Timeline for How to get by with only functorial cylindrical objects?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Aug 9, 2022 at 20:20 | comment | added | Arshak Aivazian | Taking the opportunity: of course, I have been reading you for a long time and a lot on MO, n-cafe, I have incredibly pleasant associations with your name, it's nice to chat with you :) | |
| Aug 9, 2022 at 20:20 | comment | added | Arshak Aivazian | Indeed, what is really needed is only a morphism $\alpha A \times I \to A \times 2I$ such that $i_0 \circ \alpha i_0 \circ s_0$ and $i_1 \circ \alpha = i_1 \circ s_1$. Here $i_0, i_1 \colon A \to A \times I$ compositions of inclusions $A \to A \amalg A$ with canonical embedding $A \amalg A \to A \times I$, and $s_0, s_1$ push- out arrows. In my example with a trivial model structure, such a morphism is $A \amalg A \to A \amalg A \amalg A$ (embeddings $1 \mapsto 1, 2 \mapsto 3$). I'll edit the question accordingly, thank you! | |
| Aug 9, 2022 at 17:43 | comment | added | Mike Shulman | Model categories of chain complexes, on the other hand, are generally enriched over a monoidal model category of chain complexes, and I believe the latter does have a morphism (though not an isomorphism) $\mathsf{I} \to 2\mathsf{I}$ that can implement transitivity. | |
| Aug 9, 2022 at 17:42 | comment | added | Mike Shulman | It depends on what you mean by "narrow". It's true that many natural examples are not topologically enriched. But it's known that any combinatorial model category is Quillen equivalent to a simplicially enriched one, and I expect that that could be transferred across the Qullen equivalence between simplicial sets and topological spaces to show an analogous result for topologically enriched model categories. | |
| Aug 9, 2022 at 16:38 | comment | added | Arshak Aivazian | Interesting, thanks! But topologically enriched model categories are a rather narrow class, right? For example, the model categories of chain complexes are probably not such? | |
| Aug 7, 2022 at 18:05 | history | answered | Mike Shulman | CC BY-SA 4.0 |