Quasicategories, the ∞-Categorical Comparison, and the Joyal¶
Job: develop the quasicategories / Joyal model, the model→∞ comparison (Lurie), the ∞-operads, and the presentable / combinatorial ∞ — with the Joyal vs Quillen sSet as the worked example.
Core¶
The quasicategories and the Joyal model¶
| Quasicategory / Joyal / Quillen | Description |
|---|---|
| Quasicategory | \(\mathbf{sSet}\) the Joyal model, fibrant = quasicategories (the higher categories) |
| Joyal vs Quillen | Joyal the \((\infty,1)\), Quillen the \((\infty,0)\) (the higher categories) |
The model → ∞ and the comparison¶
| Model → ∞ / \(\mathcal{M}[\mathcal{W}^{-1}]\) / Lurie | Description |
|---|---|
| Model → ∞ | \(\mathcal{M}\) the model, \(\mathcal{M}[\mathcal{W}^{-1}]\) the ∞-category (the Lurie) |
| Comparison | \(\mathrm{Ho}_\infty(\mathcal{M})\simeq\mathcal{M}[\mathcal{W}^{-1}]\) (the higher categories) |
The ∞-operads and the model¶
| ∞-operads / \(\mathcal{O}^\otimes\) / Lurie | Description |
|---|---|
| ∞-operads | \(\mathcal{O}^\otimes\) the ∞-operad (the Lurie, the higher categories) |
| Model / operad | the model the operad (the higher categories) |
The presentable / combinatorial and the ∞¶
| Presentable ∞ / combinatorial | Description |
|---|---|
| Presentable ∞ | the category theory / accessible, the higher categories |
| Combinatorial / ∞ | combinatorial model → presentable ∞ (the higher categories) |
See it¶
\(\mathbf{sSet}\) the Joyal (quasicategories) vs the Quillen (Kan) and the model → ∞. The Joyal: \(\mathbf{sSet}\) the Joyal model (fibrant = quasicategories, the higher categories). The Quillen: \(\mathbf{sSet}\) the Quillen model (fibrant = Kan, the \((\infty,0)\)). The model → ∞: \(\mathcal{M}[\mathcal{W}^{-1}]\) the ∞-category (the Lurie, the higher categories). The key point: the Joyal the quasicategory, the Quillen the Kan, the model → ∞ the Lurie.
Check¶
Explain the Joyal model structure on sSet and the comparison with the Quillen.
The Joyal: \(\mathbf{sSet}\) the Joyal model (weq = categorical equivalences, fibrant = quasicategories, the \((\infty,1)\)-categories, the higher categories). The Quillen: \(\mathbf{sSet}\) the Quillen model (weq = weak homotopy equivalences, fibrant = Kan complexes, the \((\infty,0)\) = spaces, the topology). The comparison: the Joyal the \((\infty,1)\), the Quillen the \((\infty,0)\) (the Joyal the fibrant = quasicategory, the Quillen the fibrant = Kan). The model → ∞: the \(\mathcal{M}[\mathcal{W}^{-1}]\) the ∞-category (the Lurie, the higher categories, the \(\mathrm{Ho}_\infty\simeq\mathcal{M}[\mathcal{W}^{-1}]\)). So the Joyal the \((\infty,1)\), the Quillen the \((\infty,0)\), the Lurie the comparison.
Explain the ∞-operads and the presentable ∞-categories.
The ∞-operads: the \(\mathcal{O}^\otimes\) the ∞-operad (the Lurie, the \(\otimes\) the cocartesian, the higher categories). The presentable ∞: the category theory / accessible the presentable, the higher categories the ∞. The combinatorial: the combinatorial model → presentable ∞ (the higher categories). So the ∞-operads the \(\mathcal{O}^\otimes\), the presentable the category theory.
Pitfalls¶
- The Joyal the \((\infty,1)\) (the quasicategories, the higher categories) — the Joyal the \((\infty,1)\), the Quillen the \((\infty,0)\).
- The model → ∞ (the \(\mathcal{M}[\mathcal{W}^{-1}]\), the Lurie) — the Lurie the comparison, the \(\mathrm{Ho}_\infty\simeq\mathcal{M}[\mathcal{W}^{-1}]\).
- The ∞-operads (the \(\mathcal{O}^\otimes\), the Lurie, the higher categories) — the ∞-operads the cocartesian, the Lurie.
- The presentable ∞ (the category theory / accessible, the higher categories) — the presentable the ∞, the combinatorial the model.
Refs¶
- Joyal, "The theory of quasi-categories" (2008); the Joyal model
- Lurie, Higher Topos Theory (2009); the model → ∞, the presentable ∞
- Lurie, Higher Algebra (2017); the ∞-operads, the \(E_\infty\)
- Cisinski, Higher Categories and Homotopical Algebra (2019); the comparison
Next¶
MC index · Foundations · Bousfield & Localization · Stable & Spectral · Combinatorial & Left-Determined · Higher categories · Category theory