Model structure on bounded chain complexes
tonym...@googlemail.com
Grothendieck's AB5 axiom, etc.) there exists a convenient model
structure on the category of non-negative chain complexes C(A):
- weak equivalences are homology-isomorphisms
- cofibrations are the levelwise split monos with a levelwise
projective cokernel
- and fibrations are the morphisms with are epis in all positive
levels.
This can be found in Quillen's Homotopical algebra or papers of
Christensen, Hovey, etc.
To interpret classical homological algebra in terms of homotopy theory
there have to be a connection between derived functors in the
homological sense and total derived functors in the model category
sense. I wonder if the following statement is true in general - or
even in particular cases. I am not able to find a proof in the
literature. Maybe it's obvious or even wrong:
Let A and B are two abelian categories such that model structures can
be obtained in the way described above. Let F:A-->B be a left-exact
additive functor. Is the induced functor CF:CA-->CB on chain complexes
a right Quillen-functor?
If this is the case, the total right derived functor in the model
category sense exists and there is a connection to the homological
one. I am very pleased for any hints, an idea of a proof or even a
reference.
Agusti Roig
I don't think that with your hypothesis CF is a right Quillen
functor.
Nevertheless, you can prove that CF has a derived functor as
follows.
1. But, first, you either have to change your model structure and your
boundedness conditions, or else, change your objective. I mean: with
this model structure you will be able to derive functors from the
left, not from the right.
2. Assuming you want to obtain *left* derived functors, the point is
that your functor F is *additive*. So CF preserves the classical
homotopy relation between morphisms of complexes:
g - f = dh + hd
and, as a consequence, homotopy equivalences.
3. Now some notations: let
\pi CA and \pi CB
denote the (classical) homotopy categories; that is, the categories
obtained from CA and CB identifying homotopic maps.
Let also
Ho CA and Ho CB
be the (Quillen) homotopy categories; that is, the categories obtained
from CA and CB by inverting weak equivalences (homology
isomorphisms) and let
gamma : CA ---> Ho CA and CB ---> Ho CB
denote the localizing functors.
Finally, let CA_c denote the subcategory of cofibrant complexes,
that is, complexes which are dimension-wise projective.
4. Now follow the proof of proposition 1, section 4.2, in [Q]: let
Q: CA ---> \pi CA_c
denote the well-defined functor which sends each complex to a
cofibrant model.
Since CF preserves the homotopy relation, it induces a well-defined
functor
F' : \pi CA_c ---> \pi CB .
Also gamma : CB ---> Ho CB induces a functor
\gamma': \pi CB ---> Ho CB
since, by definition, it sends weak equivalences to isomorphisms, so
you can apply lemma 8.3.4 of [H].
The composition
\gamma' F' Q : CA ---> Ho CB
sends weak equivalences to isomorphisms because, if f: K ---> L is a
weak equivalence between chain complexes, then:
4a) Qf : QK ---> QL is also a weak equivalence by the 2 out of 3
property.
4b) Since QK and QL are cofibrant complexes, a weak equivalence
between them turns out to be a homotopy equivalence; that is, an
isomorphism in \pi CA_c . (This is Whitehead theorem: [H], theorem
7.5.10, taking into account that every complex is fibrant with this
model structure.)
4c) Being a functor, \gamma'F' sends isomorphisms to isomorphisms.
As a consequence, by definition of a left derived functor, \gamma'F'Q
induces the left derived functor
LF : Ho CA ---> Ho CB
A trivial remark that maybe it's worth pointing out: essentially, what
you have done in order to define/compute the left derived functor of
F is:
LF (K) = F(QK)
That is to say: take a cofibrant model of K (a projective resolution
in classical terms) and then apply F .
References:
[Q] Quillen, Homotopical Algebra, Springer LNM 43 (1967).
[H] Hirschhorn, Model categories and their localizations, AMS Math.
Surveys and Monographs 99 (2002).
Agusti Roig