那一剑的寂寞
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operad 是个什么东西?
看到一篇 Maxim Kontsevich的论文,讲operad的,可是这个名词查都查不到,好象说operads跟VOA(vertex operator algerba)有关,还讲到什么universal algebra minicourse,这下就真的搞不懂了.
| 发表时间：2005-12-31, 03:54:52 |
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kanex
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Re: operad 是个什么东西?
新聞群組： sci.physics.research, sci.math.research
寄件人： Tom Leinster <leins...@dpmms.cam.ac.REMOVETHISBIT.uk> - 尋找此作者的訊息
日期： 10 Nov 2001 01:11:58 GMT
當地時間： 2001年11月10日(星期六) 上午9時11分
主旨： Re: operads and string theory, loop q. gravity
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zirkus wrote:
> [John Baez wrote:]
> >Of course, many people still find *operads* scary and obscure,
> >but this is sheer pig-headedness, because they are a pitifully
> >simple idea.
> Maybe people are scared because operads can seem abstract [...] Also,
> according to T. Leinster, operads can be viewed as part of higher
> dimensional category theory which is not very well understood.
I do tend to view operads as naturally a part of higher-dimensional category
theory, but this ought to make higher-dimensional category theory *easier*,
not operads *harder*! As John says, operads really are simple things.
Maybe I can explain why I see operads as "higher categorical structures".
Roughly speaking, an operad consists of some operations and a rule for
composing them. And roughly speaking, a category consists of (some objects
and) some arrows and a rule for composing them. So an operad can be seen as
a structure of the same ilk as a category. The only essential difference is
that whereas an operation (map) in a category has a single input and a single
output, an operation (element) in an operad has several inputs and a
single output.
Why *higher* categorical structures? Well, a natural picture of an operation
in an operad is a box with several input wires coming into the top and a
single output wire coming out of the bottom. Composition in an operad takes
a tree of such boxes wired together, and produces a single box as the
composite. This tree naturally occupies higher-dimensional space - that is,
more than one-dimensional space. Compare the picture of composition in a
category, where one composes a string of arrows joined together into a
(one-dimensional) line.
If the occurrence of operads in higher-dimensional category theory seems to
make them more mysterious, then maybe it's worth pointing out that sometimes
people in the subject have used generalizations of the original notion of
operad - but still kept the original name. (As zirkus points out, there are
"multiple definitions".) So if you let a paper on n-categories fall open at
the middle and see operads being discussed, then it may very well be that
these aren't operads in the traditional sense.
I actually haven't come across many people who find operads "scary and
obscure" (which maybe shows what a sheltered life I lead); in fact, it seems
to me that the idea has spread like wildfire. What does surprise me is that
they haven't generated more interest amongst category theorists, given that
they are (in my view) such natural categorical structures.
江畔何人初见月`江月何年初照人`
| 发表时间：2005-12-31, 05:56:35 |
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