# [Edinburgh-pm] Beer money

Miles Gould miles at assyrian.org.uk
Fri Jul 30 04:26:19 PDT 2010

```On Thu, Jul 29, 2010 at 10:54:38PM +0100, Ollie Saunders wrote:
> Sorry, I realized after I left the meet up today that I hadn't paid
> for my beer. I'm be happy to reimburse whoever covered for me next
> time. Thanks.

I'm pretty sure I ended up paying less than my fair share, too, so
please speak up if you got stiffed.

BTW: I've collected together my various posts on the argmunging problem
at

http://pozorvlak.livejournal.com/tag/argmungers

I didn't get around to explaining how it ties into my PhD research, but
here's an outline. An *operad* O is a sequence of sets O_0, O_1, O_2,...
(we think of O_n as a set of operators X^n -> X, for some X), equipped
with an argmunging function #. So if f : {1..m} -> {1..n} and o is in
O_m, then f#o is in O_n. Operators can be composed associatively, and
this is compatible with the argmunger in the ways you'd expect [but you,
unlike me, do not have to work out exactly what these ways are and write
them down...]. Argmunging is also compatible with function composition:
g#(f#o) = (g.f)#o.

Here's the picture to keep in mind:

That's not the usual definition of operad - it's actually a bit more
general than the usual one, and phrased differently. But isolating the
concept of argmungers allowed me to (a) play around with restricting the
munging functions in different ways, which turned out to be crucial, (b)
greatly simplify a few key proofs. So, arguing with fanboys on the
Internet is very occasionally a good idea :-)

The whole thing's at http://arxiv.org/abs/1002.0879v1, if anyone's
interested :-)

I don't have any very good links for the stuff I was trying to say about
simplices and face/degeneracy maps. This is the best I could find:

http://ncatlab.org/nlab/show/simplicial+set

The relation to associativity is explained in Example 2, but what it
doesn't mention is that every category (= directed graph with
associative composition of arrows) is entirely determined by its nerve.
So in some sense, categories are just special simplicial sets...

Miles

[1]  :-)

--
When I'm working on a problem, I never think about beauty. I think only
how to solve the problem. But when I have finished, if the solution is
not beautiful, I know it is wrong.
-- Buckminster Fuller
```