Timeline for Simplicial mapping spaces, stable $\infty$-categories, and triangles
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 14, 2019 at 16:58 | vote | accept | David White | ||
| Jun 3, 2017 at 15:27 | answer | added | Denis Nardin | timeline score: 8 | |
| Jun 3, 2017 at 15:16 | comment | added | David White | Hi Denis. I'm still confused. I take the functor map(W,-): C to sSet. But this proposition has me going to presheaves on C. The Yoneda embedding is about the assignment from W to the functor j(W) = map(W,-), but I want to know when that functor preserves limits. Maybe you'd be willing to flesh this out into a full answer? Or we can chat by email if you prefer. I'm not in any big rush. | |
| Jun 3, 2017 at 15:04 | comment | added | Denis Nardin | A reference is Proposition 5.1.3.2 in HTT (applied to the opposite category) together with the description of limits in functor categories (propoosition 5.1.2.2 in HTT). | |
| Jun 3, 2017 at 14:28 | comment | added | David White | Hi Dylan. That last comment is very helpful. | |
| Jun 3, 2017 at 14:19 | comment | added | Dylan Wilson | (On a side note, thinking about infty categories as simplicial or topologically enriched categories is good for intuition but misleading once things get more technical, especially if you're trained in various aspects of enriched category theory. For example, a limit in an infinity category has an entirely different universal property than the usual notions of 'enriched limit' since various homeomorphisms are replaced by weak equivalences). | |
| Jun 3, 2017 at 14:10 | comment | added | Dylan Wilson | This has nothing to do with enrichment- it comes from the universal property of a limit in an infty category. I'm on my phone but I'm sure skd could write the reference from HTT. (It'll probably be near the Yoneda embedding stuff?) | |
| Jun 3, 2017 at 14:07 | comment | added | David White | How do you know map(W,-) preserves limits? Are you thinking about some kind of hom-tensor duality for the enrichment (and the tensoring over sSet)? Before asking this question, I googled about such a hom-tensor adjunction and didn't find anything. | |
| Jun 3, 2017 at 14:06 | comment | added | skd | This is always true since fiber sequences are defined by pullback squares and the covariant hom is a continuous functor. (whoops, didn't see Dylan's comment above.) | |
| Jun 3, 2017 at 14:04 | comment | added | Dylan Wilson | Wait what am I talking about it's way simpler: map(W,-) always takes limits to limits. What were we worried about? | |
| Jun 3, 2017 at 14:03 | comment | added | David White | I agree that map(W,-) factors through spectra, and hence that the question I asked is equivalent to asking if this functor preserves fiber/cofiber sequences. But I still suspect that doesn't come for free. I have some concrete examples in mind about localizations. You can define them with respect to map(-,-) or with respect to internal Hom(-,-) and there are examples where those don't agree. Javier Gutierrez studies the latter definition under the name "closed localization" | |
| Jun 3, 2017 at 13:40 | history | asked | David White | CC BY-SA 3.0 |