Timeline for Is there a metamathematical $V$?
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2021 at 18:18 | comment | added | Pace Nielsen | @PeterScholze I very much appreciate your perspective, and this discussion. Thanks for helping me understand; I only use Yoneda's lemma in a very limited form in my own work. | |
| Mar 26, 2021 at 17:23 | comment | added | Peter Scholze | Sorry, my last comment wasn't very spot on. I meant that the intuitive idea of the Yoneda lemma is about embedding into functors into the category of all sets. This works only if your category in the beginning is small, of course, but that is a prerequisite anyways. I find that the idea of Yoneda is obscured if you end up choosing universes of sets and comparing the different universes. But maybe I should stop the discussion; I was just meaning to give a different perspective with this answer, one that I think is close to how I think, but I don't have strong feelings about it. | |
| Mar 26, 2021 at 15:42 | comment | added | Pace Nielsen | I just checked, and this seems to be what George Bergman does in his book "An invitation to general algebra and universal constructions." Yoneda's lemma is with respect to a given universe. etc... | |
| Mar 26, 2021 at 15:31 | comment | added | Pace Nielsen | @PeterScholze I would have thought that Yoneda's lemma poses no problem, because of "equivalences". One would show that there is stability (in terms of natural equivalences on the hom sets) when passing to larger and larger universes. I would definitely expect no problems in algebraic geometry, number theory, or algebraic topology (for the reasons given in the answers to your question). I'd think you would only lose something if you wanted to study all of ZFC+Universes itself using categories (which unnecessarily presupposes a need for categories to be foundational). Am I missing something? | |
| Mar 26, 2021 at 8:31 | comment | added | Peter Scholze | I'd argue that taking the position that there is a completed category of all sets is the default position in category theory; certainly that's the default position in algebraic geometry/number theory/algebraic topology, I'd say. The whole discussion of the Yoneda lemma (or presentable categories) in fact very much assumes this point of a completed category of all sets. | |
| Mar 26, 2021 at 3:17 | comment | added | Pace Nielsen | @theHigherGeometer I meant the "true category of all sets". I asked because in the JSL paper you recommended, it seemed to be a given. | |
| Mar 26, 2021 at 3:04 | comment | added | David Roberts♦ | @PaceNielsen "that" position being the "there is no equality on objects" position? I couldn't say for sure. Certainly there are some. Unless you mean "there is a true category of all sets", in which case, that's a bit trickier. People who are sensitive to this are also the sort of people who want to work relative to an arbitrary topos, so really a much more pluralist standpoint! | |
| Mar 25, 2021 at 23:30 | comment | added | Pace Nielsen | @theHigherGeometer I'm currently leaning towards the formalist ZFC+Universes position, where there is no category of (all) sets, just partial approximations. Are there many category theorists who take that position seriously, or do they usually just assume there really is a completed category of all of set theory? | |
| Mar 25, 2021 at 22:40 | comment | added | David Roberts♦ | @AbdelmalekAbdesselam take a well-pointed boolean autological topos with NNO satisfying: let $X$ be a non-empty set and $R\subset X\times X$ a total relation, then for all $x\in X$ there is a function $f\colon \mathbb{N}\to X$ such that $f(0)=x$, and $(f(n),f(n+1))\in X^2$ is really in $R$. There's a small amount of gloss needed here, if one wants to speak strictly in terms of ETCS, but this should be what you want. DC is a very "structural" axiom, in this form. | |
| Mar 25, 2021 at 22:00 | comment | added | David Roberts♦ | @PaceNielsen You can take the "collection of objects" of a category to be so primitive so as not to even have an equality predicate (this idea is due to Bénabou, in his only JSL paper). This changes nothing about category theory, but then you have to be a little more careful as treating objects as giving typing information about morphisms (so you aren't really using equality when talking about composability of morphisms). Then the objects do not form a class in any sense of the word. See also Makkai's FOLDS. | |
| Mar 24, 2021 at 20:19 | comment | added | Peter Scholze | So even if there is a category of all sets, this doesn't mean that there's a set/collection/class of all sets! | |
| Mar 24, 2021 at 20:19 | comment | added | Peter Scholze | Numbers can be equal. Sets can be isomorphic. Categories can be equivalent. In particular, the "set of objects of a category" isn't even a well-defined thing, as it's not invariant under equivalence; a good formalization shouldn't even allow you to talk about it. | |
| Mar 24, 2021 at 20:16 | comment | added | Pace Nielsen | I'd be very interested in your perspective of how categories are different beasts than collections. The way they've been presented to me has been as two-sorted collections, with one sort being the objects and the other sort being the arrows (satisfying certain axioms). | |
| Mar 24, 2021 at 19:37 | comment | added | Peter Scholze | So this would suggest to axiomatize the $2$-category of categories; this ought to contain the "standard category of sets" just like sets contain the "standard natural numbers". I am sure that there is work in this direction; can anybody point me to relevant literature? | |
| Mar 24, 2021 at 19:36 | comment | added | Peter Scholze | @PaceNielsen In a sense, you are absolutely right; this was the perspective I was taking when I wrote the referenced question. But I actually find the ETCSR perspective more compelling; it reflects very closely how I "want" to think about it; and I maintain that there's a conceptual leap in that case, a category is a different beast than a collection. But the analogy to PA is not perfect: Under this analogy, it seems that so far we've replaced PA by the set theory of finite sets (ZFC - Infinity, which is "equivalent" to PA). | |
| Mar 24, 2021 at 16:50 | comment | added | Pace Nielsen | In this argument, we could replace ETCSR with any other theory that is sufficiently similar to ZFC, such as NBG. Thus, your argument just looks like a reframing of the usual argument using "(proper) classes" and "hyper-classes" and so forth, but instead replacing them with (higher) categories. If I were to play devil's advocate, I'd say that the conceptual leap wasn't from numbers to set, but from numbers to "collections"; and sets, classes, and categories are all (slightly different) forms of collections. | |
| Mar 24, 2021 at 14:22 | comment | added | Peter Scholze | The expert is Mike Shulman, maybe he can answer. For now let me refer to his paper Comparing material and structural set theories that discusses many cases of this dictionary (but I believe not the one you ask about). | |
| Mar 24, 2021 at 14:09 | comment | added | Abdelmalek Abdesselam | Is there a variant of ETCSR which is equivalent to ZF plus DC only? | |
| Mar 24, 2021 at 12:40 | comment | added | Peter Scholze | More seriously, I'm not sure we're yet able to fully predict where this path would lead; rather, the idea seems to me that each new step requires a new conceptual leap, informed by actual mathematical problems (just like Cantor started to think seriously about sets because of work on Fourier series), with merely saying "(higher) category of all previous things" not actually being such a conceptual leap. | |
| Mar 24, 2021 at 12:34 | history | edited | Peter Scholze | CC BY-SA 4.0 |
added 46 characters in body
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| Mar 24, 2021 at 12:31 | comment | added | Peter Scholze | Let's leave that problem to our grand-grand-grand-children ;-). | |
| Mar 24, 2021 at 11:55 | comment | added | Student | And then $(\infty,n), (\infty, n+1), \ldots$.. and then again we run out of integer. What next? | |
| Mar 24, 2021 at 11:31 | history | answered | Peter Scholze | CC BY-SA 4.0 |