Timeline for Transporting a model category structure along a left adjoint
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| May 19, 2018 at 1:41 | history | wiki removed | Todd Trimble | ||
| May 19, 2018 at 1:39 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| May 17, 2018 at 19:45 | comment | added | Dmitri Pavlov | @AlexanderCampbell: My initial comment was about the HKRS paper cited in the original post, not the GKR paper, and what was said is true: there is no proof of the claimed result in the HKRS paper, and what remains once the wrong part is removed is Smith + Makkai-Rosický. | |
| May 17, 2018 at 15:02 | comment | added | Tim Campion | @DmitriPavlov True, maybe this answer could have been a comment, and certainly it was not as well-informed as David White's very topical answer. It was maybe a lucky guess based on Philippe's comments and the question title that perhaps a different approach might be helpful. And indeed, as I read Philippe's answer, the HKRS paper did in fact turn out to at least inspire a solution. Consider also the value of an answer not just to the OP, but to a general MO user -- particularly because of the more general title, a user is likely to find this question looking for more general information. | |
| May 17, 2018 at 14:44 | comment | added | Alexander Campbell | @DmitriPavlov And yet the theorem is true, as is everything I said in my previous comment. In the sentence you quote, "this same setting" refers to the setting of combinatorial model categories discussed in that paragraph; the main result of the paper applies in the more general setting of accessible model categories. Furthermore, they refer to "this breakthrough result" of Makkai--Rosicky in that setting, which is not to say that it is "the" breakthrough of the more general result we are discussing. | |
| May 17, 2018 at 14:36 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| May 17, 2018 at 14:30 | comment | added | Dmitri Pavlov | I also have to point out that the OP asked for a reference for the Quillen path object argument, i.e., a proof of the lifting property. The cited papers say nothing about proving the lifting property. | |
| May 17, 2018 at 14:13 | comment | added | Dmitri Pavlov | @AlexanderCampbell: Well, a theorem with a wrong proof is no theorem at all, isn't it? The correct part does amount to what I suggested. And the newer paper that you cited explicitly attributes the breakthrough to Makkai and Rosický [22] on page 3: "Very recently, [22] showed that in this same setting, left-lifted factorizations also exist". | |
| May 17, 2018 at 6:39 | comment | added | Alexander Campbell | @DmitriPavlov The main theorem in the paper Tim mentions is strictly more general than you suggest, as it applies to the more general class of accessible model categories. It may also be important to mention that the proof given in that paper is incorrect; this proof was corrected and replaced by a more elementary proof (avoiding the use of algebraic weak factorisation systems) in arXiv:1802.09889. | |
| May 16, 2018 at 17:40 | comment | added | Dmitri Pavlov | It is important to mention that this reference merely combines the Smith recognition theorem for combinatorial model categories with a recent result by Makkai and Rosický that shows that cofibrations transferred along a left adjunction are generated by a set. | |
| May 16, 2018 at 14:31 | history | answered | Tim Campion | CC BY-SA 4.0 |