Timeline for How do I check if a functor has a (left/right) adjoint?
Current License: CC BY-SA 2.5
8 events
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| Apr 5, 2017 at 1:24 | comment | added | Quique Ruiz | In Lang's third edition Algebra, p. 66, there is an explicit construction of the free group using the Freyd Adjoint Functor Theorem. It's Jacques Tits'. | |
| Mar 23, 2010 at 21:57 | comment | added | Omar Antolín-Camarena | As Tom pointed out, the GAFT does apply to proving the existence of free groups. I agree that the SAFT is not useful in very many cases, but don't let that sour you on the GAFT! For free groups, for example, I find the GAFT much more convenient than trying to construct them by hand: it's frustrating not because it's hard, but because it feels like it should be extremely easy and it's not (for example, you can try to construct the free group with reduced words but then it's harder than it should be to prove associativity, etc.). | |
| Nov 18, 2009 at 20:41 | comment | added | Greg Stevenson | @buzzard: It is probably true that our sample spaces are somewhat different. I mostly wanted to make it clear that I didn't mean the exact proof; it is probably the first place a lot of people run across the idea that "if you need to build something, take everything close enough and then beat it into submission with limits/colimits" which I think is useful. Also your example looks different to me. The first step is to build the inverse image which I think is best done in general via left Kan extension. Then one can play around and find out what one wants for sheaves of modules. | |
| Nov 18, 2009 at 18:55 | comment | added | Kevin Buzzard | Here's what this example looks like (to me). Given f:X-->Y a map of schemes, then for a sheaf F on X one does ones best to define a sheaf f_F on Y. Conversely given a sheaf G on Y one does one's best to define a sheaf f^*G on X. *Now, after making the constructions, one does some mathematics and proves that the constructions are adjoint. The point I'm trying to make is that AFT ideas do not, it seems to me, go into the constructions. The work is in checking adjointness and so in some sense the mathematics seems to be elsewhere and not AFTish. But perhaps you are thinking of other examples! | |
| Nov 18, 2009 at 18:52 | comment | added | Kevin Buzzard | Greg: I disagree with your statement "the proof of AFT is not unlike how one often goes about building these things by hand". But that might well be because we're sampling from very different sample spaces. Here's an example of adjoints that has been fundamental in my mathematical career: pushforward and pullback of sheaves of abelian groups on schemes. Here pushforward and pullback are adjoints but neither construction, it seems to me, looks (to me) at all like what goes into AFT. [continued in a sec] | |
| Nov 18, 2009 at 10:06 | comment | added | Greg Stevenson | I agree that SAFT has somewhat limited applicability and that it is often a good idea to try to actually build adjoints (as one often needs to understand them not just know they exist) and that this can be very enlightening. But I think that knowing the general categorical machinery exists is worthwhile in case one wants to use it. I also think that the feel of the proof of AFT is not unlike how one often goes about building these things by hand at least in some sense. In my opinion there are lots of general machines to which this comment and your answer applies. | |
| Nov 18, 2009 at 8:49 | comment | added | Tom Leinster | I agree with much of this. SAFT, in particular, seems to have few uses - though there are significant uses other than Stone-Cech. Regarding free groups, SAFT doesn't guarantee existence, but GAFT (General AFT = "the" AFT) does, as per my answer. | |
| Nov 18, 2009 at 8:32 | history | answered | Kevin Buzzard | CC BY-SA 2.5 |