Comments on: Joyal’s proof

By: The symmetric group, 11 « Peter Cameron's Blog

The symmetric group, 11 « Peter Cameron's Blog — Sat, 09 Apr 2011 16:57:22 +0000

[...] asked me about the probability that r random elements of Tn generate a synchronizing monoid. As I reported here, I was able to show, using Joyal’s proof of Cayley’s Theorem on trees, that for [...]

By: Peter Cameron

Peter Cameron — Thu, 26 Aug 2010 08:17:35 +0000

Yes, I first learned about it in Sydney in 1980. I visited the Pure Maths department for a term, and in a corner of my office I found a big pile of preprints of the French version of André’s paper on combinatorial formal power series. I fell under its spell straight away, especially since some of the ideas and formulae were very close to things I had come up with in the context of infinite permutation groups. We were climbing the same mountain from different sides.

Of course, I made no mention of species or categories in the post; I wanted to explain the idea as simply as possible.

By: Bob Walters

Bob Walters — Thu, 19 Aug 2010 18:00:35 +0000

I also like very much Andre Joyal’s proof. It was the example that impressed me most of the things Andre discussed in 1980, when he was writing up his paper on species in Sydney. The proof expresses the species of endomorphisms, and the species of doubly rooted trees as ‘composites’ of simpler species, the only difference being that the first contains linear orders as a ‘factor’, whereas the second has permutations as a ‘factor’. These two species are not (naturally) isomorphic but are the same at the level of counting.
I am just repeating what you said above, but it is the example which convinced me to take species seriously.