Timeline for A category with objects that are not based on sets or classes [closed]
Current License: CC BY-SA 3.0
18 events
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| Dec 18, 2012 at 4:54 | comment | added | Spice the Bird | I have wondered about something (that might be) similar. Notice that the standard example of a non-concrete category, the homotopy category, has the property that it is a localization of the category of "nice" spaces. Also, the category of "nice" spaces may be made into a concrete category, by using the forgetful functor to sets. One may then ask whether or not there is a category that does not arise as the localization of some concretizable category. | |
| Dec 17, 2012 at 22:31 | history | closed |
Andreas Blass David White Fernando Muro George Lowther Todd Trimble |
not a real question | |
| Dec 17, 2012 at 20:08 | comment | added | Omar Antolín-Camarena | A category has some sort of collection of objects and some sort of collection of morphisms. If you want an example of a category where neither of these is "a construction on sets or classes", it would help if you gave an example of a collection of things that you do not consider to be "a construction on sets or classes". If we had such a collection $C$, maybe the discrete category whose objects are $C$ (and whose morphisms are only identities) would satisfy you. | |
| Dec 17, 2012 at 17:50 | answer | added | juan | timeline score: 3 | |
| Dec 17, 2012 at 17:38 | answer | added | Vlad Patryshev | timeline score: 0 | |
| Dec 17, 2012 at 16:42 | comment | added | Fernando Muro | Is there any mathematical object at all which is not a construction on sets of classes? I don't know what to think of this question. Either it is not correctly phrased or the person who asks doesn't know anything about the foundations of mathematics. | |
| Dec 17, 2012 at 15:55 | comment | added | Todd Trimble | Maybe the OP just wants an example of a "category" (in the naive or preformal sense of the word) where the collection of morphisms is too large to be considered a class? Of course, what this is asking for depends on the details of the background set theory, but if we work in straight ZFC, and consider classes as defined by class formulas, then one cannot refer to the collection of all subclasses of the universe $V$ as forming a class. In that case, just consider the (discrete) "category" of functors from the discrete category on $V$ to the discrete category with two objects. | |
| Dec 17, 2012 at 15:03 | comment | added | Andreas Blass | Even after the OP agreed with Qiaochu's formulation, the question is still, as Qiaochu said, not easily made precise. Having no clear idea what "some construction on" (in the original question) and "constructed from" (in Qiaochu's reformulation) mean, I vote to close as "not a real question". The OP's comment on Tim Porter's answer leads me to think that he wants a category whose objects and morphism don't constitute sets or classes, which contradicts the (usual) definition of category; then the OP should tell us what alternative definition of category (s)he has in mind. | |
| Dec 17, 2012 at 12:59 | answer | added | Charles Matthews | timeline score: 10 | |
| Dec 17, 2012 at 12:58 | comment | added | Salvo Tringali | @Qiaochu: In that case, the very same article by P. Freyd mentioned by A. Mathew in his post at amathew.wordpress.com/2012/01/26/homotopy-is-not-concrete, includes another example (apart from the usual category of topological spaces and homotopy classes of continuous functions). | |
| Dec 17, 2012 at 12:57 | comment | added | jd94j39 | Ma, I think Qiaochu best explained what I am after (see the comment three above). | |
| Dec 17, 2012 at 12:54 | comment | added | Ma Ming | Are you looking for an instance of non-concrete category(See wp article en.wikipedia.org/wiki/Concrete_category)? | |
| Dec 17, 2012 at 12:46 | answer | added | Tim Porter | timeline score: 7 | |
| Dec 17, 2012 at 12:32 | comment | added | jd94j39 | That's what I'm "trying" to say Qiaochu :) | |
| Dec 17, 2012 at 12:30 | comment | added | Qiaochu Yuan | @Salvo: as far as I can tell, the OP wants to ask a question similar to "can you give me an example of a category which is not concretizable?" but something more like "can you give me an example of a category which cannot be constructed from a concretizable category?" except I am not really sure how to make this precise. | |
| Dec 17, 2012 at 12:28 | comment | added | Qiaochu Yuan | This seems like a slightly different question from the one you asked on math.SE (math.stackexchange.com/questions/260623/…), mostly because you used "are not" instead of something like "cannot be considered as." | |
| Dec 17, 2012 at 12:27 | comment | added | Salvo Tringali | The question makes no sense to me. In NBG, and axiomatic systems with a similar ontology, a category is a tuple $(\mathcal C_{\rm o}, \mathcal C_{\rm h}, s, t, c, i)$, where $\mathcal C_{\rm o}$ and $\mathcal C_{\rm h}$ are classes (and the members of a class are sets). | |
| Dec 17, 2012 at 12:20 | history | asked | jd94j39 | CC BY-SA 3.0 |