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A Proposed Formal Definition of Class and Category in DC Proof
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Dan Christensen
Oct 10, 2012, 1:45:06 AM10/10/12
to
ob is the class of sets with property P:
ALL(a):[a e ob <=> Set(a) & P(a)]
where Set is an "is a set" predicate, and P is an arbitrary property
of the sets in ob.
The class ob is a category iff...
1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
morphism)
Comments? Does this capture the essence of a category
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
ALL(a):[a e ob <=> Set(a) & P(a)]
where Set is an "is a set" predicate, and P is an arbitrary property
of the sets in ob.
The class ob is a category iff...
1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
morphism)
Comments? Does this capture the essence of a category
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Alan Eaton
Oct 10, 2012, 2:12:39 AM10/10/12
to
The objects of a category need not be sets(or any other form of collection).
The arrows (morphisms) of a category need not be functions.
Dan Christensen
Oct 10, 2012, 9:21:33 AM10/10/12
to
On Oct 10, 2:13 am, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:
> On 10/10/2012 15:45, Dan Christensen wrote:
>
>
>
>
>
>
>
>
>
> > ob is the class of sets with property P:
>
> > ALL(a):[a e ob <=> Set(a) & P(a)]
>
> > where Set is an "is a set" predicate, and P is an arbitrary property
> > of the sets in ob.
>
> > The class ob is a category iff...
>
> > 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
> > e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>
> > 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
> > morphism)
>
> > Comments? Does this capture the essence of a category
>
> > Dan
> > Download my DC Proof 2.0 software athttp://www.dcproof.com
> On 10/10/2012 15:45, Dan Christensen wrote:
>
>
>
>
>
>
>
>
>
> > ob is the class of sets with property P:
>
> > ALL(a):[a e ob <=> Set(a) & P(a)]
>
> > where Set is an "is a set" predicate, and P is an arbitrary property
> > of the sets in ob.
>
> > The class ob is a category iff...
>
> > 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
> > e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>
> > 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
> > morphism)
>
> > Comments? Does this capture the essence of a category
>
> > Dan
>
> No.
>
> The objects of a category need not be sets(or any other form of collection).
Is this a matter of opinion?
> No.
>
> The objects of a category need not be sets(or any other form of collection).
"The concept of “class” has been created to deal with 'large
collections of sets'. In
particular, we require that:
(1) the members of each class are sets,
(2) for every “property” P one can form the class of all sets with
property P.
Source: "Abstract and Concrete Categories, The Joy of Cats," p. 14
http://katmat.math.uni-bremen.de/acc/acc.pdf
> The arrows (morphisms) of a category need not be functions.
which they are defined, is central to category theory. Much of the
terminology of morphisms, as well as the intuition underlying them,
comes from concrete categories, where the objects are simply sets with
some additional structure, and morphisms are structure-preserving
functions."
Source: "Morphisms"
http://en.wikipedia.org/wiki/Morphism
So, the key distinction is concrete vs. abstract categories? What is
the difference?
Dan Christensen
Oct 10, 2012, 9:41:30 AM10/10/12
to
On Oct 10, 9:21 am, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
"In mathematics, a concrete category is a category that is equipped
with a faithful functor to the category of sets. This functor makes it
possible to think of the objects of the category as sets with
additional structure, and of its morphisms as structure-preserving
functions. Many important categories have obvious interpretations as
concrete categories, for example the category of topological spaces
and the category of groups, and trivially also the category of sets
itself. On the other hand, the homotopy category of topological spaces
is not concretizable, i.e. it does not admit a faithful functor to the
category of sets."
http://en.wikipedia.org/wiki/Morphism
Is it the case that computer science applications of category theory
theory are based on concrete categories?
wrote:
> On Oct 10, 2:13 am, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On 10/10/2012 15:45, Dan Christensen wrote:
>
> > > ob is the class of sets with property P:
>
> > > ALL(a):[a e ob <=> Set(a) & P(a)]
>
> > > where Set is an "is a set" predicate, and P is an arbitrary property
> > > of the sets in ob.
>
> > > The class ob is a category iff...
>
> > > 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
> > > e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>
> > > 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
> > > morphism)
>
> > > Comments? Does this capture the essence of a category
>
> > > Dan
> > > Download my DC Proof 2.0 software athttp://www.dcproof.com
>
> > No.
>
> > The objects of a category need not be sets(or any other form of collection).
>
> Is this a matter of opinion?
>
> "The concept of “class” has been created to deal with 'large
> collections of sets'. In
> particular, we require that:
> (1) the members of each class are sets,
> (2) for every “property” P one can form the class of all sets with
> property P.
>
> Source: "Abstract and Concrete Categories, The Joy of Cats," p. 14http://katmat.math.uni-bremen.de/acc/acc.pdf
>
>
>
>
>
>
>
>
>
> > On 10/10/2012 15:45, Dan Christensen wrote:
>
> > > ob is the class of sets with property P:
>
> > > ALL(a):[a e ob <=> Set(a) & P(a)]
>
> > > where Set is an "is a set" predicate, and P is an arbitrary property
> > > of the sets in ob.
>
> > > The class ob is a category iff...
>
> > > 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
> > > e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>
> > > 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
> > > morphism)
>
> > > Comments? Does this capture the essence of a category
>
> > > Dan
> > > Download my DC Proof 2.0 software athttp://www.dcproof.com
>
> > No.
>
> > The objects of a category need not be sets(or any other form of collection).
>
> Is this a matter of opinion?
>
> "The concept of “class” has been created to deal with 'large
> collections of sets'. In
> particular, we require that:
> (1) the members of each class are sets,
> (2) for every “property” P one can form the class of all sets with
> property P.
>
>
> > The arrows (morphisms) of a category need not be functions.
>
> "The study of morphisms and of the structures (called objects) over
> which they are defined, is central to category theory. Much of the
> terminology of morphisms, as well as the intuition underlying them,
> comes from concrete categories, where the objects are simply sets with
> some additional structure, and morphisms are structure-preserving
> functions."
>
> Source: "Morphisms"http://en.wikipedia.org/wiki/Morphism
>
> So, the key distinction is concrete vs. abstract categories? What is
> the difference?
>
Looked it up on Wiki:
> > The arrows (morphisms) of a category need not be functions.
>
> "The study of morphisms and of the structures (called objects) over
> which they are defined, is central to category theory. Much of the
> terminology of morphisms, as well as the intuition underlying them,
> comes from concrete categories, where the objects are simply sets with
> some additional structure, and morphisms are structure-preserving
> functions."
>
> Source: "Morphisms"http://en.wikipedia.org/wiki/Morphism
>
> So, the key distinction is concrete vs. abstract categories? What is
> the difference?
>
"In mathematics, a concrete category is a category that is equipped
with a faithful functor to the category of sets. This functor makes it
possible to think of the objects of the category as sets with
additional structure, and of its morphisms as structure-preserving
functions. Many important categories have obvious interpretations as
concrete categories, for example the category of topological spaces
and the category of groups, and trivially also the category of sets
itself. On the other hand, the homotopy category of topological spaces
is not concretizable, i.e. it does not admit a faithful functor to the
category of sets."
http://en.wikipedia.org/wiki/Morphism
Is it the case that computer science applications of category theory
theory are based on concrete categories?
Dan Christensen
Oct 10, 2012, 12:42:52 PM10/10/12
to
On Oct 10, 2:13 am, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:
> On 10/10/2012 15:45, Dan Christensen wrote:
>
>
>
>
>
>
>
>
>
> > ob is the class of sets with property P:
>
> > ALL(a):[a e ob <=> Set(a) & P(a)]
>
> > where Set is an "is a set" predicate, and P is an arbitrary property
> > of the sets in ob.
>
> > The class ob is a category iff...
>
> > 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
> > e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>
> > 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
> > morphism)
>
> > Comments? Does this capture the essence of a category
>
> > Dan
> > Download my DC Proof 2.0 software athttp://www.dcproof.com
>
>
>
>
>
>
>
>
>
> > ob is the class of sets with property P:
>
> > ALL(a):[a e ob <=> Set(a) & P(a)]
>
> > where Set is an "is a set" predicate, and P is an arbitrary property
> > of the sets in ob.
>
> > The class ob is a category iff...
>
> > 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
> > e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>
> > 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
> > morphism)
>
> > Comments? Does this capture the essence of a category
>
> > Dan
>
> No.
>
> The objects of a category need not be sets(or any other form of collection).
> The arrows (morphisms) of a category need not be functions.
If the arrows (morphisms) are not functions mapping elements of one
> No.
>
> The objects of a category need not be sets(or any other form of collection).
> The arrows (morphisms) of a category need not be functions.
object to another, can they be seen simply elements of a reflexive and
transitive relation on the class ob?
Alan Eaton
Oct 10, 2012, 3:22:48 PM10/10/12
to
On 11/10/2012 02:42, Dan Christensen wrote:
>
> If the arrows (morphisms) are not functions mapping elements of one
> object to another, can they be seen simply elements of a reflexive and
> transitive relation on the class ob?
>
> Dan
Only if there is at most one arrow between each pair of objects.
>
> If the arrows (morphisms) are not functions mapping elements of one
> object to another, can they be seen simply elements of a reflexive and
> transitive relation on the class ob?
>
> Dan
Dan Christensen
Oct 10, 2012, 4:56:38 PM10/10/12
to
corresponding to the ordered pairs (x,y) and (y,x). Isn't that enough
for any application?
FredJeffries
Oct 10, 2012, 8:45:49 PM10/10/12
to
On Oct 10, 6:21 am, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
wrote:
> On Oct 10, 2:13 am, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On 10/10/2012 15:45, Dan Christensen wrote:
>
> > > ob is the class of sets with property P:
>
> > > ALL(a):[a e ob <=> Set(a) & P(a)]
>
> > > where Set is an "is a set" predicate, and P is an arbitrary property
> > > of the sets in ob.
>
> > > The class ob is a category iff...
>
> > > 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
> > > e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>
> > > 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
> > > morphism)
>
> > > Comments? Does this capture the essence of a category
>
> > > Dan
> > > Download my DC Proof 2.0 software athttp://www.dcproof.com
>
> > No.
>
> > The objects of a category need not be sets(or any other form of collection).
>
> Is this a matter of opinion?
>
> "The concept of “class” has been created to deal with 'large
> collections of sets'. In
> particular, we require that:
> (1) the members of each class are sets,
> (2) for every “property” P one can form the class of all sets with
> property P.
>
> Source: "Abstract and Concrete Categories, The Joy of Cats," p. 14http://katmat.math.uni-bremen.de/acc/acc.pdf
>
>
>
>
>
>
>
>
>
> > On 10/10/2012 15:45, Dan Christensen wrote:
>
> > > ob is the class of sets with property P:
>
> > > ALL(a):[a e ob <=> Set(a) & P(a)]
>
> > > where Set is an "is a set" predicate, and P is an arbitrary property
> > > of the sets in ob.
>
> > > The class ob is a category iff...
>
> > > 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
> > > e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>
> > > 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
> > > morphism)
>
> > > Comments? Does this capture the essence of a category
>
> > > Dan
> > > Download my DC Proof 2.0 software athttp://www.dcproof.com
>
> > No.
>
> > The objects of a category need not be sets(or any other form of collection).
>
> Is this a matter of opinion?
>
> "The concept of “class” has been created to deal with 'large
> collections of sets'. In
> particular, we require that:
> (1) the members of each class are sets,
> (2) for every “property” P one can form the class of all sets with
> property P.
>
>
> > The arrows (morphisms) of a category need not be functions.
>
> "The study of morphisms and of the structures (called objects) over
> which they are defined, is central to category theory. Much of the
> terminology of morphisms, as well as the intuition underlying them,
> comes from concrete categories, where the objects are simply sets with
> some additional structure, and morphisms are structure-preserving
> functions."
>
> Source: "Morphisms"http://en.wikipedia.org/wiki/Morphism
>
> So, the key distinction is concrete vs. abstract categories? What is
> the difference?
http://en.wikipedia.org/wiki/Concrete_category
> > The arrows (morphisms) of a category need not be functions.
>
> "The study of morphisms and of the structures (called objects) over
> which they are defined, is central to category theory. Much of the
> terminology of morphisms, as well as the intuition underlying them,
> comes from concrete categories, where the objects are simply sets with
> some additional structure, and morphisms are structure-preserving
> functions."
>
> Source: "Morphisms"http://en.wikipedia.org/wiki/Morphism
>
> So, the key distinction is concrete vs. abstract categories? What is
> the difference?
FredJeffries
Oct 10, 2012, 8:47:47 PM10/10/12
to
> On Oct 10, 2:13 am, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On 10/10/2012 15:45, Dan Christensen wrote:
>
> > > ob is the class of sets with property P:
>
> > > ALL(a):[a e ob <=> Set(a) & P(a)]
>
> > > where Set is an "is a set" predicate, and P is an arbitrary property
> > > of the sets in ob.
>
> > > The class ob is a category iff...
>
> > > 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
> > > e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>
> > > 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
> > > morphism)
>
> > > Comments? Does this capture the essence of a category
>
> > > Dan
> > > Download my DC Proof 2.0 software athttp://www.dcproof.com
>
> > No.
>
> > The objects of a category need not be sets(or any other form of collection).
>
> Is this a matter of opinion?
>
> "The concept of “class” has been created to deal with 'large
> collections of sets'. In
> particular, we require that:
> (1) the members of each class are sets,
> (2) for every “property” P one can form the class of all sets with
> property P.
>
> Source: "Abstract and Concrete Categories, The Joy of Cats," p. 14http://katmat.math.uni-bremen.de/acc/acc.pdf
>
>
>
>
>
>
>
>
>
> > On 10/10/2012 15:45, Dan Christensen wrote:
>
> > > ob is the class of sets with property P:
>
> > > ALL(a):[a e ob <=> Set(a) & P(a)]
>
> > > where Set is an "is a set" predicate, and P is an arbitrary property
> > > of the sets in ob.
>
> > > The class ob is a category iff...
>
> > > 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
> > > e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>
> > > 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
> > > morphism)
>
> > > Comments? Does this capture the essence of a category
>
> > > Dan
> > > Download my DC Proof 2.0 software athttp://www.dcproof.com
>
> > No.
>
> > The objects of a category need not be sets(or any other form of collection).
>
> Is this a matter of opinion?
>
> "The concept of “class” has been created to deal with 'large
> collections of sets'. In
> particular, we require that:
> (1) the members of each class are sets,
> (2) for every “property” P one can form the class of all sets with
> property P.
>
>
> > The arrows (morphisms) of a category need not be functions.
>
> "The study of morphisms and of the structures (called objects) over
> which they are defined, is central to category theory. Much of the
> terminology of morphisms, as well as the intuition underlying them,
> comes from concrete categories, where the objects are simply sets with
> some additional structure, and morphisms are structure-preserving
> functions."
>
> Source: "Morphisms"http://en.wikipedia.org/wiki/Morphism
>
> So, the key distinction is concrete vs. abstract categories? What is
> the difference?
>
http://mathprelims.wordpress.com/2009/02/24/concrete-and-non-concrete-categories-informally/
> > The arrows (morphisms) of a category need not be functions.
>
> "The study of morphisms and of the structures (called objects) over
> which they are defined, is central to category theory. Much of the
> terminology of morphisms, as well as the intuition underlying them,
> comes from concrete categories, where the objects are simply sets with
> some additional structure, and morphisms are structure-preserving
> functions."
>
> Source: "Morphisms"http://en.wikipedia.org/wiki/Morphism
>
> So, the key distinction is concrete vs. abstract categories? What is
> the difference?
>
Dan Christensen
Oct 10, 2012, 10:54:36 PM10/10/12
to
Very useful! Thanks, Fred.
Looks like the morphisms for concrete categories must also be
surjections. So, the class ob is a CONCRETE category iff...
1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom)
& ALL(f):[f e hom <=> ALL(d):[d e a => f(d) e b]
2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]]
Alan Eaton
Oct 11, 2012, 4:51:50 AM10/11/12
to
A lack of clarity on my part. Sorry about that.
Alan Eaton
Oct 11, 2012, 6:37:35 AM10/11/12
to
On 11/10/2012 12:54, Dan Christensen wrote:
> On Oct 10, 8:47 pm, FredJeffries <fredjeffr...@gmail.com> wrote:
>> On Oct 10, 6:21 am, Dan Christensen <Dan_Christen...@sympatico.ca>
>> wrote:
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>> On Oct 10, 2:13 am, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:
>>
>>>> On 10/10/2012 15:45, Dan Christensen wrote:
>>
>>>>> ob is the class of sets with property P:
>>
>>>>> ALL(a):[a e ob <=> Set(a) & P(a)]
>>
>>>>> where Set is an "is a set" predicate, and P is an arbitrary property
>>>>> of the sets in ob.
>>
>>>>> The class ob is a category iff...
>>
>>>>> 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
>>>>> e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>>
>>>>> 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
>>>>> morphism)
>>
>>>>> Comments? Does this capture the essence of a category
>>
>>>>> Dan
>>>>> Download my DC Proof 2.0 software athttp://www.dcproof.com
>>
>>>> No.
>>
>>>> The objects of a category need not be sets(or any other form of collection).
>>
>>> Is this a matter of opinion?
>>
>>> "The concept of �class� has been created to deal with 'large
> On Oct 10, 8:47 pm, FredJeffries <fredjeffr...@gmail.com> wrote:
>> On Oct 10, 6:21 am, Dan Christensen <Dan_Christen...@sympatico.ca>
>> wrote:
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>> On Oct 10, 2:13 am, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:
>>
>>>> On 10/10/2012 15:45, Dan Christensen wrote:
>>
>>>>> ob is the class of sets with property P:
>>
>>>>> ALL(a):[a e ob <=> Set(a) & P(a)]
>>
>>>>> where Set is an "is a set" predicate, and P is an arbitrary property
>>>>> of the sets in ob.
>>
>>>>> The class ob is a category iff...
>>
>>>>> 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
>>>>> e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>>
>>>>> 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
>>>>> morphism)
>>
>>>>> Comments? Does this capture the essence of a category
>>
>>>>> Dan
>>>>> Download my DC Proof 2.0 software athttp://www.dcproof.com
>>
>>>> No.
>>
>>>> The objects of a category need not be sets(or any other form of collection).
>>
>>> Is this a matter of opinion?
>>
>>> collections of sets'. In
>>> particular, we require that:
>>> (1) the members of each class are sets,
>>> (2) for every �property� P one can form the class of all sets with
>>> particular, we require that:
>>> (1) the members of each class are sets,
>>> property P.
>>
>>> Source: "Abstract and Concrete Categories, The Joy of Cats," p. 14http://katmat.math.uni-bremen.de/acc/acc.pdf
>>
>>>> The arrows (morphisms) of a category need not be functions.
>>
>>> "The study of morphisms and of the structures (called objects) over
>>> which they are defined, is central to category theory. Much of the
>>> terminology of morphisms, as well as the intuition underlying them,
>>> comes from concrete categories, where the objects are simply sets with
>>> some additional structure, and morphisms are structure-preserving
>>> functions."
>>
>>> Source: "Morphisms"http://en.wikipedia.org/wiki/Morphism
>>
>>> So, the key distinction is concrete vs. abstract categories? What is
>>> the difference?
>>
>> http://mathprelims.wordpress.com/2009/02/24/concrete-and-non-concrete...
>
> Very useful! Thanks, Fred.
>
> Looks like the morphisms for concrete categories must also be
> surjections. So, the class ob is a CONCRETE category iff...
>
> 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom)
> & ALL(f):[f e hom <=> ALL(d):[d e a => f(d) e b]
> & ALL(d):[d e b => EXIST(g):[g e a & f(g)=d]]]] <-- New
>
> 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]]
>
> Dan
> Download my DC Proof 2.0 software at http://www.dcproof.com
>
>
The morphisms of concrete categories need not be surjections.
>>
>>> Source: "Abstract and Concrete Categories, The Joy of Cats," p. 14http://katmat.math.uni-bremen.de/acc/acc.pdf
>>
>>>> The arrows (morphisms) of a category need not be functions.
>>
>>> "The study of morphisms and of the structures (called objects) over
>>> which they are defined, is central to category theory. Much of the
>>> terminology of morphisms, as well as the intuition underlying them,
>>> comes from concrete categories, where the objects are simply sets with
>>> some additional structure, and morphisms are structure-preserving
>>> functions."
>>
>>> Source: "Morphisms"http://en.wikipedia.org/wiki/Morphism
>>
>>> So, the key distinction is concrete vs. abstract categories? What is
>>> the difference?
>>
>> http://mathprelims.wordpress.com/2009/02/24/concrete-and-non-concrete...
>
> Very useful! Thanks, Fred.
>
> Looks like the morphisms for concrete categories must also be
> surjections. So, the class ob is a CONCRETE category iff...
>
> 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom)
> & ALL(f):[f e hom <=> ALL(d):[d e a => f(d) e b]
> & ALL(d):[d e b => EXIST(g):[g e a & f(g)=d]]]] <-- New
>
> 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]]
>
> Dan
> Download my DC Proof 2.0 software at http://www.dcproof.com
>
>
A class of objects, by itself, is not a category (whether concrete or
otherwise).
For a definition of categories see:
http://katmat.math.uni-bremen.de/acc/acc.pdf
page 21
You referenced this document. Perhaps you should read more of it.
Dan Christensen
Oct 11, 2012, 9:09:55 AM10/11/12
to
On Oct 11, 6:37 am, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:
> On 11/10/2012 12:54, Dan Christensen wrote:
>
>
>
>
>
>
>
>
>
> > On Oct 10, 8:47 pm, FredJeffries <fredjeffr...@gmail.com> wrote:
> >> On Oct 10, 6:21 am, Dan Christensen <Dan_Christen...@sympatico.ca>
> >> wrote:
>
> >>> On Oct 10, 2:13 am, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:
>
> >>>> On 10/10/2012 15:45, Dan Christensen wrote:
>
> >>>>> ob is the class of sets with property P:
>
> >>>>> ALL(a):[a e ob <=> Set(a) & P(a)]
>
> >>>>> where Set is an "is a set" predicate, and P is an arbitrary property
> >>>>> of the sets in ob.
>
> >>>>> The class ob is a category iff...
>
> >>>>> 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
> >>>>> e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>
> >>>>> 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
> >>>>> morphism)
>
> >>>>> Comments? Does this capture the essence of a category
>
> >>>>> Dan
> >>>>> Download my DC Proof 2.0 software athttp://www.dcproof.com
>
> >>>> No.
>
> >>>> The objects of a category need not be sets(or any other form of collection).
>
> >>> Is this a matter of opinion?
>
> >>> "The concept of “class” has been created to deal with 'large
> On 11/10/2012 12:54, Dan Christensen wrote:
>
>
>
>
>
>
>
>
>
> > On Oct 10, 8:47 pm, FredJeffries <fredjeffr...@gmail.com> wrote:
> >> On Oct 10, 6:21 am, Dan Christensen <Dan_Christen...@sympatico.ca>
> >> wrote:
>
> >>> On Oct 10, 2:13 am, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:
>
> >>>> On 10/10/2012 15:45, Dan Christensen wrote:
>
> >>>>> ob is the class of sets with property P:
>
> >>>>> ALL(a):[a e ob <=> Set(a) & P(a)]
>
> >>>>> where Set is an "is a set" predicate, and P is an arbitrary property
> >>>>> of the sets in ob.
>
> >>>>> The class ob is a category iff...
>
> >>>>> 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom) & ALL(f):[f
> >>>>> e hom <=> ALL(d):[d e a => f(d) e b]]]] (the set of morphisms)
>
> >>>>> 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]] (the identity
> >>>>> morphism)
>
> >>>>> Comments? Does this capture the essence of a category
>
> >>>>> Dan
> >>>>> Download my DC Proof 2.0 software athttp://www.dcproof.com
>
> >>>> No.
>
> >>>> The objects of a category need not be sets(or any other form of collection).
>
> >>> Is this a matter of opinion?
>
> >>> collections of sets'. In
> >>> particular, we require that:
> >>> (1) the members of each class are sets,
> >>> (2) for every “property” P one can form the class of all sets with
> >>> particular, we require that:
> >>> (1) the members of each class are sets,
> >>> property P.
>
> >>> Source: "Abstract and Concrete Categories, The Joy of Cats," p. 14http://katmat.math.uni-bremen.de/acc/acc.pdf
>
> >>>> The arrows (morphisms) of a category need not be functions.
>
> >>> "The study of morphisms and of the structures (called objects) over
> >>> which they are defined, is central to category theory. Much of the
> >>> terminology of morphisms, as well as the intuition underlying them,
> >>> comes from concrete categories, where the objects are simply sets with
> >>> some additional structure, and morphisms are structure-preserving
> >>> functions."
>
> >>> Source: "Morphisms"http://en.wikipedia.org/wiki/Morphism
>
> >>> So, the key distinction is concrete vs. abstract categories? What is
> >>> the difference?
>
> >>http://mathprelims.wordpress.com/2009/02/24/concrete-and-non-concrete...
>
> > Very useful! Thanks, Fred.
>
> > Looks like the morphisms for concrete categories must also be
> > surjections. So, the class ob is a CONCRETE category iff...
>
> > 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom)
> > & ALL(f):[f e hom <=> ALL(d):[d e a => f(d) e b]
> > & ALL(d):[d e b => EXIST(g):[g e a & f(g)=d]]]] <-- New
>
That should be:
>
> >>> Source: "Abstract and Concrete Categories, The Joy of Cats," p. 14http://katmat.math.uni-bremen.de/acc/acc.pdf
>
> >>>> The arrows (morphisms) of a category need not be functions.
>
> >>> "The study of morphisms and of the structures (called objects) over
> >>> which they are defined, is central to category theory. Much of the
> >>> terminology of morphisms, as well as the intuition underlying them,
> >>> comes from concrete categories, where the objects are simply sets with
> >>> some additional structure, and morphisms are structure-preserving
> >>> functions."
>
> >>> Source: "Morphisms"http://en.wikipedia.org/wiki/Morphism
>
> >>> So, the key distinction is concrete vs. abstract categories? What is
> >>> the difference?
>
> >>http://mathprelims.wordpress.com/2009/02/24/concrete-and-non-concrete...
>
> > Very useful! Thanks, Fred.
>
> > Looks like the morphisms for concrete categories must also be
> > surjections. So, the class ob is a CONCRETE category iff...
>
> > 1. ALL(a):ALL(b):[a e ob & b e ob => EXIST(hom):[Set(hom)
> > & ALL(f):[f e hom <=> ALL(d):[d e a => f(d) e b]
> > & ALL(d):[d e b => EXIST(g):[g e a & f(g)=d]]]] <-- New
>
1. ALL(a):ALL(b):[a @ ob & b @ ob
=> EXIST(c):[Set(c) & ALL(f):[f @ c <=> ALL(d):[d @ a => f(d) @ b]
& EXIST(d):[d @ ob & ALL(e):[e @ d => EXIST(g):[g @ a &
f(g)=e]]]]]]
where @ = epsilon (is an element of)
Instead of restricting morphisms to surjections, I have now restricted
them to functions the codomain (range) of which are elements of ob.
> > 2. ALL(a):[a e ob => EXIST(i):ALL(b):[b e a => i(b)=b]]
>
> > Dan
>
> The morphisms of concrete categories need not be surjections.
My mistake. See my correction above.
> The morphisms of concrete categories need not be surjections.
> A class of objects, by itself, is not a category (whether concrete or
> otherwise).
>
Usually a category is given as a quadruple of objects. My presentation
doesn't lend itself to a description of this form since the only free
variable is ob. For this reason, I informally call ob a category. I am
open to suggestions for another terminology.
Frederick Williams
Oct 11, 2012, 12:25:39 PM10/11/12
to
Dan Christensen wrote:
>
> Usually a category is given as a quadruple of objects.
Really? You may, if you wish, think of categories as consisting of just
>
> Usually a category is given as a quadruple of objects.
morphisms (subject to certain axioms) with no objects.
> My presentation
> doesn't lend itself to a description of this form since the only free
> variable is ob. For this reason, I informally call ob a category. I am
> open to suggestions for another terminology.
Where are the songs of Summer?--With the sun,
Oping the dusky eyelids of the south,
Till shade and silence waken up as one,
And morning sings with a warm odorous mouth.
Alan Eaton
Oct 11, 2012, 1:23:26 PM10/11/12
to
I would suggest that you extend your presentation.
I would do something like the following:
(assuming a set based presentation (bleh!) and your Set predicate)
(assuming abreviations for pairs and triples)
Set(ob)
Set(mor)
Set(id)
Set(source)
Set(target)
Set(comp)
all p.( p e source -> some f,x.p=(f,x) )
all f,x.( (f,x) e source -> f e mor & x e ob )
all f,x1,x2.( (f,x1) e source & (f,x2) e source -> x1=x2 )
all p.( p e target -> some f,x.p=(f,x) )
all f,x.( (f,x) e target -> f e mor & x e ob )
all f,x1,x2.( (f,x1) e target & (f,x2) e target -> x1=x2 )
all f.( f e mor -> some x,y.((f,x) e source & (f,y) e target) )
all p.( p e comp -> some f,g,h.p=(f,g,h) )
all f,g,h.( (f,g,h) e comp -> f e mor & g e mor & h e mor )
all f,g,h1,h2.( (f,g,h1) e comp & (f,g,h2) e comp -> h1=h2 )
all f,g,h.(
(f,g,h) e comp -> some x,y,z.(
(f,z) e target
& (f,y) e source
& (g,y) e target
& (g,x) e source
& (h,z) e target
& (h,x) e source
)
)
all f,g,x,y,z.(
( (f,z) e target
& (f,y) e source
& (g,y) e target
& (g,x) e source
)-> some h.(
(f,g,h) e comp
& (h,z) e target
& (h,x) e source
)
)
all f,g,h,fg,gh,fgh1,fgh2.(
( (f,g,fg) e comp
& (fg,h,fgh1) e comp
& (g,h,gh) e comp
& (f,gh,fgh2) e comp
)-> fgh1=fgh2
)
all p.( p e id -> some x,f.p=(x,f) )
all x,f.( (x,f) e id -> x e ob & f e mor )
all x,f1,f2.( (x,f1) e id & (x,f2) e id -> f1=f2 )
all x.( x e ob -> some f.( (x,f) e id )
all x,f.(
(x,f) e id -> (
all g,h.((f,g,h) e comp -> g=h)
& all g,h.((g,f,h) e comp -> g=h)
)
)
all x,y.(
x e ob & y e ob -> some hom.(
Set(hom)
& all f.(f e hom == (f,x) e source & (f,y) e target)
)
)
If you insist on the objects being sets and the morphisms being
functions then:
all x.( x e ob -> Set(x) )
all f.(
f e mor -> (
Set(f)
& all p.( p e f -> some u,v.p=(u,v) )
& all u,v1,v2.( (u,v1) e f & (u,v2) e f -> v1=v2 )
& all x,y.(
(f,x) e source & (f,y) e target-> (
all u,v.( (u,v) e f -> u e x & v e y )
& all u.( u e x -> some v.(v e y & (u,v) e f) )
)
)
)
)
all f,g,h.(
(f,g,h) e comp -> all u,w.(
(u,w) e h == some v.((u,v) e g & (v,w) e f)
)
)
The above is rather restrictive though since ob is a set and thus
cannot be the class of objects for any category whose objects form
a proper class.
There are many of those.
It would be better for ob, mor, id, source, target and comp to be
predicate symbols.
Or to use a system that admits proper classes.
Or even better, a system specifically designed for category theory.
FredJeffries
Oct 11, 2012, 11:42:59 PM10/11/12
to
On Oct 11, 6:09 am, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
>
wrote:
>
> Usually a category is given as a quadruple of objects. My presentation
> doesn't lend itself to a description of this form since the only free
> variable is ob. For this reason, I informally call ob a category. I am
> open to suggestions for another terminology.
Why don't you just call it a structured set?
> doesn't lend itself to a description of this form since the only free
> variable is ob. For this reason, I informally call ob a category. I am
> open to suggestions for another terminology.
FredJeffries
Oct 26, 2012, 5:49:02 PM10/26/12
to
On Oct 11, 6:09 am, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
>
Constructs, also known as "concrete category of sets with structure"
or "categories of structured sets and structure-preserving functions
between them"
See "Theory of Mathematical Structures" By Jirí Adámek
Definition on page 5
http://books.google.com/books?id=6lFdFv8yHHUC
See also Examples 3.3 starting an page 22 of
J. Adámek, H. Herrlich and G.E. Strecker,
"Abstract and Concrete Categories"
available online
http://www.iti.cs.tu-bs.de/~adamek/AHS.pdf
Notice that after giving examples of categories that are
constructs, he gives examples "where the objects and
morphisms are _not_ structured sets and structure-preserving
functions" (his emphasis).
No, his functions are NOT all surjections even though
on page 3 of "Theory of Mathematical Structures" he
uses the term "range", but if you read the definition and
examples it is clear that he is using it as a synonym for
what has been called in these discussions "codomain".
In "Abstract and Concrete Categories" the term "codomain"
is used.
wrote:
>
> Usually a category is given as a quadruple of objects. My presentation
> doesn't lend itself to a description of this form since the only free
> variable is ob. For this reason, I informally call ob a category. I am
> open to suggestions for another terminology.
The structure you are working with seem to resemble Jiri Adamek's
> doesn't lend itself to a description of this form since the only free
> variable is ob. For this reason, I informally call ob a category. I am
> open to suggestions for another terminology.
Constructs, also known as "concrete category of sets with structure"
or "categories of structured sets and structure-preserving functions
between them"
See "Theory of Mathematical Structures" By Jirí Adámek
Definition on page 5
http://books.google.com/books?id=6lFdFv8yHHUC
See also Examples 3.3 starting an page 22 of
J. Adámek, H. Herrlich and G.E. Strecker,
"Abstract and Concrete Categories"
available online
http://www.iti.cs.tu-bs.de/~adamek/AHS.pdf
Notice that after giving examples of categories that are
constructs, he gives examples "where the objects and
morphisms are _not_ structured sets and structure-preserving
functions" (his emphasis).
No, his functions are NOT all surjections even though
on page 3 of "Theory of Mathematical Structures" he
uses the term "range", but if you read the definition and
examples it is clear that he is using it as a synonym for
what has been called in these discussions "codomain".
In "Abstract and Concrete Categories" the term "codomain"
is used.
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