\( \usepackage{mathrsfs} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\cHom}{\underline{Hom}} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\inj}{inj} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\coim}{coim} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\eq}{eq} \DeclareMathOperator{\op}{op} \DeclareMathOperator{\ob}{ob} \DeclareMathOperator{\coeq}{coeq} \DeclareMathOperator{\cone}{cone} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\Ch}{Ch} \DeclareMathOperator{\Ab}{\text{Ab}} \DeclareMathOperator{\Top}{\text{Top}} \DeclareMathOperator{\Rmod}{R\text{mod}} \DeclareMathOperator{\SSet}{\text{SSet}} \DeclareMathOperator{\sAb}{\text{sAb}} \DeclareMathOperator{\cA}{\mathcal{A}} \DeclareMathOperator{\cB}{\mathcal{B}} \DeclareMathOperator{\cC}{\mathcal{C}} \DeclareMathOperator{\cO}{\mathcal{O}} \DeclareMathOperator{\cM}{\mathcal{M}} \DeclareMathOperator{\sC}{\mathscr{C}} \DeclareMathOperator{\sF}{\mathscr{F}} \DeclareMathOperator{\sW}{\mathscr{W}} \DeclareMathOperator{\dd}{\partial} \DeclareMathOperator{\rd}{\mathrm{d}} % Rings \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\CC}{\mathbf{C}} \DeclareMathOperator{\PP}{\mathbf{P}} \DeclareMathOperator{\QQ}{\mathbf{Q}} \DeclareMathOperator{\RR}{\mathbf{R}} \DeclareMathOperator{\ZZ}{\mathbf{Z}} \)

Section5.1Model Categories

Definition5.1.1Left lifting property
A map \(i\) satisfies the left lifting property with respect to a map \(p\) if in any diagram \[ \xymatrix{ A \ar[r]^f \ar[d]_i & X \ar[d]^p\\B\ar@{..>}[ur]^h \ar[r]_g& Y } \] the map \(h\) exists making the diagram commute.

So in a category \(\cA\) a map \(P\) is projective if and only if \(0\to P\) has the left lifting property with respect to all surjections. Similarly a map \(I\) is injective if and only if \(I\to 0\) has the right lifting property with respect to all injections.

Definition5.1.2Retracts
A map \(f\colon A \to B\) is a retract if a map \(g\colon A' \to B'\) if there exists a diagram \[ \xymatrix{ A \ar@{=}@/^/[rr] \ar[r]\ar[d]_f &A' \ar[r] \ar[d]^g &A\ar[d]^f \\ B \ar@{=}@/_/[rr] \ar[r] & B'\ar[r] & B } \]
Definition5.1.3Model categories
A model category is a category \(\cM\) with three classes of maps, \(\sW\) (weak equivalences), \(\sF\) (fibrations), \(\sC\) (cofibrations). We call maps in \(\sW \cap \sF\)acyclic fibrations and those in \(\sW \cap \sC\)acyclic cofibrations. We require these categories and classes to satisfy the following:
  1. \(\cM\) has all small limits and colimits.
  2. If \(f,g,gf\) are morphisms of \(\cM\) then any two lying in \(\sW\) implies the third does also.
  3. \(\sF,\sC,\sW\) are all closed under retracts.
    1. Any map in \(\sC\) has the left lifting property with respect to any map in \(\sW\cap\sF\).
    2. Any map in \(\sF\) has the left lifting property with respect to any map in \(\sW\cap\sC\).
  5. Any map \(f\) may be functorially factored as
    1. \(f = p \circ i\) for some \(i\in \sC\), \(p\in \sF \cap \sW\).
    2. \(f = q \circ j\) for some \(j\in \sC\cap \sW\), \(q\in \sF\).
Example5.1.4
\(\Ch_{\ge 0} R\) forms a model category with \(\sW\) being the class of quasi-isomorphisms, \(\sF\) the class of chain maps surjective in strictly positive grading, and \(\sC\) bein chain maps \(f\) where \(f_n\) is injective with projective cokernel for all \(n\). This is called the projective model structure on \(\Ch_{\ge 0} R\)
Definition5.1.5Fibrant and cofibrant objects
\(A\in \cM\) is cofibrant if \(0 \to A\) is a cofibration. \(A\in \cM\) is fibrant if \(A\to 0\) is a fibration.