Re: Name of concept?

Re: Name of concept?

[Marco Grandis]

Yes, I like this symmetric presentation.
____

As to "Notice..." (where you mean ... meet im(d),  of course), I
would add the following.

The condition  dd = 0  is linked with aspects whose relevance is
often ignored.
Without it, many systems of use in homological algebra would loose
any reasonable notion of "canonical isomorphism", and one should be
extremely prudent in working with induced morphisms and Noether
isomorphisms.

Let us start (in Ab, or any abelian category) with a sublattice  L
of subobjects of a given object (necessarily modular) and consider
the subquotients having numerator and denominator in L. Then, the
canonical isomorphisms among these subquotients (induced by the
identity) are closed under composition *if and only if* L is
distributive.

- Within this restriction, being "canonically isomorphic
subquotients" has a precise meaning: there is a well-determined
canonical isomorphism linking them.

- Without this restriction, composing canonical isomorphisms can
yield different isomorphisms between two given subquotients. Working
up to canonical isomorphism, as commonly done in homological algebra,
could easily lead to errors.

(For instance, it is easy to construct such a situation for
subquotiens of  Z^2 (pairs of integers) in Ab - the classical example
of an object whose lattice of subobjects is not distributive.)

Concretely, the main systems of homological algebra giving rise to
spectral sequences (filtered differential object, filtered complex,
exact pairs, double complex) DO produce distributive lattices.
Essentially, the proof is generally based on a crucial Birkoff
theorem: the free modular lattice generated by two chains is
distributive. But all this is no longer true without assuming  dd =
0  (or something similar) in such systems: distributivity would fail.

[[   It is easy to see the role of  dd = 0  in the simplest case, the
filtered differential object.

We have a differential object  (A, d)  equipped with a consistent
filtration  (F_p A),  so that every  d(F_p A)  is contained in  F_p
A.  Then, one can prove that the sublattice of  Sub(A)  generated by
the filtration and closed under d-images and d-preimages  (written
d*)  is generated by two filtrations, the original one and

       0  -> ... d(F_p A) ...  ->   dA  ->  d*0  ->  ...   d*(F_p
A)  ...  ->  A,

using the inclusion  dA -> d*0.    ]]

Zeeman was probably the first to recognise the importance of this fact:

     E.C. Zeeman, On the filtered differential group, Ann. Math. 66
(1957), 557-585.

_____

The theory I am referring to can be seen in three papers of mine:

M.G., On distributive homological algebra, I-III, Cahiers 25 (1984),
259-301; 25 (1984), 353-379; 26 (1985), 169-213.

Recently, Francis Borceux and I have extended part of these results
to a larger setting, including Grp (groups):

F. Borceux - M. Grandis, Jordan-Holder, modularity and distributivity
in non-commutative algebra, Dip. Mat. Univ. Genova, Preprint 474 (Feb
2003).
http://www.dima.unige.it/~grandis/BGwe.dvi


Marco Grandis



On 30 Sep 2005, at 20:37, Michael Barr wrote:

Incidentally, did you know that if Z and Z' are defined so that
          a      d      a'
0 --> Z ---> C ---> C ---> Z' ---> 0
is exact, then the homology is the image (= coimage) of a'.a: Z --> Z'?
This is a triviality, but it gives a symmetric definition of homology.
Notice that it defines something even when d.d is not 0.  I guess it
is Z
mod Z meet ker(d).

Mike

Re: Name of concept?

[Michael Barr]

Incidentally, did you know that if Z and Z' are defined so that
         a      d      a'
0 --> Z ---> C ---> C ---> Z' ---> 0
is exact, then the homology is the image (= coimage) of a'.a: Z --> Z'?
This is a triviality, but it gives a symmetric definition of homology.
Notice that it defines something even when d.d is not 0.  I guess it is Z
mod Z meet ker(d).

Mike

Name of concept?

[Michael Barr]

Is there a name for the following situation:  I have a diagram of
categories and functors
                 DO(F)
          DO(A) ------> DO(B)
           |             |
           |             |
         H |             |H
           |             |
           v     F       v
           A ----------> B
It does not commute, nor is there even a 2 cell in either direction.
What I do have is illustrated below:
                DO(A)                               DO(A)
                /|\                                 /|\
               / | \                               / | \
              /  |  \                             /  |  \
            H/   |   \DO(F)                     H/   |   \DO(F)
            /    P    \                         /    P'   \
           /     |     \                       /     |     \
          v      |      v                     v      |      v
          A  <== | ==>  DO(B)                 A  ==> | <==  DO(B)
           \     |     /                       \     |     /
            \    |    /                         \    |    /
            F\   |   /H                         F\   |   /H
              \  |  /                             \  |  /
               \ | /                               \ | /
                vvv                                 vvv
                 B                                   B

and, moreover,
           P -------> HF
           |          |
           |          |
           |          |
           |          |
           |          |
           v          v
           FH ------> P'
commutes.