Holmstrom Cofibrantly generated

A model category is said to be cofibrantly generated if there are sets $I$ and $J$ of maps such that

1. The domains of the maps of I are small relative to I-cell
2. The domains of the maps of J are small relative to J-cell
3. The class of fibrations is J-inj
4. The class of trivial fibrations is I-inj

In this situtation, we refer to I as the set of generating cofibrations, and to J as the set of generating trivial cofibrations. A cofibrantly generated model category is called finitely generated if we can choose I and J so that the domains and codomains of I and J are finite relative to I-cell.

Here we have used the following abbreviations: Let I be class of morphisms in a category $C$.

1. A morphism is I-injective if if it has the RLP wrt every morphism in I. Write I-inj for the class of such morphisms.
2. A morphism is I-projective if if it has the LLP wrt every morphism in I. Write I-proj for the class of such morphisms.
3. A morphism is an I-cofibration if it has the LLP wrt every I-injective morphism. Notation: I-cofib.
4. A morphism is an I-fibration if it has the RLP wrt every I-projective morphism. Notation: I-fib.
5. If $C$ has all small colimits, define I-cell to be the class of transfinite compositions of pushouts of elements of I. Remark: I-cell is contained in I-cof.

Much more material on this is found in Hovey, Section 2.1. See for exampel Thm 2.1.19, which explains how cofibrantly generated model cats are constructed. We also get a criterion for when a functor is a Quillen functor.

For non-cofibrantly generated model cats, including examples and localization theorems, see the web page of Chorny

A cryptic note: Cofibrantly gen implies projective, and combinatorial implies injective. Does this refer to model structures on functor cats or what??

Raptis: http://front.math.ucdavis.edu/0907.2726

nLab page on Cofibrantly generated