Timeline for The homotopy category of the category of enriched categories
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 21, 2019 at 18:04 | answer | added | Kevin Carlson | timeline score: 6 | |
| Nov 19, 2019 at 12:41 | comment | added | Harry Gindi | To expand on what Kevin is saying, the statement '∞-categories are canonically schematically enriched over homotopy types' refers to being enriched over the ∞-category of homotopy types, not the homotopy category. If you take the Dwyer-Kan localization of sSet rather than the classical Gabriel-Zisman localization, this is a theorem (more or less equivalent to the proof of Quillen equivalence between quasicategories and categories enriched in Kan complexes). | |
| Nov 18, 2019 at 23:02 | comment | added | Kevin Carlson | To highlight what may be a significant misunderstanding (depending on what you mean by "schematically"), it is really not at all true that $(\infty,1)$-categories should be thought of as categories enriched over $\mathrm{Ho}\mathbf{SSet}$. | |
| Nov 18, 2019 at 14:25 | answer | added | Bewildered | timeline score: -1 | |
| Nov 18, 2019 at 13:47 | comment | added | Harry Gindi | I missed 'monoidal' oops! | |
| Nov 18, 2019 at 13:16 | answer | added | Harry Gindi | timeline score: 5 | |
| Nov 18, 2019 at 13:11 | comment | added | Frank Kong | @HarryGindi Thans for your example in the second comment. In the first comment, I suspect the definition of a monoidal model category has already included that the tensor product is a Quillen bifunctor? | |
| Nov 18, 2019 at 13:02 | comment | added | Harry Gindi | Also, I think the answer is no, since iirc there are monoids in the stable homotopy category that don't lift to E_1-algebras. This gives a counterexample to your question when you take such a homotopy monoid as a one-object category enriched in the stable homotopy category. | |
| Nov 18, 2019 at 12:59 | comment | added | Harry Gindi | You also need the model category to have a colimit-preserving tensor product satisfying Quillen's pushout-tensor axiom, or else this makes no sense. | |
| Nov 18, 2019 at 12:45 | history | asked | Frank Kong | CC BY-SA 4.0 |