Timeline for How do we compare models of ETCS?
Current License: CC BY-SA 2.5
10 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Oct 19, 2010 at 4:04 | history | edited | David Roberts♦ | CC BY-SA 2.5 |
added 47 characters in body
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| Oct 19, 2010 at 3:14 | comment | added | Todd Trimble | "Elementary" here refers to a theory which doesn't make reference to an external notion of set in the background. For example, when we refer to a small or locally small category, we refer to a background notion of set, whereas here the development is to be independent of any such prior notion. The terminology is due to Lawvere. | |
| Oct 19, 2010 at 2:48 | comment | added | Carl Mummert | I don't know anything at all about ETCS. Is it the "elementary theory" of the "category of sets", as in the model-theoretic term "elementary diagram"? | |
| Oct 19, 2010 at 1:07 | comment | added | Todd Trimble | You probably want more than preservation of finite limits since otherwise you could just map everything to the terminal object. Being a category theorist, I think "logical functor" seems like a natural choice. If that is the choice, I believe there is no weakly initial object (and I could maybe rummage up some relevant nLab pages). | |
| Oct 19, 2010 at 0:50 | comment | added | David Roberts♦ | And regarding 3: I really meant, 'is there a functor?', but this would allow silly functors like those mapping everything to the empty set. Let us say at least preserving finite limits. The flavour of set theory one is using clearly affects things like NNOs, as Joel pointed out (and I was aware of to begin with). | |
| Oct 19, 2010 at 0:48 | comment | added | David Roberts♦ | Hmm, re 4: I suppose so. But are there many arrows in this category? Is it connected? I could be completely naive and ask: is there a weakly initial object? (The constructible universe, perhaps?) | |
| Oct 19, 2010 at 0:24 | comment | added | Todd Trimble | For question 3, surely you mean not just "functor", but a "logical functor" which preserves finite limits, power objects, and natural numbers object? I'm not even sure what you mean by 4 (I mean, why not: just take objects to be ETCS categories and morphisms to be logical functors)? | |
| Oct 18, 2010 at 23:52 | answer | added | Joel David Hamkins | timeline score: 7 | |
| Oct 18, 2010 at 23:09 | history | asked | David Roberts♦ | CC BY-SA 2.5 |