/sci/ - Science & Math

what what is about?

The Cop's set is half approximately Forced Anuses.

That be set theory right their my bum tickling matey, arrr.

Bump for serious answers, I'd just like to know what the Yoneda lemma is about in layman's terms.

Come back in a year, when I took my functional analysis course.

What about the introduction of http://en.wikipedia.org/wiki/Yoneda_lemma don't you understand?

>>4729974
I have no idea what "functors" or "Cayley's theorem" or "group theory" or basically any of the links in the introduction are. Looking at the corresponding pages I understand even less, so I was hoping /sci/ could give an explanation for laypersons what this is about.

>>4729981

Oh boy. Why would you want to understand the Yoneda lemma without knowing what a functor is? That's like trying to write a book without knowing any language.
I assume troll. Or are you trying to prove something to someone? Some silly bet?

>>4729959
the phrase "It is a vast generalisation of Cayley's theorem from group theory" should be good enough.
Category theory is often about the joy of rerouting maps (funtors on mophsims) and Yoneda Lemma says that a big deal of previously abstracts maps are really in Set (the smaller set category). Just like Cayley's theorem says some previously abstract groups are subgroups of the more reasonable symmetric group, or permutations.
The left hand side is something freaky, the right hand side F(A) is something normal.
These theorems says that some abstract constructions are actually one something you already know.
Why are you interested in it?, maybe that would be a start.

>>4729981
This is a very abstract and technical result. It cannot be rephrased outside the proper setting, unlike many "down-to-Earth" results in calculus etc.

>>4729990
>Some silly bet?
That's kinda what's going on, yes. I thought this lemma was simple as it is often written that it's trivial.

>>4729994
>These theorems says that some abstract constructions are actually one something you already know.
This helps, I guess.

>>4730002

Well, people usually mean that it's proof is trivial. Although I wouldn't say it like that, it's usually just surprisinlgy easy.
However, the technical background is quite heavy. Really understanding what functors, natural transformations and hom-sets are requires an array of examples, and those are not that easy to come by. If you knew some group theory, for instance, this would all come much easier.

>>4729997
So there's no chance of me even getting a remote idea what it's about?

>>4730010
It is probably impossible to get a better understanding than >>4729994 unless you know some more mathematics. It will be a very superficial and useless understanding.

Still, unless you work with categories on a regular basis, I doubt that you will ever need this theorem. In terms of abstraction, it sits quite deep in the theory. I doubt most math majors hear about it during their undergrad years.

>>4730010

Apart from
>These theorems says that some abstract constructions are actually one something you already know.
, no, at least not in a reasonable timeframe. I had a very algebra-heavy curriculum and the Yoneda lemma became important somewhere in my third undergrad year. You could of course understand this particular result in a much shorter time, but that would be a colossal waste of time. It's important only to mathematicians working in some specific fields.

>>4730020
Could you at least tell me in what kinds of fields and ways you applied the theorem? I'm just trying to get a flavor of what it's for.

Maybe this helps:
http://math.stackexchange.com/questions/37165/can-someone-explain-the-yoneda-lemma-to-an-applied-mat
hematician

I very much like the last point brought up in that thread. Indeed it shows in a way why the result is simple to prove but still hard to understand: it's a generalization of the fact that a set is determined by its elements to arbitrary objects of a possible non-concrete category. The fact that it generalizes a "trivial" statement is what makes it in a way trivial in the end, but the fact that it's such a vast generalization to very abstract objects is what keeps it from being completely trivial. I like this viewpoint much better than all the particle accelerator talk that has gotten more attention on both math.SE and MO. Thanks for the link, although I doubt that OP will have any use for it.