Re: The omega-functor omega-category
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (8)
-
Colin McLarty -
David Leduc -
Eduardo J. Dubuc -
Fred E.J. Linton -
John Baez -
Michael Shulman -
Steve Vickers -
Vaughan Pratt