Re: Question on exact sequence

[Marco Grandis <grandis@dima.unige.it>]

Dear George (and others),

Continuing what you and others have already said, there are (at
least) two ways of extending those two lemmas.

1. Within homological algebra, there is the 'general form' of my
first msg on this point.
As I was saying, I do not like very much this extension; unless one
can find some 'meaning' for the 'generalised homologies'  H'(f,g,h)
and  H"(f,g,h)  that appear there.

2. Within homotopical algebra, one can find a deeper solution, if
more involved.
         The question is now about a triple  (f, g, alpha)  where  f,
g  are consecutive arrows and  alpha  is a nullhomotopy of their
composite  gf.  The hypothesis that we want to express is that this
'h-differential sequence' is 'h-exact' (h for homotopically, or
homotopical).
         This is studied in my paper [*] (Como 2000), in a general
setting and more particularly for a category of morphisms (Section
4); the latter is the case we are interested in for our extension. I
will only sketch what is of interest here; the interested reader can
look at [*].
         Let  D  be a category with pullbacks and pushouts, and  D'
its category of morphisms. Think of an object as a truncated chain
complex   A = (d_A: A1  -->  A0).  (In [*],  D  is also assumed to be
additive, but this is not necessary here.)
         There are obvious nullhomotopies of morphisms: inserting a
diagonal  A0  -->  B1  in a square  f: A = B. Nullhomotopies can be
whiskered with morphisms.
         Every morphism has an h-kernel (= homotopy kernel) and an h-
cokernel, defined by universal properties and constructed with the
pullback and pushout 'inside' the square.
         An object is *contractible* if its identity is
nullhomotopic; ie if its differential is iso.
         Given two consecutive morphisms  f: X  -->  A,  g: A  -->
Y,  and a nullhomotopy  alpha  of  gf,  there is a *homotopical
homology* object  H(f, g, alpha) =(w: B --> Z),  constructed via h-
kernels and h-cokernels ([*], thm 4.5).  Think of  B  as h-boundaries
and of  Z  as h-cycles.
         Say that the h-differential sequence  (f, g, alpha)  is *h-
exact* if   H(f, g, alpha)  is contractible,  ie  w  is iso.  This
yields a nullhomotopy from the h-kernel of  g  to the h-cokernel of  f.

Now (this is not written in [*]), assuming that  D  is pointed, there
is a composed nullhomotopy (by whiskering)

         Omega(Y)  -->  hKer(g)  ==>  hCok(f)  --> Sigma(X),

where:

         Omega(Y)  =  hKer(0 --> Y (sic!) )  =  (0 --> Ker(d_Y)),
         Sigma(X)  =  hCok(X --> 0)  =  (Cok(d_X) --> 0).

Thus our nullhomotopy gives the connecting morphism

         H_1(Y)) = Ker(d_Y))    -->    H_0(X) = Cok(d_X).

Exactness (of the three Omegas followed by the three Sigmas) has to
be studied.

Best
Marco

(PS. Notice that, if D is not additive, we have nullhomotopies
without homotopies!)
[*] M. Grandis, A note on exactness and stability in homotopical
algebra, Theory Appl. Categ. 9 (2001), No. 2, 17-42


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