Stable Model Categories, Spectra, and the Monoidal¶
Job: develop the stable model categories (triangulated \(\mathrm{Ho}\)), the spectra (sequential / symmetric / orthogonal), the monoidal (pushout-product), and the ring spectra / \(E_\infty\) — with the \(\mathrm{Sp}\) stable / smash as the worked example.
Core¶
The stable model categories¶
| Stable / triangulated / \(\Sigma\dashv\Omega\) | Description |
|---|---|
| Stable model category | \(\mathrm{Ho}(\mathcal{M})\) triangulated, \(\Sigma\dashv\Omega\) Quillen equivalence (the homological algebra) |
| Triangulated | the \(\mathrm{Ho}\) the triangulated, the distinguished triangles (the homological algebra) |
The spectra and the model structures¶
| Spectra / sequential / symmetric / orthogonal | Description |
|---|---|
| Spectra \(\mathrm{Sp}\) | the sequential / symmetric / orthogonal spectra (the topology) |
| Stable model | the \(\mathrm{Sp}\) the stable model, the \(\mathrm{Ho}(\mathrm{Sp})\) the stable homotopy (the topology) |
The monoidal model and the pushout-product¶
| Monoidal / pushout-product / unit / smash | Description |
|---|---|
| Monoidal model | \(\otimes\) the pushout-product axiom, unit axiom (the homological algebra) |
| Smash \(\wedge\) | the \(\wedge\) the pushout-product, the topology |
The ring spectra and the commutative¶
| Ring spectra / \(A\) / \(E_\infty\) / SAG | Description |
|---|---|
| Ring spectra | the \(A\) the monoid in \(\mathrm{Sp}\) (the higher categories) |
| \(E_\infty\) / commutative | the \(E_\infty\)-ring, the SAG / foundations |
See it¶
\(\mathrm{Sp}\) the stable model and the smash / pushout-product. The \(\mathrm{Sp}\): the stable (\(\mathrm{Ho}(\mathrm{Sp})\) the triangulated, the \(\Sigma\dashv\Omega\), the homological algebra). The smash: the \(\wedge\) the monoidal (the pushout-product axiom, the unit axiom, the homological algebra). The \(E_\infty\): the commutative ring spectra (the SAG / foundations). The key point: the \(\mathrm{Sp}\) the stable, the \(\wedge\) the monoidal, the \(E_\infty\) the SAG.
Check¶
Show that the \(\mathrm{Ho}(\mathrm{Sp})\) is triangulated and the \(\Sigma\dashv\Omega\) is the Quillen equivalence.
The \(\mathrm{Sp}\): the stable model category, the \(\mathrm{Ho}(\mathrm{Sp})\) the triangulated (the \(\Sigma\dashv\Omega\) the Quillen equivalence, the \(\Sigma\) the suspension, \(\Omega\) the loop, the \(\mathrm{Ho}(\Sigma X)\cong\mathrm{Ho}(\Omega X)\) the stable, the homological algebra). The triangulated: the \(\mathrm{Ho}(\mathrm{Sp})\) the triangulated (the distinguished triangles, the homological algebra). So the \(\mathrm{Ho}(\mathrm{Sp})\) the triangulated, the \(\Sigma\dashv\Omega\) the Quillen.
Explain the monoidal model pushout-product and the \(E_\infty\)-ring.
The monoidal model: the \(\otimes\) the pushout-product axiom (the \(f\wedge g\) the acyclic if \(f\) or \(g\) the acyclic, the homological algebra), the unit axiom (the \(I\otimes-\) the Quillen). The \(E_\infty\): the \(A\) the \(E_\infty\)-ring (the commutative monoid in \(\mathrm{Sp}\), the SAG / foundations, the higher categories). The \(\mathrm{MGL}\): the motivic homotopy / algebraic cobordism. So the monoidal the pushout-product, the \(E_\infty\) the SAG.
Pitfalls¶
- The stable the triangulated (\(\mathrm{Ho}(\mathcal{M})\), the \(\Sigma\dashv\Omega\), the homological algebra) — the stable the triangulated, the \(\Sigma\dashv\Omega\) the Quillen.
- The spectra the sequential / symmetric / orthogonal (the topology) — the symmetric the monoidal, the sequential the simple.
- The monoidal the pushout-product (the \(f\wedge g\), the unit, the homological algebra) — the pushout-product the acyclic, the unit the Quillen.
- The \(E_\infty\) the [SAG] (the spectral-algebraic-geometry/foundations) — the \(E_\infty\) the commutative, the SAG.
Refs¶
- Hovey–Shipley–Smith, "Symmetric spectra" (JAMS 2000); the symmetric / monoidal
- Schwede–Shipley, "Algebras and modules in monoidal model categories" (2000); the monoidal
- Lurie, Higher Algebra (2017); the \(E_\infty\) / the SAG
- Mandell–May–Schwede–Shipley, "Model categories of diagram spectra" (2001); the orthogonal
Next¶
MC index · Foundations · Bousfield & Localization · Quasicategories & ∞ · Combinatorial & Left-Determined · Homological algebra · SAG