2
$\begingroup$

Let $\mathcal V$ be a nice symmetric monoidal ($\infty$-)category, and consider the ($\infty$-)category $Op(\mathcal V)$ of $\mathcal V$-enriched (symmetric) operads, symmetric monoidal under the Boardman-Vogt tensor product $\otimes_{BV}$. The unit is the operad $Comm$.

Question: Is there a classification of $\mathcal V$-operads which are $\otimes_{BV}$-invertible? That is, for which operads $O \in Op(\mathcal V)$ does there exist $P \in Op(\mathcal V)$ such that $O \otimes_{BV} P = Comm$?

I'm particulary curious about the case $\mathcal V = Spaces$, and the case $\mathcal V = Vect_k$ for a field $k$ (algebraically closed of characteristic 0, if you like). My suspicion is that all invertible $\mathcal V$-operads are probably obtained from invertible objects of $\mathcal V$ via some sort of "inducing-up" procedure.

Improve this question
$\endgroup$
3
  • 5
    $\begingroup$ Are you sure the unit is Comm? I'd have thought the unit is given by the operad with $O(1)=pt$ and other operations empty. The reasoning being that a $O_1\otimes O_2$-algebra should be the same as an $O_1$-algebra in $O_2$-algebras, I.e. an object with compatible $O_1$ and $O_2 structures. Whatever the unit is, it therefore should be an operad such that everything is canonically an algebra over it. $\endgroup$
    – Achim Krause
    Commented Jul 15, 2021 at 18:49
  • 4
    $\begingroup$ It is also unclear what the BV tensor product is for $Op(Vect_k)$. It might exist for linear Hopf operads though ($\mathcal{V}$ is the category of cocommutative $k$-coalgebras). Given that you want the unit to be $Comm$, do you want instead to talk about the Hadamard tensor product? $\endgroup$
    – Pavel Safronov
    Commented Jul 15, 2021 at 19:49
  • $\begingroup$ Thanks, Achim and Pavel. I am clearly quite confused. I will try to get un-confused and revise this question at some point. $\endgroup$
    – Tim Campion ♦
    Commented Jul 20, 2021 at 13:09

0

Reset to default