Re: The omega-functor omega-category

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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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: 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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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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