Timeline for The main theorems of category theory and their applications
Current License: CC BY-SA 3.0
8 events
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| Dec 16, 2011 at 8:16 | comment | added | Andrej Bauer | More often than not the significance of a category-theoretic theorem for a particluar area will not be obvious to non-experts. And that is why people sometimes conclude that category theory is useless. So it is very important to give concrete applications, like the coalgebras above, even though a category theorists might think of it as a "trivial application". I remember that as a student I could not really see the point of Yoneda Lemma until Steve Awodey fed me sufficiently many applications. | |
| Dec 15, 2011 at 22:43 | comment | added | Todd Trimble | Yeah, okay. I think Paul said, "ideally...", whereas I was giving myself a little leeway here. I mean, let's be fair: what about monadicity theorems (as mentioned by Finn)? Has the intended application been purged of all categorical language? :-) Same question for the small object argument mentioned by Akhil. Is my answer being subjected to a different standard? It seems to me that an algebraist interested in studying coalgebras but who didn't know a lot of category theory (e.g., Gabriel-Ulmer duality) might, just might, find this sort of thing useful to know. :-) | |
| Dec 15, 2011 at 19:51 | comment | added | Martin Brandenburg | Thanks for the edit. But why should one care about products in this category when you are not interested in the category theoretic properties in the first place? What is a specific application which can be formulated without the notions of category theory? [This was also one of the requirements which Paul Siegel has mentioned in his question] | |
| Dec 15, 2011 at 17:51 | history | edited | Todd Trimble | CC BY-SA 3.0 |
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| Dec 15, 2011 at 0:54 | comment | added | Todd Trimble | Fetisov Anton: not sure what you're referring to. The "theories" here are finitely complete, and that implies they are Cauchy complete. But in any event I'm not referring to the process of trying to recover a small category from its category of presheaves (which would indeed involve Cauchy completion); I'm referring to something similar but different enough to be significant. | |
| Dec 15, 2011 at 0:18 | comment | added | Anton Fetisov | Actually, you can only recover base category up to Cauchy completion. A trivial example is two different, but Morita equivalent rings. Or do you mean something more intricate? | |
| Dec 14, 2011 at 22:08 | comment | added | Martin Brandenburg | Dear Todd, while I think this is a significant and profound answer, I don't see any application "outside of" category in it. Perhaps you know one and add it (and I delete this silly comment). It would be great if some people which have not heard so far about presentable categories (or categories at all?) read your answer and are convinced that they have to learn more about them right away ;). | |
| Dec 14, 2011 at 21:56 | history | answered | Todd Trimble | CC BY-SA 3.0 |