Timeline for Is "all categorical reasoning formally contradictory"?
Current License: CC BY-SA 2.5
20 events
| when toggle format | what | by | license | comment | |
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| May 6, 2010 at 10:27 | vote | accept | José Figueroa-O'Farrill | ||
| May 5, 2010 at 21:51 | comment | added | darij grinberg | Oh, you guys are right. The injectives are canonical, but maybe their injectivity isn't (i. e. the lifts of the morphisms aren't - they use the axiom of choice). Not sure what this means about the functors. | |
| May 5, 2010 at 20:08 | answer | added | Dan Doel | timeline score: 17 | |
| May 5, 2010 at 19:26 | answer | added | François G. Dorais | timeline score: 20 | |
| May 5, 2010 at 18:56 | comment | added | BCnrd | @Mariano: One has to be careful. I only claimed canonicity in a sense I defined (no "choices"), but didn't claim functoriality in a good sense: that construction you have in mind is most surely not an additive functor, so it's not really a useful kind of functoriality. Grothendieck points out the same issue in his Tohoku paper. Anyway, so at least the actual complex is constructed in a definite controlled manner. We can then play with manipulations of Hom sets since we grant set theory (and Grothendieck does nontrivial set theory with Hom sets, like transfinite induction). | |
| May 5, 2010 at 18:20 | comment | added | algori | Added "set-theory" tag. Hope that's ok. | |
| May 5, 2010 at 18:19 | history | edited | algori |
added the "set theory tag"
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| May 5, 2010 at 17:59 | comment | added | José Figueroa-O'Farrill | Actually the PDF file I just linked contains an "abstract" with more details! | |
| May 5, 2010 at 17:57 | comment | added | José Figueroa-O'Farrill | There was a mini-workshop in February 2009 at Oberwolfach on Category Theory. The report is here: mfo.de/programme/schedule/2009/08a/OWR_2009_08.pdf and it mentions Cartier, but has no detail about the proposal: "and last but not least a new proposal for a logical foundation of category theory by Pierre Cartier. The basic ideas of Cartier’s proposal can be traced back to the beginnings of his Bourbaki membership in the early fifties and corresponding discussions of the foundations of category theory, partly visible in the online collections of the Bourbaki archives." | |
| May 5, 2010 at 17:27 | comment | added | François G. Dorais | Does anyone know what Cartier is referring to at the end of the paragraph? "I have just proposed a possible program to give a solid basis to categories at a meeting in Oberwolfach." | |
| May 5, 2010 at 16:38 | comment | added | Mariano Suárez-Álvarez | @darij: look at the usual proof that there are enough injectives in a category of modules: the resolution that can be derived from that proof is completely functorial. | |
| May 5, 2010 at 14:53 | comment | added | darij grinberg | Sure about injective resolutions? I know about projectives. As far as I know, injective resolutions require the axiom of choice, and the axiom of choice over classes is something even more evil than the axiom of choice alone... | |
| May 5, 2010 at 13:59 | comment | added | BCnrd | @darij: modules have canonical (="choice-free") free resolutions using generating set of all elements. And if one goes back to ways of making injectives, can say same for injective resolutions (ditto in various sheaf categories). These are things which everyone has to come to terms with in their own way; I have no idea what an "anafunctor" is, and don't see the need for anything extra. Anyway, in categorical stuff infinitely many things do get used all at once: derived categories, for example. Computers aren't a good test; they don't do anything related to "infinite" issues Cartier raises. | |
| May 5, 2010 at 13:57 | comment | added | Gerald Edgar | This is not unheard of. Many years ago, a much older senior professor described something he was telling his students, where first you take the direct product of all groups, then continue with a construction somehow. And "category theory" was his justification that this was OK. I gave him some comment about solution-set conditions, but I don't think it registered. | |
| May 5, 2010 at 13:19 | answer | added | algori | timeline score: 6 | |
| May 5, 2010 at 13:02 | comment | added | darij grinberg | On the other hand, if one works naively, lots of the reasoning one makes is wrong. For instance, is Tor(-, -) a bifunctor from Mod to Mod, in the sense that the Tor of two modules is a module? Well, one can choose a resolution and get a module... but not canonically. The best one can get is an isomorphism class of modules with canonical isomorphisms. Does this still justify calling Tor(-, -) a bifunctor? I don't know. But (please correct me if this is wrong) if one is just a bit more careful, one comes up with the notion of anafunctor, and Tor makes sense again... | |
| May 5, 2010 at 12:57 | comment | added | darij grinberg | I can't believe category theory is contradictory, as there are programming languages like Haskell based upon it, and computers usually aren't used in fields where one can derive contradictions... One never really uses a whole incredible large category like Vec or Set; most often the theorems one wants to prove are about finitely many objects & morphisms, and during the proof one may use finitely many constructions such as direct products, quotients, compositioins... things that are already well-defined in a nice, finite "category stump" with all the unnecessary stuff cut off. | |
| May 5, 2010 at 10:53 | comment | added | user2330 | The interview goes on like that : "Grothendieck provided a partial solution in terms of universes but a revolution of the foundations similar to what Cauchy and Weierstrass did for analysis is still to arrive." | |
| May 5, 2010 at 10:52 | answer | added | Kevin Buzzard | timeline score: 37 | |
| May 5, 2010 at 10:16 | history | asked | José Figueroa-O'Farrill | CC BY-SA 2.5 |