The n-Category Café

I’ve noticed spam has been getting much cleverer of late. They pay some attention to what’s been said in the thread.

The requested URL /~brunov/publications/MinimalModel.pdf was not found on this server.

Someone is very witty…

I will let Bruno know that the world is clamouring for his preprint.

At first I thought Career Test was joking that ‘BV-operad’ stands for ‘Bruno Valette operad’!

Ha! That’s surely a movie waiting to be made.

Your wish is granted.

Your wish is granted.

Thanks. That now answers CareerTest’s question: the talk will not be about current research, but will be an introduction to homotopy operad theory.

By the way, I tend to disagree, if you allow and if that makes sense, with the title of the talk, which apparently is

Algebra + homotopy = higher structure .

It seems to me that Algebra + homotopy gives higher algebra and that there are higher structures that want to be called higher geometry instead of higher algebra. But maybe the idea is that one reflects the other (which it does).

Also, I think it is not quite fair to say that the mathematical notion that models homotopical algebra is that of an operad (by which is implicitly meant an $\infty$-operad, really, presented by the standard model structure on operads enriched in simplicial sets/topological spaces or in chain complexes for the simpler cases). There is instead a trinity

1. . $\infty$-operad

2. . $\infty$-monad

3. . $\infty$-algebraic theory

in that there is, if not quite an equivalence, large overlap between the three notions and their $\infty$-algebras (just as in 1-category theory).

Sometimes it is useful to pass back and forth between these three notions. For instance $E_\infty$-algebras (rings) are traditionally maybe mostly conceived as $\infty$-algebras over (a cofibrant resolution of the) commutative operad, but there is also a simple and useful incarnation of the corresponding $(\infty,1)$-Lawvere theory (here): It is in fact just a $(2,1)$-theory: namely the $(2,1)$-category $Span(FinSet)$ of spans of finite sets. From that one sees immediately that the free $E_\infty$-algebra in $\infty Grpd$ is the union of the delooping of all the symmetric groups $\Sigma_n$

$$\begin{aligned} Free_{E_\infty}(\ast) &\simeq Span(FinSet)(\ast,\ast) \\ & \simeq Core(FinSet) \\ & \simeq \coprod_{n \in \mathbb{N}} \mathbf{B}\Sigma_n \end{aligned} \,.$$

In order to get the same insight with operads, one has to construct explicitly the Barrat-Eccles-resolution of $Comm$. Maybe not a big deal either, but I think the structure of the free $E_\infty$-algebra is more immediate in the perspective of the $(\infty,1)$-algebraic theory (well, maybe one could say that this is a tautology, since like algebraic theories, these are by their very nature controled by the free objects).

Finally a question: I wasn’t aware that there is any application of operad theory to Gröbner bases. (But then, I have only faint recollection of Gröbner base theory in the first place.) What is the relation?

Urs wrote:

the talk will not be about current research, but will be an introduction to homotopy operad theory.

Right. And this gives me an opportunity to say a bit more about the purpose of this seminar series.

Scotland’s mathematical community is not enormous, but there are actually quite a lot of people with some kind of interest in category theory, and they’re extremely varied. A lot of them are not primarily mathematicians, or not in mathematics departments: they’re computer scientists, or at least in computer science departments. People with this background probably know a lot about type theory and logic, but might have only a hazy idea of what, say, homology is. (That’s not true of all these computer scientists, but I think it is true of some of them.)

On the other hand, you’ve got algebraists and algebraic geometers and topologists who use things like derived categories or operads. They might have literally never heard of the subject called “type theory”, and might have only a hazy idea of what, say, first order logic is.

So our speakers have a tough job. We try to mix together the different communities, and we try to arrange speakers with broad appeal. It’s probably true that some of the most successful talks have been of a survey-like nature. Giving a talk at our seminar is something like giving a colloquium, but to an audience who all know some category theory.