Timeline for Maximal Cisinski model structure on simplicial sets
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 8, 2017 at 17:53 | comment | added | Daniel Gerigk | According to Remarque 1.3.15 in [Cis] we have $\mathbf{An}_{\mathfrak{L}} = \operatorname{l}(\operatorname{r}(\Lambda'_L(\mathcal{M})))$, where $\Lambda'_L(\mathcal{M})$ is the set of monomorphisms of the form $L \times \partial \Delta^n \cup \{\epsilon\} \times \Delta^n \to L \times \Delta^n$ with $n \in \mathbb{N}$ and $\epsilon \in \{0,1\}$. | |
| Jul 29, 2017 at 17:15 | comment | added | Andrea Gagna | I edited the last paragraph trying to answer better to your questions. Anyway, the minimal model structure is indeed strictly smaller than Joyal's for properness reasons and every quasi-category is fibrant in the minimal structure. | |
| Jul 29, 2017 at 17:11 | history | edited | Andrea Gagna | CC BY-SA 3.0 |
Add some details
|
| Jul 29, 2017 at 15:25 | comment | added | Valery Isaev | A less trivial example is that every fibrant object must have the right lifting property with respect to the maps ${\Lambda'}^k_n \to \Delta'_n$, where $\Delta'_n$ is the nerve of the trivial groupoid on $n$ objects and ${\Lambda'}^k_n$ is the obvious colimit of $\Delta'_{n-1}$. Now, the question is whether this condition is sufficient. | |
| Jul 29, 2017 at 14:35 | comment | added | Valery Isaev | I was referring to these constructions in the first paragraph of the post. The question is whether we can prove that $\mathbf{W}(\mathrm{Sp})$ is strictly smaller than $\mathbf{W}$. The second question is about an explicit description of fibrant objects (and weak equivalence). I would not be surprised if there is no explicit description for them, but maybe we can prove at least some of their properties. A simple example of such a property is that every contractible Kan complex is fibrant. It would be nice to see other such properties. | |
| Jul 29, 2017 at 14:01 | history | edited | Andrea Gagna | CC BY-SA 3.0 |
deleted 2 characters in body
|
| Jul 29, 2017 at 13:52 | history | answered | Andrea Gagna | CC BY-SA 3.0 |