Timeline for Would it be possible to propose a satisfying categorical definition for the notion of basis?
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| May 16, 2023 at 10:11 | history | edited | Contactomorph | CC BY-SA 4.0 |
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| Apr 25, 2023 at 19:42 | history | edited | Contactomorph | CC BY-SA 4.0 |
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| Apr 25, 2023 at 9:09 | comment | added | N. Virgo | You might be interested in this paper, which is a different category-theoretic definition of basis. (I've only skimmed it and noted that it sounded interesting - I don't know how it relates to your idea.) Bart Jacobs - Bases as Coalgebras (2013). | |
| Apr 24, 2023 at 13:42 | comment | added | Adam Přenosil | It bears stating explicitly that the definition you gave is simply the definition of a free object over a set of generators. (The answers sort of hint at this, but it should be stated explicitly, since it seems like the OP is not aware of this.) | |
| Apr 23, 2023 at 18:11 | history | became hot network question | |||
| Apr 23, 2023 at 14:18 | answer | added | Simon Henry | timeline score: 9 | |
| Apr 23, 2023 at 14:17 | answer | added | Dave Benson | timeline score: 17 | |
| Apr 23, 2023 at 12:47 | comment | added | Yemon Choi | I am not an expert, but it looks to me like this notion has a good chance of being appropriate whenever the forgetful functor from your category to the category of sets-and-functions "has good behaviour" in some sense. For instance k-Vect is monadic over Set, if I recall correctly. I am not sure if the notion above is good for R-Mod when R is a more general ring, since "minimal generating sets need not be independent" (szyzygies etc) | |
| Apr 23, 2023 at 10:48 | comment | added | Peter LeFanu Lumsdaine | The main question, which I read as essentially “What category-theoretic notions of basis have been studied?”, is a good one, and I hope will get good answers by people who know the topic well. But your friend’s opinion that the notion you give is “not appropriate” is rather subjective, and not really either “correct” or incorrect — as you note, this notion subsumes many classical notions of basis, but fails to capture other notions. So it’s appropriate for some purposes, but not all. | |
| S Apr 23, 2023 at 10:10 | review | First questions | |||
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| S Apr 23, 2023 at 10:10 | history | asked | Contactomorph | CC BY-SA 4.0 |