophical" discussion.
Last time I inadvertently used the word "evil" in a question and you
know what happened.

I have just asked another dummy question on horizontal composition in
a 2-category. Please teach me some mathematics.


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