I = { A --> B }
J = { U --> V }
such that the domains A of the maps in I are small relative to the
I-cellular maps, the domains U of the maps in J are small relative to
the J-cellular maps, the fibrations are the maps that have RLP with
respect to the maps in J, and the trivial fibrations are the maps that
have RLP with respect to the maps in I. It follows that, in a
cofibrantly generated model category, the cofibrations are the
retracts of I-cellular maps, and the trivial cofibrations are the
retracts of J-cellular maps. We note that the weak equivalences may be
characterized as the maps that can be factored as the composition of a
trivial cofibration followed by a trivial fibration. However, this
characterization of weak equivalences is not very useful and it is
always necessary to have some other definition of weak equivalences in
order to verify the model category axioms.
We further oulined the proof that the category of topological spaces and continuous maps is a cofibrantly generated model category with generators
I = { S^{n-1} --> D^n | n non-negative integer}
J = { D^n --> D^n x [0,1] | n non-negative integer}
and with weak equivalences the maps f : X --> Y such that the induced
map
f_* : \pi_n(X,x) --> \pi_n(Y,f(x))
is an isomorphism, for all non-negative integers n and all x in
X.
The definition of a fibration was given by Serre in his thesis. He showed that for a fibration f : X --> Y there exists a spectral sequence
E^2 = H_*(Y; H_*(F)) ==> H_*(X)
from the homology of Y with coefficients in the (local coefficient
system given by the) homology of the fiber F = f^{-1}(y) and
converging to the homology of X. It is this spectral sequence and
others similar to it that makes it possible to evaluate the homology
of practically every space.