terminology in definitions of limits
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.
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
Peter E observed that 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
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.
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
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
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
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
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
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
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
participants (10)
-
Charles Wells -
Colin McLarty -
Eduardo J. Dubuc -
John Baez -
mail.btinternet.com -
Michael Barr -
Paul Taylor -
peasthope＠shaw.ca -
Richard Garner -
Vaughan Pratt