Timeline for Are fibered categories fibrant objects in some model structure on Cat/C?
Current License: CC BY-SA 4.0
21 events
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| Nov 12, 2020 at 12:51 | comment | added | Praphulla Koushik | @MikeShulman Is that so? I do not know, your comment looks both positive and negative at the same time :) | |
| Nov 11, 2020 at 18:00 | comment | added | Mike Shulman | Also BTW, your last sentence presents a false dichotomy. Even if there is no model structure in which the fibered categories are the fibrant objects, that doesn't mean the terms are entirely unrelated. (-: | |
| Nov 11, 2020 at 2:31 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
added 4 characters in body
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| Nov 11, 2020 at 2:30 | comment | added | Praphulla Koushik | @MikeShulman yes, I mean fibered categories are the fibrant objects. I should have written it more clearly. Editing it now. | |
| Nov 11, 2020 at 2:28 | comment | added | Praphulla Koushik | @AlexanderCampbell βIs any retract in πππ/πΆ of a fibered category over πΆ a fibered category over πΆβ looks more than a comment and less than an answer :) Thanks, did not thought about it. I will think and respond. | |
| Nov 11, 2020 at 2:02 | comment | added | Mike Shulman | BTW, I presume that you meant to ask for a model structure in which fibered categories are the fibrant objects; i.e., such that a category over $C$ is fibered iff it is fibrant. There is of course a model structure in which fibered categories are fibrant objects, but not every fibrant object is a fibered category: the slice model structure induced from the canonical model structure on $\mathit{Cat}$. | |
| Nov 11, 2020 at 1:57 | comment | added | Mike Shulman | One concrete issue, related to @TimCampion's first comment, is that the "inclusion" of fibered categories into $\mathit{Cat}/C$ is not full: the "natural" choice of morphism between fibered categories is a functor that preserves cartesian arrows. This is not generally what happens for the fibrant objects in a model category. | |
| Nov 10, 2020 at 23:39 | comment | added | Alexander Campbell | Is any retract in $\mathbf{Cat}/C$ of a fibered category over $C$ a fibered category over $C$? | |
| Nov 10, 2020 at 15:40 | comment | added | Praphulla Koushik | @TimCampion Oh, thank you. I will see that paper :) | |
| Nov 10, 2020 at 15:27 | comment | added | Tim Campion | The Barwick and Shah paper is meant to simply address the idea that there is some relationship between different notions of "fibration", including model-categorical ones, as well as categorical fibrations, i.e. fibered categories. | |
| Nov 10, 2020 at 15:25 | comment | added | Praphulla Koushik | @TimCampion Hi. I do not completely understand which question does the paper by Barwick and Shah answers :O May be you are assuming I know more things :) can you please elaborate a little bit. | |
| Nov 10, 2020 at 15:18 | comment | added | Praphulla Koushik | @AdittyaChaudhuri Thanks for the second reference. First reference I saw long back. Did not read it after that :D Your comment might be sufficient motivation to read that paper.. They fix a Grothendieck topology on $\mathcal{C}$, which might be very different from the category with out such choice. | |
| Nov 10, 2020 at 15:16 | comment | added | Praphulla Koushik | @DavidWhite Hi. I also think the answer is βyesβ. Hope some one responds. :) | |
| Nov 10, 2020 at 15:09 | comment | added | Tim Campion | The inclusion of fibered categories over $C$ into $Cat/C$ has a left 2-adjoint given by $(D \to C) \mapsto D \downarrow C$, and in fact is 2-monadic. Even better, the resulting 2-monad is lax-idempotent. If the 2-monad were idempotent, we'd have a reflective subcategory, and thus a model structure in a sort of degenerate way. As it is, I'm not sure... For the $\infty$-categorical context, see Barwick and Shah. | |
| Nov 10, 2020 at 14:57 | comment | added | Adittya Chaudhuri | Also check arxiv.org/pdf/1806.06129.pdf (Categorical notions of fibration) by Loregian and Riehl | |
| Nov 10, 2020 at 14:48 | comment | added | Adittya Chaudhuri | Please check arxiv.org/pdf/math/0110247.pdf (A homotopy theory for stacks by Sharon Hollander) | |
| Nov 10, 2020 at 14:43 | comment | added | David White | I think the terminology represents "two unrelated terms that sound the same" since both are kinda inspired by fibrations in topological spaces (like, for every point in the base space there is a fiber above it). That said, the answer to your question could still be "yes" and I'm curious to see if anyone knows. | |
| Nov 10, 2020 at 14:24 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
added 5 characters in body; edited title
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| Nov 10, 2020 at 14:24 | history | undeleted | Praphulla Koushik | ||
| Nov 10, 2020 at 14:23 | history | deleted | Praphulla Koushik | via Vote | |
| Nov 10, 2020 at 14:15 | history | asked | Praphulla Koushik | CC BY-SA 4.0 |