Model categories : Part 2 (Definitions)

Before we move to the notion of "a model structure on a category", we need to recall (or introduce) some definitions. Definition : Let \(\mathcal{C}\) be a category. An object \(X\) of \(\mathcal{C}\) is said to be a retract of an object \(\mathcal{C}\) if there exists arrows \(X\xrightarrow{f} Y\xrightarrow{g} X\) such that, the composition \(g\circ f:X\rightarrow X\) is equal to the identity arrow \(1_X:X\rightarrow X\). Definition : Let \(\mathcal{C}\) be a category. We define the morphism category of \(\mathcal{C}\), denoted by \(\text{Map}(\mathcal{C})\) whose

• objects are the arrows of \(\mathcal{C}\),
• morphisms are commutative diagrams in \(\mathcal{C}\).

Definition : Let \(\mathcal{C}\) be a category. A morphism \(f\) in \(\mathcal{C}\) is said to be a retract of a morphism \(g\) in \(\mathcal{C}\), if, \(f\) is a retract of \(g\), when both \(f\) and \(g\) are seen as objects of \(\text{Map}(\mathcal{C})\). Definition : Let \(\mathcal{C}\) be a category. Let \(i:A\rightarrow B\) and \(p:X\rightarrow Y\) be morphisms in \(\mathcal{C}\). We say that \(i\) has the left lifting property with respect to \(p\) or \(p\) has the right lifting property with respect to \(i\) if, for every commutative diagram there exists an arrow \(h:B\rightarrow X\) such that \(h\circ i=f\) and \(p\circ h=g\). Definition : Let \(\mathcal{C}\) be a category. A model structure on \(\mathcal{C}\) consists of the following data :

1. a subcategory of \(\mathcal{C}\) called “weak equivalences”,
2. a subcategory of \(\mathcal{C}\) called “fibrations”,
3. a subcategory of \(\mathcal{C}\) called “cofibrations”,

satisfying certain conditions:

1. If \(f,g\) are morphisms of \(\mathcal{C}\) such that \(gf\) is defined and two of \(f,g,gf\) are "weak equivalences" then so is the third.
2. A retract of a "weak equivalece" is a "weak equivalence".
3. A retract of a "fibration" is a "fibration".
4. A retract of a "cofibration" is a "cofibration".
5. factorizing property (part \(1\)) of arrows of \(\mathcal{C}\) : for each \(f\) in \(\text{Mor}(\mathcal{C})\), there exists a cofibration \(i\) and a traivial fibration \(q\) such that \(f=qi\).
6. factorizing property (part \(2\)) of arrows of \(\mathcal{C}\) : for each \(f\) in \(\text{Mor}(\mathcal{C})\), there exists a fibration \(p\) and a trivial cofibration \(j\) such that \(f=pj\).
7. Any commutative diagram of the type has lifting property if either \(i\) or \(p\) is a "weak equivalence".

Definition : A model category is defined to be a category that has

1. all small limits,
2. all small colimits,
3. a model structure in \(\mathcal{C}\).

Construction of new model categories from old model categories:

1. Let \(\mathcal{C}\) and \(\mathcal{D}\) be model categories. Defining the collection of fibrations (cofibrations, weak equivalences) as pairs \((f,g)\) where both \(f\) and \(g\) are fibrations (cofibrations, weak equivalences) defined a model structure on the product category \(\mathcal{C}\times \mathcal{D}\). This model category is called the product model category produced from model categories \(\mathcal{C}\) and \(\mathcal{D}\).