Timeline for Cofibrations of functors
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Feb 13, 2019 at 17:08 | comment | added | Reid Barton | If you restrict to left Quillen functors and take weak equivalences to be the natural transformations which are weqs on cofibrant objects (the usual criterion to induce an isomorphism between left derived functors), then it seems to be a cofibration category provided that M is tractable (to better control the acyclic cofibrations) and N is enriched or something so that you have a way to build cylinder objects (I don't see why we should expect rlp(cofibrations) to consist of weak equivalences). Not sure whether one can get by with less. | |
| Feb 13, 2019 at 15:40 | comment | added | Mike Shulman | I suppose in light of the fact that the left adjoint <blank> cofibrant functors are the left Quillen functors, one might think of "Quillen cofibration" or "left Quillen transformation" for the general notion. | |
| Feb 13, 2019 at 15:19 | comment | added | Mike Shulman | A pity. What about something weaker like a cofibration category? | |
| Feb 13, 2019 at 15:06 | comment | added | Reid Barton | I don't remember whether I ever constructed an explicit counterexample but I'm almost certain the answer is generally no. I couldn't find any plausible definition of the weak equivalences between left adjoints which are not Quillen functors (i.e., cofibrant). | |
| Feb 13, 2019 at 14:34 | comment | added | Mike Shulman | I don't suppose (in the left adjoints case) these are actually the cofibrations of a model structure on ${\rm Fun}^L(\cal M,N)$? There's an obvious choice of the acyclic cofibrations too: those such that the same pushout corner map is an acyclic cofibration for any cofibration $i$. | |
| Feb 13, 2019 at 14:26 | comment | added | Reid Barton | (I suppose that's not so much a negative result as a lack of a result, but never mind...) | |
| Feb 13, 2019 at 14:18 | comment | added | Reid Barton | Ah, well, I have only negative results there: I haven't seen this class of maps considered in the literature before (which doesn't mean it's not there), and I just call them "cofibrations". | |
| Feb 13, 2019 at 8:27 | comment | added | Mike Shulman | Thanks, that's useful to know. Presumably when $\cal M,N$ are only accessible, the analogous lifting result for accessible wfs also applies. However I am also interested in the notion when $S,T$ are not left adjoints, and what I'm mainly looking for is a name for the concept and possibly a reference. | |
| Feb 13, 2019 at 6:52 | history | edited | Reid Barton | CC BY-SA 4.0 |
fixed nonsense
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| Feb 13, 2019 at 6:41 | history | answered | Reid Barton | CC BY-SA 4.0 |