Timeline for Ultracategories with one object
Current License: CC BY-SA 4.0
48 events
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| May 1, 2022 at 13:37 | comment | added | user480841 | @MaximeRamzi I see! | |
| May 1, 2022 at 13:11 | comment | added | Maxime Ramzi | For ultra preordered set, I think this is the only reasonable definition and I had the same in mind. | |
| May 1, 2022 at 12:49 | comment | added | user480841 | @MaximeRamzi How would you define an ultra-preordered set? I would define it naively as an ultracategory whose underlying category happens to be a preorder. (Just as a would define an ultramonoid to be an ultracategory whose underlying category happens to have exactly one object.) Are you here using a different definition too when you say you haven't found a simple characterization of ultra-preordered sets? | |
| May 1, 2022 at 12:46 | comment | added | user480841 | @MaximeRamzi Thank you for your repeated help. I learned a lot from our conversation! | |
| May 1, 2022 at 12:45 | vote | accept | user480841 | ||
| Apr 30, 2022 at 16:26 | comment | added | Maxime Ramzi | I think, but haven't worked the details, that an ultrafunctor $*\to \mathcal M$ should be the sane thing as an object $x\in \mathcal M$ such that all the transformations from (4) in Definition 7 are isomorphisms when evaluated at the constant $x$-valued $J$-indexed family. Does that condition make sense ? | |
| Apr 30, 2022 at 16:22 | comment | added | Maxime Ramzi | To define pointed ultracategories in the style of definition 7, I would have to think about the translation from 1 to 7. What I can say ahead of doing this is that it would definitely be of the form " an $x$ in $\mathcal M$ satisfying such and such property" . So the data of an object + some condition | |
| Apr 30, 2022 at 16:21 | comment | added | Maxime Ramzi | To answer your question, I don't think the two notions are equivalent for ultracategories. I call $\mathcal M_*$ what you would call $\mathcal M$, but that's just an artifact of Definition 1 vs 7. For your question about (2,1) vs 1, yes, part of my answer is the claim that this (2,1)-category is in fact equivalent to a 1-category - just as in the usual comparison of monoids and pointed categories. | |
| Apr 30, 2022 at 14:24 | comment | added | user480841 | @MaximeRamzi As for what the "main" questions of my post were, maybe I shouldn't have said that the $\mathrm{End}$ questions were the only main questions. The question what an ultramonoid is was also a major question, which you answered totally fine using your definition of ultramonoid. Maybe there isn't a nice answer with my definition of ultramonoid, so maybe that's the best one can say! | |
| Apr 30, 2022 at 14:20 | comment | added | user480841 | @MaximeRamzi Despite our misunderstanding of what the other means by "ultramonoid" (so that technically your answer does not answer the questions I intended to ask) let me say a big thank you to you, because understanding your perspective has been a great opportunity for me to learn some new terminology, and I appreciate your result that ultramonoids in your sense are monoids in compact Hausdorff topological spaces very much! :-) | |
| Apr 30, 2022 at 14:11 | comment | added | user480841 | @MaximeRamzi can "$x$ is an ultrafunctor from $1$ to an ultracategory $\mathcal M$" be characterized as "$x$ is an object of $\mathcal M$ with ... such that ..." similar to how one can characterize a point $1\to C$ of a category $C$ as being the same as an object of that category $C$? | |
| Apr 30, 2022 at 14:06 | comment | added | user480841 | @MaximeRamzi How would you define the term "pointed ultracategory" in the style of Definition 7 which I prefer? | |
| Apr 30, 2022 at 14:05 | comment | added | user480841 | @MaximeRamzi Do you take the homotopy category of that (2,1)-category or is the situation similar to the situation with categories instead of ultracategories in which it happens by accident that the 2-category of pointed categories with one object up to isomorphism is equivalent to a 1-category? | |
| Apr 30, 2022 at 14:05 | comment | added | user480841 | @MaximeRamzi In your original portion of the answer you use the phrase "the category of ultramonoids". I asked: "How do you define the category of ultramonoids?" Then in the clarifying comments you added that "I define the (2,1)-category of ultramonoids as a full subcategory of the category of pointed ultracategories (in the sense I described earlier)." But here you only say what the (2,1)-category of ultramonoids is and not what the 1-category is. | |
| Apr 30, 2022 at 13:55 | comment | added | user480841 | @MaximeRamzi What do you mean by $*\to \mathcal M_*$ in your answer? Is it correct that you use the following definition of "ultramonoid": an ultramonoid is a pointed ultracategory $(\mathcal M, x\colon 1\to\mathcal M)$ such that the ultrafunctor $x$ is essentially surjective? Maybe it makes sense to call the whole thing $\mathcal M_*$, but you talk about $*\to \mathcal M_∗$ being "essentially surjective" -- this only makes sense when deleting the $*$ from $\mathcal M_*$ to me. | |
| Apr 30, 2022 at 13:48 | comment | added | user480841 | @MaximeRamzi Comparison: These two definitions are equivalent, in the sense that one gets the same 1-category of monoids (no matter which definition one chooses). Now my question is: if we talk about ultracategories instead of categories, are these two definitions equivalent too? I guess if not, that's the origin of our misunderstanding: I tried to use a definition of "ultramonoid" along the lines of the second definition, whereas you used a definition of "ultramonoid" along the lines of the first definition. | |
| Apr 30, 2022 at 13:45 | comment | added | user480841 | @MaximeRamzi There are two definitions of "monoid". First definition: A monoid is a pointed category with one object up to isomorphism. Since pointed categories form a 2-category by definition, this yields a 2-category of monoids, but it turns out that this 2-category of monoids is equivalent to a 1-category. Second definition: A monoid is a small category with exactly one object. Here, "category" refers to an object of the 1-category of all small categories. Hence monoids in that sense form a 1-category too. | |
| Apr 30, 2022 at 12:54 | comment | added | user480841 | @MaximeRamzi Let me give two definitions: a category with one object is a category that has exactly one object; an ultracategory with one object is an ultracategory that has exactly one object. | |
| Apr 30, 2022 at 12:53 | comment | added | user480841 | @MaximeRamzi "it's unclear to me what "a category with one object" means" -> how are you then able to claim that "a monoid is not the same as a category with one object"? | |
| Apr 30, 2022 at 12:09 | comment | added | user480841 | @DavidRoberts Thanks! | |
| Apr 26, 2022 at 21:15 | comment | added | David Roberts♦ | @user480841 I meant what Maxime called strongly connected, not the wp definition, which my clarification was meant to indicate. I put that query since monoids are equivalent to pointed categories with a single isomorphism class of objects, not just pointed categories. | |
| Apr 26, 2022 at 16:50 | history | edited | Maxime Ramzi | CC BY-SA 4.0 |
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| Apr 26, 2022 at 16:38 | history | edited | Maxime Ramzi | CC BY-SA 4.0 |
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| Apr 26, 2022 at 16:32 | comment | added | Maxime Ramzi | @user480841 I tried to answer your further questions in an edit. | |
| Apr 26, 2022 at 16:32 | history | edited | Maxime Ramzi | CC BY-SA 4.0 |
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| Apr 26, 2022 at 16:06 | comment | added | user480841 | @MaximeRamzi "I'm not a big fan of "connected" for categories, maybe "strongly connected" would work." What is a strongly connected category? Tried the nLab, but didn't find anything. | |
| Apr 26, 2022 at 15:59 | comment | added | user480841 | @DavidRoberts I did some research. Do you mean en.wikipedia.org/wiki/Connected_category? (However, then I don't understand your "as in, one isomorphism class".) | |
| Apr 26, 2022 at 15:53 | comment | added | user480841 | @MaximeRamzi Let me also add that (in my view) the two main questions of my post were 1) for which structures $\mathbf A$ the monoids $\mathrm{End}(\mathbf A)$ and $\mathrm{End}'(\mathbf A)$ carry the structure of an ultramonoid and 2) to what extent each ultrafunctor $F\colon\mathcal M\to\mathcal N$ induces for each $M\in\mathcal M$ (sufficiently nice, say) an ultrafunctor $\mathrm{End}(M)\to\mathrm{End}(F(M))$. Can you say anything about these questions? Do ultrafunctors preserves sufficiently nice objects after all? | |
| Apr 26, 2022 at 15:49 | comment | added | user480841 | @DavidRoberts When is a pointed category connected and what do you mean by the question "Connected pointed categories?"? | |
| Apr 26, 2022 at 15:47 | comment | added | user480841 | As I understand interpret your third last paragraph (I don't really understand all the words you are using: point, locally cartesian, comparison morphism, $\beta$, $^{fr}$, ...) you are saying that if $\mathcal M$ is an ultracategory and $M$ is a sufficiently nice object of $\mathcal M$, then $\mathrm{End}(M)$ can be considered as an ultramonoid. Can you say more concretely in the case $\mathcal M=$ the ultracategory of models of some-first order theory (with elementary embeddings as morphisms) when an object $M\in\mathcal M$ (i.e., a model) is sufficiently nice? | |
| Apr 26, 2022 at 15:42 | comment | added | user480841 | What do you mean by "point"? (Of an ultracategory, say. You also talk about points of $\mathcal M_\ast$, but I don't even know what type of object that is.) | |
| Apr 26, 2022 at 15:37 | comment | added | user480841 | How do you define the category of ultramonoids? Are the morphisms ultrafunctors or isomorphism classes of ultrafunctors? | |
| Apr 26, 2022 at 15:33 | comment | added | user480841 | I have never seen this notation $*_{\beta I}\to \mathcal M_{\beta I}$. Would you mind recalling the notation you are using here? In my post I specifically referred to Definition 7 and not Definition 1. I don't even know what a Grothendieck fibration is. | |
| Apr 26, 2022 at 15:31 | comment | added | user480841 | "In particular, I'll interpret your question as [...]" What do you mean by that? Which question do you interpret? Do you use the same definition of "ultramonoid" as I do or not? | |
| Apr 26, 2022 at 15:28 | comment | added | user480841 | Are you saying the 1-category of monoids is equivalent to the 2-category of pointed categories? Or that the 2-category of monoids is equivalent to the 2-category of pointed categories? What is the 2-category of monoids? | |
| Apr 26, 2022 at 15:27 | comment | added | user480841 | @MaximeRamzi Thanks for your partial answer! You argue that a monoid is not the same as a category with one object, but is the same as a pointed category with one object. What is a pointed category (nLab says it's a category with a zero object -- but this doesn't seem to be related to what you are saying) and how do you define the 2-category of pointed categories? | |
| Apr 20, 2022 at 7:08 | comment | added | Maxime Ramzi | @DavidRoberts : sorry I did not answer because I was trying to fix the mistake in my answer (which I think I managed to, for monoids). I'm not a big fan of "connected" for categories, maybe "strongly connected" would work. | |
| Apr 20, 2022 at 7:08 | comment | added | Maxime Ramzi | @KevinArlin : it turns out I could fix my answer for the monoid case (a pointed ultracategory allows me to say more things than just an ultracategory with a point in $\mathcal M_*$). Hopefully with the new version of the answer, it is clearer what doesn't work for categories ! If not, I can add some more explicit words | |
| Apr 20, 2022 at 7:05 | history | edited | Maxime Ramzi | CC BY-SA 4.0 |
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| Apr 20, 2022 at 6:49 | history | undeleted | Maxime Ramzi | ||
| Apr 20, 2022 at 6:41 | history | deleted | Maxime Ramzi | via Vote | |
| Apr 20, 2022 at 6:38 | comment | added | Maxime Ramzi | Even for monoids it starts getting wrong, because you're allowed some noninvertible transformations. Also, "pointed" becomes a more subtle structure. | |
| Apr 20, 2022 at 6:33 | history | edited | Maxime Ramzi | CC BY-SA 4.0 |
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| Apr 20, 2022 at 6:31 | comment | added | Maxime Ramzi | @KevinArlin : I was wondering about that too, and it seems I made a mistake - I didn't read the definition of ultracategory carefully enough. They are locally cartesian.fibrations, and only certain locally cartesian edges are closed under composition, not all of them. In particular most of what I said is not proved properly/potentially wrong | |
| Apr 20, 2022 at 6:23 | history | edited | Maxime Ramzi | CC BY-SA 4.0 |
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| Apr 20, 2022 at 5:41 | comment | added | Kevin Carlson | I don’t fully understand this stuff, but is it too obvious to be worth including any tip about why ultracategories are not in general simply categories in compact Hausdorff spaces, if indeed they are not? | |
| Apr 20, 2022 at 0:18 | comment | added | David Roberts♦ | Connected pointed categories? (as in, one isomorphism class) | |
| Apr 19, 2022 at 19:50 | history | answered | Maxime Ramzi | CC BY-SA 4.0 |