Combinatorial, Left-Determined, and Vopěnka¶
Job: develop the combinatorial model categories (cofibrantly generated + locally presentable), the left-determined (Raptis / Rosický), the Vopěnka principle, and the transfer / existence — with the combinatorial / Vopěnka as the worked example.
Core¶
The combinatorial model categories¶
| Combinatorial / cofibrantly generated / locally presentable | Description |
|---|---|
| Combinatorial | cofibrantly generated + locally presentable (the category theory / accessible) |
| Small object argument | the cofibrantly generated (the category theory) |
The left-determined model categories¶
| Left-determined / \(\mathcal{W}\) / Raptis–Rosický | Description |
|---|---|
| Left-determined | cofibrations determined by \(\mathcal{W}\) (the Raptis / Rosický) |
| Accessible | the category theory / accessible |
The Vopěnka and the combinatorial¶
| Vopěnka / reflection / combinatorial | Description |
|---|---|
| Vopěnka principle | every class has a reflection (the category theory / accessible) |
| Vopěnka / combinatorial | Vopěnka \(\Rightarrow\) combinatorial the cofibrantly generated |
The transfer and the existence¶
| Transfer / \(\mathcal{M}\to\mathcal{N}\) / existence | Description |
|---|---|
| Transfer | \(\mathcal{M}\to\mathcal{N}\) the transferred model (the category theory / monads) |
| Existence | the transferred model the existence (the category theory) |
See it¶
Combinatorial model and the Vopěnka / left-determined. The combinatorial: the cofibrantly generated + locally presentable (the category theory / accessible). The Vopěnka: the every class the reflection (the category theory). The left-determined: the cofibrations determined by the \(\mathcal{W}\) (the Raptis / Rosický). The key point: the combinatorial the cofibrantly generated + locally presentable, the Vopěnka the reflection, the left-determined the \(\mathcal{W}\).
Check¶
Explain the combinatorial model categories and the cofibrantly generated.
The combinatorial: the cofibrantly generated (the \(I, J\) the generating cofibrations / acyclic cofibrations, the small object argument the factorizations, the category theory / accessible) + the locally presentable (the cocomplete + accessible, the Adámek–Rosický). The Vopěnka: the every class the reflection (the category theory, the Vopěnka \(\Rightarrow\) the combinatorial the cofibrantly generated). The left-determined: the cofibrations determined by \(\mathcal{W}\) (the Raptis / Rosický, the \(\mathcal{W}\) the weak equivalences the determine the cofibrations). So the combinatorial the cofibrantly generated + locally presentable, the Vopěnka the reflection, the left-determined the \(\mathcal{W}\).
Explain the transfer of model structures and the existence.
The transfer: the \(\mathcal{M}\) the model, the \(\mathcal{N}\) the category, the \(F:\mathcal{M}\to\mathcal{N}\) the adjunction, the transferred model on \(\mathcal{N}\) (the \(\mathcal{W}_\mathcal{N}=F(\mathcal{W}_\mathcal{M})\), the fibrations \(F(\mathrm{Fib}_\mathcal{M})\), the category theory / monads). The existence: the transferred model the exists iff the \(F\) the left adjoint + the acyclic cofibrations the transferred (the category theory). The Crans / the Bayeh–Sarazola: the transferred model the existence the conditions. So the transfer the \(\mathcal{M}\to\mathcal{N}\), the existence the conditions.
Pitfalls¶
- The combinatorial the cofibrantly generated + locally presentable (the category theory / accessible) — the combinatorial the cofibrantly generated, the locally presentable.
- The left-determined the \(\mathcal{W}\) (the Raptis / Rosický) — the \(\mathcal{W}\) the cofibrations, the left-determined.
- The Vopěnka the reflection (the category theory, the large-cardinal) — the Vopěnka the reflection, the combinatorial.
- The transfer the existence (the category theory / monads) — the transfer the \(\mathcal{M}\to\mathcal{N}\), the existence the conditions.
Refs¶
- Dugger, "Combinatorial model categories have presentations" (2001); the combinatorial
- Raptis–Rosický, "On the accessibility of the homotopy category" (2015); the left-determined
- Adámek–Rosický, Locally Presentable and Accessible Categories (1994); the Vopěnka
- Bayeh–Sarazola–Tennent, "A sufficient condition for the existence of a transferred model structure" (2021)
Next¶
MC index · Foundations · Bousfield & Localization · Stable & Spectral · Quasicategories & ∞ · Category theory · Category theory / monads