Model Categories


Paris

What follows is an accessible introduction to model categories

$f$ is a `retract` of $g$ iff $pq = rs = 1$.

$$ \begin{xy} \xymatrix{ A\ar[r]^s\ar[d]^f & C\ar[r]^r\ar[d]^g & A\ar[d]^f \\ B\ar[r]^q & D\ar[r]^p & B } \end{xy} $$

A `functorial factorization` is an ordered pair $(\alpha, \beta)$ of functors $\texttt{Map } \mathscr{C} \rightarrow \texttt{Map } \mathscr{C}$ such that $f = \beta(f) \circ \alpha(f)$ for all $f \in \mathscr{C}$. In particular, the domain of $\alpha(f)$ is the domain of $f$, the codomain of $\alpha(f)$ is the domain of $\beta(f)$, and the codomain of $\beta(f)$ is the codomain of $f$.

$i$ has `left-lifting property` with respect to $p$ if there is a lift $h : B \rightarrow X$ such that $hi = f$ and $ph = g$.

$$ \begin{xy} \xymatrix{ A\ar[r]^f\ar[d]^i & X\ar[d]^p \\ B\ar[r]^g\ar@{.>}[ru]^h & Y \\ } \end{xy} $$

A `model structure` on $\mathscr{C}$ is three subcategories of $\mathscr{C}$ called weak equivalences, cofibrations, and fibrations, and two functorial factorizations $(\alpha, \beta)$ and $(\gamma, \delta)$ satisfying the following properties:

  1. (2-out-of-3) If $f$ and $g$ are morphisms of $\mathscr{C}$ such that $gf$ is defined, and two of $f$, $g$ and $gf$ are weak equivalences, then so is the third.
  2. (Retracts) If $f$ and $g$ are morphisms of $\mathscr{C}$ such that $f$ is the retract of $g$, and $g$ is a weak equivalence, cofibration, or fibration, then so is $f$.
  3. (Lifting) Define a map to be a trivial fibration if it is both a fibration and a weak equivalence; similarly, define trivial cofibration. Then, trivial cofibrations have left-lifting property with respect to fibrations, and cofibrations have left-lifting property with respect to trivial fibrations.
  4. (Factorization) For any morphism $f$, $\alpha(f)$ is a cofibration, $\beta(f)$ is a trivial fibration, $\gamma(f)$ is a trivial cofibration, and $\delta(f)$ is a fibration.

A `model category` is a category $\mathscr{C}$ with all small limits and colimits together with a model structure on $\mathscr{C}$.