Accessible, Locally Presentable, and Vopěnka¶
Job: develop the accessible categories (\(\kappa\)-accessible / \(\kappa\)-filtered), the locally presentable (Adámek–Rosický), the combinatorial model categories, and the Vopěnka principle — with the \(\mathbf{Set}\) / \(R\)-Mod locally presentable as the worked example.
Core¶
The accessible categories¶
| Accessible / \(\kappa\) / \(\kappa\)-presentable / \(\kappa\)-filtered | Description |
|---|---|
| \(\kappa\)-accessible | has \(\kappa\)-filtered colimits, generated by \(\kappa\)-presentable objects |
| \(\kappa\)-presentable | \(\mathcal{C}(X,-)\) preserves \(\kappa\)-filtered colimits |
| \(\kappa\)-filtered | every \(\kappa\)-small diagram has a cocone |
The locally presentable¶
| Locally presentable / cocomplete / Adámek–Rosický | Description |
|---|---|
| Locally presentable | cocomplete + accessible (the Adámek–Rosický) |
| Examples | \(\mathbf{Set}\), \(\mathbf{Grp}\), \(R\)-Mod, AG |
The combinatorial / the small¶
| Combinatorial / cofibrantly generated / model categories | Description |
|---|---|
| Combinatorial model category | cofibrantly generated + locally presentable (the model categories) |
| Small / combinatorial | the model categories, the higher categories |
The Vopěnka principle and the reflection¶
| Vopěnka / reflection / large-cardinal / accessible | Description |
|---|---|
| Vopěnka principle | every proper class of objects has a morphism between two (the large-cardinal) |
| Reflection | Vopěnka \(\Rightarrow\) accessible full subcategory reflection |
See it¶
\(\mathbf{Set}\) and \(R\)-Mod the locally presentable and the combinatorial. The \(\mathbf{Set}\): the locally presentable (the \(\omega\)-presentable, the cocomplete, the Adámek–Rosický). The \(R\)-Mod: the locally presentable (the \(\omega\)-presentable, the cocomplete). The combinatorial: the model categories the cofibrantly generated + locally presentable (the higher categories). The key point: the \(\mathbf{Set}\) / \(R\)-Mod the locally presentable, the combinatorial the model categories.
Check¶
Show that \(\mathbf{Set}\) is locally presentable, and explain the \(\omega\)-presentable.
The \(\mathbf{Set}\): the cocomplete (all small colimits), the \(\omega\)-accessible (every set is the \(\omega\)-filtered colimit of its finite subsets, the finite sets the \(\omega\)-presentable). The locally presentable: the cocomplete + \(\omega\)-accessible (the Adámek–Rosický). The \(\omega\)-presentable: the \(\mathcal{C}(X,-)\) preserves the \(\omega\)-filtered colimits (the finite sets, the \(\mathrm{Hom}(X,-)\) the finite \(X\) the preserves). So the \(\mathbf{Set}\) the locally presentable, the \(\omega\)-presentable.
Explain the Vopěnka principle and the reflection, and the large-cardinal.
The Vopěnka: the every proper class of objects (in a locally presentable category) has a morphism between two — the large-cardinal principle (the Vopěnka, the equiconsistent with the huge cardinals). The reflection: the Vopěnka \(\Rightarrow\) the accessible full subcategory reflection (the every class the reflective, the Adámek–Rosický). The combinatorial: the model categories the cofibrantly generated + locally presentable (the higher categories, the Vopěnka the combinatorial). So the Vopěnka the large-cardinal, the reflection the accessible.
Pitfalls¶
- The \(\kappa\)-accessible the \(\kappa\)-filtered (the \(\kappa\)-presentable, the \(\kappa\)-filtered colimits) — the accessible the \(\kappa\), the presentable the cocomplete.
- The locally presentable the cocomplete (the cocomplete + accessible, the Adámek–Rosický) — the locally presentable the cocomplete, the accessible.
- The combinatorial the cofibrantly generated (the model categories, the locally presentable) — the combinatorial the cofibrantly generated, the higher categories.
- The Vopěnka the large-cardinal (the proper class, the reflection) — the Vopěnka the large-cardinal, the reflection the accessible.
Refs¶
- Adámek–Rosický, Locally Presentable and Accessible Categories (1994)
- Makkai–Paré, Accessible Categories (1989); the Vopěnka
- Lurie, Higher Topos Theory (2009); the combinatorial / the higher categories
- Beke, "Sheafifiable homotopy model categories" (2000); the combinatorial
Next¶
CT index · Foundations · Monads & Algebras · Monoidal & Enriched · Categorical Logic & Topoi · Model categories · Higher categories