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.
Example 4: The category I is the empty category. Then there is a unique functor from I to any other category C. The colimit of this functor is the initial object of C.
Example 5: The category I has a terminal object *. Then the colimit of an I-diagram X is X(*).
We next began the discussion of cofibrantly generated model categories. Let C be a category that has all small colimits, and let I be a class of morphisms in C. We defined small objects relative to I, I-injective maps, I-cofibrations, and I-cellular maps. We showed that every I-cellular map is an I-cofibration. See the note for Lecture 7 for the definitions.