Timeline for Between abstract and concrete: What's the right way to think of specific categories?
Current License: CC BY-SA 2.5
19 events
| when toggle format | what | by | license | comment | |
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| Jan 23, 2010 at 14:56 | comment | added | Hans-Peter Stricker | In the final end: of course. | |
| Jan 22, 2010 at 20:38 | comment | added | Qiaochu Yuan | Then you agree that category theory shouldn't be bound by the properties of sets! | |
| Jan 22, 2010 at 18:27 | comment | added | Hans-Peter Stricker | And I thought of category theory as a kind of unifying theory, why then not to think in a unified manner about its objects? | |
| Jan 22, 2010 at 18:00 | comment | added | Hans-Peter Stricker | I definitely did not expect this question to be silly, but if you think so, I have to live with that. | |
| Jan 22, 2010 at 16:06 | comment | added | Mariano Suárez-Álvarez | @Hans: trying to think about everything the same way is not "trying to be unreasonably rigorous", as there is no rigorousness in it at all, but being a bit silly :) | |
| Jan 22, 2010 at 14:36 | comment | added | Qiaochu Yuan | No, that's not what I meant. You started with a poset of structured sets, regarded it as an abstract poset, and asked if we could recover the original structured sets. You did not specify what kind of structure you wanted; it is totally conceivable that the poset you described could be recovered as the poset of some set of groups under inclusion, for example. | |
| Jan 22, 2010 at 14:13 | comment | added | Hans-Peter Stricker | Ah! I had in mind any standard representation of an unlabelled graph, e.g. an adjacency matrix (factored out by isomorphism). But that's off-topic here. | |
| Jan 22, 2010 at 14:01 | comment | added | Qiaochu Yuan | That's not the part I didn't understand; you weren't clear about what kind of representations you wanted of the abstract thing. | |
| Jan 22, 2010 at 13:58 | comment | added | Hans-Peter Stricker | @Qiaochu: Your last sentence in your edit convinces me. By the way, that's exactly what I had in mind yesterday when I said ""forget the inner structure of the objects" (what you claimed not to understand?). | |
| Jan 22, 2010 at 13:55 | comment | added | Hans-Peter Stricker | @Charles: I will think about this, maybe I try to be unreasonabily rigorous. | |
| Jan 22, 2010 at 13:45 | comment | added | Charles Siegel | Why should we have to think about everything the same way? We don't even think of all groups the same way! See mathoverflow.net/questions/2551/… | |
| Jan 22, 2010 at 13:22 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Jan 22, 2010 at 12:26 | comment | added | Hans-Peter Stricker | In any case, there should be one way of thinking of all categories, and if (2) is only possible for some categories, we have to take (1) for all, so even if (2) is appealing for concrete categories, we have to avoid it, right? | |
| Jan 22, 2010 at 12:21 | comment | added | Hans-Peter Stricker | @Qiaochu: Even if there is no forgetful functor from a homotopy category to Set, one can think of the homotopy classes as classes of (functions as) sets, or am I wrong? | |
| Jan 22, 2010 at 12:19 | comment | added | Qiaochu Yuan | @Hans: Perhaps this nLab entry is relevant: ncatlab.org/nlab/show/cocomplete+well-pointed+topos | |
| Jan 22, 2010 at 12:10 | comment | added | Harry Gindi | @Qiaochu, we specifically want an adjoint pair of functors rather than just a particular forgetful functor. | |
| Jan 22, 2010 at 12:07 | comment | added | Harry Gindi | @Hans: To my knowledge, every definition of a category uses some of the theory of sets to develop the basic results in category theory. As I said before, and as you can read the discussion on the nLab, ETCC is not very good at all. I am not aware of a single useful result from ETCC that doesn't also appear as a consequene of ETCS. | |
| Jan 22, 2010 at 12:01 | comment | added | Hans-Peter Stricker | Would you agree then, that "the category of sets" is a convenient shortcut for "the (abstract) category that is isomorphic to the class of all sets with all functions between them". Or doesn't this make sense? | |
| Jan 22, 2010 at 11:48 | history | answered | Qiaochu Yuan | CC BY-SA 2.5 |