Timeline for A model category of abelian categories?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 27, 2015 at 10:55 | comment | added | Zhen Lin | One needs a little bit more than chosen (co)limits to define strict abelian categories. However, §8 of the paper does sketch an extension which might be relevant. | |
| May 27, 2015 at 10:07 | comment | added | Dmitri Pavlov | @ZhenLin: It seems that a paper by Kelly and Lack “On the monadicity of categories with chosen colimits” does establish the desired monadicity. | |
| May 21, 2015 at 0:42 | comment | added | David Roberts♦ | @TheoJohnson-Freyd The paper is at arxiv.org/abs/1402.7108 | |
| May 14, 2015 at 13:10 | answer | added | john | timeline score: 6 | |
| May 12, 2015 at 8:14 | answer | added | Alexander Campbell | timeline score: 4 | |
| May 12, 2015 at 3:09 | comment | added | Alexander Campbell | I would expect there to be a finitary 2-monad $T$ on Cat with $\mathfrak{M} = T\text{-Alg}_s$, so your first two questions would be answered in the affirmative by Steve Lack's Homotopy-theoretic aspects of 2-monads. | |
| May 12, 2015 at 2:42 | comment | added | David Roberts♦ | @TheoJohnson-Freyd well, let me finish editing it first :-) It will most definitely be on the arXiv before the publisher gets their hands on it. | |
| May 12, 2015 at 2:30 | comment | added | Theo Johnson-Freyd | @DavidRoberts No chance at updating the arXiv version to match the final refereed version? In any case, I hope you post an answer here recapping your comments and with some links to your upcoming paper. | |
| May 12, 2015 at 0:41 | comment | added | David Roberts♦ | Ah, that makes it even more likely! This sort of thing happens, as you know, with categories of fibrant objects; the issue is that non-invertible 2-cells haven't as far as I am aware been treated from such a viewpoint. I'd be happy to keep looking at this, but I haven't the time right now. Feel free to email me. | |
| May 12, 2015 at 0:27 | comment | added | Zhen Lin | Interesting. If the answer to Q1 and Q2 are both yes, then I would expect to get some kind of calculus of fractions. (Notice that the purported model structure in Q1 has the property that all objects are fibrant.) | |
| May 12, 2015 at 0:14 | comment | added | David Roberts♦ | I have a paper accepted with Applied Categorical Structures that contains the relevant material backing up these comments. The arXiv version is outdated, so I hesitate to link to it. | |
| May 12, 2015 at 0:11 | comment | added | David Roberts♦ | ...to get the result. One only needs to check the condition that for every functor $F$ in $\mathfrak{A}$ there is a weak equivalence $W$ and a functor $G$ in $\mathfrak{M}$ such that $U(G) \simeq F\circ U(W)$, for $U\colon \mathfrak{M}\to \mathfrak{A}$ the functor you give. A similar result is true for either weak 2-functors or monoidal functors, so I expect something similar here. Makkai's anafunctors are also worth mentioning here. | |
| May 12, 2015 at 0:07 | comment | added | David Roberts♦ | I could probably jump direct to question 3, using Pronk's bicategorical localisation technology. I presume weak equivalences in $\mathcal{M}$ are ess. surj. fully faithful functors. If it is true that for any such $f\colon A \to B$, one can find a surjective on objects, ff functor $c\colon C\to B$ in $\mathcal{M}$ (say from a freely generated version of $B$ or something), a functor $C\to A$ and a nat iso in the resulting triangle, the one gets a bicategory of fractions. Then one can apply Pronk's comparison theorem (proposition 24 in numdam.org/item?id=CM_1996__102_3_243_0) ... | |
| May 12, 2015 at 0:00 | comment | added | Zhen Lin | Question 3 is the most interesting part, and questions 1 and 2 seem like the natural way to proceed. | |
| May 11, 2015 at 23:59 | comment | added | David Roberts♦ | Are you most interested in question 3 itself, or do you really want to go via questions 1 and 2? | |
| May 11, 2015 at 22:50 | history | edited | Zhen Lin | CC BY-SA 3.0 |
added 38 characters in body
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| May 11, 2015 at 22:10 | history | asked | Zhen Lin | CC BY-SA 3.0 |