Timeline for Combinatorial proof that some model categories are monoidal/enriched?
Current License: CC BY-SA 3.0
6 events
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| Mar 12, 2018 at 19:07 | comment | added | Tim Campion | Oh -- one more thing I wanted to say -- it seems that given a Cisinski-Olschok model structure with symmetric monoidal closed $\otimes$, the most natural way to prove that $\Lambda(S)$ is in the cofibrant closure of $S$ is often to prove more generally that $f \hat \otimes g$ is in the cofibrant closure of $S$ for $f \in S$ and $g$ a cofibration. So showing that $S$ is a pseudo-generating set and showing that the model structure is monoidal are done all at once. | |
| Mar 11, 2018 at 18:34 | comment | added | Simon Henry | Well I would indeed prefer to find references, but that is still good to know. | |
| Mar 11, 2018 at 18:27 | comment | added | Tim Campion | So for example, it should be straightforward to churn out combinatorial proofs for existence and cartesianness of model structures on marked simplicial sets over a base, (iterated) complete Segal spaces, etc. But I suppose if your main concern is to be able to quote something, this isn't much help... | |
| Mar 11, 2018 at 18:27 | comment | added | Tim Campion | I have some handwritten notes where I do this for complicial sets -- it's basically just a few more things to check on top of the quasicategory case. I don't know of anything in the literature. But I would say "most" combinatorial model categories in practice are naturally described via Cisinski-Olschok theory -- the only real restriction is that you need every object cofibrant, since the functorial cylinder is always there in practice. | |
| Mar 11, 2018 at 18:01 | comment | added | Simon Henry | This is indeed true, but I already know that. At this point, I'm really just looking for examples to which applies this kind of machinery, i.e. finitely presentable categories where we do have a direct proof that (at least some cases) of the pushout-product are verified. | |
| Mar 11, 2018 at 17:06 | history | answered | Tim Campion | CC BY-SA 3.0 |