Timeline for Could groups be used instead of sets as a foundation of mathematics?
Current License: CC BY-SA 4.0
30 events
| when toggle format | what | by | license | comment | |
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| Nov 15, 2022 at 17:18 | comment | added | Akiva Weinberger | A fun puzzle is to ask how far you can get using cyclically ordered sets as your primitive. Can you construct unordered pairs? triples? 4-tuples? 5-tuples? | |
| Nov 15, 2022 at 9:57 | history | edited | David Roberts♦ |
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| Nov 15, 2022 at 6:40 | vote | accept | Oscar Cunningham | ||
| Nov 15, 2022 at 4:17 | answer | added | Mohammad Golshani | timeline score: 7 | |
| Nov 14, 2022 at 2:08 | answer | added | Martin Brandenburg | timeline score: 13 | |
| Feb 14, 2020 at 7:59 | vote | accept | Oscar Cunningham | ||
| Nov 14, 2022 at 6:32 | |||||
| Feb 13, 2020 at 19:53 | comment | added | godelian | Excellent question! | |
| Feb 13, 2020 at 10:32 | comment | added | Oscar Cunningham | @ToddTrimble That's a nice result! Is there always a way to define the comonad within the categorical language? | |
| Feb 12, 2020 at 23:03 | comment | added | Todd Trimble | As for which categories of algebras are such that the free functor $Set \to Alg$ is comonadic: these are quite general. See this section in the nLab: ncatlab.org/nlab/show/… | |
| Feb 12, 2020 at 22:04 | comment | added | Martin Brandenburg | Oups. :-) $\mbox{}$ | |
| Feb 12, 2020 at 22:03 | answer | added | Simon Henry | timeline score: 38 | |
| Feb 12, 2020 at 21:10 | comment | added | Oscar Cunningham | @MartinBrandenburg That sounds like it answers my question. | |
| Feb 12, 2020 at 19:10 | comment | added | Martin Brandenburg | This follows from a more general theorem about the classification of comonoid objects in $\mathsf{Mon}$ via $E$-systems, see Bergman's "Invitation to general algebra", Section 9.6. | |
| Feb 12, 2020 at 17:16 | comment | added | Martin Brandenburg | @SimonHenry Free groups are examples of cogroups internal to $\mathsf{Grp}$, and the cogroup maps probably preserve the generators. | |
| Feb 12, 2020 at 16:15 | comment | added | Simon Henry | Also: One can characterize sets as certain co-algebras in abelian group (as the free abelian group with their "diagonal" coalgebra structure). But I don't know if the tensor product of abelian group can be characterized abstractely. | |
| Feb 12, 2020 at 15:16 | comment | added | Simon Henry | ... In theory this should be possible as Set is comonadic over both groups and abelian groups... If one could characterize abstractly this comonad that would prove the result immediately. One can also try a more elementary approach: I think Free groups can be characterized as the projective objects, but this is not enough as that does not fix the set of generators, but maybe there is a structure definable in the language that can do that ? | |
| Feb 12, 2020 at 15:11 | comment | added | Simon Henry | Two comments: 1) the category of groups does have some interesting categorical structures, that have been abstracted and studied a lot by people working on semi-abelian categories, proto-modular categories, etc... One of them is that you can give purely categorically a definition of the automorphism group of a group. Aut(G) is universal for "group action on G", and a "group action of H on G" can be defined as a split exact sequence connecting G and H. 2) Have you try to instead recover the category of sets as the full subcategory of free groups and morphisms preserving generators ?... | |
| Feb 12, 2020 at 15:02 | comment | added | Oscar Cunningham | @JoelDavidHamkins When I say that $\sf{ETCS+R}$ is biinterpretable with $\sf ZFC$, I mean exactly that. (You have to get the language right for it to work out perfectly; morphisms have equality but objects don't.) In the questions at the end I ask for biinterpretability, then mutual interpretability, then equiconsistency, which I believe are in decreasing order of strength. Proving any of those would be interesting, but biinterpretability would be best. (Conversely disproving any of those would be interesting but disproving equiconsistency would be best.) | |
| Feb 12, 2020 at 14:56 | comment | added | Joel David Hamkins | Could you confirm that you really mean to refer to bi-interpretability in the question, as opposed to mutual interpretability? | |
| Feb 12, 2020 at 12:43 | history | edited | Oscar Cunningham | CC BY-SA 4.0 |
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| Feb 12, 2020 at 12:03 | comment | added | user44143 | @PaulTaylor, I hope that my answer mitigates two of your concerns. We can start to axiomatize the category of groups without axiomatizing their carriers, using axioms for the existence of groups like $1$, $\mathbb{Z}$ and $Q$ (the quarternion group) as defined there, an axiom that any two groups have a direct product, etc. It also means that if we can interpret ZFC in the category of abelian groups, then we can interpret it in the category of all groups too. | |
| Feb 12, 2020 at 3:36 | answer | added | user44143 | timeline score: 7 | |
| Feb 11, 2020 at 22:28 | comment | added | Paul Taylor | I apologise for being unfriendly earlier. Apart from limits and colimits, the category of groups does not have any interesting categorical structure (so far as I am aware) that could stand in for function spaces or powersets. Abelian groups would be a better bet, since they form a symmetric monoidal closed category; maybe you could characterise the symmetric tensor algebra functor, which would be the start of something more interesting. | |
| Feb 11, 2020 at 10:26 | comment | added | Paul Taylor | As you can see from my other contributions to this site, I am not going to defend set theory. Indeed I said carriers not sets. But I challenge you to axiomatise (the category of) groups in whatever foundational system you like without first axiomatising their carriers. | |
| Feb 11, 2020 at 10:22 | comment | added | Oscar Cunningham | @PaulTaylor The idea would be to demonstrate that the choice of sets as foundations was arbitrary (or at least motivated only be ease-of-use rather than necessity). Alternatively, if you want to show that the choice of sets wasn't arbitrary then you could prove that $\sf{ETCG+R}$ isn't biinterpretable with $\sf ZFC$. | |
| Feb 11, 2020 at 9:47 | comment | added | Paul Taylor | What intuition would you be trying to capture by using groups as foundations? Also, I cannot imagine how you would axiomatise groups without first axiomatising their carriers. Groupoids and Homotopy Type Theory, well that would be a different matter. | |
| Feb 10, 2020 at 7:46 | history | edited | Oscar Cunningham | CC BY-SA 4.0 |
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| Feb 10, 2020 at 7:40 | comment | added | Oscar Cunningham | @cody I don't see what being a topos has to do with biinterpretability. For example the category of topological spaces isn't a topos, but one can interpret $\sf{ETCS+R}$ in $\sf{ETCTS+R}$, by looking at the full subcategory on the discrete spaces. One identifies the discrete spaces in the categorical language by saying that they are precisely the objects such that any morphism into them which is both monic and epic is an isomorphism. | |
| Feb 9, 2020 at 20:49 | comment | added | cody | Is the category of Groups is not even a topos, which makes bi-interpretability very iffy (certainly very strange). | |
| Feb 9, 2020 at 18:33 | history | asked | Oscar Cunningham | CC BY-SA 4.0 |