Timeline for What precisely Is "Categorification"?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Dec 16, 2009 at 8:49 | comment | added | Harry Gindi | Yes, the first application of half-sharp yields a separated presheaf. In fact, this is exactly the same for stacks, but the notion is messed up by the terminology. In fact, categories fibered in groupoids should be called prestacks, while prestacks should be called separated prestacks. However, it's too late in the game to change it now. | |
| Nov 11, 2009 at 4:45 | comment | added | Alex Hoffnung | 1) Thanks for clarifying your meaning in the first issue. 2) Sorry, you are right. In fact, sheafification would be pretty bad name if it weren't idempotent. | |
| Nov 11, 2009 at 4:19 | comment | added | Pete L. Clark | BTW, if I am not mistaken, sheafification in a general Grothendieck topologies is the name for a functor # which is idempotent. You may mean that in order to get this functor you have to do twice the same process (often denoted by "half a sharp") which in the topological case gives you the sheafification. Anyway, to rephrase, categorification is not an adjoint functor! | |
| Nov 11, 2009 at 4:14 | comment | added | Pete L. Clark | Now you are looking for more meaning in my answer than I had intended when writing it, I'm afraid. The word "systematic" can be omitted if you like. What I think I meant was that, since categorification is not restricted to any one specific mathematical context, it is in some respects a philosophy or an "ism" -- i.e., use of the term connotes the idea that it is beneficial to look for more categorical formulations of mathematical ideas in many mathematical contexts ("systematically"). | |
| Nov 11, 2009 at 3:13 | comment | added | Qiaochu Yuan | I think "idempotent" should just be read as "behaves like a completion," e.g. on an object which is already "complete" it should do nothing. As for "systematic," perhaps that should be read as "unique"? | |
| Nov 11, 2009 at 2:16 | comment | added | Alex Hoffnung | On another note, what role does idempotent play in this explanation? Sheafification is not an idempotent functor for general Grothendieck topologies. It is still a systematic process. So should I read systematic as functorial? | |
| Nov 11, 2009 at 1:43 | comment | added | Ben Webster♦ | Sorry if I misunderstood. I am still a bit confused then about what you mean by systematic. Are you referring to an internalization or enrichment process? (not actually Ben, but rather Alex Hoffnung). | |
| Nov 10, 2009 at 19:02 | comment | added | Pete L. Clark | I don't think there is a typo in my response, and I did not intend systematic to be in contrast to creative. (Also I said "one aspect of"...) You are right to point out that categorification is a meta-mathematical term that has no one precise meaning: it is not, for instance, an idempotent functor like sheafification or groupification. | |
| Nov 10, 2009 at 16:06 | history | answered | Alex Hoffnung | CC BY-SA 2.5 |