Bousfield Localization, Cellular, and Simplicial Model Categories¶
Job: develop the Bousfield localization (\(L_{\mathcal{S}}\)), the cellular model categories (Hirschhorn), the simplicial model categories (mapping spaces), and the framed / cosimplicial — with the \(L_{K(n)}\mathrm{Sp}\) as the worked example.
Core¶
The Bousfield localization¶
| Bousfield / \(L_{\mathcal{S}}\) / \(\mathcal{S}\)-local | Description |
|---|---|
| Left Bousfield \(L_{\mathcal{S}}\mathcal{M}\) | \(\mathcal{S}\)-local equivalences, \(\mathcal{S}\)-local fibrant objects (the localization) |
| \(\mathcal{S}\)-local object | \(X\) with \(\mathrm{Map}(A,X)\) weq for all \(A\in\mathcal{S}\) |
The cellular model categories¶
| Cellular / cofibrations / small object argument | Description |
|---|---|
| Cellular (Hirschhorn) | cofibrations are cellular, small object argument builds factorizations |
| Proper | cellular \(\Rightarrow\) left/right proper (the topology) |
The simplicial model categories¶
| Simplicial / \(\mathrm{Map}(X,Y)\) / \(\infty\) | Description |
|---|---|
| Simplicial model category | \(\mathrm{Map}(X,Y)\in\mathbf{sSet}\), the simplicial enrichment (the higher categories) |
| Mapping space | \(\pi_0\mathrm{Map}(X,Y)\cong\mathrm{Ho}(X,Y)\), \(\pi_*\) the higher (the higher categories) |
The framed / cosimplicial¶
| Framed / cosimplicial / Hovey | Description |
|---|---|
| Framed model category | cosimplicial / simplicial frames, the \(\mathrm{Ho}\) the frames (the Hovey) |
| Frames / \(\infty\) | the higher categories the \(\infty\) |
See it¶
Bousfield localization of spectra at \(K(n)\) and the cellular / simplicial. The Bousfield: \(L_{K(n)}\mathrm{Sp}\) (the \(K(n)\)-local spectra, the motivic homotopy / chromatic, the spectral sequences / Adams–Novikov). The cellular: the spectra the cellular (the Hirschhorn, the small object argument, the topology). The simplicial: the \(\mathrm{Sp}\) the simplicial (the \(\mathrm{Map}(X,Y)\in\mathbf{sSet}\), the higher categories). The key point: the Bousfield \(L_{K(n)}\) the localization, the cellular / simplicial the spectra.
Check¶
Explain the left Bousfield localization and the \(\mathcal{S}\)-local objects.
The left Bousfield \(L_{\mathcal{S}}\mathcal{M}\): the \(\mathcal{S}\)-local equivalences (the \(f:X\to Y\) with \(\mathrm{Map}(Y,Z)\to\mathrm{Map}(X,Z)\) weq for all \(\mathcal{S}\)-local \(Z\)), the \(\mathcal{S}\)-local fibrant (the \(Z\) with \(\mathrm{Map}(A,Z)\) weq for all \(A\in\mathcal{S}\)). The \(L_{\mathcal{S}}\): the \(\mathcal{S}\)-local equivalences the new weak equivalences, the \(\mathcal{S}\)-local the new fibrant (the Bousfield). The \(K(n)\): the \(L_{K(n)}\mathrm{Sp}\) the \(K(n)\)-local spectra (the motivic homotopy / chromatic). So the Bousfield the \(\mathcal{S}\)-local, the \(K(n)\) the spectra.
Explain the simplicial model category and the mapping space.
The simplicial: the \(\mathcal{M}\) the simplicial (the \(\mathrm{Map}(X,Y)\in\mathbf{sSet}\) the simplicial enrichment, the axioms the SM7, the higher categories). The mapping space: the \(\pi_0\mathrm{Map}(X,Y)\cong\mathrm{Ho}(X,Y)\) (the homotopy classes, the topology), the \(\pi_i\mathrm{Map}(X,Y)\) the higher (the higher categories). The frames: the cosimplicial / simplicial frames (the Hovey, the \(\mathrm{Ho}\) the frames). So the simplicial the \(\mathrm{Map}\), the frames the \(\mathrm{Ho}\).
Pitfalls¶
- The Bousfield the left / right (the \(L_{\mathcal{S}}\) the left, the \(R_{\mathcal{S}}\) the right) — the left the cofibrant, the right the fibrant.
- The cellular the small object argument (the Hirschhorn, the cofibrations the cellular) — the cellular the small, the proper.
- The simplicial the SM7 (the higher categories, the \(\mathrm{Map}\)) — the simplicial the enrichment, the \(\pi_*\) the higher.
- The framed / cosimplicial (the Hovey, the higher categories) — the frames the \(\mathrm{Ho}\), the \(\infty\).
Refs¶
- Bousfield, "Localization of spectra with respect to homology" (Topology 1979); the \(L_{\mathcal{S}}\)
- Hirschhorn, Model Categories and Their Localizations (2003); the cellular
- Hovey, Model Categories (1999); the simplicial / the framed
- Goerss–Jardine, Simplicial Homotopy Theory (1999); the SM7
Next¶
MC index · Foundations · Stable & Spectral · Quasicategories & ∞ · Combinatorial & Left-Determined · Higher categories · Topology