Re: Snake lemma-Five lemma

In article <1126125442.536304.237210@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Ed Hook <hook@xxxxxxxxxxxx> wrote:
>
>Arturo Magidin wrote:

>> > Is anyone else struck by just how _wrongheaded_ this exercise
>> > is ?? The direct proof of the Five Lemma is a pleasant (even
>> > soothing) exercise in diagram-chasing -- to smack it with the
>> > Snake Lemma just seems *ugly* ...
>>
>> But diagram-chasing requires the objects to be sets and the maps to be
>> set-maps. Invoking the Snake-Lemma gives you an element-free proof of
>> the Five Lemma, which is itself something that might be useful.
>>
>
> Well ... clearly ... I wasn't thinking along those lines.
> Given that desire for greater generality/applicability, I
> guess I was a little harsh ... :-)

Most of the time I would agree with you; in at least part (b) I think,
as you did, the Snake Lemma is not unreasonable. If you will be
working mostly/exclusively with set-based categories, then there is
certainly no point in searching for, if you'll pardon the pun,
pointless generality.

> But that raises a question in my mind: is there an "element-
> free" proof of the Snake Lemma ??

Ah, I was afraid of that question... I thought Bourbaki or Mac Lane
would have one, but neither does; Bourbaki restricts it to commutative
groups, Mac Lane has it as an exercise on diagram chasing (talk about
irony, there!). So my answer is... I don't know.

>> Proving some
>> of these results without having to fall back on element-chasing may
>> itself be a worthwhile exercise.

> So I hope that you'll answer my question up above ...

You'll note my use of "may". In some instances, not having to rely on
specific elements and element chasing may be worthwhile; just as
proving properties of vector spaces ->without<- invoking the fact that
every vector space can be regarded to as a set of ordered tuples with
the usual operations, may be a worthwhile exercise.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.

Relevant Pages

  • Re: Definition of measurability
    ... Arturo Magidin wrote: ... (among other possible choices of generating sets). ... exercise to show that: ... it is enough to show that the inverse image of elements of a basis are ...
    (sci.math)
  • Re: Definition of measurability
    ... Arturo Magidin wrote: ... (among other possible choices of generating sets). ... exercise to show that: ... If P is a collection of subsets of Y, Ais the sigma-algebra ...
    (sci.math)
  • Re: continuity (mistake?)
    ... Consider the sequence given by ... / 1/n if n is odd ... you spoiled the exercise :-( ... Arturo Magidin ...
    (sci.math)
Loading