A (work in progress) translation of Joyal’s original paper on species

tl;dr: I’m working on an English translation, with additional commentary, of Joyal’s 1981 paper introducing the concept of combinatorial species. Collaboration and feedback welcome!

Back when I was writing my PhD thesis on combinatorial species, I was aware that André Joyal’s original papers introducing combinatorial species are written in French, which I don’t read. I figured this was no big deal, since there is plenty of secondary literature on species in English (most notably Bergeron et al., which, though originally written in French, has been translated into English by Margaret Readdy). But at some point I asked a question on MathOverflow to which I hadn’t been able to find an answer, and was told that the answer was already in one of Joyal’s original papers!

So I set out to try to read Joyal’s original papers in French (there are two in particular: Une théorie combinatoire des séries formelles, and Foncteurs analytiques et espèces de structures), and found out that it was actually possible since (a) they are mathematics papers, not high literature; (b) I already understand a lot of the mathematics; and (c) these days, there are many easily accessible digital tools to help with the task of translation.

However, although it was possible for me to read them, it was still hard work, and for someone without my background in combinatorics it would be very tough going—which is a shame since the papers are really very beautiful. So I decided to do something to help make the papers and their ideas more widely accessible. In particular, I’m making an English translation of the papers¹—or at least of the first one, for now—interspersed with my own commentary to fill in more background, give additional examples, make connections to computation and type theory, or offer additional perspective. I hope it will be valuable to those in the English-speaking mathematics and computer science communities who want to learn more about species or gain more appreciation for a beautiful piece of mathematical history.

This is a long-term project, and not a high priority at the moment; I plan to work on it slowly but steadily. I’ve only worked on the first paper so far, and I’m at least far enough along that I’m not completely embarrassed to publicize it (but not much more than that). I decided to publicize my effort now, instead of waiting until I’m done, for several reasons: first, it may be a very long time before I’m really “done”, and some people may find it helpful or interesting before it gets to that point. Second, I would welcome collaboration, whether in the form of help with the translation itself, editing or extending the commentary, or simply offering feedback on early drafts or fixing typos. You can find an automatically updated PDF with the latest draft here, and the github repo is here. There are also simple instructions for compiling the paper yourself (using stack) should you want to do that.