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Proposed Formal Definition of a Concrete Category
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More options Oct 11 2012, 1:31 pm
Newsgroups: sci.logic, sci.math
From: Dan Christensen <Dan_Christen...@sympatico.ca>
Date: Thu, 11 Oct 2012 10:31:51 -0700 (PDT)
Local: Thurs, Oct 11 2012 1:31 pm
Subject: Proposed Formal Definition of a Concrete Category
"The categories that naturally come to mind when trying to think up
examples of a category (such as sets with functions, groups with
homomorphisms, topological spaces with continuous maps, …) are
sometimes called concrete categories.  These are categories where the
objects are sets, usually with some additional structure (group
structure, a topology, etc.), and the morphisms are well-defined
functions between those sets that preserve the structure."

Source: "Concrete and Non-Concrete Categories (Informally)"
http://mathprelims.wordpress.com/2009/02/24/concrete-and-non-concrete...

Formally:

1  ALL(a):[a @ ob <=> P(a)]

where
@ = epsilon (is an element of)
ob = the class whose elements have property P

2  ALL(a):ALL(b):[a @ ob & b @ ob => EXIST(hom):[ALL(f):[f @ hom
<=> ALL(c):[c @ a => f(c) @ b]   <-- f is a function
& ALL(c):[c @ b => EXIST(d):[d @ a & f(d)=c]]]]     <-- f is a
surjection

where
hom = the set (class?) of all morphisms (surjections) from a to b

3  ALL(a):[a @ ob => EXIST(i):ALL(b):[b @ a => i(b)=b]]  <-- identity
morphism

where
i = the required identity morphism for each element of ob

The required composition of morphisms is just usual composition of
functions.

Dan

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More options Oct 11 2012, 2:01 pm
Newsgroups: sci.logic, sci.math
From: Alan Eaton <alan.dennis.ea...@gmail.com>
Date: Fri, 12 Oct 2012 04:01:11 +1000
Local: Thurs, Oct 11 2012 2:01 pm
Subject: Re: Proposed Formal Definition of a Concrete Category
On 12/10/2012 03:31, Dan Christensen wrote:

The morphisms need not be surjections.
Why are you back on that horse again?

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More options Oct 11 2012, 3:57 pm
Newsgroups: sci.logic, sci.math
From: Dan Christensen <Dan_Christen...@sympatico.ca>
Date: Thu, 11 Oct 2012 12:57:00 -0700 (PDT)
Subject: Re: Proposed Formal Definition of a Concrete Category
On Oct 11, 2:01 pm, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:

You are right -- I do keep vacillating. Apologies in advance.

I have been trying to piece together the essence of category theory
from several, not necessarily consistent informal accounts. Nothing
seems to be exactly what I was looking for. The above represents my
latest thinking on this matter.

"In mathematics, a morphism is an abstraction derived from structure-
preserving mappings between two mathematical structures. The notion of
morphism recurs in much of contemporary mathematics. In set theory,
morphisms are functions; in linear algebra, linear transformations; in
group theory, group homomorphisms; in topology, continuous functions,
and so on.
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....

"There are two operations which are defined on every morphism, the
domain (or source) and the codomain (or target). If a morphism f has
domain X and codomain Y, we write f : X -> Y. Thus a morphism is
represented by an arrow from its domain to its codomain. The
collection of all morphisms from X to Y is denoted homC(X,Y) or simply
hom(X, Y) and called the hom-set between X and Y."

Source: "Morphisms" http://en.wikipedia.org/wiki/Morphism

In category theory (for concrete categories), the "arrow" seems to
connect the domain and codomain of some morphism that preserves some
property (P in my axioms). We want to refer to those arrows
collectively. We want all of those ordered pairs of sets (in ob) that
are "connected" by one or more morphisms.

Given a pair of sets x and y in ob, how do we specify the morphisms
that connect them? Let x be the domain, and y the codomain of these
morphisms. These morphisms are a subset of all functions mapping the
elements of set x to the elements of set y. We want ONLY the
surjections, because a function mapping x to y that is not a
surjection cannot be a morphism connecting x and y, with x as the
domain and y as the codomain. Your thoughts?

Dan

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More options Oct 11 2012, 11:37 pm
Newsgroups: sci.logic, sci.math
From: Dan Christensen <Dan_Christen...@sympatico.ca>
Date: Thu, 11 Oct 2012 20:37:19 -0700 (PDT)
Local: Thurs, Oct 11 2012 11:37 pm
Subject: Re: Proposed Formal Definition of a Concrete Category
Another, more "visual" approach making use of an "Arrow" predicate...

Define the class ob:

1  ALL(a):[a @ ob <=> P(a)]

where
@ = epsilon (is an element of)

Define the Arrow predicate: For all a and b in ob, Arrow(a,b) means
there exists a surjection from a to b

2  ALL(a):ALL(b):[a @ ob & b @ ob
=> [Arrow(a,b) <=> EXIST(f):[ALL(c):[c @ a => f(c) @ b]
& ALL(c):[c @ b => EXIST(d):[d @ a & f(d)=c]]]]]

The Arrow predicate is reflexive on ob

3  ALL(a):[a @ ob => Arrow(a,a)]

Dan

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More options Oct 12 2012, 4:40 am
Newsgroups: sci.logic, sci.math
From: Alan Eaton <alan.dennis.ea...@gmail.com>
Date: Fri, 12 Oct 2012 18:40:11 +1000
Local: Fri, Oct 12 2012 4:40 am
Subject: Re: Proposed Formal Definition of a Concrete Category
On 12/10/2012 05:57, Dan Christensen wrote:

> On Oct 11, 2:01 pm, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:

<snipped for brevity>

The arrow does not '...connect the domain and codomain of some
morphism...'. The arrow *is* the morphism.

> Given a pair of sets x and y in ob, how do we specify the morphisms
> that connect them? Let x be the domain, and y the codomain of these
> morphisms. These morphisms are a subset of all functions mapping the
> elements of set x to the elements of set y. We want ONLY the
> surjections, because a function mapping x to y that is not a
> surjection cannot be a morphism connecting x and y, with x as the
> domain and y as the codomain. Your thoughts?

> Dan

Perhaps an explicit example is required.

Consider the class of vector spaces over R(the real numbers).

Let objects = the class of vector spaces over R.

Let morphisms = the class of R-linear maps between the objects.

For any two objects X and Y, let hom(X,Y) be the class of
all R-linear maps from X to Y.

For any object X, let id(X) be the R-linear identity map on X.

For any two morphisms that are composable as R-linear maps let
their composition be their composition as R-linear maps.

The above definitions of objects, morphisms, hom, id and composition
collectively satisfy the definition of a category.

Consider a particular pair of vector spaces over R:

X = R^3
Y = R^2

and a particular map between them:

f:X->Y
f((x,y,z)) = (x,0)

then we have:

X and Y are objects in this category because they are vector
spaces over R.

f is a morphism in this category because f is an R-linear map
between the objects X and Y.

f is an element of hom(X,Y) because f is an R-linear map from X to Y.

f is not injective.
f is not surjective.

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 Discussion subject changed to "test" by LudovicoVan
More options Oct 12 2012, 12:54 pm
Newsgroups: sci.logic, sci.math
From: "LudovicoVan" <ju...@diegidio.name>
Date: Fri, 12 Oct 2012 17:54:34 +0100
Local: Fri, Oct 12 2012 12:54 pm
Subject: Re: test
"Dan Christensen" <Dan_Christen...@sympatico.ca> wrote in message

> test

failed

-LV

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 Discussion subject changed to "Proposed Formal Definition of a Concrete Category" by Dan Christensen
More options Oct 12 2012, 4:54 pm
Newsgroups: sci.logic, sci.math
From: Dan Christensen <Dan_Christen...@sympatico.ca>
Date: Fri, 12 Oct 2012 13:54:24 -0700 (PDT)
Local: Fri, Oct 12 2012 4:54 pm
Subject: Re: Proposed Formal Definition of a Concrete Category
On Oct 11, 11:37 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:

Developing this idea a little further, I define Category and Arrow
predicates:

Define: Category

1  ALL(a):[Category(a)
<=> ALL(b):[b @ a => EXIST(f):ALL(c):[c @ b => f(c)=c]]]

Define: Arrow

2  ALL(ob):ALL(a):ALL(b):[Arrow(ob,a,b) <=> a @ ob & b @ ob
& EXIST(f):[ALL(c):[c @ a => f(c) @ b]
& ALL(c):[c @ b => EXIST(d):[d @ a & f(d)=c]]]]

It is then easy to prove the Arrow relation is reflexive:

ALL(x):[Category(x) => ALL(a):[a @ x => Arrow(x,a,a)]]

The Arrow relation is transitive:

ALL(x):[Category(x) => ALL(a):ALL(b):ALL(c):[a @ x & b @ x & c @ x
=> [Arrow(x,a,b) & Arrow(x,b,c) => Arrow(x,a,c)]]]

Dan

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More options Oct 12 2012, 8:44 pm
Newsgroups: sci.logic, sci.math
From: Alan Eaton <alan.dennis.ea...@gmail.com>
Date: Sat, 13 Oct 2012 10:44:47 +1000
Local: Fri, Oct 12 2012 8:44 pm
Subject: Re: Proposed Formal Definition of a Concrete Category
On 13/10/2012 06:54, Dan Christensen wrote:

<snip>

> Developing this idea a little further, I define Category and Arrow
> predicates:

> Define: Category

> 1  ALL(a):[Category(a)
>     <=> ALL(b):[b @ a => EXIST(f):ALL(c):[c @ b => f(c)=c]]]

> Define: Arrow

> 2  ALL(ob):ALL(a):ALL(b):[Arrow(ob,a,b) <=> a @ ob & b @ ob
>     & EXIST(f):[ALL(c):[c @ a => f(c) @ b]
>     & ALL(c):[c @ b => EXIST(d):[d @ a & f(d)=c]]]]

Still obsessing over surjective functions?

> It is then easy to prove the Arrow relation is reflexive:

> ALL(x):[Category(x) => ALL(a):[a @ x => Arrow(x,a,a)]]

> The Arrow relation is transitive:

> ALL(x):[Category(x) => ALL(a):ALL(b):ALL(c):[a @ x & b @ x & c @ x
> => [Arrow(x,a,b) & Arrow(x,b,c) => Arrow(x,a,c)]]]

> Dan

In category theory, 'arrow' and 'morphism' are synonyms.

If your intention was for the relation Arrow to indicate the
presence of arrows between objects of a category then
consider the following:

dom(f) =df {x| some y.((x,y) e f)}

rng(f) =df {y| some x.((x,y) e f)}

f:func =df (
all p.( p e f -> some x,y.( p=(x,y) ) )
& all x,y,z.( (x,y) e f & (x,z) e f -> y=z )
)

f:x->y =df ( f:func & dom(f)=x & rng(f) c y )

0 = {}
1 = {0}
2 = {0,1}

object = {1,2}

arrow = {f| some x,y.(x e object & y e object & f:x->y) }

Let composition of arrows be the usual composition of functions.
For any objects x,y let hom(x,y) = {f| f:x->y}
For any object x let id(x) = {(a,a)| a e x}

We have, then, a category. A very simple one.

According to your definitions we have:

Category(object)
~Arrow(object, 1, 2)

but we have:

{(0,0)} e hom(1,2)

Arrow(object, 1, 2)

If your intention for the relation Arrow was otherwise then either
you are misusing the terminology or you are confused as to what a
category is.

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More options Oct 13 2012, 12:14 am
Newsgroups: sci.logic, sci.math
From: Dan Christensen <Dan_Christen...@sympatico.ca>
Date: Fri, 12 Oct 2012 21:14:46 -0700 (PDT)
Local: Sat, Oct 13 2012 12:14 am
Subject: Re: Proposed Formal Definition of a Concrete Category
On Oct 12, 8:44 pm, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:

It is true that I am improvising to some extent and using "category"
is a slightly different way than is usual in mathematics. It seems
unavoidable, however. I will summarize my thinking on morphisms, etc.
Perhaps you can tell me where you think I am wrong.

Every class of objects is completely determined by a property that is
common to all elements of that class.

Morphisms are structure-preserving functions. Suppose, for example,
that we have a class X with some defined structure, that is it has
some property P that defines certain relationships between elements of
X. If a function maps each element of X to EACH of the elements of
another class Y that also has property P, then that function is a
morphism with respect to P.

Consider the class C of all classes that have the some common property
P. A function mapping any element of C to another element of C is a
morphism with respect to P. And that function must be a surjection. An
arrow from A to B indicates the existence of a morphism wrt P.

Now, an identity function on any element of C is, of course, a
morphism wrt P. Therefore, our system of arrows must also be
reflexive. Using ordinary composition of functions, we can prove that
our system of arrows must also be transitive.

Dan

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More options Oct 13 2012, 12:19 am
Newsgroups: sci.logic, sci.math
From: Graham Cooper <grahamcoop...@gmail.com>
Date: Fri, 12 Oct 2012 21:19:29 -0700 (PDT)
Local: Sat, Oct 13 2012 12:19 am
Subject: Re: Proposed Formal Definition of a Concrete Category
On Oct 13, 2:14 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:

go for the hat trick and add in symmetric and you get a partition!

Herc

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More options Oct 13 2012, 6:32 am
Newsgroups: sci.logic, sci.math
From: Alan Eaton <alan.dennis.ea...@gmail.com>
Date: Sat, 13 Oct 2012 20:32:14 +1000
Local: Sat, Oct 13 2012 6:32 am
Subject: Re: Proposed Formal Definition of a Concrete Category
On 13/10/2012 14:14, Dan Christensen wrote:

> On Oct 12, 8:44 pm, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:
>> On 13/10/2012 06:54, Dan Christensen wrote:

>> <snip>

<snip>

Your deviant term usage is avoidable.

> Every class of objects is completely determined by a property that is
> common to all elements of that class.

> Morphisms are structure-preserving functions. Suppose, for example,
> that we have a class X with some defined structure, that is it has
> some property P that defines certain relationships between elements of
> X. If a function maps each element of X to EACH of the elements of
> another class Y that also has property P, then that function is a
> morphism with respect to P.

> Consider the class C of all classes that have the some common property
> P. A function mapping any element of C to another element of C is a
> morphism with respect to P. And that function must be a surjection. An
> arrow from A to B indicates the existence of a morphism wrt P.

There is an arrow from A to B for each morphism from A to B because
each arrow is a morphism and each morphism is an arrow.

As I said already: in category theory, 'arrow' and 'morphism' are synonyms.

> Now, an identity function on any element of C is, of course, a
> morphism wrt P. Therefore, our system of arrows must also be
> reflexive. Using ordinary composition of functions, we can prove that
> our system of arrows must also be transitive.

> Dan

I gave you an explicit example of a category that is small enough
for the details to be worked out by hand.

hom(1,1) = { {(0,0)} }
hom(1,1) contains a single morphism. It is a surjection.

hom(1,2) = { {(0,0)}, {(0,1)} }
hom(1,2) contains two morphisms, neither of which is a surjection.

hom(2,1) = { {(0,0),(1,0)} }
hom(2,1) contains a single morphism. It is a surjection.

hom(2,2) = {
{(0,0),(1,0)},
{(0,1),(1,0)},
{(0,0),(1,1)},
{(0,1),(1,1)}

}

hom(2,2) contains four morphisms, two of which are not surjections.

Two more examples follow.

In the category Set:

Every object is a set.
Every set is an object.

Every morphism is a total function from one set to another.
Every total function from one set to another is a morphism.

For any pair of sets X and Y, hom(X,Y) is the class of all
total functions from X to Y.

Composition of morphisms is function composition.

For any set X, hom(X,pow(X)) is non-empty and contains no surjections.
(where pow(X) is the power set of X)

For any partially ordered set (P, <=) there is a category such that:

The objects are the elements of P.

The morphisms are the pairs (p,q) such that p <= q.

For pair of morphisms ((p,q),(q,r)), the composition is (p,r).

None of the morphisms are functions.
Therefore, none of the morphisms are surjections.

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More options Oct 13 2012, 10:10 am
Newsgroups: sci.logic, sci.math
From: Dan Christensen <Dan_Christen...@sympatico.ca>
Date: Sat, 13 Oct 2012 07:10:21 -0700 (PDT)
Local: Sat, Oct 13 2012 10:10 am
Subject: Re: Proposed Formal Definition of a Concrete Category
On Oct 13, 6:32 am, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:

Maybe I should use another name for my category predicate. How about

So, we agree.

Good.

> In the category Set:

>    Every object is a set.
>    Every set is an object.

>    Every morphism is a total function from one set to another.
>    Every total function from one set to another is a morphism.

Yes, they preserve the property of being a set.

>    For any pair of sets X and Y, hom(X,Y) is the class of all
>    total functions from X to Y.

>    Composition of morphisms is function composition.

>    For any set X, hom(X,pow(X)) is non-empty and contains no surjections.
>    (where pow(X) is the power set of X)

> For any partially ordered set (P, <=) there is a category such that:

>    The objects are the elements of P.

>    The morphisms are the pairs (p,q) such that p <= q.

Morphisms are structure-preserving functions. No structure is being
preserved. These are not even functions. You have only a relation on
P.

>    For pair of morphisms ((p,q),(q,r)), the composition is (p,r).

>    None of the morphisms are functions.
>    Therefore, none of the morphisms are surjections.

There are no morphisms.

Dan

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More options Oct 13 2012, 11:20 am
Newsgroups: sci.logic, sci.math
From: Frederick Williams <freddywilli...@btinternet.com>
Date: Sat, 13 Oct 2012 16:20:24 +0100
Local: Sat, Oct 13 2012 11:20 am
Subject: Re: Proposed Formal Definition of a Concrete Category

If you look up the definition of category you will see that Alan Eaton's
example satisfies it.

--
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.

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More options Oct 13 2012, 12:28 pm
Newsgroups: sci.logic, sci.math
From: "Jesse F. Hughes" <je...@phiwumbda.org>
Date: Sat, 13 Oct 2012 12:24:01 -0400
Local: Sat, Oct 13 2012 12:24 pm
Subject: Re: Proposed Formal Definition of a Concrete Category

Dan Christensen <Dan_Christen...@sympatico.ca> writes:
> Morphisms are structure-preserving functions. Suppose, for example,
> that we have a class X with some defined structure, that is it has
> some property P that defines certain relationships between elements of
> X. If a function maps each element of X to EACH of the elements of
> another class Y that also has property P, then that function is a
> morphism with respect to P.

That's not the definition of morphism[1].  In fact, I'm not sure *what*
your last sentence is supposed to mean, but I suspect you're trying to
require that morphisms are onto.

Take, for example, the category Monoid.  A monoid is a tuple

<S, e, *>

where e in S and *: S x S -> S is associative.

A morphism

f: <S, e, *> -> <S', e', *'>

is a set function f:S -> S' such that

f(e) = e'

f(s * t) = f(s) *' f(t)

There is *NO REQUIREMENT THAT f BE ONTO*.

Example: Let S = {0}, e = 0 and *:S x S -> S be the identity.  It is
trivial to note that <S, e, *> and <N, 0, +> are both monoids (where N
is the set of natural numbers, of course).

It is also obvious that the map f:S -> N mapping 0 to 0 is a monoid
homomorphism (i.e., a morphism in the category Monoid).  This map is not
onto.

Footnotes:
[1]  I don't see that your notion of structure is coherent either.

--
"And I'll reinforce the point that you are an enemy of humanity, that
my predecessors are people like Gauss, Euler, Newton, Archimedes and
others who you are spitting upon as you do it to me by trying to keep
their discipline trashed as it is now." -- James S. Harris

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More options Oct 13 2012, 12:55 pm
Newsgroups: sci.logic, sci.math
From: Alan Eaton <alan.dennis.ea...@gmail.com>
Date: Sun, 14 Oct 2012 02:54:58 +1000
Local: Sat, Oct 13 2012 12:54 pm
Subject: Re: Proposed Formal Definition of a Concrete Category
On 14/10/2012 00:10, Dan Christensen wrote:

> On Oct 13, 6:32 am, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:
>> On 13/10/2012 14:14, Dan Christensen wrote:
>>> It is true that I am improvising to some extent and using "category"
>>> is a slightly different way than is usual in mathematics. It seems
>>> unavoidable, however. I will summarize my thinking on morphisms, etc.
>>> Perhaps you can tell me where you think I am wrong.

>> Your deviant term usage is avoidable.

> Maybe I should use another name for my category predicate. How about

Call it whatever you like.
Are you going to change the thread title as well?
My concern is your misapprehension of categories. Renaming the predicate

What is good? That there are more examples following?

I notice that you did not comment on the fact that there are
non-surjective morphisms in the preceding example.

I notice, again, that you did not comment on the fact that there are
non-surjective morphisms in the preceding example.

>> For any partially ordered set (P, <=) there is a category such that:

>>     The objects are the elements of P.

>>     The morphisms are the pairs (p,q) such that p <= q.

> Morphisms are structure-preserving functions. No structure is being
> preserved. These are not even functions. You have only a relation on
> P.

The morphisms of a category need not be functions.
I indicated, below, that that is the case in this example.

That some categories have objects without non-trivial structure and
morphisms that are not functions is the reason that some people
(myself included) prefer the term 'arrow' over 'morphism'.

>>     For pair of morphisms ((p,q),(q,r)), the composition is (p,r).

>>     None of the morphisms are functions.
>>     Therefore, none of the morphisms are surjections.

> There are no morphisms.

Yes there are.

As Frederick indicated, the previous example satisfies the definition
of a category.

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More options Oct 13 2012, 3:55 pm
Newsgroups: sci.logic, sci.math
From: Dan Christensen <Dan_Christen...@sympatico.ca>
Date: Sat, 13 Oct 2012 12:55:40 -0700 (PDT)
Local: Sat, Oct 13 2012 3:55 pm
Subject: Re: Proposed Formal Definition of a Concrete Category
On Oct 13, 12:55 pm, Alan Eaton <alan.dennis.ea...@gmail.com> wrote:

OK.

It's not your fault -- I just couldn't make sense of the previous
example.

Interesting point, but isn't a morphism wrt to some property (or
structure) P just a transformation that preserves property P? That is
my intuitive sense of it. You cannot transform any set into its power
set since, in a transformation, one element is transforms into exactly
one other element. Can you then have morphism from a set to its power
set?

> >> For any partially ordered set (P, <=) there is a category such that:

> >>     The objects are the elements of P.

> >>     The morphisms are the pairs (p,q) such that p <= q.

> > Morphisms are structure-preserving functions. No structure is being
> > preserved. These are not even functions. You have only a relation on
> > P.

> The morphisms of a category need not be functions.

An odd notion of "morphism" to my mind.

> I indicated, below, that that is the case in this example.

> That some categories have objects without non-trivial structure and
> morphisms that are not functions is the reason that some people
> (myself included) prefer the term 'arrow' over 'morphism'.

> >>     For pair of morphisms ((p,q),(q,r)), the composition is (p,r).

> >>     None of the morphisms are functions.
> >>     Therefore, none of the morphisms are surjections.

> > There are no morphisms.

> Yes there are.

A morphism has to mean something more than a single ordered pair. It
has to include the notion of a structure preserving transformation,
does it not? Otherwise, you are just talking about a binary relation
defined on a class of objects that may be nothing more atomic
individuals.

> As Frederick indicated, the previous example satisfies the definition
> of a category.

The usual definitions of category theory seem rather vague to me. It
seems to me that there are too many words, and too much hand waving. I
am open to suggestions -- something stated in the notation of first-
order logic and set theory, something of the form of the axioms in my
original posting if that's not asking too much.

Dan

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More options Oct 13 2012, 4:13 pm
Newsgroups: sci.logic, sci.math
From: "Jesse F. Hughes" <je...@phiwumbda.org>
Date: Sat, 13 Oct 2012 16:12:02 -0400
Local: Sat, Oct 13 2012 4:12 pm
Subject: Re: Proposed Formal Definition of a Concrete Category

Dan Christensen <Dan_Christen...@sympatico.ca> writes:
> Interesting point, but isn't a morphism wrt to some property (or
> structure) P just a transformation that preserves property P? That is
> my intuitive sense of it. You cannot transform any set into its power
> set since, in a transformation, one element is transforms into exactly
> one other element. Can you then have morphism from a set to its power
> set?

The function f:N -> P(N) mapping n to {n} is a morphism in the category
Set.

So is the function g:N -> P(N) mapping n to N \ {n}.

And the function mapping n to N.

And the function mapping n to n u {0,...,47}.

And so on.

You see that *sets* of elements are elements of the powerset.  Thus, a
function S -> P(S) takes elements of S to subsets of S -- because
subsets of S *are* elements of P(S).

--
Jesse F. Hughes

"My name is Apusta Malusta Cadeau and I fight bad guys.  And I'm a
knight."  -- A. M. Cadeau (nee Quincy P. Hughes), age 4

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More options Oct 13 2012, 5:31 pm
Newsgroups: sci.logic, sci.math
From: Dan Christensen <Dan_Christen...@sympatico.ca>
Date: Sat, 13 Oct 2012 14:31:31 -0700 (PDT)
Local: Sat, Oct 13 2012 5:31 pm
Subject: Re: Proposed Formal Definition of a Concrete Category
On Oct 13, 4:13 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

As I understand, a morphism on is a structure-preserving
transformation. I don't see the point of category theory otherwise.

Dan

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More options Oct 13 2012, 5:43 pm
Newsgroups: sci.logic, sci.math
From: "Jesse F. Hughes" <je...@phiwumbda.org>
Date: Sat, 13 Oct 2012 17:42:52 -0400
Local: Sat, Oct 13 2012 5:42 pm
Subject: Re: Proposed Formal Definition of a Concrete Category

In the case of the category Set, there is no structure to be preserved.
It is a degenerate case.

In any case, whether you see the point or not, you ought to be looking
at actual textbooks and working through their examples rather than just
trying to draw inferences from your own intuitions of what category
theory should be.

Because, you see, people have already defined terms like "category" and
even "concrete category" and you don't have to guess what these ought to
mean.  You can learn what they actually do mean.

So, start with the definition of category, learn what a functor is,
learn what it means for a functor to be faithful and then you can figure
out what a concrete category is.  It's a category C with a faithful
functor C -> Set.  (Oops!  You'll have to learn what the category Set
is, too.)

That's how you learn what existing terminology means.  Much simpler than
just guessing.

--
Jesse F. Hughes
"If the car stops and you're not getting out, then you have to start
it again."        -- Quincy P. Hughes (age 3) on his father's skills
with a manual transmission.

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More options Oct 13 2012, 7:34 pm
Newsgroups: sci.logic, sci.math
From: Rotwang <sg...@hotmail.co.uk>
Date: Sun, 14 Oct 2012 00:34:42 +0100
Local: Sat, Oct 13 2012 7:34 pm
Subject: Re: Proposed Formal Definition of a Concrete Category
On 13/10/2012 22:31, Dan Christensen wrote:

Clearly, then, it doesn't have one. All the mathematicians who think
they've found applications of category theory as usually understood must
have been mistaken. There can't possibly be any application of
categories whose so-called morphisms are something other than
structure-preserving transformations, because any such application would
be immediately apparent to you, Dan Christensen, upon sort-of reading
several informal explanations of what a category is.

Don't let Jesse or Alan or Frederick or any of those other mediocre
minds who were foolish enough to waste their time finding out the actual
definition of 'category' tell you otherwise - mathematical terms such as
'morphism' mean exactly what you guess them to mean, and any
mathematician or mathematical textbook that says they mean something
else is wrong.

--
I have made a thing that superficially resembles music:

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More options Oct 13 2012, 8:07 pm
Newsgroups: sci.logic, sci.math
From: Graham Cooper <grahamcoop...@gmail.com>
Date: Sat, 13 Oct 2012 17:07:57 -0700 (PDT)
Local: Sat, Oct 13 2012 8:07 pm
Subject: Re: Proposed Formal Definition of a Concrete Category
On Oct 14, 9:34 am, Rotwang <sg...@hotmail.co.uk> wrote:

How many of these texts have computer programs that support these
definitions?

It's very hard to find computer generated definitions that will fit
into a formal parser.

IMO group theory and category theory look like AI reruns of Search
Operands and Semantic Networks.

the ole

X -isa-> Y

Beware the HARD PROBLEM!

Herc

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More options Oct 13 2012, 11:37 pm
Newsgroups: sci.logic, sci.math
From: Dan Christensen <Dan_Christen...@sympatico.ca>
Date: Sat, 13 Oct 2012 20:37:19 -0700 (PDT)
Local: Sat, Oct 13 2012 11:37 pm
Subject: Re: Proposed Formal Definition of a Concrete Category
On Oct 13, 8:07 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

Odd, isn't it? I found that even for something as ubiquitous as
Peano's Axioms, it took me quite a bit of tinkering to piece together
a workable and truly formal set of axioms. I don't know if my
definition of a concrete category here is "correct" or workable, but
the experts here seem content with the verbose and, to me, vague
axioms/definitions usually given as the basis for category theory.

Could there actually be anything to the popular (mis?)perception:
"Abstract nonsense is a popular term used by mathematicians to
describe certain kinds of arguments and concepts in category theory?"

Source: "Introduction to Category Theory"
http://en.wikiversity.org/wiki/Introduction_to_Category_Theory

Dan

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More options Oct 14 2012, 12:03 am
Newsgroups: sci.logic, sci.math
From: "Jesse F. Hughes" <je...@phiwumbda.org>
Date: Sun, 14 Oct 2012 00:01:29 -0400
Local: Sun, Oct 14 2012 12:01 am
Subject: Re: Proposed Formal Definition of a Concrete Category

Yes, it really is remarkable that you have greater insights in every
mathematical topic than the experts have.  And the definition of
category really is remarkably complex, isn't it?

Part of your problem is that you want to leap straight for the concept
of concrete category, but it would be best if you learned the simpler
concepts first, namely, category in its usual generality.  That is the
easier concept to understand.

On the other hand, you could instead pretend to be revolutionizing the
field with your half-baked ideas free from the clutter of actual
understanding.  That's probably more fun.  Do that.

But if you want to actually *learn* what a category is, I'd be happy to
help.

> Could there actually be anything to the popular (mis?)perception:
> "Abstract nonsense is a popular term used by mathematicians to
> describe certain kinds of arguments and concepts in category theory?"

> Source: "Introduction to Category Theory"
> http://en.wikiversity.org/wiki/Introduction_to_Category_Theory

Category theory is abstract nonsense by design, but it's darned useful
abstract nonsense.

--
Jesse F. Hughes

"Wikipedia [...] is not a passive observer but a active and aggressive
fascist dictator of nicknames." -- Archimedes ("Arky") Plutonium

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More options Oct 14 2012, 12:32 am
Newsgroups: sci.logic, sci.math
From: Dan Christensen <Dan_Christen...@sympatico.ca>
Date: Sat, 13 Oct 2012 21:32:10 -0700 (PDT)
Local: Sun, Oct 14 2012 12:32 am
Subject: Re: Proposed Formal Definition of a Concrete Category
On Oct 13, 5:43 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

There is a rudimentary structure defined by set membership.

> In any case, whether you see the point or not, you ought to be looking
> at actual textbooks and working through their examples rather than just
> trying to draw inferences from your own intuitions of what category
> theory should be.

I have been looking for a while, for a truly formal definition of the
concept of a category. Unable to find one ready-made, I have been
trying to piece together my own from various informal definitions (see
original posting). Call it a high intolerance for ambiguity in things
mathematical.

> Because, you see, people have already defined terms like "category" and
> even "concrete category" and you don't have to guess what these ought to
> mean.  You can learn what they actually do mean.

I shouldn't have to "learn" what they actually mean. Ideally, in
mathematics, we should have a formal, stand-alone definition. Apart
from some well chosen, informal commentary, that is all the "meaning"
you should need. Why can't anyone provide such a definition? Is it
impossible in the case of category theory?

Dan

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More options Oct 14 2012, 12:45 am
Newsgroups: sci.logic, sci.math
From: Dan Christensen <Dan_Christen...@sympatico.ca>
Date: Sat, 13 Oct 2012 21:45:56 -0700 (PDT)
Local: Sun, Oct 14 2012 12:45 am
Subject: Re: Proposed Formal Definition of a Concrete Category
On Oct 14, 12:03 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

So complex that a formal definition seems to elude all the experts
here.

> Part of your problem is that you want to leap straight for the concept
> of concrete category, but it would be best if you learned the simpler
> concepts first, namely, category in its usual generality.  That is the
> easier concept to understand.

The formal definition of a category then...

> On the other hand, you could instead pretend to be revolutionizing the
> field with your half-baked ideas free from the clutter of actual
> understanding.

You don't like my formal definition? It could well be flawed, so let's
see yours. No words, no hand waving, just mathematical symbols if you
don't mind. My tiny brain can't seem to handle much more than that.

Dan