Timeline for The "concreteness preorder" on categories
Current License: CC BY-SA 4.0
12 events
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| Jan 1, 2019 at 22:17 | comment | added | John Baez | @FoscoLoregian - thanks! Ivan di Liberti's talk ends with the question: is there any (locally small) category that does not admit a faithful functor to the homotopy category of CW complexes? It's amazing that this is, or at least was, an open question. If anyone solves it, I hope they let me know! | |
| Dec 27, 2018 at 17:19 | history | edited | John Baez | CC BY-SA 4.0 |
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| Dec 26, 2018 at 21:58 | comment | added | Benjamin Steinberg | @IvanDiLiberti sciencedirect.com/science/article/pii/0022404987901071 | |
| Dec 26, 2018 at 18:35 | comment | added | Ivan Di Liberti | @BenjaminSteinberg can you link something?! | |
| Dec 26, 2018 at 15:48 | comment | added | Benjamin Steinberg | People in semigroup theory have looked at the related pre-order $C\prec D$ of $C$ is a quotient of a category $C'$ with a faithful functor to $D$. | |
| Dec 25, 2018 at 21:21 | comment | added | Tim Campion | There is also work by Pultr and Trnkova on the preorder on categories given by full embeddability. Keyword "universal categories". | |
| Dec 25, 2018 at 20:35 | comment | added | Tim Campion | To clarify: the question only makes sense if "category" means "locally small category". Kucera showed that every category is a localization of a concrete category. Velebilova gave conditions under which a localization of a concrete category is concrete. Perhaps thinking about her conditions could give a stratification of categories by concreteness. Freyd also gave necessary and sufficient conditions for concreteness in "On the concreteness of certain categories". | |
| Dec 25, 2018 at 13:22 | comment | added | Simon Henry | It seems likely to me that there is a functors from $(\infty,n)$-categories to spaces faithful on the homotopy categories. For example sending a category to the disjoint union of its spaces of $k$-cells for $k \leqslant n$, | |
| Dec 25, 2018 at 12:11 | comment | added | fosco | @JohnBaez: Ivan recently gave a talk at MPIM on this subject: math.muni.cz/~diliberti/Talk/Golem.pdf and has a section in it on his research plan (§1 of math.muni.cz/~diliberti/memoir.pdf ) | |
| Dec 25, 2018 at 11:58 | comment | added | fosco | @JohnBaez: see also mathoverflow.net/questions/270904/faithful-earths-of-categories It is a nice Christmas present that you are entering a similar circle of thought! | |
| Dec 25, 2018 at 11:41 | comment | added | Simon Henry | you can look at : arxiv.org/abs/1704.00303 but as far as I know, not much more is known | |
| Dec 25, 2018 at 7:57 | history | asked | John Baez | CC BY-SA 4.0 |