Example 1: The set of objects in I is the set of non-negative integers and there is a unique morphism from n to m if m is greater than or equal to n. An I-diagram is then a the same as a sequence of maps in C
X_0 --> X_1 --> X_2 --> X_3 --> ....
If C is the category of sets and if all maps in the sequence are
inclusions of a subset, then the colimit is equal to the union of the
sets X_i. Similarly, if C is the category of R-modules and if all maps
in the sequence are inclusions of a submodule, then the colimit is the
union of the R-modules X_i.
Example 2: The category I is discrete in the sense I has only identity morphisms. Then an I-diagram is the same as a set of objects of C indexed by the set I. In this case the colimit is called the coproduct of the objects X_i. If C is the category of sets, then the coproduct of the sets X_i is the disjoint union of the sets X_i, and if C is the category of R-modules, then the coproduct of the R-modules X_i is the direct sum of the R-modules X_i.
Example 3: The category I has three objects 0, 1, and 2 and two non-identity morphisms 0 --> 1 and 0 --> 2. Then an I-diagram in C consists of three objects X_0, X_1, and X_2 and two maps X_0 --> X_1 and X_0 --> X_2. In this case, colimit = pushout.
We next began the construction of a model structure on the category whose objects are the bounded below chain complexes of left modules over a ring R and whose morphisms are the chain maps. Let S^{n-1} be the chain complex
... --> 0 --> 0 --> R --> 0 --> ...
with R located in degree n-1, and let D^n be the chain complex
... --> 0 --> R === R --> 0 --> ...
with R located in degrees n and n-1. There is a chain map S^{n-1} -->
D^n that is given by the identity map R = R in degree n-1. We consider
the two sets of maps
I = { S^{n-1} --> D^n | n integer }
J = { 0 ----> D^n | n integer }
We then defined weak equivalences to be the maps f : X --> Y such that
the induced map f_* : H_*(X) --> H_*(Y) on homology is an isomorphism;
we defined fibrations to be the maps that have the right lifting
property (abbreviated RLP) with respect to the maps in J. And we
defined cofibrations to the maps that have the left lifting
property (abbreviated LLP) with respect to the maps that are both
fibrations and weak equivalences. We showed that the maps that are
both fibrations and weak equivalences are the maps that have RLP with
respect to the maps in I. We then introduced the notion of an
I-cellular map f : A --> B. This means that B can be written as the
colimit of a sequence of inclusions
A = B_0 --> B_1 --> B_2 --> .... --> B
where B_m is obtained from B_{m-1} as pushout along a direct sum of
maps in I. We showed that every I-cellular map is a cofibration. We
then showed that every map f : X --> Y can be factored as the
composition f = p j of an I-cellular map j and a map p that is both a
fibration and a weak equivalence. The proof was by Quillen's small
object argument. Finally, we proved that every cofibration is a
retract of an I-cellular map. In this sense the maps in I generate all
cofibration. We say that I is a set of generating cofibrations.
We plan to complete the proof next time and to give some examples of calculations of Ext-groups.
For reading see MacLane: Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, for colimits and Hovey for the model structure on the category of chain complexes.