Re: terminology in definitions of limits

Re: terminology in definitions of limits

[mail.btinternet.com]

Without getting into discussion of the `game' aspect, I feel category
theorists should speak out against the epsilon-delta approach to limits as
against the neighbourhood
                       f(M) \subseteq N
approach, where the notation easily describes  the pictures. The
epsilon-delta approach is in terms of measurement of a neighbourhood, i.e.
one step away from the neighbourhood, and less actual (I almost wrote
`real'!), and students find that step difficult.  The utility of
epsilon-delta is in terms of calculation, rather than geometry and
structure.

The  `only measurable things are real' approach is based on the notion that
numbers are the most important aspect of science, rather than one tool to
investigate  structure.

Ronnie

Re: terminology in definitions of limits

[Eduardo J. Dubuc]

of course, by choice (and many times without choice), there are lots of
functions \delta = f(\epsilon). It is a good question to see when there is a
continous such "f".

e.d.




Charles Wells wrote:
> Calculus teachers do something similar when they make an epsilon-delta proof
> into a game:  The opponent picks an epsilon (the test object) and you have
> to come up with a delta.
> There is one big difference between epsilon-delta proofs and limits.  To
> show that something is a limit you have to find, for each test object, the
> unique arrow specified by the definition of limit.  Thus you are producing a
> function (indeed, a bijection).   The delta for a given epsilon is not unique,
> and so there is no natural function giving a delta for each epsilon.  I am
> pretty sure this makes epsilon-delta proofs harder for non-talented students
> than proving something is a limit.  I know some calculus teachers talk about
> there being a function that takes epsilon to delta, but I suspect it is a
> mistake to bring that up.
>
> Charles Wells
>
> On Wed, Jan 21, 2009 at 1:34 AM, Vaughan Pratt <pratt@cs.stanford.edu>wrote:
>
>> Colin McLarty wrote:
>>
>>> I often call them "test objects" in talking with students (by analogy
>>> with "test particles" in General Relativity).  I don't think I have ever
>>> done it in print.  But I did use "T" as the typical name of such an
>>> object in my book.
>>>
>>> I am curious to know what others think.
>>>
>> From a game-theoretic standpoint one can be either taking the test or
>> administering it.  Both sides call it the test, showing that the name is
>> stable under perp (change of team).
>>
>> However that's not to say that "test" gives a helpful perspective in
>> either case.  A right adjoint defined by its adjunction is simply a
>> specification of *all* homsets to it, and dually, in the case of left
>> adjoints, of all the homsets from it.  What you're calling a "test"
>> object there is for me merely the variable being universally quantified
>> over in the definition of "all."
>>
>> Whether a student is going to find it helpful thinking of a universally
>> quantified variable as a "test object" is going to be less a question of
>> what the student thinks about that perspective than what the teacher
>> thinks about it and whether they can convey their point of view.  The
>> mathematically talented student who immediately sees it is merely being
>> universally quantified over may be more puzzled than helped.
>>
>> But then how many of us are so lucky as to have a significant number of
>> mathematically talented students in our classes?
>>
>> Vaughan
>>
>>
>>
>>
>
>

Re: terminology in definitions of limits

[Richard Garner]

I have always used the phrase "test object" in a slightly
different sense. Namely, to refer to a tractably small
collection of objects that one may use, not only to detect,
but also to calculate some right adjoint. Thus in Set, one
may take the terminal object; in Set/X, the elements 1-->X;
in Cat, the ordinals 1, 2 and 3; in presheaf categories, the
representables; and so on. The best case is that these test
objects are colimit dense, since then your calculations
always yield a right adjoint as soon as the functor you start
with preserves colimits.

Richard

Re: terminology in definitions of limits

[Michael Barr]

This is getting peripheral to the main point.  AS far as I recall, I
thought of T as a test object.  As for epsilon-delta, Bishop required that
delta be prescribed as a constructible function of epsilon in order that a
function be continuous.  He required that the convergence be uniform on
every closed interval, so that this function on a closed interval was
independent of the points in the interval.

Michael

On Wed, 21 Jan 2009, Charles Wells wrote:

> Calculus teachers do something similar when they make an epsilon-delta proof
> into a game:  The opponent picks an epsilon (the test object) and you have
> to come up with a delta.
> There is one big difference between epsilon-delta proofs and limits.  To
> show that something is a limit you have to find, for each test object, the
> unique arrow specified by the definition of limit.  Thus you are producing a
> function (indeed, a bijection).   The delta for a given epsilon is not unique,
> and so there is no natural function giving a delta for each epsilon.  I am
> pretty sure this makes epsilon-delta proofs harder for non-talented students
> than proving something is a limit.  I know some calculus teachers talk about
> there being a function that takes epsilon to delta, but I suspect it is a
> mistake to bring that up.
>
> Charles Wells
>

Re: terminology in definitions of limits

[John Baez]

Dear Categorists -

On Tue, Jan 20, 2009 at 11:34 PM, Vaughan Pratt <pratt@cs.stanford.edu>wrote:


> Colin McLarty wrote:
>
>> I often call them "test objects" in talking with students (by analogy with
>> "test particles" in General Relativity).  I don't think I have ever done it
>> in print.
>
>

> From a game-theoretic standpoint one can be either taking the test or
> administering it.   [..]  What you're calling a "test" object there is for
> me merely the variable being universally quantified over in the definition
> of "all."


 When I teach limits I call Colin's "test object" a "competitor" to the true
limit, or "pretender to the throne", and describe the universal property as
saying "whatever you can do, I can do better".

This game-theoretic approach to universal properties becomes more
interesting when dealing with n-categorical weak limits: the two players
take turns making moves.  First the proponent picks a cone, then the
challenger picks a cone, then the proponent picks a map between cones, then
the challenger picks a map between cones, then the proponent picks a map
between maps between cones, etc..

This idea is important in opetopic n-categories, and there's also an
omega-categorical version - a nice discussion appears starting at the bottom
of page 32 of this paper by Makkai:

http://www.math.mcgill.ca/makkai/equivalence/equivinpdf/equivalence.pdf

"The Hero has to answer each move of the Challenger [...] If Hero can keep
it up forever, he wins; otherwise he loses."

Best,
jb

Re: terminology in definitions of limits

[Charles Wells]

Calculus teachers do something similar when they make an epsilon-delta proof
into a game:  The opponent picks an epsilon (the test object) and you have
to come up with a delta.
There is one big difference between epsilon-delta proofs and limits.  To
show that something is a limit you have to find, for each test object, the
unique arrow specified by the definition of limit.  Thus you are producing a
function (indeed, a bijection).   The delta for a given epsilon is not unique,
and so there is no natural function giving a delta for each epsilon.  I am
pretty sure this makes epsilon-delta proofs harder for non-talented students
than proving something is a limit.  I know some calculus teachers talk about
there being a function that takes epsilon to delta, but I suspect it is a
mistake to bring that up.

Charles Wells

On Wed, Jan 21, 2009 at 1:34 AM, Vaughan Pratt <pratt@cs.stanford.edu>wrote:

>
> Colin McLarty wrote:
>
>> I often call them "test objects" in talking with students (by analogy
>> with "test particles" in General Relativity).  I don't think I have ever
>> done it in print.  But I did use "T" as the typical name of such an
>> object in my book.
>>
>> I am curious to know what others think.
>>
>
> From a game-theoretic standpoint one can be either taking the test or
> administering it.  Both sides call it the test, showing that the name is
> stable under perp (change of team).
>
> However that's not to say that "test" gives a helpful perspective in
> either case.  A right adjoint defined by its adjunction is simply a
> specification of *all* homsets to it, and dually, in the case of left
> adjoints, of all the homsets from it.  What you're calling a "test"
> object there is for me merely the variable being universally quantified
> over in the definition of "all."
>
> Whether a student is going to find it helpful thinking of a universally
> quantified variable as a "test object" is going to be less a question of
> what the student thinks about that perspective than what the teacher
> thinks about it and whether they can convey their point of view.  The
> mathematically talented student who immediately sees it is merely being
> universally quantified over may be more puzzled than helped.
>
> But then how many of us are so lucky as to have a significant number of
> mathematically talented students in our classes?
>
> Vaughan
>
>
>
>


-- 
professional website: http://www.cwru.edu/artsci/math/wells/home.html
blog: http://www.gyregimble.blogspot.com/
abstract math website: http://www.abstractmath.org/MM//MMIntro.htm
personal website:  http://www.abstractmath.org/Personal/index.html

Re: terminology in definitions of limits

[Vaughan Pratt]

Colin McLarty wrote:
> I often call them "test objects" in talking with students (by analogy
> with "test particles" in General Relativity).  I don't think I have ever
> done it in print.  But I did use "T" as the typical name of such an
> object in my book.
>
> I am curious to know what others think.

 From a game-theoretic standpoint one can be either taking the test or
administering it.  Both sides call it the test, showing that the name is
stable under perp (change of team).

However that's not to say that "test" gives a helpful perspective in
either case.  A right adjoint defined by its adjunction is simply a
specification of *all* homsets to it, and dually, in the case of left
adjoints, of all the homsets from it.  What you're calling a "test"
object there is for me merely the variable being universally quantified
over in the definition of "all."

Whether a student is going to find it helpful thinking of a universally
quantified variable as a "test object" is going to be less a question of
what the student thinks about that perspective than what the teacher
thinks about it and whether they can convey their point of view.  The
mathematically talented student who immediately sees it is merely being
universally quantified over may be more puzzled than helped.

But then how many of us are so lucky as to have a significant number of
mathematically talented students in our classes?

Vaughan

Re: terminology in definitions of limits

[Colin McLarty]

I often call them "test objects" in talking with students (by analogy
with "test particles" in General Relativity).  I don't think I have ever
done it in print.  But I did use "T" as the typical name of such an
object in my book.

I am curious to know what others think.

best, Colin

----- Original Message -----
From: categories@mta.ca
Date: Tuesday, January 20, 2009 11:01 am
Subject: categories: terminology in definitions of limits
To: categories@mta.ca

> Folk,
>
> Each definition of a limit which I've
> seen contains something I would describe
> as a "probe object" or "test object".  The
> definition of map object in L&S page 313
> for example, has X with a criterion asserted
> for every object X in the category.
>
> Is there any sense in my terminology?
>
> Thanks,        ... Peter E.
>
>
>
>

Re: terminology in definitions of limits

[Paul Taylor]

Peter E observed that
> each definition of a limit which I've seen contains something
> I would describe as a "probe object" or "test object"
although I am not sure whether his question is about the name for
this (for which either of his suggestions is reasonable), or what.

Limits are, of course, examples of right adjoints, and the situation
that Peter describes is a case of the adjoint correspondence

   (considered as a trivial diagram) test object  ----->  diagram
   ==============================================================
                            test object ------>  limit of diagram

So the left adjoint is a "forgetful" functor,  which takes the test
object and considers it as a trivial diagram,  ie with identities
as edges.

Giving the test object a "name" in the sense of an English word
is not such a big deal.

However, I would argue that it is important to give it a "name"
in the sense of using a particular letter uniformly for it.

For this purpose, I propose the Greek letter capital Gamma.

The reason for this choice is that the same role is played in
symbolic logic by the "context",  ie the collection of parameters,
along with their types and hypotheses,  that occurs in any
mathematical statement.   In type theory, the letter Gamma is
traditionally and uniformly used for this purpose.

(Can some type or proof theorist tell me who introduced or
established this convention?)

Indeed, I use this convention both for this test object and for
other parts of the anatomy of an adjunction systematically throughout
my book, "Practical Foundations of Mathematics"  (CUP, 1999).

In so far as there was a previous convention in category theory for
the name of this object, it was "U".  This came from sheaf theory,
where, by the Yoneda lemma, we need only consider maps from
hom(-,U), where U belongs to the base category.  This category was
primordially the lattice of open subsets of a topological space,
so the convention came from that of using "U" for an open set.
I believe that German-speaking authors were responsible for this,
though I don't know what German word it was that began with U.

Speaking of sheaf theory,  when and to whom was it first apparent
that the category of sheaves depends only on the lattice of open
sets, and not on the points of a topological space?

Paul Taylor
www.PaulTaylor.EU
pt09 @ PaulTaylor.EU

terminology in definitions of limits

[peasthope]

Folk,

Each definition of a limit which I've 
seen contains something I would describe 
as a "probe object" or "test object".  The 
definition of map object in L&S page 313 
for example, has X with a criterion asserted 
for every object X in the category.

Is there any sense in my terminology?

Thanks,        ... Peter E.