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Section 12.6 Examples

Example 12.13. The classical model structure on \(\mathbf{Top}\).

Equip \(\mathbf{Top}\) with the model structure in which a map \(f \colon X \to Y\) is:
  • a weak equivalence if \(f_* \colon \pi_n(X, x_0) \to \pi_n(Y, f(x_0))\) is a bijection for all \(n\) and all \(x_0\text{;}\)
  • a cofibration if it is the inclusion of a relative CW complex, or a retract of such a map;
  • a fibration if it is a Serre fibration.
Cofibrant objects are retracts of CW complexes; cofibrant replacement is CW approximation (Theorem 9.17). Every object is fibrant. The homotopy category \(\mathrm{Ho}(\mathbf{Top})\) is the classical homotopy category, equivalent to the category of CW complexes and homotopy classes of maps.

Example 12.14. The classical model structure on \(\mathbf{sSet}\).

The category \(\mathbf{sSet}\) of simplicial sets carries a model structure in which:
  • a map is a weak equivalence if its geometric realization is a weak equivalence of topological spaces;
  • cofibrations are the levelwise injections (monomorphisms);
  • fibrations are Kan fibrations: maps \(f \colon X \to Y\) with the right lifting property against the horn inclusions
    for all \(0 \leq k \leq n\text{.}\)
The fibrant objects are precisely the Kan complexes, and every simplicial set is cofibrant.

Example 12.15. Realization vs. singular complex.

Geometric realization and the singular-complex functor form an adjunction
\begin{equation*} |\,\cdot\,| \colon \mathbf{sSet} \rightleftarrows \mathbf{Top} \,\colon\, \mathrm{Sing}, \end{equation*}
which is a Quillen equivalence. Consequently \(\mathrm{Ho}(\mathbf{Top}) \simeq \mathrm{Ho}(\mathbf{sSet})\text{:}\) simplicial sets and topological spaces are interchangeable as foundations for homotopy theory.

Example 12.16. Dold–Kan.

The Dold–Kan correspondence between simplicial abelian groups and non-negatively graded chain complexes,
\begin{equation*} \mathrm{Ch}_{\geq 0}(\mathbb{Z}) \;\rightleftarrows\; \mathbf{sAb}, \end{equation*}
is a Quillen equivalence between the projective model structure of Example 12.8 and the model structure on \(\mathbf{sAb}\) inherited from \(\mathbf{sSet}\text{.}\)

Remark 12.17.

Model categories make ordinary \(1\)-categorical machinery sufficient to capture homotopical content, but the underlying limits and colimits are computed in the strict \(1\)-category, not homotopy-coherently. To handle homotopy-coherent limits/colimits, and for many other modern purposes, one is led to the framework of \(\infty\)-categories.
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