Timeline for Axioms for the category of groups
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
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| Nov 9, 2022 at 14:36 | answer | added | Martin Brandenburg | timeline score: 2 | |
| Oct 27, 2022 at 11:05 | comment | added | მამუკა ჯიბლაძე | @DavidRoberts Thank you, it helps in the sense that all these are very interesting sources. On the other hand, I do not see how any of the axioms proposed there can be formulated in a category. I see how to do it with the axiomatization by Joyal & Moerdijk of classes of small-fibred maps, though. | |
| Oct 27, 2022 at 8:39 | comment | added | David Roberts♦ | @მამუკაჯიბლაძე McLarty cites Osius' 1975 "Logical and set-theoretical tools in elementary topoi" in that 2008 thread as being an old source for the claim (presented via an alternative but apparently equivalent axiom scheme) that ETCS plus a replacement-style schema is in one sense equivalent to ZFC. He also points to his own 2004 "Exploring Categorical Structuralism". Does this help? | |
| Oct 27, 2022 at 8:31 | comment | added | David Roberts♦ | @მამუკაჯიბლაძე do you mean that one can augment the ETCS axioms to something equivalent to ZFC? There's work of McLarty, and also of Shulman, but also others. When I say "in the same style", I was being imprecise. What I meant was not just by appeal to something literally out of ZFC, but something that feels like category theory. There's a bunch of discussion here: golem.ph.utexas.edu/category/2021/07/large_sets_125.html including stating the various versions | |
| Oct 27, 2022 at 7:35 | comment | added | მამუკა ჯიბლაძე | @DavidRoberts I browsed through several mentions of replacement here (including Andrej Bauer's 2013 and your 2015 questions about it) and on nLab, but could not pin down a definite reference for what you are saying. It must be hidden somewhere in a 2008 cat-dist thread, could you provide a link? | |
| Oct 24, 2022 at 23:53 | comment | added | David Roberts♦ | And, definitely, ETCS can replace ZFC for all mathematics that doesn't require instances of replacement, as ETCS is equivalent in strength to BZC. One can augment the original ETCS axioms (in the same style) to recover a system equivalent to ZFC itself. 2/2 | |
| Oct 24, 2022 at 23:51 | comment | added | David Roberts♦ | @JesseElliott Essentially is doing having lifting, here. And, you'll note, Lawvere says "with the additional... axiom of completeness". If you have a category of sets already, then you can talk about completeness with respect to colimits indexed by sets in it, and so pin down the category of sets up to equivalence, which is what Lawvere is meaning. But if you are talking about the first-order axiom system ETCS, it pins down the category of sets about as uniquely as the ZFC axioms pins down the universe $V$ of sets. 1/ | |
| Oct 24, 2022 at 23:16 | comment | added | Jesse Elliott | @DavidRoberts. Why, then, does Lawvere say on the first page of his paper, "There is essentially only one category which satisfies these eight axioms together with the additional (nonelementary) axiom of completeness, namely, the category of sets and mappings"? And can ETCS replace ZFC as a foundation for pre-1950 mathematics? | |
| Oct 24, 2022 at 11:42 | comment | added | David Roberts♦ | @Jesse no, but you have to specify what you mean by "the category Set". For instance, taking the ETCS axioms doesn't give you a unique category, since any model of ZFC will give rise to a category of sets satisfying those axioms. You could take an inner model, or a forcing extension, or an end extension or whatever, and get very different categories. Even just on the category side, you can start with Set and build a new category satisfying the same ETCS axioms, but they are not equivalent. | |
| Oct 24, 2022 at 1:17 | vote | accept | Madeleine Birchfield | ||
| Oct 23, 2022 at 23:54 | comment | added | Jesse Elliott | Are the categories Set and Group unique up to equivalence? How does this relate to the fact models of ZFC are not unique, e.g, some satisfy CH and even GCH while others do not? GCH can be expressed categorically. | |
| Oct 23, 2022 at 19:49 | comment | added | Kevin Carlson | @DavidRoberts Sure, I've posted it. | |
| Oct 23, 2022 at 19:48 | answer | added | Kevin Carlson | timeline score: 10 | |
| Oct 23, 2022 at 8:32 | comment | added | David Roberts♦ | @KevinArlin can you write an answer summarising the axioms, pointing to the MSE post? | |
| Oct 22, 2022 at 23:14 | history | edited | YCor | CC BY-SA 4.0 |
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| Oct 22, 2022 at 22:31 | comment | added | Kevin Carlson | The most relevant answer I’ve seen appeared on MSE a few years back: math.stackexchange.com/questions/2332425/… | |
| Oct 22, 2022 at 13:23 | comment | added | Oscar Cunningham | See also: Could groups be used instead of sets as a foundation of mathematics? | |
| Oct 22, 2022 at 11:27 | comment | added | მამუკა ჯიბლაძე | One non-interesting possibility is to use a theorem of Kan that the category of cogroups in the category of groups is equivalent to the category of sets. So the axioms might be finite coproducts + the ETCS axioms on the category of internal cogroups. What seems to me interesting is which kind of categories are equivalent to the category of internal groups in the category of their internal cogroups. | |
| Oct 22, 2022 at 8:49 | history | became hot network question | |||
| Oct 22, 2022 at 7:19 | comment | added | David Roberts♦ | It's a semiabelian category, to start. One could wonder what special properties is has in addition that singles it out among those. | |
| Oct 22, 2022 at 1:50 | answer | added | Qiaochu Yuan | timeline score: 13 | |
| Oct 22, 2022 at 0:45 | history | asked | Madeleine Birchfield | CC BY-SA 4.0 |