This week I was at the Young Topologists Meeting! It’s gotten even bigger than two years ago, as there were more than 180 participants this year. The conference was quite interesting, and Copenhagen is a really nice city! The main theme was homological stability, about which I have learned a lot. The organizers should be applauded, because I can’t imagine how hard it must have been to plan a conference this big.

Now that I’ve done all my (math-related) travelling for the summer, I hope I’ll be able to post actual content here…

Maybe you (I don’t know who reads this anyway) haven’t noticed, but this blog has a new URL: blog.erad.fr! I splurged and got a domain name. The blog is (for now) still accessible via the old GitHub URL, but this may change at some point… Maybe more details on this later.

Update: And now the old Github URL shouldn’t work anymore. Hehe.

Last week I was invited by Thomas Willwacher to ETH Zürich for a few days, during which I also had the opportunity to give a talk at the “Talks in Mathematical Physics” seminar. It was a very interesting few days, and I’m very grateful for this invitation!

It also explains why I’ve been slacking off a bit with this blog – my teaching for the year is done and I’m traveling a lot in a short period…

I just came back from the Mathematisches Forschungsinstitut Oberwolfach!

I was there for the “Factorization Algebras and Functorial Field Theories” workshop. It was an incredible experience. The talks were all, of course, very interesting, and I learned a lot about QFT. But more than that, I learned a lot speaking to the people about various research topics – several people made several very interesting remarks to me, and others explained to me various points to mathematics that had remained obscure to me until this point.

The ambiance at the institute is amazing too. There were researchers from all around the world, and we were all there without hardly any external distractions: we all ate at the same times, slept in the same buildings… It was rather relaxed (like most mathematical conferences I went to, to be fair), and I particularly liked this informal session on Wednesday between 8PM and 10PM, during which anyone could walk up to the board and give a 5 min talk about whatever they wanted. It was really interesting to learn about all this research.

Of course I’m very grateful to the organizers for giving me the opportunity to come, in particular to Owen Gwilliam (who I think is the one who invited me).

The purpose of this post is to record the definition of $\infty$-operads and explain why it works like that. For this I’m using Lurie’s definition of $\infty$-operads; there is also a definition by Cisinski–Moerdijk–Weiss using dendroidal sets, about which I might talk later.
Indeed, the definition on an $\infty$-operad is a bit mysterious taken “as-is” – see [HA, §2.1.1.10]. My goal is to explain how to reach this definition, mostly for my own sake. Most of what follows is taken either from the book Higher Algebra, the $n$Lab, or the semester-long workshop about hgiher category theory in Lille in 2015.
The first thing to explain would be Cartesian (and coCartesian) morphisms. They are generalizations of Grothendieck fibrations from ordinary category theory. The basic idea of a Grothendieck fibration $p : E \to B$ is that the fibers $E_b = p^{-1}(b)$ depend contravariantly on $b$, i.e. given a morphism $f : b \to b'$, there exists a lift (a functor) $\bar{f} : E_{b'} \to E_b$. The definition of a Grothendieck fibration is exactly what’s needed for all this to work correctly.